Research Article Active Constrained Layer Damping of Smart...

18
Hindawi Publishing Corporation Journal of Composites Volume 2013, Article ID 824163, 17 pages http://dx.doi.org/10.1155/2013/824163 Research Article Active Constrained Layer Damping of Smart Skew Laminated Composite Plates Using 1–3 Piezoelectric Composites R. M. Kanasogi and M. C. Ray Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721302, India Correspondence should be addressed to M. C. Ray; [email protected] Received 20 January 2013; Accepted 5 April 2013 Academic Editor: Hui Shen Shen Copyright © 2013 R. M. Kanasogi and M. C. Ray. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper deals with the analysis of active constrained layer damping (ACLD) of smart skew laminated composite plates. e constraining layer of the ACLD treatment is composed of the vertically/obliquely reinforced 1–3 piezoelectric composites (PZCs). A finite element model has been developed for accomplishing the task of the active constrained layer damping of skew laminated symmetric and antisymmetric cross-ply and antisymmetric angle-ply composite plates integrated with the patches of such ACLD treatment. Both in-plane and out-of-plane actuations by the constraining layer of the ACLD treatment have been utilized for deriving the finite element model. e analysis revealed that the vertical actuation dominates over the in-plane actuation. Particular emphasis has been placed on investigating the performance of the patches when the orientation angle of the piezoelectric fibers of the constraining layer is varied in the two mutually orthogonal vertical planes. Also, the effects of varying the skew angle of the substrate laminated composite plates and different boundary conditions on the performance of the patches have been studied. e analysis reveals that the vertically and the obliquely reinforced 1–3 PZC materials should be used for achieving the best control authority of ACLD treatment, as the boundary conditions of the smart skew laminated composite plates are simply supported and clamped-clamped, respectively. 1. Introduction Extensive research on the use of piezoelectric materials for making distributed actuators and sensors of light weight flexible smart structures has been carried out during the past several years [1–16]. e distributed piezoelectric actuators and sensors are either mounted on or embedded into the host flexible light weight structures. When they are activated with proper control voltage, the resulting structures attain self-controlling and self-sensing capabilities. Such flexible structures having built-in mechanism for self-controlling and self-sensing capabilities are customarily called “smart structures.” In most of the work on smart structures, the distributed actuators were considered to be made of the existing monolithic piezoelectric materials. e magnitudes of the piezoelectric coefficients of the existing monolithic piezoelectric materials are very small. Hence, the distributed actuators made of these materials need large control voltage for satisfactory control of smart structures. e further research on the efficient use of these low-control authority monolithic piezoelectric materials led to the development of the active constrained layer damping (ACLD) treatment [17]. e ACLD treatment consists of a constraining layer made of the piezoelectric materials and a constrained viscoelastic layer. e flexural vibration control by the constrained layer damping treatment is attributed to the dissipation of energy in the constrained viscoelastic layer due to its transverse shear deformations. e constraining layer of the activated ACLD treatment increases the transverse shear deformations of the viscoelastic constrained layer over its passive counterpart resulting in improved damping of the host structures. e control voltage required to cause transverse shear defor- mations in the low-stiff constrained viscoelastic layer of the ACLD treatment is compatible with the low-control authority of the monolithic piezoelectric materials. Hence, the piezoelectric materials perform much better to attenuate the vibrations of smart structures when they are used for the constraining layer of the ACLD treatment than when

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Hindawi Publishing CorporationJournal of CompositesVolume 2013 Article ID 824163 17 pageshttpdxdoiorg1011552013824163

Research ArticleActive Constrained Layer Damping of Smart Skew LaminatedComposite Plates Using 1ndash3 Piezoelectric Composites

R M Kanasogi and M C Ray

Department of Mechanical Engineering Indian Institute of Technology Kharagpur 721302 India

Correspondence should be addressed to M C Ray mcraymechiitkgpernetin

Received 20 January 2013 Accepted 5 April 2013

Academic Editor Hui Shen Shen

Copyright copy 2013 R M Kanasogi and M C Ray This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper deals with the analysis of active constrained layer damping (ACLD) of smart skew laminated composite plates Theconstraining layer of the ACLD treatment is composed of the verticallyobliquely reinforced 1ndash3 piezoelectric composites (PZCs)A finite element model has been developed for accomplishing the task of the active constrained layer damping of skew laminatedsymmetric and antisymmetric cross-ply and antisymmetric angle-ply composite plates integrated with the patches of such ACLDtreatment Both in-plane and out-of-plane actuations by the constraining layer of the ACLD treatment have been utilized forderiving the finite elementmodelThe analysis revealed that the vertical actuation dominates over the in-plane actuation Particularemphasis has been placed on investigating the performance of the patches when the orientation angle of the piezoelectric fibers ofthe constraining layer is varied in the two mutually orthogonal vertical planes Also the effects of varying the skew angle of thesubstrate laminated composite plates and different boundary conditions on the performance of the patches have been studied Theanalysis reveals that the vertically and the obliquely reinforced 1ndash3 PZC materials should be used for achieving the best controlauthority of ACLD treatment as the boundary conditions of the smart skew laminated composite plates are simply supported andclamped-clamped respectively

1 Introduction

Extensive research on the use of piezoelectric materials formaking distributed actuators and sensors of light weightflexible smart structures has been carried out during the pastseveral years [1ndash16] The distributed piezoelectric actuatorsand sensors are either mounted on or embedded into thehost flexible light weight structures When they are activatedwith proper control voltage the resulting structures attainself-controlling and self-sensing capabilities Such flexiblestructures having built-in mechanism for self-controllingand self-sensing capabilities are customarily called ldquosmartstructuresrdquo In most of the work on smart structures thedistributed actuators were considered to be made of theexisting monolithic piezoelectric materials The magnitudesof the piezoelectric coefficients of the existing monolithicpiezoelectric materials are very small Hence the distributedactuators made of these materials need large control voltagefor satisfactory control of smart structures The further

research on the efficient use of these low-control authoritymonolithic piezoelectric materials led to the development ofthe active constrained layer damping (ACLD) treatment [17]The ACLD treatment consists of a constraining layer madeof the piezoelectric materials and a constrained viscoelasticlayer The flexural vibration control by the constrained layerdamping treatment is attributed to the dissipation of energyin the constrained viscoelastic layer due to its transverse sheardeformations The constraining layer of the activated ACLDtreatment increases the transverse shear deformations of theviscoelastic constrained layer over its passive counterpartresulting in improved damping of the host structures Thecontrol voltage required to cause transverse shear defor-mations in the low-stiff constrained viscoelastic layer ofthe ACLD treatment is compatible with the low-controlauthority of the monolithic piezoelectric materials Hencethe piezoelectric materials perform much better to attenuatethe vibrations of smart structures when they are used forthe constraining layer of the ACLD treatment than when

2 Journal of Composites

they are directly bonded to the flexible host structures Ifthe constraining piezoelectric layer of the ACLD treatmentis not activated with the control voltage the treatment causesthe standard passive constrained layer damping of the smartstructure Thus the ACLD treatment being under operationprovides the attributes of both passive and active dampingsand acts as an inbuilt fail-safemechanism Since its inceptionextensive research has been carried out to investigate theperformance of the ACLD treatment for active damping ofsmart structures [18ndash25]

Piezoelectric composite (PZC) materials have beenemerged as the new class of smart materials Such PZCmaterials are composed of piezoelectric fiber reinforcementsand epoxy matrix These PZC materials provide wide rangeof effective material properties good conformability andstrength integrity [26] Among the various PZC materialsstudied by the researchers the vertically and the obliquelyreinforced 1ndash3 PZCmaterials are commercially available [27]and are being effectively used for underwater transducersmedical imaging applications and high frequency ultrasonictransducers [26]The constructional feature of a laminamadeof the vertically reinforced 1ndash3 PZC material is that thepiezoelectric fibers are vertically aligned across the thicknessof the lamina In case of the obliquely reinforced 1ndash3 PZCthe piezoelectric fibers are obliquely aligned in the verticalplane across the thickness of the lamina These PZCs arecharacterized by improvedmechanical performance electro-mechanical coupling characteristics and acoustic impedancematching over the existingmonolithic piezoelectricmaterials[26] Research on PZC materials is mainly concerned withthe micromechanical analysis of these materials [28ndash34]Recently Ray and his coworkers [35ndash37] investigated theperformance of these 1ndash3 PZC materials for active dampingof linear and nonlinear vibrations of composite beams platesand shells

Skew laminated composite plates are widely used inengineering applications such as aircraft wings and marineindustries and are highly prone to undergoing vibrationsResearchers extensively investigated the free vibrational char-acteristics of such skew plates For example Krishnan andDeshpande [38] carried out the free vibration analyses of can-tilevered skew isotropic platesand three-layered symmetriccross-ply laminates by deriving two finite element modelsKrishna Reddy and Palaninathan [39] developed a generalhigh precision triangular plate bending finite element for thefree vibration analysis of laminated skew plates Babu andKant [40] developed shear deformable finite element modelsfor the buckling analysis of skew composite plates and panelsGarg et al [41] developed a simple C∘ isoparametric finiteelement model based on a higher order shear deformationtheory for the free vibration analysis of isotropic orthotropicand layered anisotropic composite and sandwich skew lami-nates

The review of the existing literature on the skew platesreveals that the attention has not yet been focused on investi-gating the active control of vibrations of skew laminated com-posite plates using piezoelectric composites In this paperauthors intend to investigate the active constrained layerdamping (ACLD) of skew laminated composite plates For

such investigation three-dimensional analysis of ACLD ofskew laminated composite plates integrated with the patchesof ACLD treatment has been carried out by the finite elementmethod The constraining layer of the ACLD treatment isconsidered to be made of the vertically or the obliquelyreinforced 1ndash3 PZC materials Particular emphasis has beenplaced on investigating the effect of variation of piezoelectricfiber orientation angle on the performance of the ACLDpatches for controlling the vibrations of the skew laminatedcomposite plates

2 Finite Element Model of Smart SkewLaminated Composite Plate

Figure 1(a) illustrates the schematic diagram of a smartskew laminated composite plate composed of 119873 numberof orthotropic layers The length and the skew width ofthe plate are denoted by 119886 and 119887 respectively The topsurface of the plate is integrated with the skewed patches ofthe ACLD treatment The constraining layer of the ACLDtreatment is made of the verticallyobliquely reinforced 1ndash3 piezoelectric composite (PZC) material The notations ℎℎ119901 and ℎV represent the thicknesses of the substrate skew

laminated plate the piezoelectric composite layer and theviscoelastic layer respectively The skew angle of the plate isdenoted by 120572 The middle plane of the substrate compositeplate is considered as the reference plane The origin of thelaminate coordinate system (119909119910119911) is located at one cornerof the reference plane such that the lines 119910 = 0 and 119910 =

119887 cos120572 represent the two opposite boundaries of the substrateskew composite plates while the lines 119909 = 119910 tan120572 and119909 = 119886 + 119910 tan120572 describe the two opposite skewed edgesof the plate Denoting by 119896 (= 1 2 3 119873 + 2) the layernumber of any layer of the overall plate the thicknesscoordinates 119911 of the top and the bottom surfaces of any(119896th) layer are represented by ℎ

119896+1and ℎ

119896 respectively The

fiber orientation angle in any layer of the substrate plate inthe plane (119909119910) of the lamina with respect to the laminatecoordinate system is denoted by 120579 The piezoelectric fibers inthe constraining layer made of the obliquely reinforced 1ndash3PZC material are coplanar with the vertical 119909119911- or 119910119911-planemaking an angle 120595 with respect to the 119911-axis as shown inFigure 1(b) If the value of120595 is 0∘ the layer becomes verticallyreinforced 1ndash3 PZC layer The overall skew composite platebeing studied here is thin and consequently the first ordershear deformation theory (FSDT) can be used to model theaxial displacements in all the layers of the overall plate InFigure 2 the kinematics of axial deformations of the overallplate based on the FSDT has been illustrated Displayed inthis figure the variables 119906

0and V

0represent the generalized

translational displacement of a point (119909 119910) on the referenceplane (119911 = 0) along 119909- and 119910-directions respectively 120579

119909

120601119909 and 120574

119909denote the generalized rotations of the normal to

the middle planes of the substrate plate the viscoelastic layerand the 1ndash3 PZC layer respectively about the 119910-axis while120579119910 120601

119910and 120574

119910represent the generalized rotations of the same

about the 119909-axis respectively According to the kinematics ofdeformations illustrated in Figure 2 the axial displacements

Journal of Composites 3

120572119884998400

119883998400

119885998400 119885

ℎ2ℎ

1198864 1198862

1198874

1198872

Patch 2Skew laminated composite plate

1198864

1198874

Viscoelastic layer

Patch 1

119883

119884

120572 Vertically reinforced 1ndash3 PZC layer

(a)

119911

119910 119910

119909 119909

Surface electrode Surface electrode

Piezoelectric fiber Epoxy matrix

Epoxy matrixPiezoelectric fiber

119911

120595 120595

(b)

Figure 1 (a) Schematic representation of a skew laminated composite plate integrated with the patches of ACLD treatment composed ofverticallyobliquely reinforced 1ndash3 piezoelectric composite constraining layer (b) Schematic diagram of layers of obliquely reinforced 1ndash3piezoelectric composite

Viscoelastic layer Viscoelastic layerSkew laminated plate

119911

ℎ2 ℎ2

ℎ ℎ

Undeformed transverse section Undeformed transverse section

Deformed transverse

119910

sectionDeformed transverse

section

ℎ ℎ

119911

120579119909

120579119910

119909

1199060

120574119910

120601119910

120601119909

120574119909

Skew laminated plate

ℎ1199011ndash3 PZC layer

1199080

0

1ndash3 PZC layer

1199080

Figure 2 Kinematics of deformations

119906 and V at any point in any layer of the overall plate along 119909-and 119910-directions respectively can be expressed as

119906 (119909 119910 119911 119905) = 1199060(119909 119910 119905) + (119911 minus ⟨119911 minus

ℎ

2⟩)120579

119909(119909 119910 119905)

+ (⟨119911 minusℎ

2⟩ minus ⟨119911 minus ℎ

119873+2⟩) 120601

119909(119909 119910 119905)

+ ⟨119911 minus ℎ119873+2

⟩ 120574119909(119909 119910 119905)

(1)

V (119909 119910 119911 119905) = V0(119909 119910 119905) + (119911 minus ⟨119911 minus

ℎ

2⟩)120579

119910(119909 119910 119905)

+ (⟨119911 minusℎ

2⟩ minus ⟨119911 minus ℎ

119873+2⟩) 120601

119910(119909 119910 119905)

+ ⟨119911 minus ℎ119873+2

⟩ 120574119910(119909 119910 119905)

(2)

in which a function within the bracket ⟨sdot⟩ represents anappropriate singularity function which satisfies the continu-ity of displacements between two adjacent continua Since thetransverse actuation of the constraining layer of the ACLDtreatment will be used for the flexural vibration controlof the plate the transverse normal strain in the overallplate must be considered in the model Hence as the plateconsidered here is thin the transverse displacements (119908) atany point in the substrate plate the viscoelastic layer and

4 Journal of Composites

the 1ndash3 PZC layer are assumed to be linearly varying acrosstheir thicknesses Thus similar to the axial displacement thetransverse displacements at any point in the overall plate canbe expressed as

119908 (119909 119910 119911 119905) = 1199080(119909 119910 119905) + (119911 minus ⟨119911 minus

ℎ

2⟩)120579

119911(119909 119910 119905)

+ (⟨119911 minusℎ

2⟩ minus ⟨119911 minus ℎ

119873+2⟩) 120601

119911(119909 119910 119905)

+ ⟨119911 minus ℎ119873+2

⟩ 120574119911(119909 119910 119905)

(3)

in which1199080refers to the transverse displacement at any point

on the reference plane and 120579119911 120601

119911 and 120574

119911are the generalized

displacements representing the gradients of the transversedisplacement in the substrate plate the viscoelastic layer andthe 1ndash3 PZC layer respectively with respect to the thicknesscoordinate (119911)

For the ease of analysis the generalized displacementvariables are grouped into the following two vectors

119889119905 = [119906

0V0

1199080]119879

119889119903 = [120579

119909120579119910

120579119911

120601119909

120601119910

120601119911

120574119909

120574119910

120574119911]119879

(4)

In order to implement the selective integration rule forcomputing the element stiffness matrices corresponding tothe transverse shear deformations the state of strain at any

point in the overall plate is divided into the following twostrain vectors 120576

119887 and 120576

119904

120576119887 = [120576

119909120576119910

120576119909119910

120576119911]119879

120576119904 = [120576

119909119911120576119910119911

]119879 (5)

in which 120576119909 120576

119910 and 120576

119911are the normal strains along 119909-

119910- and 119911-directions respectively 120576119909119910

is the in-plane shearstrain 120576

119909119911and 120576

119910119911are the transverse shear strains Using the

displacement fields given by (1)ndash(3) and the linear strain-displacement relations the vectors 120576

119887119888

120576119887V and 120576

119887119901

defining the state of in-plane and transverse normal strainsat any point in the substrate skew composite plate the vis-coelastic layer and the active constraining layer respectivelycan be expressed as

120576119887119888

= 120576119887119905 + [119885

1] 120576

119887119903 120576

119887V = 120576

119887119905 + [119885

2] 120576

119887119903

120576119887119901

= 120576119887119905 + [119885

3] 120576

119887119903

(6)

Similarly the vectors 120576119904119888

120576119904V and 120576

119904119901

defining the stateof transverse shear strains at any point in the substrate com-posite plate the viscoelastic layer and the active constraininglayer respectively can be expressed as

120576119904119888

= 120576119904119905 + [119885

4] 120576

119904119903 120576

119904V = 120576

119904119905 + [119885

5] 120576

119904119903

120576119904119901

= 120576119904119905 + [119885

6] 120576

119904119903

(7)

The various matrices appearing in (6) and (7) have beendefined in the Appendix while the generalized strain vectorsare given by

120576119887119905 = [

1205971199060

120597119909

120597V0

120597119910

1205971199060

120597119910+

120597V0

1205971199090]

119879

120576119904119905 = [

1205971199080

120597119909

1205971199080

120597119910]

119879

120576119887119903 = [

120597120579119909

120597119909

120597120579119910

120597119910

120597120579119909

120597119910+

120597120579119910

120597119909120579119911

120597120601119909

120597119909

120597120601119910

120597119910

120597120601119909

120597119910+

120597120601119910

120597119909120601119911

120597120574119909

120597119909

120597120574119910

120597119910

120597120574119909

120597119910+

120597120574119910

120597119909120574119911

]

119879

120576119904119903 = [120579

119909120579119910

120601119909

120601119910

120574119909

120574119910

120597120579119911

120597119909

120597120579119911

120597119910

120597120601119911

120597119909

120597120601119911

120597119910

120597120574119911

120597119909

120597120574119911

120597119910]

119879

(8)

Similar to the strain vectors given by (5) the state of stressesat any point in the overall plat is described by the followingstress vectors

120590119887 = [120590

119909120590119910

120590119909119910

120590119911]119879

120590119904 = [120590

119909119911120590119910119911

]119879

(9)

where 120590119909 120590

119910 and 120590

119911are the normal stresses along 119909- 119910-

and 119911-directions respectively 120590119909119910is the in-plane shear stress

120590119909119911

and 120590119910119911

are the transverse shear stresses The constitutive

relations for the material of any orthotropic layer of thesubstrate plate are given by

120590119896

119887

= [119862119896

119887

] 120576119896

119887

120590119896

119904

= [119862119896

119904

] 120576119896

119904

(119896 = 1 2 3 119873)

(10)

Journal of Composites 5

where the elastic coefficient matrices are

[119862119896

119887

] =

[[[[[[[[[[[[[[

[

119862119896

11

119862119896

12

119862119896

16

119862119896

13

119862119896

12

119862119896

22

119862119896

26

119862119896

23

119862119896

16

119862119896

26

119862119896

66

119862119896

36

119862119896

13

119862119896

23

119862119896

36

119862119896

33

]]]]]]]]]]]]]]

]

[119862119896

119904

] =[[

[

119862119896

55

119862119896

45

119862119896

45

119862119896

44

]]

]

(11)

and 119862119896

119894119895

(119894 119895 = 1 2 3 6) are the transformed elasticcoefficients with respect to the reference coordinate systemThematerial of the viscoelastic layer is assumed to be linearlyviscoelastic and homogenous isotropic and is modeled byusing the complex modulus approach Thus the shear mod-ulus119866 and Youngrsquos modulus 119864 of the viscoelastic material aregiven by

119866 = 1198661015840

(1 + 119894120578) 119864 = 2119866 (1 + ]) (12)

in which 1198661015840 is the storage modulus ] is Poissonrsquos ratio and

120578 is the loss factor at a particular operating temperature andfrequency Employing the complex modulus approach theconstitutive relations for the material of the viscoelastic layer(119896 = 119873 + 1) can also be represented by (10) with 119862

119896

119894119895

(119894 119895 =

1 2 3 6) being the complex elastic constants [29 30]The constraining PZC layer will be subjected to the appliedelectric field (119864

119911) acting across its thickness (ie along the

119911-direction) only Accordingly the constitutive relations forthe 1ndash3 PZC material with respect to the laminate coordinatesystem (119909119910119911) can be expressed as

120590119896

119887

= [119862119896

119887

] 120576119896

119887

+ [119862119896

119887119904

] 120576119896

119904

minus 119890119887 119864

119911

120590119896

119904

= [119862119896

119887119904

] 120576119896

119887

+ [119862119896

119904

] 120576119896

119904

minus 119890119904 119864

119911

119863119911= 119890

119887119879

120576119896

119887

+ 119890119904119879

120576119896

119904

+ 12057633119864119911

119896 = 119873 + 2

(13)

Here 119863119911represents the electric displacement along the 119911-

direction and 12057633

is the dielectric constant It may be notedfrom the previous form of the constitutive relations that thetransverse shear strains are coupled with the in-plane normalstrains due to the orientation of piezoelectric fibers in the

vertical 119909119911- or 119910119911-plane and the corresponding couplingelastic constant matrices [119862

119873+2

119887119904

] are given by

[119862119873+2

119887119904

] =

[[[[[[[[[[

[

119862119873+2

15

0

119862119873+2

25

0

0 119862119873+2

46

119862119873+2

35

0

]]]]]]]]]]

]

or

[119862119873+2

119887119904

] =

[[[[[[[[[[

[

0 119862119873+2

14

0 119862119873+2

24

119862119873+2

56

0

0 119862119873+2

34

]]]]]]]]]]

]

(14)

as the piezoelectric fibers are coplanar with the vertical 119909119911-or 119910119911-plane respectively Note that if the fibers are coplanarwith both 119909119911- and 119910119911-planes (ie 120595 = 0

∘) this couplingmatrix becomes a nullmatrix Also the piezoelectric constantmatrices 119890

119887 and 119890

119904 appearing in (13) contain the following

transformed effective piezoelectric coefficients of the 1ndash3PZC

119890119887 = [119890

3111989032

11989036

11989033]119879

119890119904 = [119890

3511989034]119879

(15)

The total potential energy 119879119901and the kinetic energy 119879

119896of the

overall plateACLD system are given by [35]

119879119901

=1

2[

119873+2

sum

119896=1

intΩ

(120576119896

119887

119879

120590119896

119887

+ 120576119903

119904

119879

120590119896

119904

) 119889Ω

minusintΩ

119863119911119864119911119889Ω] minus int

119860

119889119879

119891 119889119860

119879119896=

1

2

119873+2

sum

119896=1

intΩ

120588119896

(2

+ V2 + 2

) 119889Ω

(16)

in which 120588119896 is the mass density of the 119896th layer 119891 is the

externally applied surface traction acting over a surface area119860 andΩ represents the volume of the concerned layer Sincethe plates under study are considered to be thin the rotaryinertia of the overall plate has been neglected in estimatingthe kinetic energy The overall plate is discretized by eight-noded isoparametric quadrilateral elements Following (4)the generalized displacement vectors associated with the 119894th(119894 = 1 2 3 8) node of the element can be written as

119889119905119894 = [119906

0119894V0119894

1199080119894]119879

119889119903119894 = [120579

119909119894120579119910119894

120579119911119894

120601119909119894

120601119910119894

120601119911119894

120574119909119894

120574119910119894

120574119911119894]119879

(17)

6 Journal of Composites

Thus the generalized displacement vectors at any pointwithin the element can be expressed in terms of the nodalgeneralized displacement vectors 119889

119890

119905

and 119889119890

119903

as follows

119889119890

119905

= [119873119905] 119889

119890

119905

119889119890

119903

= [119873119903] 119889

119890

119903

(18)

in which

119889119890

119905

= [119889119890

1199051

119879

119889119890

1199052

119879

sdot sdot sdot 119889119890

1199058

119879

]119879

119889119890

119903

= [119889119890

1199031

119879

119889119890

1199032

119879

sdot sdot sdot 119889119890

1199038

119879

]119879

[119873119905] = [119873

11990511198731199052

sdot sdot sdot 1198731199058]119879

[119873119903] = [119873

11990311198731199032

sdot sdot sdot 1198731199038]119879

119873119905119894

= 119899119894119868119905 119873

119903119894= 119899

119894119868119903

(19)

while 119868119905and 119868

119903are (3times3) and (9times 9) identitymatrices respec-

tively and 119899119894is the shape function of natural coordinates

associatedwith the 119894th nodeMaking use of the relations givenby (6)ndash(8) and (18) the strain vectors at any point within theelement can be expressed in terms of the nodal generalizeddisplacement vectors as follows

120576119887119888

= [119861119905119887] 119889

119890

119905

+ [1198851] [119861

119903119887] 119889

119890

119903

120576119887V = [119861

119905119887] 119889

119890

119905

+ [1198852] [119861

119903119887] 119889

119890

119903

120576119887119901

= [119861119905119887] 119889

119890

119905

+ [1198853] [119861

119903119887] 119889

119890

119903

120576119904119888

= [119861119905119904] 119889

119890

119905

+ [1198854] [119861

119903119904] 119889

119890

119903

120576119904V = [119861

119905119904] 119889

119890

119905

+ [1198855] [119861

119903119904] 119889

119890

119903

120576119904119901

= [119861119905119904] 119889

119890

119905

+ [1198856] [119861

119903119904] 119889

119890

119903

(20)

in which the nodal strain-displacement matrices [119861119905119887] [119861

119903119887]

[119861119905119904] and [119861

119903119904] are given by

[119861119905119887] = [119861

11990511988711198611199051198872

sdot sdot sdot 1198611199051198878

]

[119861119903119887] = [119861

11990311988711198611199031198872

sdot sdot sdot 1198611199031198878

]

[119861119905119904] = [119861

11990511990411198611199051199042

sdot sdot sdot 1198611199051199048

]

[119861119903119904] = [119861

11990311990411198611199031199042

sdot sdot sdot 1198611199031199048

]

(21)

The submatrices of [119861119905119887] [119861

119903119887] [119861

119905119904] and [119861

119903119904] have been

explicitly presented in the Appendix On substitution of (9)ndash(13) and (20) into (16) the total potential energy 119879

119890

119901

and the

kinetic energy 119879119890

119896

of a typical element augmented with theACLD treatment can be expressed as

119879119890

119901

=1

2[119889

119890

119905

119879

[119870119890

119905119905

] 119889119890

119905

+ 119889119890

119905

119879

[119870119890

119905119903

] 119889119890

119903

+ 119889119890

119903

119879

[119870119890

119905119903

]119879

times 119889119890

119905

+ 119889119890

119903

119879

[119870119890

119903119903

] 119889119890

119903

minus 2119889119890

119905

119879

119865119890

119905119901

119881

minus 2119889119890

119903

119879

119865119890

119903119901

119881 minus 2119889119890

119905

119879

119865119890

minus 12057633

1198812

ℎ119901

]

(22)

119879119890

119896

=1

2 119889

119890

119905

119879

[119872119890

] 119889119890

119905

(23)

In (22)119881 represents the voltage difference applied across thethickness of the PZC layer The elemental stiffness matrices[119870

119890

119905119905

] [119870119890

119905119903

] and [119870119890

119903119903

] the elemental electroelastic couplingvectors 119865

119890

119905119901

and 119865119890

119903119901

the elemental load vector 119865119890

andthe elemental mass matrix [119872

119890

] appearing in (22) and (23)are given by

[119870119890

119905119905

] = [119870119890

119905119887

] + [119870119890

119905119904

] + [119870119890

119905119887119904

]119901119887

+ [119870119890

119905119887119904

]119901119904

[119870119890

119905119903

] = [119870119890

119905119903119887

] + [119870119890

119905119903119904

]

+1

2([119870

119890

119905119903119887119904

]119901119887

+ [119870119890

119903119905119887119904

]119879

119901119887

+ [119870119890

119905119903119887119904

]119901119904

+ [119870119890

119903119905119887119904

]119879

119901119904

)

[119870119890

119903119905

] = [119870119890

119905119903

]119879

[119870119890

119903119903

] = [119870119890

119903119903119887

] + [119870119890

119903119903119904

] + [119870119890

119903119903119887119904

]119901119887

+ [119870119890

119903119903119887119904

]119901119904

119865119890

119905119901

= 119865119890

119905119887

119901

+ 119865119890

119905119904

119901

119865119890

119903119901

= 119865119890

119903119887

119901

+ 119865119890

119903119904

119901

119865119890

= int

119886119890

0

int

119887119890

0

[119873119905]119879

119891 119889119909 119889119910

[119872119890

] = int

119886119890

0

int

119887119890

0

119898[119873]119879

[119873] 119889119909 119889119910 119898 =

119873+2

sum

119896=1

120588119896

(ℎ119896+1

minus ℎ119896)

(24)

where

[119870119890

119905119887

] = int119860

[119861119905119887]119879

([119863119905119887] + [119863

119905119887]V + [119863

119905119887]119901

) [119861119905119887] 119889119909 119889119910

[119870119890

119905119904

] = int119860

[119861119905119904]119879

([119863119905119904] + [119863

119905119904]V + [119863

119905119904]119901

) [119861119905119904] 119889119909 119889119910

[119870119890

119905119887119904

]119901119887

= int119860

[119861119905119887]119879

[119863119905119887119904

]119901

[119861119905119904] 119889119909 119889119910

[119870119890

119905119887119904

]119901119904

= int119860

[119861119905119904]119879

[119863119905119887119904

]119901

[119861119905119887] 119889119909 119889119910

[119870119890

119905119903119887

] = int119860

[119861119905119903119887

]119879

([119863119905119903119887

] + [119863119905119903119887

]V + [119863119905119903119887

]119901

)

times [119861119905119903119887

] 119889119909 119889119910

Journal of Composites 7

[119870119890

119905119903119887119904

]119901119887

= int119860

[119861119905119887]119879

[119863119905119903119887119904

]119901

[119861119903119904] 119889119909 119889119910

[119870119890

119903119905119887119904

]119901119887

= int119860

[119861119903119887]119879

[119863119903119905119887119904

]119901

[119861119905119904] 119889119909 119889119910

[119870119890

119905119903119887119904

]119901119904

= int119860

[119861119905119904]119879

[119863119903119905119887119904

]119901

[119861119903119887] 119889119909 119889119910

[119870119890

119903119905119887119904

]119901119904

= int119860

[119861119903119904]119879

[119863119905119903119887119904

]119901

[119861119905119887] 119889119909 119889119910

[119870119890

119905119903119904

] = int119860

[119861119905119903119904

]119879

([119863119905119903119904

] + [119863119905119903119904

]V + [119863119905119903119904

]119901

)

times [119861119905119903119904

] 119889119909 119889119910

[119870119890

119903119903119887

] = int119860

[119861119903119903119887

]119879

([119863119903119903119887

] + [119863119903119903119887

]V + [119863119903119903119887

]119901

)

times [119861119903119903119887

] 119889119909 119889119910

[119870119890

119903119903119904

] = int119860

[119861119903119903119904

]119879

([119863119903119903119904

] + [119863119903119903119904

]V + [119863119903119903119904

]119901

)

times [119861119903119903119904

] 119889119909 119889119910

[119870119890

119903119903119887119904

]119901119887

= int119860

[119861119903119887]119879

[119863119903119903119887119904

]119901

[119861119903119904] 119889119909 119889119910

[119870119890

119903119903119887119904

]119901119904

= int119860

[119861119903119904]119879

[119863119903119903119887119904

]119901

[119861119903119887] 119889119909 119889119910

119865119890

119905119887

119901

= int119860

[119861119905119887]119879

119863119905119887119901

119889119909 119889119910

119865119890

119905119904

119901

= int119860

[119861119905119904]119879

119863119905119904119901

119889119909 119889119910

119865119890

119903119904

119901

= int119860

[119861119903119904]119879

119863119903119904119901

119889119909 119889119910

(25)

The various rigidity matrices and the rigidity vectors forelectro-elastic coupling appearing in the previous elementalmatrices are given by

[119863119905119887] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] 119889119911

[119863119905119903119887

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] [1198851] 119889119911

[119863119903119903119887

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[1198851]119879

[119862119896

119887

] [1198851] 119889119911

[119863119905119904] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119904

] 119889119911

[119863119905119903119904

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] [1198854] 119889119911

[119863119903119903119904

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[1198854]119879

[119862119896

119887

] [1198854] 119889119911

[119863119905119887]V = ℎV [119862

119873+1

119887

]

[119863119905119903119887

]V = int

ℎ119873+2

ℎ119873+1

[119862119873+1

119887

] [1198852] 119889119911

[119863119903119903119887

]V = int

ℎ119873+2

ℎ119873+1

[1198852]119879

[119862119873+1

119887

] [1198852] 119889119911

[119863119905119904]V = ℎV [119862

119873+1

119904

]

[119863119905119903119904

]V = int

ℎ119873+2

ℎ119873+1

[119862119873+1

119904

] [1198855] 119889119911

[119863119903119903119904

]V = int

ℎ119873+2

ℎ119873+1

[1198855]119879

[119862119873+1

119887

] [1198855] 119889119911

[119863119905119887]119901

= ℎ119901[119862

119873+2

119887

]

[119863119905119903119887

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887

] [1198853] 119889119911

[119863119903119903119887

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+2

119887

] [1198853] 119889119911

[119863119905119904]119901

= ℎ119901[119862

119873+2

119904

]

[119863119905119903119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119904

] [1198856] 119889119911

[119863119903119903119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198856]119879

[119862119873+2

119887

] [1198856] 119889119911

[119863119905119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887119904

] 119889119911

[119863119905119903119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887119904

] [1198856] 119889119911

[119863119903119905119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+2

119887119904

] 119889119911

[119863119903119903119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+3

119887119904

] [1198856] 119889119911

119863119905119887119901

= int

ℎ119873+3

ℎ119873+2

minus119890119887

ℎ119901

119889119911

119863119903119887119901

= int

ℎ119873+3

ℎ119873+2

minus[119885

3]119879

119890119887

ℎ119901

119889119911

119863119905119904119901

= int

ℎ119873+3

ℎ119873+2

minus119890119904

ℎ119901

119889119911

119863119903119904119901

= int

ℎ119873+3

ℎ119873+2

minus[119885

6]119879

119890119904

ℎ119901

119889119911

(26)

8 Journal of Composites

Applying the dynamic version of the virtual work principle[32] the following open-loop equations of motion for anelement of the overall plate integrated with the ACLDtreatment are obtained

[119872119890

] 119889119890

119905

+ [119870119890

119905119905

] 119889119890

119905

+ [119870119890

119905119903

] 119889119890

119903

= 119865119890

119905119901

119881 + 119865119890

[119870119890

119905119903

]119879

119889119890

119905

+ [119870119890

119903119903

] 119889119890

119903

= 119865119890

119903119901

119881

(27)

Since the elastic constant matrix of the viscoelastic layer iscomplex the stiffness matrices of an element integrated withthe ACLD treatment are complex In case of an elementnot integrated with the ACLD treatment the electro-elasticcoupling matrices become null vectors and the elementalstiffnessmatrices are to be computed without considering thepiezoelectric and viscoelastic layers It should also be notedthat as the stiffness matrices associated with the transverseshear strains are derived separately one can employ theselective integration rule in a straight forward manner toavoid shear locking problem in case of thin plates Therestrained boundary conditions at the skew edges of theplate are such that the displacements of the points locatedat the skew edges are to be restrained along 119909

1015840- 1199101015840- and

1199111015840-directions (Figure 1(a)) Hence in order to impose theboundary conditions in a straight forward manner thegeneralized displacement vectors of a point lying on the skewedge are to be transformed as follows

119889119905 = [119871

119905] 119889

1015840

119905

119889119903 = [119871

119903] 119889

1015840

119903

(28)

where 1198891015840

119905

and 1198891015840

119903

are the generalized displacement vectorsof the point with respect to 119909

1015840

1199101015840

1199111015840 coordinate system and are

given by

1198891015840

119905

= [1199061015840

119900

V1015840119900

1199081015840

119900

]119879

1198891015840

119903

= [1205791015840

119909

1205791015840

119910

1205791015840

119911

1206011015840

119909

1206011015840

119910

1206011015840

119911

1205741015840

119911

1205741015840

119910

1205741015840

119911]119879

(29)

Also the transformation matrices [119871119905] and [119871

119903] are given by

[119871119905] = [

[

119888 119904 0

minus119904 119888 0

0 0 1

]

]

[119871119903] =

[[[[[[[[[[[[

[

119888 119904 0 0 0 0 0 0 0

minus119904 119888 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 119888 119904 0 0 0 0

0 0 0 minus119904 119888 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 119888 119904 0

0 0 0 0 0 0 minus119904 119888 0

0 0 0 0 0 0 0 0 1

]]]]]]]]]]]]

]

(30)

in which 119888 = cos120572 and 119904 = sin120572Thus the elementalmatricesof the element containing the nodes lying on the skew edgeare to be augmented as follows

[119870119890

119905119905

] = [1198791]119879

[119870119890

119905119905

] [1198791] [119870

119890

119905119903

] = [1198791]119879

[119870119890

119905119903

] [1198792]

[119870119890

119903119903

] = [1198792]119879

[119870119890

119903119903

] [1198792] [119872

119890

] = [1198791]119879

[119872119890

] [1198791]

(31)

The forms of the transformation matrices [1198791] and [119879

2] are

given by

[1198791] =

[[[[[[[[[[[

[

[119871119905] 119900 119900 119900 119900 119900 119900 119900

119900 [119871119905] 119900 119900 119900 119900 119900 119900

119900 119900 [119871119905] 119900 119900 119900 119900 119900

119900 119900 119900 [119871119905] 119900 119900 119900 119900

119900 119900 119900 119900 [119871119905] 119900 119900 119900

119900 119900 119900 119900 119900 [119871119905] 119900 119900

119900 119900 119900 119900 119900 119900 [119871119905] 119900

119900 119900 119900 119900 119900 119900 119900 [119871119905]

]]]]]]]]]]]

]

[1198792] =

[[[[[[[[[[[

[

[119871119903] 119900 119900 119900 119900 119900 119900 119900

119900 [119871119903] 119900 119900 119900 119900 119900 119900

119900 119900 [119871119903] 119900 119900 119900 119900 119900

119900 119900 119900 [119871119903] 119900 119900 119900 119900

119900 119900 119900 119900 [119871119903] 119900 119900 119900

119900 119900 119900 119900 119900 [119871119903] 119900 119900

119900 119900 119900 119900 119900 119900 [119871119903] 119900

119900 119900 119900 119900 119900 119900 119900 [119871119903]

]]]]]]]]]]]

]

(32)

in which 119900 and 119900 are (3 times 3) and (9 times 9) null matricesrespectively It may be noted that in case of a node ofsuch element which does not lie on the skew edge thegeneralized displacement vectors of this node with respect to119909119910119911 coordinate system are not required to be transformedHence thematrices [119871

119905] and [119871

119903] corresponding to suchnode

turn out to be (3times3) and (9times9) identitymatrices respectivelyThe elemental equations of motion are now assembled

to obtain the open-loop global equations of motion of theoverall skew plate as follows

[119872] + [119870119905119905] 119883 + [119870

119905119903] 119883

119903 =

119902

sum

119895=1

119865119895

119905119901

119881119895

+ 119865

[119870119905119903]119879

119883 + [119870119903119903] 119883

119903 =

119902

sum

119895=1

119865119895

119903119901

119881119895

(33)

where [119870119905119905] [119870

119905119903] and [119870

119903119903] are the global stiffness matrices

of the overall skew laminated composite plate [119872] is theglobal mass matrix 119865

119905119901 and 119865

119903119901 are the global electro-

elastic coupling vectors 119883 and 119883119903 are the global nodal

generalized displacement vectors 119865 is the global nodalforce vector 119902 is the number of patches and 119881

119895 is thevoltage applied to the 119895th patch Since the elemental stiffnessmatrices of an element augmented with the ACLD treatmentare complex the global stiffness matrices become complexand the energy dissipation characteristics of the overallplate are attributed to the imaginary part of these matricesHence the global equations of motion as derived previouslyalso represent the modeling of the passive (uncontrolled)constrained layer damping of the substrate plate when theconstraining layer is not subjected to any control voltagefollowing a derivative control law

Journal of Composites 9

3 Closed-Loop Model

In the active control approach the active constraining layerof each patch is activated with a control voltage negativelyproportional to the transverse velocity of a point Thus thecontrol voltage for each patch can be expressed in terms of thederivatives of the global nodal degrees of freedom as follows

119881119895

= minus119896119895

119889

= minus119896119895

119889

[119880119895

119905

] minus 119896119895

119889

(ℎ

2) [119880

119895

119903

] 119903 (34)

where [119880119895

119905

] and [119880119895

119903

] are the unit vectors describing thelocation of the velocity sensor for the 119895th patch and 119896

119895

119889

isthe control gain for the 119895th patch Substituting (34) into(33) the final equations of motion governing the closed-loop dynamics of the overall skew laminated compositeplateACLD system can be obtained as follows

[119872] + [119870119905119905] 119883 + [119870

119905119903] 119883

119903 +

119898

sum

119895=1

119896119895

119889

119865119895

119905119901

[119880119895

119905

]

+

119898

sum

119895=1

119896119895

119889

(ℎ

2) 119865

119895

119905119901

[119880119895

119903

] 119903 = 119865

[119870119905119903]119879

119883 + [119870119903119903] 119883

119903 +

119898

sum

119895=1

119896119895

119889

119865119895

119905119901

[119880119895

119905

]

+

119898

sum

119895=1

119896119895

119889

(ℎ

2) 119865

119895

119905119901

[119880119895

119903

] 119903 = 0

(35)

4 Results and Discussion

In this section the numerical results obtained by the finiteelement model derived in the previous section have beenpresented Symmetric as well as antisymmetric cross-plyand antisymmetric angle-ply laminated skew substrate platesintegrated with the two patches of ACLD treatment areconsidered for presenting the numerical results As shown inFigure 1 the locations of the patches have been selected insuch a way that the energy dissipation corresponding to thefirst fewmodes becomesmaximum PZT-5Hspur compositewith 60 fiber volume fraction has been considered for thematerial of the constraining layer of the ACLD treatmentUsing themicromechanicsmodel and thematerial propertiesof the constituent phases [35] the effective elastic andpiezoelectric material properties of the 1ndash3 PZC with respectto the principle material coordinate system are computed asfollows

11986211

= 929GPa 11986212

= 618GPa 11986213

= 605GPa

11986233

= 3544GPa 11986223

= 11986213 119862

44= 158GPa

11986266

= 154GPa 11986255

= 11986244

11989031

= minus01902Cm2

11989032

= 11989031

11989033

= 184107Cm2

11989024

= 0004Cm2

11989015

= 11989024

(36)

The material of the orthotropic layers of the substrate platesis a graphiteepoxy composite and its material properties are[42]

119864119871= 172GPa 119864

119871

119864119879

= 25 119866119871119879

= 05119864119879

119866119879119879

= 02119864119879 ]

119871119879= ]

119879119879= 025

(37)

in which the symbols have the usual meaning Unless oth-erwise mentioned the aspect ratio (119886ℎ) and the thicknessof the skew substrate plates are considered as 100 and0003m respectively while the orthotropic layers of theskew substrate plates are of equal thickness Also the lengthand the skew sides of the substrate are equal (ie 119886 =

119887) The loss factor of the viscoelastic layer considered fornumerical results remains invariant [20] within a broadband (0ndash400Hz) of frequency range and the values of thecomplex shear modulus Poisson ratio and the density ofthe viscoelastic layer are considered as 20(1 + 119894)MNmminus2049 and 1140 kgmminus3 respectively [20] The thicknesses ofthe viscoelastic layer and the 1ndash3 PZC layer are consideredas 508120583m and 250 120583m respectively The value of the shearcorrection factor is used as 56 Three points of Gaussianintegration rule are considered for computing the elementmatrices corresponding to the bending deformations whiletwo points of that are used for computing the elementmatrices corresponding to transverse shear deformationsThe simply supported boundary conditions at the edges of theoverall plate are invoked as follows

at 119909 = 119910 tan120572 119909 = 119886 + 119910 tan120572

V1015840119900

= 1199081015840

119900

= 1205791015840

119910

= 1206011015840

119910

= 1205741015840

119910

= 1205791015840

119911

= 1205741015840

119911

= 0

at 119910 = 0 119910 = 119887 cos120572

119906119900= 119908

119900= 120579

119909= 120601

119909= 120574

119909= 120579

119911= 120574

119911= 0

(38)

To verify the validity of the present finite elementmodel (FEM) the natural frequencies of the skew laminatedcomposite plates integrated with the inactivated patches ofnegligible thickness are first computed and subsequentlycompared with the existing analytical results [41] of theidentical plates without being integrated with the patchesSuch comparisons for the simply supported and clamped-clamped skew laminated plates are presented in Tables 1 and2 respectively It may be observed from these tables that theresults are in excellent agreement validating the model of theplate derived here

In order to assess the performance of the verti-callyobliquely reinforced 1ndash3 PZC as the material of thedistributed actuators of smart skew laminated compositeplates frequency responses for ACLD of the substrate skewlaminated composite plates are investigated To compute the

10 Journal of Composites

Table 1 Comparison of fundamental natural frequencies (120596) of simply supported skew laminated composite plates integratedwith the patchesof negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 15699 28917 37325 14829 24656 32522 17974 33351 33351Present FEM 15635 24383 35033 15076 24380 32254 18493 33359 33370

15∘ Reference [41] 16874 30458 39600 15741 25351 30270 18313 32490 36724Present FEM 16571 29840 36505 15796 25775 29892 18675 32075 35810

30∘ Reference [41] 20840 34023 46997 18871 29372 34489 20270 34431 42361Present FEM 19596 31690 46796 18226 29585 32357 19894 32365 43208

45∘ Reference [41] 28925 41906 54149 25609 33126 40617 25609 33131 42772Present FEM 24811 44875 53289 22996 34773 44889 23194 34870 45009

lowast

120596 = (120596119886

2

120587

2

ℎ)radic120588119864119879 120596 is the circular natural frequency

Table 2 Comparison of fundamental natural frequencies (120596) of clamped-clamped skew laminated composite plates integrated with thepatches of negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 23687 35399 41122 22990 37880 37880 22119 37339 37339Present FEM 23201 34769 44102 23315 36531 36545 22433 36000 36012

15∘ Reference [41] 24663 36255 43418 23809 37516 40785 23099 36997 40438Present FEM 23699 34821 44049 23741 35856 38401 23049 35346 38092

30∘ Reference [41] 27921 39557 50220 26666 39851 47227 26325 39549 52107Present FEM 25366 35696 44734 25240 41943 45373 24945 36113 50932

45∘ Reference [41] 34739 47129 58789 33015 46290 58423 33015 46290 58423Present FEM 28665 47074 54562 28377 47614 56620 28377 47614 55162

frequency response functions the skew laminated compositeplates are harmonically excited by a force of amplitude 1Newton applied to a point (1198862 (1198864) cos120572 ℎ2) on the skewsubstrate platesThe control voltage supplied to the constrain-ing layer of the first patch is negatively proportional to thevelocity of the point (1198862 (1198864) cos120572 ℎ2) and that applied tothe constraining layer of the second patch is negatively pro-portional to the velocity of the point (1198862 (31198864) cos120572 ℎ2)

Figures 3 4 and 5 illustrate the frequency responsefunctions for the transverse displacement of a point of asimply supported symmetric cross-ply (0∘90∘0∘) plate with15∘ 30∘ and 45∘ skew angles respectively while the fiberorientation angle (120595) in the constraining layer is 0∘ Thesefigures display both uncontrolled and controlled responsesand clearly show that the active constraining layer made ofthe vertically reinforced 1ndash3 PZC material being studied heresignificantly attenuates the amplitude of vibrations enhanc-ing the damping characteristics of the overall laminatedskew composite plates over the passive damping (uncon-trolled) Similar responses are also obtained for antisym-metric cross-ply (0∘90∘0∘90∘) and antisymmetric angle-ply (45∘minus45∘45∘minus45∘) skew laminated composite plates

However for brevity these results are not presented hereThemaximum values of the control voltages required to computethe controlled responses presented in Figures 3ndash5 have beenfound to be quite low and are illustrated in Figure 6 in caseof the symmetric cross-ply (0∘90∘0∘) plate with skew angle15∘ only In order to examine the contribution of the verticalactuation in improving the damping characteristics of theskew laminated composite plates as demonstrated in Figures3ndash5 active control responses of the symmetric cross-plylaminated composite plate with 15∘ skew angle are plotted inFigure 7with andwithout considering the value of 119890

33and 119890

31

It may be noted that when the values of 11989031is zero and that of

11989033is nonzero the vertical actuation of the active constraining

layer of the ACLD treatment is responsible for increasing thetransverse shear deformations of the viscoelastic constrainedlayer over the passive counterpart resulting in improveddamping of the smart plate over its passive damping Onthe other hand if 119890

33is zero and 119890

31is nonzero the in-

plane actuation of the active constraining layer causes thetransverse shear deformation of the viscoelastic core of theACLD treatment leading to the active damping of the smartplate It is evident from Figure 7 that the contribution of the

Journal of Composites 11

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 3 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 4 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 30∘)

vertical actuation by the constraining vertically reinforced 1ndash3 PZC layer is significantly larger than that of the in-planeactuation by the same for controlling the modes displayedin Figure 7 Though not shown here similar results are alsoobtained for antisymmetric cross-ply and angle-ply skewplates

Figures 8 9 and 10 demonstrate the effect of variationof piezoelectric fiber orientation angle on the performanceof the patches for improving the damping characteristicsof simply supported symmetric cross-ply (0∘90∘0∘)

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 5 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 45∘)

0

5

10

15

20

25

30

35

40

45

50

Con

trol v

olta

ge (V

olt)

0 100 200 300 400 500 600Frequency (Hz)

119870119889 = 600

119870119889 = 1200

Figure 6 Frequency response functions for the control voltages foractive damping of the simply supported skew symmetric cross-ply(0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

antisymmetric cross-ply (0∘90∘0∘90∘) and angle-ply(45∘minus45∘45∘minus45∘) substrate skew composite plates with30∘ skew angle respectively while the orientation angle (120595)

of the piezoelectric fibers is varied in the vertical 119909119911-planeThese figures illustrate that the attenuating capability of thepatches becomesmaximumwhen the value of the orientationangle (120595) of the fibers in the 119909119911-plane is 0∘ Similar effectis also obtained when the fiber orientation angle (120595) in theconstraining layer of the ACLD treatment is varied in thevertical 119910119911-plane as illustrated in Figures 11 12 and 13 forthe symmetric cross-ply (0∘90∘0∘) antisymmetric cross-ply(0∘90∘0∘90∘) and angle-ply (45∘minus45∘45∘minus45∘) substrate

12 Journal of Composites

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

11989031 = minus01902 11989033 = 184107

11989031 = minus01902 11989033 = 011989031 = 0 11989033 = 184107

w(a2(b4)co

s120572h2

) (m

)

Figure 7 Frequency response functions demonstrating the con-tribution of vertical actuation on the controlled responses forthe transverse displacement 119908 (1198862 (1198864) cos120572 ℎ2) of the skewsymmetric cross-ply (0∘90∘0∘) plate (119870

119889

= 600 119886ℎ = 100 120595 = 0∘

120572 = 15∘)

0 100 200 300 400 500 600Frequency (Hz)

0

02

04

06

08

1

12times10minus4

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 8 Effect of variation of piezoelectric fiber orientation angle(120595) on the controlled responses of a simply supported symmetricskew cross-ply (0∘90∘0∘) plate when the piezoelectric fibers of theconstraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

skew composite plates with 30∘ skew angle respectivelyIn this regard it should be noted that for investigating theeffect of variation of piezoelectric fiber orientation angle(120595) on the performance of the ACLD patches the valueof 120595 has been smoothly varied to compute the controlledresponses However for the sake of clarity in the plots theresponses corresponding to the four specific values of 120595

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 9 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 10 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

have been presented such that the optimum performanceof the patches can be evaluated The significant effect of thevariation of the piezoelectric fiber orientation angle in thevertical 119909119911- and 119910119911-planes has been noticed when all theboundaries of the overall plates are clamped as shown inFigures 14ndash19 When the piezoelectric fibers are coplanarwith the vertical 119909119911-plane the frequency response functionsplotted in Figures 14 15 and 16 reveal that the control

Journal of Composites 13

0

02

04

06

08

1

12

14

16times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 11 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 12 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=600 119886ℎ = 100 120572 = 30

∘)

authority of the patches becomes maximum for improvingthe damping characteristics of the clamped-clamped skewcross-ply and angle-ply substrate plates if the value of thepiezoelectric fiber orientation angle (120595) is 30∘ The same isalso found to be true if the piezoelectric fibers are coplanarwith the vertical 119910119911-plane as shown in Figures 17 18 and 19for the previously mentioned skew plates Similar responsesare also found for the effect of piezoelectric fiber orientation

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 13 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers of the constraining layer are coplanar with the119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 14 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600119886ℎ = 100120572 = 30

∘)

angle on the active damping of other skew laminated plateswith different skew angles However for the sake of brevitythey are not presented here

5 Conclusions

In this paper a study has been carried out to investi-gate the performance of the ACLD treatment for active

14 Journal of Composites

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 15 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 16 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

damping of smart skew laminated composite plates whenverticallyobliquely reinforced 1ndash3 PZC materials are usedas the materials for the constraining layer of the ACLDtreatment A layerwise FSDT-based finite element modelhas been developed to describe the dynamics of the skewlaminated composite plates integrated with the patches ofACLD treatment Unlike the existing finite element modelsof smart structures integrated with ACLD treatment the

200 250 300 350 400 450 500 550 600 650 7000

02

04

06

08

1

12

14

16

18

2times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 17 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

200 300 400 500 600 700 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 18 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

derivation of the present finite element model includes thetransverse deformations of the substrate skew laminatedcomposite plate the constrained viscoelastic layer and theconstraining 1ndash3 PZC layer of the ACLD treatment along thethickness (ie 119911) direction such that both vertical and in-plane actuations by the constraining layer of the patches canbe utilized for active damping of the plates The frequency

Journal of Composites 15

0

05

1

15

2

25

3

35

4times10minus5

200 300 400 500 600 700 800Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 19 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

responses of the symmetric cross-ply antisymmetric cross-ply and angle-ply skew composite plates indicate that theactive constraining layer of the ACLD treatment being madeof the vertically reinforced 1ndash3 PZC material significantlyenhances the damping characteristics of the plates over thepassive damping The analysis revealed that if the verticallyreinforced 1ndash3 PZCmaterial is used for the constraining layerof the ACLD treatment then the contribution of the verticalactuation by the constraining layer alone for improving theactive damping characteristics of the smart skew laminatedcomposite plate is significantly larger than that due to thein-plane actuation by the constraining layer alone It isimportant to note from the present investigation that thevariation of the orientation angle of the piezoelectric fibersof the constraining layer in the vertical 119909119911- and 119910119911-planesas well as the skew angle and boundary conditions of theoverall plates significantly affects the performance of thepatches For the simply supported laminated substrate platesthe performance of the patches becomes maximumwhen thepiezoelectric fiber orientation angle (120595) is 0∘ while in caseof the clamped-clamped laminated skew substrate plates themaximum control authority of the patches is achieved whenthe value of 120595 is 30∘ irrespective of the cases in which thefibers are oriented in 119909119911- and 119910119911-planes

Appendix

In (6) and (7) the matrices [1198851] [119885

2] [119885

3] [119885

4] [119885

5] and

[1198856] are given by

[1198851] = [[119885

1] 0 0]

[1198852] = [(

ℎ

2) 119868 [119885

2] 0]

[1198853] = [(

ℎ

2) 119868 ℎV119868 [119885

3]]

[1198854] = [119868 0 0 119911119868 0 0]

[1198855] = [0 119868 0 (

ℎ

2) 119868 (119911 minus

ℎ

2) 119868 0]

[1198856] = [0 0 119868 (

ℎ

2) 119868 ℎV119868 (119911 minus ℎ

119873+2) 119868]

(A1)

in which

[1198851] =

[[[

[

119911 0 0 0

0 119911 0 0

0 0 119911 0

0 0 0 1

]]]

]

[1198852] =

[[[[[[[[

[

(119911 minusℎ

2) 0 0 0

0 (119911 minusℎ

2) 0 0

0 0 (119911 minusℎ

2) 0

0 0 0 1

]]]]]]]]

]

[1198853] =

[[[

[

(119911 minus ℎ119873+2

) 0 0 0

0 (119911 minus ℎ119873+2

) 0 0

0 0 (119911 minus ℎ119873+2

) 0

0 0 0 1

]]]

]

[119868] =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

[119868] = [1 0

0 1]

[0] = [0 0

0 0] [0] = [

0 0

0 0]

(A2)

The various submatrices 119861119905119887119894 119861

119903119887119894 and 119861

119903119904119894 appearing in (21)

are given by

119861119905119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 0

]]]]]]]]]]]

]

119861119905119904119894

=

[[[[

[

0 0120597120578

119894

120597119909

0 0120597120578

119894

120597119910

]]]]

]

16 Journal of Composites

119861119903119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 1

]]]]]]]]]]]

]

119861119903119887119894

=

[[[[[[

[

119861119903119887119894

0

0

0 119861119903119887119894

0

0

0 119861119903119887119894

]]]]]]

]

119861119903119904119894

=

[[[[[[[[[[[[[[[[[[

[

119868 0 0

0

119868 0

0 0

119868

119861119903119904119894

0 0

0 119861119903119904119894

0

0 0 119861119903119904119894

]]]]]]]]]]]]]]]]]]

]

119868= [1 0 0

0 1 0]

(A3)

wherein

0 and 0 are the (3 times 3) and (2 times 3) null matricesrespectively

References

[1] T Bailey and J E Hubbard ldquoDistributedpiezoelectric polymeractive vibration control of a cantilever beamrdquo Journal of Guid-ance Control and Dynamics vol 8 no 5 pp 605ndash611 1985

[2] E F Crawley and J de Luis ldquoUse of piezoelectric actuators aselements of intelligent structuresrdquo AIAA Journal vol 25 no 10pp 1373ndash1385 1987

[3] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988

[4] C K Lee W W Chiang and O Sulivan ldquoPiezoelectric modalsensoractuator pairs for critical active damping vibrationcontrolrdquo Journal of the Acoustical Society of America vol 90 no1 pp 374ndash384 1991

[5] S Hanagud M W Obal and A J Calise ldquoOptimal vibrationcontrol by the use of piezoceramic sensors and actuatorsrdquoJournal of Guidance Control and Dynamics vol 15 no 5 pp1199ndash1206 1992

[6] Y Gu R L Clark C R Fuller and A C Zander ldquoExperimentson active control of plate vibration using piezoelectric actuatorsand polyvinylidene fluoride (PVDF) modal sensorsrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 116 no3 pp 303ndash308 1994

[7] A Baz and S Poh ldquoOptimal vibration control with modalpositive position feedbackrdquo Optimal Control Applications andMethods vol 17 no 2 pp 141ndash149 1996

[8] A Kugi K Schlacher and H Irschik ldquoInfinite-dimensionalcontrol of nonlinear beam vibrations by piezoelectric actuatorand sensor layersrdquo Nonlinear Dynamics vol 19 no 1 pp 71ndash911999

[9] D Sun L Tong and D Wang ldquoVibration control of platesusing discretely distributed piezoelectric quasi-modal actua-torssensorsrdquo AIAA Journal vol 39 no 9 pp 1766ndash1772 2001

[10] G Caruso S Galeani and L Menini ldquoActive vibration controlof an elastic plate using multiple piezoelectric sensors andactuatorsrdquo SimulationModelling Practice andTheory vol 11 no5-6 pp 403ndash419 2003

[11] X Zhang and A G Erdman ldquoOptimal placement of piezo-electric sensors and actuators for controlled flexible linkagemechanismsrdquo Journal of Vibration and Acoustics Transactionsof the ASME vol 128 no 2 pp 256ndash260 2006

[12] G Gatti M J Brennan and P Gardonio ldquoActive dampingof a beam using a physically collocated accelerometer andpiezoelectric patch actuatorrdquo Journal of Sound and Vibrationvol 303 no 3ndash5 pp 798ndash813 2007

[13] M K Kwak S Heo and M Jeong ldquoDynamic modellingand active vibration controller design for a cylindrical shellequipped with piezoelectric sensors and actuatorsrdquo Journal ofSound and Vibration vol 321 no 3ndash5 pp 510ndash524 2009

[14] I Bruant L Gallimard and S Nikoukar ldquoOptimal piezoelectricactuator and sensor location for active vibration control usinggenetic algorithmrdquo Journal of Sound and Vibration vol 329 no10 pp 1615ndash1635 2010

[15] Y Fu J Wang and Y Mao ldquoNonlinear vibration and activecontrol of functionally graded beams with piezoelectric sensorsand actuatorsrdquo Journal of Intelligent Material Systems andStructures vol 22 no 18 pp 2093ndash2102 2011

[16] F Chen M Hong M Song and H Cui ldquoOptimal control ofa beam with discontinuously distributed piezoelectric sensorsand actuatorsrdquo Journal of Marine Science and Application vol11 no 1 pp 44ndash51 2012

[17] B Azvine G R Tomlinson and R J Wynne ldquoUse of activeconstrained-layer damping for controlling resonant vibrationrdquoSmart Materials and Structures vol 4 no 1 pp 1ndash6 1995

[18] A Baz and J Ro ldquoVibration control of plates with activeconstrained layer dampingrdquo SmartMaterials and Structures vol5 no 3 pp 272ndash280 1996

[19] C Chantalakhana and R Stanway ldquoActive constrained layerdamping of clamped-clamped plate vibrationsrdquo Journal ofSound and Vibration vol 241 no 5 pp 755ndash777 2001

[20] J Ro and A Baz ldquoOptimum placement and control ofactive constrained layer damping using modal strain energyapproachrdquo JVCJournal of Vibration and Control vol 8 no 6pp 861ndash876 2002

[21] M C Ray and J N Reddy ldquoOptimal control of thin circularcylindrical laminated composite shells using active constrainedlayer damping treatmentrdquo Smart Materials and Structures vol13 no 1 pp 64ndash72 2004

[22] C M A Vasques B R Mace P Gardonio and J D RodriguesldquoArbitrary active constrained layer damping treatments onbeams finite element modelling and experimental validationrdquoComputers and Structures vol 84 no 22-23 pp 1384ndash14012006

[23] F M Li K Kishimoto K Wang Z B Chen and W HHuang ldquoVibration control of beams with active constrainedlayer dampingrdquo Smart Materials and Structures vol 17 no 6Article ID 065036 2008

[24] L Yuan Y Xiang Y Huang and J Lu ldquoA semi-analyticalmethod and the circumferential dominant modal control ofcircular cylindrical shells with active constrained layer dampingtreatmentrdquo SmartMaterials and Structures vol 19 no 2 ArticleID 025010 2010

[25] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

Journal of Composites 17

[26] W A Smith and B A Auld ldquoModeling 1ndash3 composite piezo-electrics thickness-mode oscillationsrdquo IEEE Transactions onUltrasonics Ferroelectrics and Frequency Control vol 38 no 1pp 40ndash47 1991

[27] PiezocompositesMaterial Systems Inc 543 Great Road Little-ton MA 01460

[28] Y Benveniste and G J Dvorak ldquoUniform fields and universalrelations in piezoelectric compositesrdquo Journal of the Mechanicsand Physics of Solids vol 40 no 6 pp 1295ndash1312 1992

[29] M L Dunn and M Taya ldquoMicromechanics predictions ofthe effective electroelastic moduli of piezoelectric compositesrdquoInternational Journal of Solids and Structures vol 30 no 2 pp161ndash175 1993

[30] J Aboudi ldquoMicromechanical prediction of the effective coeffi-cients of thermo-piezoelectric multiphase compositesrdquo Journalof Intelligent Material Systems and Structures vol 9 no 9 pp713ndash722 1999

[31] X Ruan T W Chou A Safari and S C Danforth ldquoA 3-Dconnectivity model for effective piezoelectric properties of yarncompositesrdquo Journal of Composite Materials vol 36 no 14 pp1693ndash1708 2002

[32] A Baz and A Tempia ldquoActive piezoelectric damping compos-itesrdquo Sensors and Actuators A vol 112 no 2-3 pp 340ndash3502004

[33] H Berger S Kari U Gabbert et al ldquoUnit cell models ofpiezoelectric fiber composites for numerical and analytical cal-culation of effective propertiesrdquo SmartMaterials and Structuresvol 15 no 2 pp 451ndash458 2006

[34] M Sakthivel and A Arockiarajan ldquoAn analytical model forpredicting thermo-electro-mechanical response of 1ndash3 piezo-electric compositesrdquo Computational Materials Science vol 48no 4 pp 759ndash767 2010

[35] M C Ray and A K Pradhan ldquoOn the use of verticallyreinforced 1ndash3 piezoelectric composites for hybrid damping oflaminated composite platesrdquo Mechanics of Advanced Materialsand Structures vol 14 no 4 pp 245ndash261 2007

[36] M C Ray and A K Pradhan ldquoPerformance of vertically andobliquely reinforced 1ndash3 piezoelectric composites for activedamping of laminated composite shellsrdquo Journal of Sound andVibration vol 315 no 4-5 pp 816ndash835 2008

[37] S K Sarangi and M C Ray ldquoSmart damping of geometricallynonlinear vibrations of laminated composite beams using verti-cally reinforced 1ndash3 piezoelectric compositesrdquo Smart Materialsand Structures vol 19 no 7 Article ID 075020 2010

[38] A Krishnan and J V Deshpande ldquoVibration of skew laminatesrdquoJournal of Sound and Vibration vol 153 no 2 pp 351ndash358 1992

[39] A R Krishna Reddy and R Palaninathan ldquoFree vibration ofskew laminatesrdquo Computers and Structures vol 70 no 4 pp415ndash423 1999

[40] C S Babu and T Kant ldquoTwo shear deformable finite elementmodels for buckling analysis of skew fibre-reinforced compositeand sandwich panelsrdquo Composite Structures vol 46 no 2 pp115ndash124 1999

[41] A K Garg R K Khare and T Kant ldquoFree vibration of skewfiber-reinforced composite and sandwich laminates using ashear deformable finite element modelrdquo Journal of SandwichStructures and Materials vol 8 no 1 pp 33ndash53 2006

[42] J N Reddy Mechanics of Laminated Composites Plates Theoryand Analysis CRC press Boca Raton Fla USA 1996

Submit your manuscripts athttpwwwhindawicom

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materials

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Journal ofNanomaterials

Page 2: Research Article Active Constrained Layer Damping of Smart …downloads.hindawi.com/archive/2013/824163.pdf · 2019-07-31 · Research Article Active Constrained Layer Damping of

2 Journal of Composites

they are directly bonded to the flexible host structures Ifthe constraining piezoelectric layer of the ACLD treatmentis not activated with the control voltage the treatment causesthe standard passive constrained layer damping of the smartstructure Thus the ACLD treatment being under operationprovides the attributes of both passive and active dampingsand acts as an inbuilt fail-safemechanism Since its inceptionextensive research has been carried out to investigate theperformance of the ACLD treatment for active damping ofsmart structures [18ndash25]

Piezoelectric composite (PZC) materials have beenemerged as the new class of smart materials Such PZCmaterials are composed of piezoelectric fiber reinforcementsand epoxy matrix These PZC materials provide wide rangeof effective material properties good conformability andstrength integrity [26] Among the various PZC materialsstudied by the researchers the vertically and the obliquelyreinforced 1ndash3 PZCmaterials are commercially available [27]and are being effectively used for underwater transducersmedical imaging applications and high frequency ultrasonictransducers [26]The constructional feature of a laminamadeof the vertically reinforced 1ndash3 PZC material is that thepiezoelectric fibers are vertically aligned across the thicknessof the lamina In case of the obliquely reinforced 1ndash3 PZCthe piezoelectric fibers are obliquely aligned in the verticalplane across the thickness of the lamina These PZCs arecharacterized by improvedmechanical performance electro-mechanical coupling characteristics and acoustic impedancematching over the existingmonolithic piezoelectricmaterials[26] Research on PZC materials is mainly concerned withthe micromechanical analysis of these materials [28ndash34]Recently Ray and his coworkers [35ndash37] investigated theperformance of these 1ndash3 PZC materials for active dampingof linear and nonlinear vibrations of composite beams platesand shells

Skew laminated composite plates are widely used inengineering applications such as aircraft wings and marineindustries and are highly prone to undergoing vibrationsResearchers extensively investigated the free vibrational char-acteristics of such skew plates For example Krishnan andDeshpande [38] carried out the free vibration analyses of can-tilevered skew isotropic platesand three-layered symmetriccross-ply laminates by deriving two finite element modelsKrishna Reddy and Palaninathan [39] developed a generalhigh precision triangular plate bending finite element for thefree vibration analysis of laminated skew plates Babu andKant [40] developed shear deformable finite element modelsfor the buckling analysis of skew composite plates and panelsGarg et al [41] developed a simple C∘ isoparametric finiteelement model based on a higher order shear deformationtheory for the free vibration analysis of isotropic orthotropicand layered anisotropic composite and sandwich skew lami-nates

The review of the existing literature on the skew platesreveals that the attention has not yet been focused on investi-gating the active control of vibrations of skew laminated com-posite plates using piezoelectric composites In this paperauthors intend to investigate the active constrained layerdamping (ACLD) of skew laminated composite plates For

such investigation three-dimensional analysis of ACLD ofskew laminated composite plates integrated with the patchesof ACLD treatment has been carried out by the finite elementmethod The constraining layer of the ACLD treatment isconsidered to be made of the vertically or the obliquelyreinforced 1ndash3 PZC materials Particular emphasis has beenplaced on investigating the effect of variation of piezoelectricfiber orientation angle on the performance of the ACLDpatches for controlling the vibrations of the skew laminatedcomposite plates

2 Finite Element Model of Smart SkewLaminated Composite Plate

Figure 1(a) illustrates the schematic diagram of a smartskew laminated composite plate composed of 119873 numberof orthotropic layers The length and the skew width ofthe plate are denoted by 119886 and 119887 respectively The topsurface of the plate is integrated with the skewed patches ofthe ACLD treatment The constraining layer of the ACLDtreatment is made of the verticallyobliquely reinforced 1ndash3 piezoelectric composite (PZC) material The notations ℎℎ119901 and ℎV represent the thicknesses of the substrate skew

laminated plate the piezoelectric composite layer and theviscoelastic layer respectively The skew angle of the plate isdenoted by 120572 The middle plane of the substrate compositeplate is considered as the reference plane The origin of thelaminate coordinate system (119909119910119911) is located at one cornerof the reference plane such that the lines 119910 = 0 and 119910 =

119887 cos120572 represent the two opposite boundaries of the substrateskew composite plates while the lines 119909 = 119910 tan120572 and119909 = 119886 + 119910 tan120572 describe the two opposite skewed edgesof the plate Denoting by 119896 (= 1 2 3 119873 + 2) the layernumber of any layer of the overall plate the thicknesscoordinates 119911 of the top and the bottom surfaces of any(119896th) layer are represented by ℎ

119896+1and ℎ

119896 respectively The

fiber orientation angle in any layer of the substrate plate inthe plane (119909119910) of the lamina with respect to the laminatecoordinate system is denoted by 120579 The piezoelectric fibers inthe constraining layer made of the obliquely reinforced 1ndash3PZC material are coplanar with the vertical 119909119911- or 119910119911-planemaking an angle 120595 with respect to the 119911-axis as shown inFigure 1(b) If the value of120595 is 0∘ the layer becomes verticallyreinforced 1ndash3 PZC layer The overall skew composite platebeing studied here is thin and consequently the first ordershear deformation theory (FSDT) can be used to model theaxial displacements in all the layers of the overall plate InFigure 2 the kinematics of axial deformations of the overallplate based on the FSDT has been illustrated Displayed inthis figure the variables 119906

0and V

0represent the generalized

translational displacement of a point (119909 119910) on the referenceplane (119911 = 0) along 119909- and 119910-directions respectively 120579

119909

120601119909 and 120574

119909denote the generalized rotations of the normal to

the middle planes of the substrate plate the viscoelastic layerand the 1ndash3 PZC layer respectively about the 119910-axis while120579119910 120601

119910and 120574

119910represent the generalized rotations of the same

about the 119909-axis respectively According to the kinematics ofdeformations illustrated in Figure 2 the axial displacements

Journal of Composites 3

120572119884998400

119883998400

119885998400 119885

ℎ2ℎ

1198864 1198862

1198874

1198872

Patch 2Skew laminated composite plate

1198864

1198874

Viscoelastic layer

Patch 1

119883

119884

120572 Vertically reinforced 1ndash3 PZC layer

(a)

119911

119910 119910

119909 119909

Surface electrode Surface electrode

Piezoelectric fiber Epoxy matrix

Epoxy matrixPiezoelectric fiber

119911

120595 120595

(b)

Figure 1 (a) Schematic representation of a skew laminated composite plate integrated with the patches of ACLD treatment composed ofverticallyobliquely reinforced 1ndash3 piezoelectric composite constraining layer (b) Schematic diagram of layers of obliquely reinforced 1ndash3piezoelectric composite

Viscoelastic layer Viscoelastic layerSkew laminated plate

119911

ℎ2 ℎ2

ℎ ℎ

Undeformed transverse section Undeformed transverse section

Deformed transverse

119910

sectionDeformed transverse

section

ℎ ℎ

119911

120579119909

120579119910

119909

1199060

120574119910

120601119910

120601119909

120574119909

Skew laminated plate

ℎ1199011ndash3 PZC layer

1199080

0

1ndash3 PZC layer

1199080

Figure 2 Kinematics of deformations

119906 and V at any point in any layer of the overall plate along 119909-and 119910-directions respectively can be expressed as

119906 (119909 119910 119911 119905) = 1199060(119909 119910 119905) + (119911 minus ⟨119911 minus

ℎ

2⟩)120579

119909(119909 119910 119905)

+ (⟨119911 minusℎ

2⟩ minus ⟨119911 minus ℎ

119873+2⟩) 120601

119909(119909 119910 119905)

+ ⟨119911 minus ℎ119873+2

⟩ 120574119909(119909 119910 119905)

(1)

V (119909 119910 119911 119905) = V0(119909 119910 119905) + (119911 minus ⟨119911 minus

ℎ

2⟩)120579

119910(119909 119910 119905)

+ (⟨119911 minusℎ

2⟩ minus ⟨119911 minus ℎ

119873+2⟩) 120601

119910(119909 119910 119905)

+ ⟨119911 minus ℎ119873+2

⟩ 120574119910(119909 119910 119905)

(2)

in which a function within the bracket ⟨sdot⟩ represents anappropriate singularity function which satisfies the continu-ity of displacements between two adjacent continua Since thetransverse actuation of the constraining layer of the ACLDtreatment will be used for the flexural vibration controlof the plate the transverse normal strain in the overallplate must be considered in the model Hence as the plateconsidered here is thin the transverse displacements (119908) atany point in the substrate plate the viscoelastic layer and

4 Journal of Composites

the 1ndash3 PZC layer are assumed to be linearly varying acrosstheir thicknesses Thus similar to the axial displacement thetransverse displacements at any point in the overall plate canbe expressed as

119908 (119909 119910 119911 119905) = 1199080(119909 119910 119905) + (119911 minus ⟨119911 minus

ℎ

2⟩)120579

119911(119909 119910 119905)

+ (⟨119911 minusℎ

2⟩ minus ⟨119911 minus ℎ

119873+2⟩) 120601

119911(119909 119910 119905)

+ ⟨119911 minus ℎ119873+2

⟩ 120574119911(119909 119910 119905)

(3)

in which1199080refers to the transverse displacement at any point

on the reference plane and 120579119911 120601

119911 and 120574

119911are the generalized

displacements representing the gradients of the transversedisplacement in the substrate plate the viscoelastic layer andthe 1ndash3 PZC layer respectively with respect to the thicknesscoordinate (119911)

For the ease of analysis the generalized displacementvariables are grouped into the following two vectors

119889119905 = [119906

0V0

1199080]119879

119889119903 = [120579

119909120579119910

120579119911

120601119909

120601119910

120601119911

120574119909

120574119910

120574119911]119879

(4)

In order to implement the selective integration rule forcomputing the element stiffness matrices corresponding tothe transverse shear deformations the state of strain at any

point in the overall plate is divided into the following twostrain vectors 120576

119887 and 120576

119904

120576119887 = [120576

119909120576119910

120576119909119910

120576119911]119879

120576119904 = [120576

119909119911120576119910119911

]119879 (5)

in which 120576119909 120576

119910 and 120576

119911are the normal strains along 119909-

119910- and 119911-directions respectively 120576119909119910

is the in-plane shearstrain 120576

119909119911and 120576

119910119911are the transverse shear strains Using the

displacement fields given by (1)ndash(3) and the linear strain-displacement relations the vectors 120576

119887119888

120576119887V and 120576

119887119901

defining the state of in-plane and transverse normal strainsat any point in the substrate skew composite plate the vis-coelastic layer and the active constraining layer respectivelycan be expressed as

120576119887119888

= 120576119887119905 + [119885

1] 120576

119887119903 120576

119887V = 120576

119887119905 + [119885

2] 120576

119887119903

120576119887119901

= 120576119887119905 + [119885

3] 120576

119887119903

(6)

Similarly the vectors 120576119904119888

120576119904V and 120576

119904119901

defining the stateof transverse shear strains at any point in the substrate com-posite plate the viscoelastic layer and the active constraininglayer respectively can be expressed as

120576119904119888

= 120576119904119905 + [119885

4] 120576

119904119903 120576

119904V = 120576

119904119905 + [119885

5] 120576

119904119903

120576119904119901

= 120576119904119905 + [119885

6] 120576

119904119903

(7)

The various matrices appearing in (6) and (7) have beendefined in the Appendix while the generalized strain vectorsare given by

120576119887119905 = [

1205971199060

120597119909

120597V0

120597119910

1205971199060

120597119910+

120597V0

1205971199090]

119879

120576119904119905 = [

1205971199080

120597119909

1205971199080

120597119910]

119879

120576119887119903 = [

120597120579119909

120597119909

120597120579119910

120597119910

120597120579119909

120597119910+

120597120579119910

120597119909120579119911

120597120601119909

120597119909

120597120601119910

120597119910

120597120601119909

120597119910+

120597120601119910

120597119909120601119911

120597120574119909

120597119909

120597120574119910

120597119910

120597120574119909

120597119910+

120597120574119910

120597119909120574119911

]

119879

120576119904119903 = [120579

119909120579119910

120601119909

120601119910

120574119909

120574119910

120597120579119911

120597119909

120597120579119911

120597119910

120597120601119911

120597119909

120597120601119911

120597119910

120597120574119911

120597119909

120597120574119911

120597119910]

119879

(8)

Similar to the strain vectors given by (5) the state of stressesat any point in the overall plat is described by the followingstress vectors

120590119887 = [120590

119909120590119910

120590119909119910

120590119911]119879

120590119904 = [120590

119909119911120590119910119911

]119879

(9)

where 120590119909 120590

119910 and 120590

119911are the normal stresses along 119909- 119910-

and 119911-directions respectively 120590119909119910is the in-plane shear stress

120590119909119911

and 120590119910119911

are the transverse shear stresses The constitutive

relations for the material of any orthotropic layer of thesubstrate plate are given by

120590119896

119887

= [119862119896

119887

] 120576119896

119887

120590119896

119904

= [119862119896

119904

] 120576119896

119904

(119896 = 1 2 3 119873)

(10)

Journal of Composites 5

where the elastic coefficient matrices are

[119862119896

119887

] =

[[[[[[[[[[[[[[

[

119862119896

11

119862119896

12

119862119896

16

119862119896

13

119862119896

12

119862119896

22

119862119896

26

119862119896

23

119862119896

16

119862119896

26

119862119896

66

119862119896

36

119862119896

13

119862119896

23

119862119896

36

119862119896

33

]]]]]]]]]]]]]]

]

[119862119896

119904

] =[[

[

119862119896

55

119862119896

45

119862119896

45

119862119896

44

]]

]

(11)

and 119862119896

119894119895

(119894 119895 = 1 2 3 6) are the transformed elasticcoefficients with respect to the reference coordinate systemThematerial of the viscoelastic layer is assumed to be linearlyviscoelastic and homogenous isotropic and is modeled byusing the complex modulus approach Thus the shear mod-ulus119866 and Youngrsquos modulus 119864 of the viscoelastic material aregiven by

119866 = 1198661015840

(1 + 119894120578) 119864 = 2119866 (1 + ]) (12)

in which 1198661015840 is the storage modulus ] is Poissonrsquos ratio and

120578 is the loss factor at a particular operating temperature andfrequency Employing the complex modulus approach theconstitutive relations for the material of the viscoelastic layer(119896 = 119873 + 1) can also be represented by (10) with 119862

119896

119894119895

(119894 119895 =

1 2 3 6) being the complex elastic constants [29 30]The constraining PZC layer will be subjected to the appliedelectric field (119864

119911) acting across its thickness (ie along the

119911-direction) only Accordingly the constitutive relations forthe 1ndash3 PZC material with respect to the laminate coordinatesystem (119909119910119911) can be expressed as

120590119896

119887

= [119862119896

119887

] 120576119896

119887

+ [119862119896

119887119904

] 120576119896

119904

minus 119890119887 119864

119911

120590119896

119904

= [119862119896

119887119904

] 120576119896

119887

+ [119862119896

119904

] 120576119896

119904

minus 119890119904 119864

119911

119863119911= 119890

119887119879

120576119896

119887

+ 119890119904119879

120576119896

119904

+ 12057633119864119911

119896 = 119873 + 2

(13)

Here 119863119911represents the electric displacement along the 119911-

direction and 12057633

is the dielectric constant It may be notedfrom the previous form of the constitutive relations that thetransverse shear strains are coupled with the in-plane normalstrains due to the orientation of piezoelectric fibers in the

vertical 119909119911- or 119910119911-plane and the corresponding couplingelastic constant matrices [119862

119873+2

119887119904

] are given by

[119862119873+2

119887119904

] =

[[[[[[[[[[

[

119862119873+2

15

0

119862119873+2

25

0

0 119862119873+2

46

119862119873+2

35

0

]]]]]]]]]]

]

or

[119862119873+2

119887119904

] =

[[[[[[[[[[

[

0 119862119873+2

14

0 119862119873+2

24

119862119873+2

56

0

0 119862119873+2

34

]]]]]]]]]]

]

(14)

as the piezoelectric fibers are coplanar with the vertical 119909119911-or 119910119911-plane respectively Note that if the fibers are coplanarwith both 119909119911- and 119910119911-planes (ie 120595 = 0

∘) this couplingmatrix becomes a nullmatrix Also the piezoelectric constantmatrices 119890

119887 and 119890

119904 appearing in (13) contain the following

transformed effective piezoelectric coefficients of the 1ndash3PZC

119890119887 = [119890

3111989032

11989036

11989033]119879

119890119904 = [119890

3511989034]119879

(15)

The total potential energy 119879119901and the kinetic energy 119879

119896of the

overall plateACLD system are given by [35]

119879119901

=1

2[

119873+2

sum

119896=1

intΩ

(120576119896

119887

119879

120590119896

119887

+ 120576119903

119904

119879

120590119896

119904

) 119889Ω

minusintΩ

119863119911119864119911119889Ω] minus int

119860

119889119879

119891 119889119860

119879119896=

1

2

119873+2

sum

119896=1

intΩ

120588119896

(2

+ V2 + 2

) 119889Ω

(16)

in which 120588119896 is the mass density of the 119896th layer 119891 is the

externally applied surface traction acting over a surface area119860 andΩ represents the volume of the concerned layer Sincethe plates under study are considered to be thin the rotaryinertia of the overall plate has been neglected in estimatingthe kinetic energy The overall plate is discretized by eight-noded isoparametric quadrilateral elements Following (4)the generalized displacement vectors associated with the 119894th(119894 = 1 2 3 8) node of the element can be written as

119889119905119894 = [119906

0119894V0119894

1199080119894]119879

119889119903119894 = [120579

119909119894120579119910119894

120579119911119894

120601119909119894

120601119910119894

120601119911119894

120574119909119894

120574119910119894

120574119911119894]119879

(17)

6 Journal of Composites

Thus the generalized displacement vectors at any pointwithin the element can be expressed in terms of the nodalgeneralized displacement vectors 119889

119890

119905

and 119889119890

119903

as follows

119889119890

119905

= [119873119905] 119889

119890

119905

119889119890

119903

= [119873119903] 119889

119890

119903

(18)

in which

119889119890

119905

= [119889119890

1199051

119879

119889119890

1199052

119879

sdot sdot sdot 119889119890

1199058

119879

]119879

119889119890

119903

= [119889119890

1199031

119879

119889119890

1199032

119879

sdot sdot sdot 119889119890

1199038

119879

]119879

[119873119905] = [119873

11990511198731199052

sdot sdot sdot 1198731199058]119879

[119873119903] = [119873

11990311198731199032

sdot sdot sdot 1198731199038]119879

119873119905119894

= 119899119894119868119905 119873

119903119894= 119899

119894119868119903

(19)

while 119868119905and 119868

119903are (3times3) and (9times 9) identitymatrices respec-

tively and 119899119894is the shape function of natural coordinates

associatedwith the 119894th nodeMaking use of the relations givenby (6)ndash(8) and (18) the strain vectors at any point within theelement can be expressed in terms of the nodal generalizeddisplacement vectors as follows

120576119887119888

= [119861119905119887] 119889

119890

119905

+ [1198851] [119861

119903119887] 119889

119890

119903

120576119887V = [119861

119905119887] 119889

119890

119905

+ [1198852] [119861

119903119887] 119889

119890

119903

120576119887119901

= [119861119905119887] 119889

119890

119905

+ [1198853] [119861

119903119887] 119889

119890

119903

120576119904119888

= [119861119905119904] 119889

119890

119905

+ [1198854] [119861

119903119904] 119889

119890

119903

120576119904V = [119861

119905119904] 119889

119890

119905

+ [1198855] [119861

119903119904] 119889

119890

119903

120576119904119901

= [119861119905119904] 119889

119890

119905

+ [1198856] [119861

119903119904] 119889

119890

119903

(20)

in which the nodal strain-displacement matrices [119861119905119887] [119861

119903119887]

[119861119905119904] and [119861

119903119904] are given by

[119861119905119887] = [119861

11990511988711198611199051198872

sdot sdot sdot 1198611199051198878

]

[119861119903119887] = [119861

11990311988711198611199031198872

sdot sdot sdot 1198611199031198878

]

[119861119905119904] = [119861

11990511990411198611199051199042

sdot sdot sdot 1198611199051199048

]

[119861119903119904] = [119861

11990311990411198611199031199042

sdot sdot sdot 1198611199031199048

]

(21)

The submatrices of [119861119905119887] [119861

119903119887] [119861

119905119904] and [119861

119903119904] have been

explicitly presented in the Appendix On substitution of (9)ndash(13) and (20) into (16) the total potential energy 119879

119890

119901

and the

kinetic energy 119879119890

119896

of a typical element augmented with theACLD treatment can be expressed as

119879119890

119901

=1

2[119889

119890

119905

119879

[119870119890

119905119905

] 119889119890

119905

+ 119889119890

119905

119879

[119870119890

119905119903

] 119889119890

119903

+ 119889119890

119903

119879

[119870119890

119905119903

]119879

times 119889119890

119905

+ 119889119890

119903

119879

[119870119890

119903119903

] 119889119890

119903

minus 2119889119890

119905

119879

119865119890

119905119901

119881

minus 2119889119890

119903

119879

119865119890

119903119901

119881 minus 2119889119890

119905

119879

119865119890

minus 12057633

1198812

ℎ119901

]

(22)

119879119890

119896

=1

2 119889

119890

119905

119879

[119872119890

] 119889119890

119905

(23)

In (22)119881 represents the voltage difference applied across thethickness of the PZC layer The elemental stiffness matrices[119870

119890

119905119905

] [119870119890

119905119903

] and [119870119890

119903119903

] the elemental electroelastic couplingvectors 119865

119890

119905119901

and 119865119890

119903119901

the elemental load vector 119865119890

andthe elemental mass matrix [119872

119890

] appearing in (22) and (23)are given by

[119870119890

119905119905

] = [119870119890

119905119887

] + [119870119890

119905119904

] + [119870119890

119905119887119904

]119901119887

+ [119870119890

119905119887119904

]119901119904

[119870119890

119905119903

] = [119870119890

119905119903119887

] + [119870119890

119905119903119904

]

+1

2([119870

119890

119905119903119887119904

]119901119887

+ [119870119890

119903119905119887119904

]119879

119901119887

+ [119870119890

119905119903119887119904

]119901119904

+ [119870119890

119903119905119887119904

]119879

119901119904

)

[119870119890

119903119905

] = [119870119890

119905119903

]119879

[119870119890

119903119903

] = [119870119890

119903119903119887

] + [119870119890

119903119903119904

] + [119870119890

119903119903119887119904

]119901119887

+ [119870119890

119903119903119887119904

]119901119904

119865119890

119905119901

= 119865119890

119905119887

119901

+ 119865119890

119905119904

119901

119865119890

119903119901

= 119865119890

119903119887

119901

+ 119865119890

119903119904

119901

119865119890

= int

119886119890

0

int

119887119890

0

[119873119905]119879

119891 119889119909 119889119910

[119872119890

] = int

119886119890

0

int

119887119890

0

119898[119873]119879

[119873] 119889119909 119889119910 119898 =

119873+2

sum

119896=1

120588119896

(ℎ119896+1

minus ℎ119896)

(24)

where

[119870119890

119905119887

] = int119860

[119861119905119887]119879

([119863119905119887] + [119863

119905119887]V + [119863

119905119887]119901

) [119861119905119887] 119889119909 119889119910

[119870119890

119905119904

] = int119860

[119861119905119904]119879

([119863119905119904] + [119863

119905119904]V + [119863

119905119904]119901

) [119861119905119904] 119889119909 119889119910

[119870119890

119905119887119904

]119901119887

= int119860

[119861119905119887]119879

[119863119905119887119904

]119901

[119861119905119904] 119889119909 119889119910

[119870119890

119905119887119904

]119901119904

= int119860

[119861119905119904]119879

[119863119905119887119904

]119901

[119861119905119887] 119889119909 119889119910

[119870119890

119905119903119887

] = int119860

[119861119905119903119887

]119879

([119863119905119903119887

] + [119863119905119903119887

]V + [119863119905119903119887

]119901

)

times [119861119905119903119887

] 119889119909 119889119910

Journal of Composites 7

[119870119890

119905119903119887119904

]119901119887

= int119860

[119861119905119887]119879

[119863119905119903119887119904

]119901

[119861119903119904] 119889119909 119889119910

[119870119890

119903119905119887119904

]119901119887

= int119860

[119861119903119887]119879

[119863119903119905119887119904

]119901

[119861119905119904] 119889119909 119889119910

[119870119890

119905119903119887119904

]119901119904

= int119860

[119861119905119904]119879

[119863119903119905119887119904

]119901

[119861119903119887] 119889119909 119889119910

[119870119890

119903119905119887119904

]119901119904

= int119860

[119861119903119904]119879

[119863119905119903119887119904

]119901

[119861119905119887] 119889119909 119889119910

[119870119890

119905119903119904

] = int119860

[119861119905119903119904

]119879

([119863119905119903119904

] + [119863119905119903119904

]V + [119863119905119903119904

]119901

)

times [119861119905119903119904

] 119889119909 119889119910

[119870119890

119903119903119887

] = int119860

[119861119903119903119887

]119879

([119863119903119903119887

] + [119863119903119903119887

]V + [119863119903119903119887

]119901

)

times [119861119903119903119887

] 119889119909 119889119910

[119870119890

119903119903119904

] = int119860

[119861119903119903119904

]119879

([119863119903119903119904

] + [119863119903119903119904

]V + [119863119903119903119904

]119901

)

times [119861119903119903119904

] 119889119909 119889119910

[119870119890

119903119903119887119904

]119901119887

= int119860

[119861119903119887]119879

[119863119903119903119887119904

]119901

[119861119903119904] 119889119909 119889119910

[119870119890

119903119903119887119904

]119901119904

= int119860

[119861119903119904]119879

[119863119903119903119887119904

]119901

[119861119903119887] 119889119909 119889119910

119865119890

119905119887

119901

= int119860

[119861119905119887]119879

119863119905119887119901

119889119909 119889119910

119865119890

119905119904

119901

= int119860

[119861119905119904]119879

119863119905119904119901

119889119909 119889119910

119865119890

119903119904

119901

= int119860

[119861119903119904]119879

119863119903119904119901

119889119909 119889119910

(25)

The various rigidity matrices and the rigidity vectors forelectro-elastic coupling appearing in the previous elementalmatrices are given by

[119863119905119887] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] 119889119911

[119863119905119903119887

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] [1198851] 119889119911

[119863119903119903119887

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[1198851]119879

[119862119896

119887

] [1198851] 119889119911

[119863119905119904] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119904

] 119889119911

[119863119905119903119904

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] [1198854] 119889119911

[119863119903119903119904

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[1198854]119879

[119862119896

119887

] [1198854] 119889119911

[119863119905119887]V = ℎV [119862

119873+1

119887

]

[119863119905119903119887

]V = int

ℎ119873+2

ℎ119873+1

[119862119873+1

119887

] [1198852] 119889119911

[119863119903119903119887

]V = int

ℎ119873+2

ℎ119873+1

[1198852]119879

[119862119873+1

119887

] [1198852] 119889119911

[119863119905119904]V = ℎV [119862

119873+1

119904

]

[119863119905119903119904

]V = int

ℎ119873+2

ℎ119873+1

[119862119873+1

119904

] [1198855] 119889119911

[119863119903119903119904

]V = int

ℎ119873+2

ℎ119873+1

[1198855]119879

[119862119873+1

119887

] [1198855] 119889119911

[119863119905119887]119901

= ℎ119901[119862

119873+2

119887

]

[119863119905119903119887

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887

] [1198853] 119889119911

[119863119903119903119887

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+2

119887

] [1198853] 119889119911

[119863119905119904]119901

= ℎ119901[119862

119873+2

119904

]

[119863119905119903119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119904

] [1198856] 119889119911

[119863119903119903119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198856]119879

[119862119873+2

119887

] [1198856] 119889119911

[119863119905119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887119904

] 119889119911

[119863119905119903119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887119904

] [1198856] 119889119911

[119863119903119905119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+2

119887119904

] 119889119911

[119863119903119903119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+3

119887119904

] [1198856] 119889119911

119863119905119887119901

= int

ℎ119873+3

ℎ119873+2

minus119890119887

ℎ119901

119889119911

119863119903119887119901

= int

ℎ119873+3

ℎ119873+2

minus[119885

3]119879

119890119887

ℎ119901

119889119911

119863119905119904119901

= int

ℎ119873+3

ℎ119873+2

minus119890119904

ℎ119901

119889119911

119863119903119904119901

= int

ℎ119873+3

ℎ119873+2

minus[119885

6]119879

119890119904

ℎ119901

119889119911

(26)

8 Journal of Composites

Applying the dynamic version of the virtual work principle[32] the following open-loop equations of motion for anelement of the overall plate integrated with the ACLDtreatment are obtained

[119872119890

] 119889119890

119905

+ [119870119890

119905119905

] 119889119890

119905

+ [119870119890

119905119903

] 119889119890

119903

= 119865119890

119905119901

119881 + 119865119890

[119870119890

119905119903

]119879

119889119890

119905

+ [119870119890

119903119903

] 119889119890

119903

= 119865119890

119903119901

119881

(27)

Since the elastic constant matrix of the viscoelastic layer iscomplex the stiffness matrices of an element integrated withthe ACLD treatment are complex In case of an elementnot integrated with the ACLD treatment the electro-elasticcoupling matrices become null vectors and the elementalstiffnessmatrices are to be computed without considering thepiezoelectric and viscoelastic layers It should also be notedthat as the stiffness matrices associated with the transverseshear strains are derived separately one can employ theselective integration rule in a straight forward manner toavoid shear locking problem in case of thin plates Therestrained boundary conditions at the skew edges of theplate are such that the displacements of the points locatedat the skew edges are to be restrained along 119909

1015840- 1199101015840- and

1199111015840-directions (Figure 1(a)) Hence in order to impose theboundary conditions in a straight forward manner thegeneralized displacement vectors of a point lying on the skewedge are to be transformed as follows

119889119905 = [119871

119905] 119889

1015840

119905

119889119903 = [119871

119903] 119889

1015840

119903

(28)

where 1198891015840

119905

and 1198891015840

119903

are the generalized displacement vectorsof the point with respect to 119909

1015840

1199101015840

1199111015840 coordinate system and are

given by

1198891015840

119905

= [1199061015840

119900

V1015840119900

1199081015840

119900

]119879

1198891015840

119903

= [1205791015840

119909

1205791015840

119910

1205791015840

119911

1206011015840

119909

1206011015840

119910

1206011015840

119911

1205741015840

119911

1205741015840

119910

1205741015840

119911]119879

(29)

Also the transformation matrices [119871119905] and [119871

119903] are given by

[119871119905] = [

[

119888 119904 0

minus119904 119888 0

0 0 1

]

]

[119871119903] =

[[[[[[[[[[[[

[

119888 119904 0 0 0 0 0 0 0

minus119904 119888 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 119888 119904 0 0 0 0

0 0 0 minus119904 119888 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 119888 119904 0

0 0 0 0 0 0 minus119904 119888 0

0 0 0 0 0 0 0 0 1

]]]]]]]]]]]]

]

(30)

in which 119888 = cos120572 and 119904 = sin120572Thus the elementalmatricesof the element containing the nodes lying on the skew edgeare to be augmented as follows

[119870119890

119905119905

] = [1198791]119879

[119870119890

119905119905

] [1198791] [119870

119890

119905119903

] = [1198791]119879

[119870119890

119905119903

] [1198792]

[119870119890

119903119903

] = [1198792]119879

[119870119890

119903119903

] [1198792] [119872

119890

] = [1198791]119879

[119872119890

] [1198791]

(31)

The forms of the transformation matrices [1198791] and [119879

2] are

given by

[1198791] =

[[[[[[[[[[[

[

[119871119905] 119900 119900 119900 119900 119900 119900 119900

119900 [119871119905] 119900 119900 119900 119900 119900 119900

119900 119900 [119871119905] 119900 119900 119900 119900 119900

119900 119900 119900 [119871119905] 119900 119900 119900 119900

119900 119900 119900 119900 [119871119905] 119900 119900 119900

119900 119900 119900 119900 119900 [119871119905] 119900 119900

119900 119900 119900 119900 119900 119900 [119871119905] 119900

119900 119900 119900 119900 119900 119900 119900 [119871119905]

]]]]]]]]]]]

]

[1198792] =

[[[[[[[[[[[

[

[119871119903] 119900 119900 119900 119900 119900 119900 119900

119900 [119871119903] 119900 119900 119900 119900 119900 119900

119900 119900 [119871119903] 119900 119900 119900 119900 119900

119900 119900 119900 [119871119903] 119900 119900 119900 119900

119900 119900 119900 119900 [119871119903] 119900 119900 119900

119900 119900 119900 119900 119900 [119871119903] 119900 119900

119900 119900 119900 119900 119900 119900 [119871119903] 119900

119900 119900 119900 119900 119900 119900 119900 [119871119903]

]]]]]]]]]]]

]

(32)

in which 119900 and 119900 are (3 times 3) and (9 times 9) null matricesrespectively It may be noted that in case of a node ofsuch element which does not lie on the skew edge thegeneralized displacement vectors of this node with respect to119909119910119911 coordinate system are not required to be transformedHence thematrices [119871

119905] and [119871

119903] corresponding to suchnode

turn out to be (3times3) and (9times9) identitymatrices respectivelyThe elemental equations of motion are now assembled

to obtain the open-loop global equations of motion of theoverall skew plate as follows

[119872] + [119870119905119905] 119883 + [119870

119905119903] 119883

119903 =

119902

sum

119895=1

119865119895

119905119901

119881119895

+ 119865

[119870119905119903]119879

119883 + [119870119903119903] 119883

119903 =

119902

sum

119895=1

119865119895

119903119901

119881119895

(33)

where [119870119905119905] [119870

119905119903] and [119870

119903119903] are the global stiffness matrices

of the overall skew laminated composite plate [119872] is theglobal mass matrix 119865

119905119901 and 119865

119903119901 are the global electro-

elastic coupling vectors 119883 and 119883119903 are the global nodal

generalized displacement vectors 119865 is the global nodalforce vector 119902 is the number of patches and 119881

119895 is thevoltage applied to the 119895th patch Since the elemental stiffnessmatrices of an element augmented with the ACLD treatmentare complex the global stiffness matrices become complexand the energy dissipation characteristics of the overallplate are attributed to the imaginary part of these matricesHence the global equations of motion as derived previouslyalso represent the modeling of the passive (uncontrolled)constrained layer damping of the substrate plate when theconstraining layer is not subjected to any control voltagefollowing a derivative control law

Journal of Composites 9

3 Closed-Loop Model

In the active control approach the active constraining layerof each patch is activated with a control voltage negativelyproportional to the transverse velocity of a point Thus thecontrol voltage for each patch can be expressed in terms of thederivatives of the global nodal degrees of freedom as follows

119881119895

= minus119896119895

119889

= minus119896119895

119889

[119880119895

119905

] minus 119896119895

119889

(ℎ

2) [119880

119895

119903

] 119903 (34)

where [119880119895

119905

] and [119880119895

119903

] are the unit vectors describing thelocation of the velocity sensor for the 119895th patch and 119896

119895

119889

isthe control gain for the 119895th patch Substituting (34) into(33) the final equations of motion governing the closed-loop dynamics of the overall skew laminated compositeplateACLD system can be obtained as follows

[119872] + [119870119905119905] 119883 + [119870

119905119903] 119883

119903 +

119898

sum

119895=1

119896119895

119889

119865119895

119905119901

[119880119895

119905

]

+

119898

sum

119895=1

119896119895

119889

(ℎ

2) 119865

119895

119905119901

[119880119895

119903

] 119903 = 119865

[119870119905119903]119879

119883 + [119870119903119903] 119883

119903 +

119898

sum

119895=1

119896119895

119889

119865119895

119905119901

[119880119895

119905

]

+

119898

sum

119895=1

119896119895

119889

(ℎ

2) 119865

119895

119905119901

[119880119895

119903

] 119903 = 0

(35)

4 Results and Discussion

In this section the numerical results obtained by the finiteelement model derived in the previous section have beenpresented Symmetric as well as antisymmetric cross-plyand antisymmetric angle-ply laminated skew substrate platesintegrated with the two patches of ACLD treatment areconsidered for presenting the numerical results As shown inFigure 1 the locations of the patches have been selected insuch a way that the energy dissipation corresponding to thefirst fewmodes becomesmaximum PZT-5Hspur compositewith 60 fiber volume fraction has been considered for thematerial of the constraining layer of the ACLD treatmentUsing themicromechanicsmodel and thematerial propertiesof the constituent phases [35] the effective elastic andpiezoelectric material properties of the 1ndash3 PZC with respectto the principle material coordinate system are computed asfollows

11986211

= 929GPa 11986212

= 618GPa 11986213

= 605GPa

11986233

= 3544GPa 11986223

= 11986213 119862

44= 158GPa

11986266

= 154GPa 11986255

= 11986244

11989031

= minus01902Cm2

11989032

= 11989031

11989033

= 184107Cm2

11989024

= 0004Cm2

11989015

= 11989024

(36)

The material of the orthotropic layers of the substrate platesis a graphiteepoxy composite and its material properties are[42]

119864119871= 172GPa 119864

119871

119864119879

= 25 119866119871119879

= 05119864119879

119866119879119879

= 02119864119879 ]

119871119879= ]

119879119879= 025

(37)

in which the symbols have the usual meaning Unless oth-erwise mentioned the aspect ratio (119886ℎ) and the thicknessof the skew substrate plates are considered as 100 and0003m respectively while the orthotropic layers of theskew substrate plates are of equal thickness Also the lengthand the skew sides of the substrate are equal (ie 119886 =

119887) The loss factor of the viscoelastic layer considered fornumerical results remains invariant [20] within a broadband (0ndash400Hz) of frequency range and the values of thecomplex shear modulus Poisson ratio and the density ofthe viscoelastic layer are considered as 20(1 + 119894)MNmminus2049 and 1140 kgmminus3 respectively [20] The thicknesses ofthe viscoelastic layer and the 1ndash3 PZC layer are consideredas 508120583m and 250 120583m respectively The value of the shearcorrection factor is used as 56 Three points of Gaussianintegration rule are considered for computing the elementmatrices corresponding to the bending deformations whiletwo points of that are used for computing the elementmatrices corresponding to transverse shear deformationsThe simply supported boundary conditions at the edges of theoverall plate are invoked as follows

at 119909 = 119910 tan120572 119909 = 119886 + 119910 tan120572

V1015840119900

= 1199081015840

119900

= 1205791015840

119910

= 1206011015840

119910

= 1205741015840

119910

= 1205791015840

119911

= 1205741015840

119911

= 0

at 119910 = 0 119910 = 119887 cos120572

119906119900= 119908

119900= 120579

119909= 120601

119909= 120574

119909= 120579

119911= 120574

119911= 0

(38)

To verify the validity of the present finite elementmodel (FEM) the natural frequencies of the skew laminatedcomposite plates integrated with the inactivated patches ofnegligible thickness are first computed and subsequentlycompared with the existing analytical results [41] of theidentical plates without being integrated with the patchesSuch comparisons for the simply supported and clamped-clamped skew laminated plates are presented in Tables 1 and2 respectively It may be observed from these tables that theresults are in excellent agreement validating the model of theplate derived here

In order to assess the performance of the verti-callyobliquely reinforced 1ndash3 PZC as the material of thedistributed actuators of smart skew laminated compositeplates frequency responses for ACLD of the substrate skewlaminated composite plates are investigated To compute the

10 Journal of Composites

Table 1 Comparison of fundamental natural frequencies (120596) of simply supported skew laminated composite plates integratedwith the patchesof negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 15699 28917 37325 14829 24656 32522 17974 33351 33351Present FEM 15635 24383 35033 15076 24380 32254 18493 33359 33370

15∘ Reference [41] 16874 30458 39600 15741 25351 30270 18313 32490 36724Present FEM 16571 29840 36505 15796 25775 29892 18675 32075 35810

30∘ Reference [41] 20840 34023 46997 18871 29372 34489 20270 34431 42361Present FEM 19596 31690 46796 18226 29585 32357 19894 32365 43208

45∘ Reference [41] 28925 41906 54149 25609 33126 40617 25609 33131 42772Present FEM 24811 44875 53289 22996 34773 44889 23194 34870 45009

lowast

120596 = (120596119886

2

120587

2

ℎ)radic120588119864119879 120596 is the circular natural frequency

Table 2 Comparison of fundamental natural frequencies (120596) of clamped-clamped skew laminated composite plates integrated with thepatches of negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 23687 35399 41122 22990 37880 37880 22119 37339 37339Present FEM 23201 34769 44102 23315 36531 36545 22433 36000 36012

15∘ Reference [41] 24663 36255 43418 23809 37516 40785 23099 36997 40438Present FEM 23699 34821 44049 23741 35856 38401 23049 35346 38092

30∘ Reference [41] 27921 39557 50220 26666 39851 47227 26325 39549 52107Present FEM 25366 35696 44734 25240 41943 45373 24945 36113 50932

45∘ Reference [41] 34739 47129 58789 33015 46290 58423 33015 46290 58423Present FEM 28665 47074 54562 28377 47614 56620 28377 47614 55162

frequency response functions the skew laminated compositeplates are harmonically excited by a force of amplitude 1Newton applied to a point (1198862 (1198864) cos120572 ℎ2) on the skewsubstrate platesThe control voltage supplied to the constrain-ing layer of the first patch is negatively proportional to thevelocity of the point (1198862 (1198864) cos120572 ℎ2) and that applied tothe constraining layer of the second patch is negatively pro-portional to the velocity of the point (1198862 (31198864) cos120572 ℎ2)

Figures 3 4 and 5 illustrate the frequency responsefunctions for the transverse displacement of a point of asimply supported symmetric cross-ply (0∘90∘0∘) plate with15∘ 30∘ and 45∘ skew angles respectively while the fiberorientation angle (120595) in the constraining layer is 0∘ Thesefigures display both uncontrolled and controlled responsesand clearly show that the active constraining layer made ofthe vertically reinforced 1ndash3 PZC material being studied heresignificantly attenuates the amplitude of vibrations enhanc-ing the damping characteristics of the overall laminatedskew composite plates over the passive damping (uncon-trolled) Similar responses are also obtained for antisym-metric cross-ply (0∘90∘0∘90∘) and antisymmetric angle-ply (45∘minus45∘45∘minus45∘) skew laminated composite plates

However for brevity these results are not presented hereThemaximum values of the control voltages required to computethe controlled responses presented in Figures 3ndash5 have beenfound to be quite low and are illustrated in Figure 6 in caseof the symmetric cross-ply (0∘90∘0∘) plate with skew angle15∘ only In order to examine the contribution of the verticalactuation in improving the damping characteristics of theskew laminated composite plates as demonstrated in Figures3ndash5 active control responses of the symmetric cross-plylaminated composite plate with 15∘ skew angle are plotted inFigure 7with andwithout considering the value of 119890

33and 119890

31

It may be noted that when the values of 11989031is zero and that of

11989033is nonzero the vertical actuation of the active constraining

layer of the ACLD treatment is responsible for increasing thetransverse shear deformations of the viscoelastic constrainedlayer over the passive counterpart resulting in improveddamping of the smart plate over its passive damping Onthe other hand if 119890

33is zero and 119890

31is nonzero the in-

plane actuation of the active constraining layer causes thetransverse shear deformation of the viscoelastic core of theACLD treatment leading to the active damping of the smartplate It is evident from Figure 7 that the contribution of the

Journal of Composites 11

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 3 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 4 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 30∘)

vertical actuation by the constraining vertically reinforced 1ndash3 PZC layer is significantly larger than that of the in-planeactuation by the same for controlling the modes displayedin Figure 7 Though not shown here similar results are alsoobtained for antisymmetric cross-ply and angle-ply skewplates

Figures 8 9 and 10 demonstrate the effect of variationof piezoelectric fiber orientation angle on the performanceof the patches for improving the damping characteristicsof simply supported symmetric cross-ply (0∘90∘0∘)

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 5 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 45∘)

0

5

10

15

20

25

30

35

40

45

50

Con

trol v

olta

ge (V

olt)

0 100 200 300 400 500 600Frequency (Hz)

119870119889 = 600

119870119889 = 1200

Figure 6 Frequency response functions for the control voltages foractive damping of the simply supported skew symmetric cross-ply(0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

antisymmetric cross-ply (0∘90∘0∘90∘) and angle-ply(45∘minus45∘45∘minus45∘) substrate skew composite plates with30∘ skew angle respectively while the orientation angle (120595)

of the piezoelectric fibers is varied in the vertical 119909119911-planeThese figures illustrate that the attenuating capability of thepatches becomesmaximumwhen the value of the orientationangle (120595) of the fibers in the 119909119911-plane is 0∘ Similar effectis also obtained when the fiber orientation angle (120595) in theconstraining layer of the ACLD treatment is varied in thevertical 119910119911-plane as illustrated in Figures 11 12 and 13 forthe symmetric cross-ply (0∘90∘0∘) antisymmetric cross-ply(0∘90∘0∘90∘) and angle-ply (45∘minus45∘45∘minus45∘) substrate

12 Journal of Composites

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

11989031 = minus01902 11989033 = 184107

11989031 = minus01902 11989033 = 011989031 = 0 11989033 = 184107

w(a2(b4)co

s120572h2

) (m

)

Figure 7 Frequency response functions demonstrating the con-tribution of vertical actuation on the controlled responses forthe transverse displacement 119908 (1198862 (1198864) cos120572 ℎ2) of the skewsymmetric cross-ply (0∘90∘0∘) plate (119870

119889

= 600 119886ℎ = 100 120595 = 0∘

120572 = 15∘)

0 100 200 300 400 500 600Frequency (Hz)

0

02

04

06

08

1

12times10minus4

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 8 Effect of variation of piezoelectric fiber orientation angle(120595) on the controlled responses of a simply supported symmetricskew cross-ply (0∘90∘0∘) plate when the piezoelectric fibers of theconstraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

skew composite plates with 30∘ skew angle respectivelyIn this regard it should be noted that for investigating theeffect of variation of piezoelectric fiber orientation angle(120595) on the performance of the ACLD patches the valueof 120595 has been smoothly varied to compute the controlledresponses However for the sake of clarity in the plots theresponses corresponding to the four specific values of 120595

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 9 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 10 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

have been presented such that the optimum performanceof the patches can be evaluated The significant effect of thevariation of the piezoelectric fiber orientation angle in thevertical 119909119911- and 119910119911-planes has been noticed when all theboundaries of the overall plates are clamped as shown inFigures 14ndash19 When the piezoelectric fibers are coplanarwith the vertical 119909119911-plane the frequency response functionsplotted in Figures 14 15 and 16 reveal that the control

Journal of Composites 13

0

02

04

06

08

1

12

14

16times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 11 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 12 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=600 119886ℎ = 100 120572 = 30

∘)

authority of the patches becomes maximum for improvingthe damping characteristics of the clamped-clamped skewcross-ply and angle-ply substrate plates if the value of thepiezoelectric fiber orientation angle (120595) is 30∘ The same isalso found to be true if the piezoelectric fibers are coplanarwith the vertical 119910119911-plane as shown in Figures 17 18 and 19for the previously mentioned skew plates Similar responsesare also found for the effect of piezoelectric fiber orientation

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 13 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers of the constraining layer are coplanar with the119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 14 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600119886ℎ = 100120572 = 30

∘)

angle on the active damping of other skew laminated plateswith different skew angles However for the sake of brevitythey are not presented here

5 Conclusions

In this paper a study has been carried out to investi-gate the performance of the ACLD treatment for active

14 Journal of Composites

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 15 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 16 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

damping of smart skew laminated composite plates whenverticallyobliquely reinforced 1ndash3 PZC materials are usedas the materials for the constraining layer of the ACLDtreatment A layerwise FSDT-based finite element modelhas been developed to describe the dynamics of the skewlaminated composite plates integrated with the patches ofACLD treatment Unlike the existing finite element modelsof smart structures integrated with ACLD treatment the

200 250 300 350 400 450 500 550 600 650 7000

02

04

06

08

1

12

14

16

18

2times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 17 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

200 300 400 500 600 700 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 18 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

derivation of the present finite element model includes thetransverse deformations of the substrate skew laminatedcomposite plate the constrained viscoelastic layer and theconstraining 1ndash3 PZC layer of the ACLD treatment along thethickness (ie 119911) direction such that both vertical and in-plane actuations by the constraining layer of the patches canbe utilized for active damping of the plates The frequency

Journal of Composites 15

0

05

1

15

2

25

3

35

4times10minus5

200 300 400 500 600 700 800Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 19 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

responses of the symmetric cross-ply antisymmetric cross-ply and angle-ply skew composite plates indicate that theactive constraining layer of the ACLD treatment being madeof the vertically reinforced 1ndash3 PZC material significantlyenhances the damping characteristics of the plates over thepassive damping The analysis revealed that if the verticallyreinforced 1ndash3 PZCmaterial is used for the constraining layerof the ACLD treatment then the contribution of the verticalactuation by the constraining layer alone for improving theactive damping characteristics of the smart skew laminatedcomposite plate is significantly larger than that due to thein-plane actuation by the constraining layer alone It isimportant to note from the present investigation that thevariation of the orientation angle of the piezoelectric fibersof the constraining layer in the vertical 119909119911- and 119910119911-planesas well as the skew angle and boundary conditions of theoverall plates significantly affects the performance of thepatches For the simply supported laminated substrate platesthe performance of the patches becomes maximumwhen thepiezoelectric fiber orientation angle (120595) is 0∘ while in caseof the clamped-clamped laminated skew substrate plates themaximum control authority of the patches is achieved whenthe value of 120595 is 30∘ irrespective of the cases in which thefibers are oriented in 119909119911- and 119910119911-planes

Appendix

In (6) and (7) the matrices [1198851] [119885

2] [119885

3] [119885

4] [119885

5] and

[1198856] are given by

[1198851] = [[119885

1] 0 0]

[1198852] = [(

ℎ

2) 119868 [119885

2] 0]

[1198853] = [(

ℎ

2) 119868 ℎV119868 [119885

3]]

[1198854] = [119868 0 0 119911119868 0 0]

[1198855] = [0 119868 0 (

ℎ

2) 119868 (119911 minus

ℎ

2) 119868 0]

[1198856] = [0 0 119868 (

ℎ

2) 119868 ℎV119868 (119911 minus ℎ

119873+2) 119868]

(A1)

in which

[1198851] =

[[[

[

119911 0 0 0

0 119911 0 0

0 0 119911 0

0 0 0 1

]]]

]

[1198852] =

[[[[[[[[

[

(119911 minusℎ

2) 0 0 0

0 (119911 minusℎ

2) 0 0

0 0 (119911 minusℎ

2) 0

0 0 0 1

]]]]]]]]

]

[1198853] =

[[[

[

(119911 minus ℎ119873+2

) 0 0 0

0 (119911 minus ℎ119873+2

) 0 0

0 0 (119911 minus ℎ119873+2

) 0

0 0 0 1

]]]

]

[119868] =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

[119868] = [1 0

0 1]

[0] = [0 0

0 0] [0] = [

0 0

0 0]

(A2)

The various submatrices 119861119905119887119894 119861

119903119887119894 and 119861

119903119904119894 appearing in (21)

are given by

119861119905119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 0

]]]]]]]]]]]

]

119861119905119904119894

=

[[[[

[

0 0120597120578

119894

120597119909

0 0120597120578

119894

120597119910

]]]]

]

16 Journal of Composites

119861119903119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 1

]]]]]]]]]]]

]

119861119903119887119894

=

[[[[[[

[

119861119903119887119894

0

0

0 119861119903119887119894

0

0

0 119861119903119887119894

]]]]]]

]

119861119903119904119894

=

[[[[[[[[[[[[[[[[[[

[

119868 0 0

0

119868 0

0 0

119868

119861119903119904119894

0 0

0 119861119903119904119894

0

0 0 119861119903119904119894

]]]]]]]]]]]]]]]]]]

]

119868= [1 0 0

0 1 0]

(A3)

wherein

0 and 0 are the (3 times 3) and (2 times 3) null matricesrespectively

References

[1] T Bailey and J E Hubbard ldquoDistributedpiezoelectric polymeractive vibration control of a cantilever beamrdquo Journal of Guid-ance Control and Dynamics vol 8 no 5 pp 605ndash611 1985

[2] E F Crawley and J de Luis ldquoUse of piezoelectric actuators aselements of intelligent structuresrdquo AIAA Journal vol 25 no 10pp 1373ndash1385 1987

[3] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988

[4] C K Lee W W Chiang and O Sulivan ldquoPiezoelectric modalsensoractuator pairs for critical active damping vibrationcontrolrdquo Journal of the Acoustical Society of America vol 90 no1 pp 374ndash384 1991

[5] S Hanagud M W Obal and A J Calise ldquoOptimal vibrationcontrol by the use of piezoceramic sensors and actuatorsrdquoJournal of Guidance Control and Dynamics vol 15 no 5 pp1199ndash1206 1992

[6] Y Gu R L Clark C R Fuller and A C Zander ldquoExperimentson active control of plate vibration using piezoelectric actuatorsand polyvinylidene fluoride (PVDF) modal sensorsrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 116 no3 pp 303ndash308 1994

[7] A Baz and S Poh ldquoOptimal vibration control with modalpositive position feedbackrdquo Optimal Control Applications andMethods vol 17 no 2 pp 141ndash149 1996

[8] A Kugi K Schlacher and H Irschik ldquoInfinite-dimensionalcontrol of nonlinear beam vibrations by piezoelectric actuatorand sensor layersrdquo Nonlinear Dynamics vol 19 no 1 pp 71ndash911999

[9] D Sun L Tong and D Wang ldquoVibration control of platesusing discretely distributed piezoelectric quasi-modal actua-torssensorsrdquo AIAA Journal vol 39 no 9 pp 1766ndash1772 2001

[10] G Caruso S Galeani and L Menini ldquoActive vibration controlof an elastic plate using multiple piezoelectric sensors andactuatorsrdquo SimulationModelling Practice andTheory vol 11 no5-6 pp 403ndash419 2003

[11] X Zhang and A G Erdman ldquoOptimal placement of piezo-electric sensors and actuators for controlled flexible linkagemechanismsrdquo Journal of Vibration and Acoustics Transactionsof the ASME vol 128 no 2 pp 256ndash260 2006

[12] G Gatti M J Brennan and P Gardonio ldquoActive dampingof a beam using a physically collocated accelerometer andpiezoelectric patch actuatorrdquo Journal of Sound and Vibrationvol 303 no 3ndash5 pp 798ndash813 2007

[13] M K Kwak S Heo and M Jeong ldquoDynamic modellingand active vibration controller design for a cylindrical shellequipped with piezoelectric sensors and actuatorsrdquo Journal ofSound and Vibration vol 321 no 3ndash5 pp 510ndash524 2009

[14] I Bruant L Gallimard and S Nikoukar ldquoOptimal piezoelectricactuator and sensor location for active vibration control usinggenetic algorithmrdquo Journal of Sound and Vibration vol 329 no10 pp 1615ndash1635 2010

[15] Y Fu J Wang and Y Mao ldquoNonlinear vibration and activecontrol of functionally graded beams with piezoelectric sensorsand actuatorsrdquo Journal of Intelligent Material Systems andStructures vol 22 no 18 pp 2093ndash2102 2011

[16] F Chen M Hong M Song and H Cui ldquoOptimal control ofa beam with discontinuously distributed piezoelectric sensorsand actuatorsrdquo Journal of Marine Science and Application vol11 no 1 pp 44ndash51 2012

[17] B Azvine G R Tomlinson and R J Wynne ldquoUse of activeconstrained-layer damping for controlling resonant vibrationrdquoSmart Materials and Structures vol 4 no 1 pp 1ndash6 1995

[18] A Baz and J Ro ldquoVibration control of plates with activeconstrained layer dampingrdquo SmartMaterials and Structures vol5 no 3 pp 272ndash280 1996

[19] C Chantalakhana and R Stanway ldquoActive constrained layerdamping of clamped-clamped plate vibrationsrdquo Journal ofSound and Vibration vol 241 no 5 pp 755ndash777 2001

[20] J Ro and A Baz ldquoOptimum placement and control ofactive constrained layer damping using modal strain energyapproachrdquo JVCJournal of Vibration and Control vol 8 no 6pp 861ndash876 2002

[21] M C Ray and J N Reddy ldquoOptimal control of thin circularcylindrical laminated composite shells using active constrainedlayer damping treatmentrdquo Smart Materials and Structures vol13 no 1 pp 64ndash72 2004

[22] C M A Vasques B R Mace P Gardonio and J D RodriguesldquoArbitrary active constrained layer damping treatments onbeams finite element modelling and experimental validationrdquoComputers and Structures vol 84 no 22-23 pp 1384ndash14012006

[23] F M Li K Kishimoto K Wang Z B Chen and W HHuang ldquoVibration control of beams with active constrainedlayer dampingrdquo Smart Materials and Structures vol 17 no 6Article ID 065036 2008

[24] L Yuan Y Xiang Y Huang and J Lu ldquoA semi-analyticalmethod and the circumferential dominant modal control ofcircular cylindrical shells with active constrained layer dampingtreatmentrdquo SmartMaterials and Structures vol 19 no 2 ArticleID 025010 2010

[25] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

Journal of Composites 17

[26] W A Smith and B A Auld ldquoModeling 1ndash3 composite piezo-electrics thickness-mode oscillationsrdquo IEEE Transactions onUltrasonics Ferroelectrics and Frequency Control vol 38 no 1pp 40ndash47 1991

[27] PiezocompositesMaterial Systems Inc 543 Great Road Little-ton MA 01460

[28] Y Benveniste and G J Dvorak ldquoUniform fields and universalrelations in piezoelectric compositesrdquo Journal of the Mechanicsand Physics of Solids vol 40 no 6 pp 1295ndash1312 1992

[29] M L Dunn and M Taya ldquoMicromechanics predictions ofthe effective electroelastic moduli of piezoelectric compositesrdquoInternational Journal of Solids and Structures vol 30 no 2 pp161ndash175 1993

[30] J Aboudi ldquoMicromechanical prediction of the effective coeffi-cients of thermo-piezoelectric multiphase compositesrdquo Journalof Intelligent Material Systems and Structures vol 9 no 9 pp713ndash722 1999

[31] X Ruan T W Chou A Safari and S C Danforth ldquoA 3-Dconnectivity model for effective piezoelectric properties of yarncompositesrdquo Journal of Composite Materials vol 36 no 14 pp1693ndash1708 2002

[32] A Baz and A Tempia ldquoActive piezoelectric damping compos-itesrdquo Sensors and Actuators A vol 112 no 2-3 pp 340ndash3502004

[33] H Berger S Kari U Gabbert et al ldquoUnit cell models ofpiezoelectric fiber composites for numerical and analytical cal-culation of effective propertiesrdquo SmartMaterials and Structuresvol 15 no 2 pp 451ndash458 2006

[34] M Sakthivel and A Arockiarajan ldquoAn analytical model forpredicting thermo-electro-mechanical response of 1ndash3 piezo-electric compositesrdquo Computational Materials Science vol 48no 4 pp 759ndash767 2010

[35] M C Ray and A K Pradhan ldquoOn the use of verticallyreinforced 1ndash3 piezoelectric composites for hybrid damping oflaminated composite platesrdquo Mechanics of Advanced Materialsand Structures vol 14 no 4 pp 245ndash261 2007

[36] M C Ray and A K Pradhan ldquoPerformance of vertically andobliquely reinforced 1ndash3 piezoelectric composites for activedamping of laminated composite shellsrdquo Journal of Sound andVibration vol 315 no 4-5 pp 816ndash835 2008

[37] S K Sarangi and M C Ray ldquoSmart damping of geometricallynonlinear vibrations of laminated composite beams using verti-cally reinforced 1ndash3 piezoelectric compositesrdquo Smart Materialsand Structures vol 19 no 7 Article ID 075020 2010

[38] A Krishnan and J V Deshpande ldquoVibration of skew laminatesrdquoJournal of Sound and Vibration vol 153 no 2 pp 351ndash358 1992

[39] A R Krishna Reddy and R Palaninathan ldquoFree vibration ofskew laminatesrdquo Computers and Structures vol 70 no 4 pp415ndash423 1999

[40] C S Babu and T Kant ldquoTwo shear deformable finite elementmodels for buckling analysis of skew fibre-reinforced compositeand sandwich panelsrdquo Composite Structures vol 46 no 2 pp115ndash124 1999

[41] A K Garg R K Khare and T Kant ldquoFree vibration of skewfiber-reinforced composite and sandwich laminates using ashear deformable finite element modelrdquo Journal of SandwichStructures and Materials vol 8 no 1 pp 33ndash53 2006

[42] J N Reddy Mechanics of Laminated Composites Plates Theoryand Analysis CRC press Boca Raton Fla USA 1996

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 3: Research Article Active Constrained Layer Damping of Smart …downloads.hindawi.com/archive/2013/824163.pdf · 2019-07-31 · Research Article Active Constrained Layer Damping of

Journal of Composites 3

120572119884998400

119883998400

119885998400 119885

ℎ2ℎ

1198864 1198862

1198874

1198872

Patch 2Skew laminated composite plate

1198864

1198874

Viscoelastic layer

Patch 1

119883

119884

120572 Vertically reinforced 1ndash3 PZC layer

(a)

119911

119910 119910

119909 119909

Surface electrode Surface electrode

Piezoelectric fiber Epoxy matrix

Epoxy matrixPiezoelectric fiber

119911

120595 120595

(b)

Figure 1 (a) Schematic representation of a skew laminated composite plate integrated with the patches of ACLD treatment composed ofverticallyobliquely reinforced 1ndash3 piezoelectric composite constraining layer (b) Schematic diagram of layers of obliquely reinforced 1ndash3piezoelectric composite

Viscoelastic layer Viscoelastic layerSkew laminated plate

119911

ℎ2 ℎ2

ℎ ℎ

Undeformed transverse section Undeformed transverse section

Deformed transverse

119910

sectionDeformed transverse

section

ℎ ℎ

119911

120579119909

120579119910

119909

1199060

120574119910

120601119910

120601119909

120574119909

Skew laminated plate

ℎ1199011ndash3 PZC layer

1199080

0

1ndash3 PZC layer

1199080

Figure 2 Kinematics of deformations

119906 and V at any point in any layer of the overall plate along 119909-and 119910-directions respectively can be expressed as

119906 (119909 119910 119911 119905) = 1199060(119909 119910 119905) + (119911 minus ⟨119911 minus

ℎ

2⟩)120579

119909(119909 119910 119905)

+ (⟨119911 minusℎ

2⟩ minus ⟨119911 minus ℎ

119873+2⟩) 120601

119909(119909 119910 119905)

+ ⟨119911 minus ℎ119873+2

⟩ 120574119909(119909 119910 119905)

(1)

V (119909 119910 119911 119905) = V0(119909 119910 119905) + (119911 minus ⟨119911 minus

ℎ

2⟩)120579

119910(119909 119910 119905)

+ (⟨119911 minusℎ

2⟩ minus ⟨119911 minus ℎ

119873+2⟩) 120601

119910(119909 119910 119905)

+ ⟨119911 minus ℎ119873+2

⟩ 120574119910(119909 119910 119905)

(2)

in which a function within the bracket ⟨sdot⟩ represents anappropriate singularity function which satisfies the continu-ity of displacements between two adjacent continua Since thetransverse actuation of the constraining layer of the ACLDtreatment will be used for the flexural vibration controlof the plate the transverse normal strain in the overallplate must be considered in the model Hence as the plateconsidered here is thin the transverse displacements (119908) atany point in the substrate plate the viscoelastic layer and

4 Journal of Composites

the 1ndash3 PZC layer are assumed to be linearly varying acrosstheir thicknesses Thus similar to the axial displacement thetransverse displacements at any point in the overall plate canbe expressed as

119908 (119909 119910 119911 119905) = 1199080(119909 119910 119905) + (119911 minus ⟨119911 minus

ℎ

2⟩)120579

119911(119909 119910 119905)

+ (⟨119911 minusℎ

2⟩ minus ⟨119911 minus ℎ

119873+2⟩) 120601

119911(119909 119910 119905)

+ ⟨119911 minus ℎ119873+2

⟩ 120574119911(119909 119910 119905)

(3)

in which1199080refers to the transverse displacement at any point

on the reference plane and 120579119911 120601

119911 and 120574

119911are the generalized

displacements representing the gradients of the transversedisplacement in the substrate plate the viscoelastic layer andthe 1ndash3 PZC layer respectively with respect to the thicknesscoordinate (119911)

For the ease of analysis the generalized displacementvariables are grouped into the following two vectors

119889119905 = [119906

0V0

1199080]119879

119889119903 = [120579

119909120579119910

120579119911

120601119909

120601119910

120601119911

120574119909

120574119910

120574119911]119879

(4)

In order to implement the selective integration rule forcomputing the element stiffness matrices corresponding tothe transverse shear deformations the state of strain at any

point in the overall plate is divided into the following twostrain vectors 120576

119887 and 120576

119904

120576119887 = [120576

119909120576119910

120576119909119910

120576119911]119879

120576119904 = [120576

119909119911120576119910119911

]119879 (5)

in which 120576119909 120576

119910 and 120576

119911are the normal strains along 119909-

119910- and 119911-directions respectively 120576119909119910

is the in-plane shearstrain 120576

119909119911and 120576

119910119911are the transverse shear strains Using the

displacement fields given by (1)ndash(3) and the linear strain-displacement relations the vectors 120576

119887119888

120576119887V and 120576

119887119901

defining the state of in-plane and transverse normal strainsat any point in the substrate skew composite plate the vis-coelastic layer and the active constraining layer respectivelycan be expressed as

120576119887119888

= 120576119887119905 + [119885

1] 120576

119887119903 120576

119887V = 120576

119887119905 + [119885

2] 120576

119887119903

120576119887119901

= 120576119887119905 + [119885

3] 120576

119887119903

(6)

Similarly the vectors 120576119904119888

120576119904V and 120576

119904119901

defining the stateof transverse shear strains at any point in the substrate com-posite plate the viscoelastic layer and the active constraininglayer respectively can be expressed as

120576119904119888

= 120576119904119905 + [119885

4] 120576

119904119903 120576

119904V = 120576

119904119905 + [119885

5] 120576

119904119903

120576119904119901

= 120576119904119905 + [119885

6] 120576

119904119903

(7)

The various matrices appearing in (6) and (7) have beendefined in the Appendix while the generalized strain vectorsare given by

120576119887119905 = [

1205971199060

120597119909

120597V0

120597119910

1205971199060

120597119910+

120597V0

1205971199090]

119879

120576119904119905 = [

1205971199080

120597119909

1205971199080

120597119910]

119879

120576119887119903 = [

120597120579119909

120597119909

120597120579119910

120597119910

120597120579119909

120597119910+

120597120579119910

120597119909120579119911

120597120601119909

120597119909

120597120601119910

120597119910

120597120601119909

120597119910+

120597120601119910

120597119909120601119911

120597120574119909

120597119909

120597120574119910

120597119910

120597120574119909

120597119910+

120597120574119910

120597119909120574119911

]

119879

120576119904119903 = [120579

119909120579119910

120601119909

120601119910

120574119909

120574119910

120597120579119911

120597119909

120597120579119911

120597119910

120597120601119911

120597119909

120597120601119911

120597119910

120597120574119911

120597119909

120597120574119911

120597119910]

119879

(8)

Similar to the strain vectors given by (5) the state of stressesat any point in the overall plat is described by the followingstress vectors

120590119887 = [120590

119909120590119910

120590119909119910

120590119911]119879

120590119904 = [120590

119909119911120590119910119911

]119879

(9)

where 120590119909 120590

119910 and 120590

119911are the normal stresses along 119909- 119910-

and 119911-directions respectively 120590119909119910is the in-plane shear stress

120590119909119911

and 120590119910119911

are the transverse shear stresses The constitutive

relations for the material of any orthotropic layer of thesubstrate plate are given by

120590119896

119887

= [119862119896

119887

] 120576119896

119887

120590119896

119904

= [119862119896

119904

] 120576119896

119904

(119896 = 1 2 3 119873)

(10)

Journal of Composites 5

where the elastic coefficient matrices are

[119862119896

119887

] =

[[[[[[[[[[[[[[

[

119862119896

11

119862119896

12

119862119896

16

119862119896

13

119862119896

12

119862119896

22

119862119896

26

119862119896

23

119862119896

16

119862119896

26

119862119896

66

119862119896

36

119862119896

13

119862119896

23

119862119896

36

119862119896

33

]]]]]]]]]]]]]]

]

[119862119896

119904

] =[[

[

119862119896

55

119862119896

45

119862119896

45

119862119896

44

]]

]

(11)

and 119862119896

119894119895

(119894 119895 = 1 2 3 6) are the transformed elasticcoefficients with respect to the reference coordinate systemThematerial of the viscoelastic layer is assumed to be linearlyviscoelastic and homogenous isotropic and is modeled byusing the complex modulus approach Thus the shear mod-ulus119866 and Youngrsquos modulus 119864 of the viscoelastic material aregiven by

119866 = 1198661015840

(1 + 119894120578) 119864 = 2119866 (1 + ]) (12)

in which 1198661015840 is the storage modulus ] is Poissonrsquos ratio and

120578 is the loss factor at a particular operating temperature andfrequency Employing the complex modulus approach theconstitutive relations for the material of the viscoelastic layer(119896 = 119873 + 1) can also be represented by (10) with 119862

119896

119894119895

(119894 119895 =

1 2 3 6) being the complex elastic constants [29 30]The constraining PZC layer will be subjected to the appliedelectric field (119864

119911) acting across its thickness (ie along the

119911-direction) only Accordingly the constitutive relations forthe 1ndash3 PZC material with respect to the laminate coordinatesystem (119909119910119911) can be expressed as

120590119896

119887

= [119862119896

119887

] 120576119896

119887

+ [119862119896

119887119904

] 120576119896

119904

minus 119890119887 119864

119911

120590119896

119904

= [119862119896

119887119904

] 120576119896

119887

+ [119862119896

119904

] 120576119896

119904

minus 119890119904 119864

119911

119863119911= 119890

119887119879

120576119896

119887

+ 119890119904119879

120576119896

119904

+ 12057633119864119911

119896 = 119873 + 2

(13)

Here 119863119911represents the electric displacement along the 119911-

direction and 12057633

is the dielectric constant It may be notedfrom the previous form of the constitutive relations that thetransverse shear strains are coupled with the in-plane normalstrains due to the orientation of piezoelectric fibers in the

vertical 119909119911- or 119910119911-plane and the corresponding couplingelastic constant matrices [119862

119873+2

119887119904

] are given by

[119862119873+2

119887119904

] =

[[[[[[[[[[

[

119862119873+2

15

0

119862119873+2

25

0

0 119862119873+2

46

119862119873+2

35

0

]]]]]]]]]]

]

or

[119862119873+2

119887119904

] =

[[[[[[[[[[

[

0 119862119873+2

14

0 119862119873+2

24

119862119873+2

56

0

0 119862119873+2

34

]]]]]]]]]]

]

(14)

as the piezoelectric fibers are coplanar with the vertical 119909119911-or 119910119911-plane respectively Note that if the fibers are coplanarwith both 119909119911- and 119910119911-planes (ie 120595 = 0

∘) this couplingmatrix becomes a nullmatrix Also the piezoelectric constantmatrices 119890

119887 and 119890

119904 appearing in (13) contain the following

transformed effective piezoelectric coefficients of the 1ndash3PZC

119890119887 = [119890

3111989032

11989036

11989033]119879

119890119904 = [119890

3511989034]119879

(15)

The total potential energy 119879119901and the kinetic energy 119879

119896of the

overall plateACLD system are given by [35]

119879119901

=1

2[

119873+2

sum

119896=1

intΩ

(120576119896

119887

119879

120590119896

119887

+ 120576119903

119904

119879

120590119896

119904

) 119889Ω

minusintΩ

119863119911119864119911119889Ω] minus int

119860

119889119879

119891 119889119860

119879119896=

1

2

119873+2

sum

119896=1

intΩ

120588119896

(2

+ V2 + 2

) 119889Ω

(16)

in which 120588119896 is the mass density of the 119896th layer 119891 is the

externally applied surface traction acting over a surface area119860 andΩ represents the volume of the concerned layer Sincethe plates under study are considered to be thin the rotaryinertia of the overall plate has been neglected in estimatingthe kinetic energy The overall plate is discretized by eight-noded isoparametric quadrilateral elements Following (4)the generalized displacement vectors associated with the 119894th(119894 = 1 2 3 8) node of the element can be written as

119889119905119894 = [119906

0119894V0119894

1199080119894]119879

119889119903119894 = [120579

119909119894120579119910119894

120579119911119894

120601119909119894

120601119910119894

120601119911119894

120574119909119894

120574119910119894

120574119911119894]119879

(17)

6 Journal of Composites

Thus the generalized displacement vectors at any pointwithin the element can be expressed in terms of the nodalgeneralized displacement vectors 119889

119890

119905

and 119889119890

119903

as follows

119889119890

119905

= [119873119905] 119889

119890

119905

119889119890

119903

= [119873119903] 119889

119890

119903

(18)

in which

119889119890

119905

= [119889119890

1199051

119879

119889119890

1199052

119879

sdot sdot sdot 119889119890

1199058

119879

]119879

119889119890

119903

= [119889119890

1199031

119879

119889119890

1199032

119879

sdot sdot sdot 119889119890

1199038

119879

]119879

[119873119905] = [119873

11990511198731199052

sdot sdot sdot 1198731199058]119879

[119873119903] = [119873

11990311198731199032

sdot sdot sdot 1198731199038]119879

119873119905119894

= 119899119894119868119905 119873

119903119894= 119899

119894119868119903

(19)

while 119868119905and 119868

119903are (3times3) and (9times 9) identitymatrices respec-

tively and 119899119894is the shape function of natural coordinates

associatedwith the 119894th nodeMaking use of the relations givenby (6)ndash(8) and (18) the strain vectors at any point within theelement can be expressed in terms of the nodal generalizeddisplacement vectors as follows

120576119887119888

= [119861119905119887] 119889

119890

119905

+ [1198851] [119861

119903119887] 119889

119890

119903

120576119887V = [119861

119905119887] 119889

119890

119905

+ [1198852] [119861

119903119887] 119889

119890

119903

120576119887119901

= [119861119905119887] 119889

119890

119905

+ [1198853] [119861

119903119887] 119889

119890

119903

120576119904119888

= [119861119905119904] 119889

119890

119905

+ [1198854] [119861

119903119904] 119889

119890

119903

120576119904V = [119861

119905119904] 119889

119890

119905

+ [1198855] [119861

119903119904] 119889

119890

119903

120576119904119901

= [119861119905119904] 119889

119890

119905

+ [1198856] [119861

119903119904] 119889

119890

119903

(20)

in which the nodal strain-displacement matrices [119861119905119887] [119861

119903119887]

[119861119905119904] and [119861

119903119904] are given by

[119861119905119887] = [119861

11990511988711198611199051198872

sdot sdot sdot 1198611199051198878

]

[119861119903119887] = [119861

11990311988711198611199031198872

sdot sdot sdot 1198611199031198878

]

[119861119905119904] = [119861

11990511990411198611199051199042

sdot sdot sdot 1198611199051199048

]

[119861119903119904] = [119861

11990311990411198611199031199042

sdot sdot sdot 1198611199031199048

]

(21)

The submatrices of [119861119905119887] [119861

119903119887] [119861

119905119904] and [119861

119903119904] have been

explicitly presented in the Appendix On substitution of (9)ndash(13) and (20) into (16) the total potential energy 119879

119890

119901

and the

kinetic energy 119879119890

119896

of a typical element augmented with theACLD treatment can be expressed as

119879119890

119901

=1

2[119889

119890

119905

119879

[119870119890

119905119905

] 119889119890

119905

+ 119889119890

119905

119879

[119870119890

119905119903

] 119889119890

119903

+ 119889119890

119903

119879

[119870119890

119905119903

]119879

times 119889119890

119905

+ 119889119890

119903

119879

[119870119890

119903119903

] 119889119890

119903

minus 2119889119890

119905

119879

119865119890

119905119901

119881

minus 2119889119890

119903

119879

119865119890

119903119901

119881 minus 2119889119890

119905

119879

119865119890

minus 12057633

1198812

ℎ119901

]

(22)

119879119890

119896

=1

2 119889

119890

119905

119879

[119872119890

] 119889119890

119905

(23)

In (22)119881 represents the voltage difference applied across thethickness of the PZC layer The elemental stiffness matrices[119870

119890

119905119905

] [119870119890

119905119903

] and [119870119890

119903119903

] the elemental electroelastic couplingvectors 119865

119890

119905119901

and 119865119890

119903119901

the elemental load vector 119865119890

andthe elemental mass matrix [119872

119890

] appearing in (22) and (23)are given by

[119870119890

119905119905

] = [119870119890

119905119887

] + [119870119890

119905119904

] + [119870119890

119905119887119904

]119901119887

+ [119870119890

119905119887119904

]119901119904

[119870119890

119905119903

] = [119870119890

119905119903119887

] + [119870119890

119905119903119904

]

+1

2([119870

119890

119905119903119887119904

]119901119887

+ [119870119890

119903119905119887119904

]119879

119901119887

+ [119870119890

119905119903119887119904

]119901119904

+ [119870119890

119903119905119887119904

]119879

119901119904

)

[119870119890

119903119905

] = [119870119890

119905119903

]119879

[119870119890

119903119903

] = [119870119890

119903119903119887

] + [119870119890

119903119903119904

] + [119870119890

119903119903119887119904

]119901119887

+ [119870119890

119903119903119887119904

]119901119904

119865119890

119905119901

= 119865119890

119905119887

119901

+ 119865119890

119905119904

119901

119865119890

119903119901

= 119865119890

119903119887

119901

+ 119865119890

119903119904

119901

119865119890

= int

119886119890

0

int

119887119890

0

[119873119905]119879

119891 119889119909 119889119910

[119872119890

] = int

119886119890

0

int

119887119890

0

119898[119873]119879

[119873] 119889119909 119889119910 119898 =

119873+2

sum

119896=1

120588119896

(ℎ119896+1

minus ℎ119896)

(24)

where

[119870119890

119905119887

] = int119860

[119861119905119887]119879

([119863119905119887] + [119863

119905119887]V + [119863

119905119887]119901

) [119861119905119887] 119889119909 119889119910

[119870119890

119905119904

] = int119860

[119861119905119904]119879

([119863119905119904] + [119863

119905119904]V + [119863

119905119904]119901

) [119861119905119904] 119889119909 119889119910

[119870119890

119905119887119904

]119901119887

= int119860

[119861119905119887]119879

[119863119905119887119904

]119901

[119861119905119904] 119889119909 119889119910

[119870119890

119905119887119904

]119901119904

= int119860

[119861119905119904]119879

[119863119905119887119904

]119901

[119861119905119887] 119889119909 119889119910

[119870119890

119905119903119887

] = int119860

[119861119905119903119887

]119879

([119863119905119903119887

] + [119863119905119903119887

]V + [119863119905119903119887

]119901

)

times [119861119905119903119887

] 119889119909 119889119910

Journal of Composites 7

[119870119890

119905119903119887119904

]119901119887

= int119860

[119861119905119887]119879

[119863119905119903119887119904

]119901

[119861119903119904] 119889119909 119889119910

[119870119890

119903119905119887119904

]119901119887

= int119860

[119861119903119887]119879

[119863119903119905119887119904

]119901

[119861119905119904] 119889119909 119889119910

[119870119890

119905119903119887119904

]119901119904

= int119860

[119861119905119904]119879

[119863119903119905119887119904

]119901

[119861119903119887] 119889119909 119889119910

[119870119890

119903119905119887119904

]119901119904

= int119860

[119861119903119904]119879

[119863119905119903119887119904

]119901

[119861119905119887] 119889119909 119889119910

[119870119890

119905119903119904

] = int119860

[119861119905119903119904

]119879

([119863119905119903119904

] + [119863119905119903119904

]V + [119863119905119903119904

]119901

)

times [119861119905119903119904

] 119889119909 119889119910

[119870119890

119903119903119887

] = int119860

[119861119903119903119887

]119879

([119863119903119903119887

] + [119863119903119903119887

]V + [119863119903119903119887

]119901

)

times [119861119903119903119887

] 119889119909 119889119910

[119870119890

119903119903119904

] = int119860

[119861119903119903119904

]119879

([119863119903119903119904

] + [119863119903119903119904

]V + [119863119903119903119904

]119901

)

times [119861119903119903119904

] 119889119909 119889119910

[119870119890

119903119903119887119904

]119901119887

= int119860

[119861119903119887]119879

[119863119903119903119887119904

]119901

[119861119903119904] 119889119909 119889119910

[119870119890

119903119903119887119904

]119901119904

= int119860

[119861119903119904]119879

[119863119903119903119887119904

]119901

[119861119903119887] 119889119909 119889119910

119865119890

119905119887

119901

= int119860

[119861119905119887]119879

119863119905119887119901

119889119909 119889119910

119865119890

119905119904

119901

= int119860

[119861119905119904]119879

119863119905119904119901

119889119909 119889119910

119865119890

119903119904

119901

= int119860

[119861119903119904]119879

119863119903119904119901

119889119909 119889119910

(25)

The various rigidity matrices and the rigidity vectors forelectro-elastic coupling appearing in the previous elementalmatrices are given by

[119863119905119887] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] 119889119911

[119863119905119903119887

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] [1198851] 119889119911

[119863119903119903119887

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[1198851]119879

[119862119896

119887

] [1198851] 119889119911

[119863119905119904] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119904

] 119889119911

[119863119905119903119904

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] [1198854] 119889119911

[119863119903119903119904

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[1198854]119879

[119862119896

119887

] [1198854] 119889119911

[119863119905119887]V = ℎV [119862

119873+1

119887

]

[119863119905119903119887

]V = int

ℎ119873+2

ℎ119873+1

[119862119873+1

119887

] [1198852] 119889119911

[119863119903119903119887

]V = int

ℎ119873+2

ℎ119873+1

[1198852]119879

[119862119873+1

119887

] [1198852] 119889119911

[119863119905119904]V = ℎV [119862

119873+1

119904

]

[119863119905119903119904

]V = int

ℎ119873+2

ℎ119873+1

[119862119873+1

119904

] [1198855] 119889119911

[119863119903119903119904

]V = int

ℎ119873+2

ℎ119873+1

[1198855]119879

[119862119873+1

119887

] [1198855] 119889119911

[119863119905119887]119901

= ℎ119901[119862

119873+2

119887

]

[119863119905119903119887

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887

] [1198853] 119889119911

[119863119903119903119887

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+2

119887

] [1198853] 119889119911

[119863119905119904]119901

= ℎ119901[119862

119873+2

119904

]

[119863119905119903119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119904

] [1198856] 119889119911

[119863119903119903119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198856]119879

[119862119873+2

119887

] [1198856] 119889119911

[119863119905119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887119904

] 119889119911

[119863119905119903119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887119904

] [1198856] 119889119911

[119863119903119905119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+2

119887119904

] 119889119911

[119863119903119903119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+3

119887119904

] [1198856] 119889119911

119863119905119887119901

= int

ℎ119873+3

ℎ119873+2

minus119890119887

ℎ119901

119889119911

119863119903119887119901

= int

ℎ119873+3

ℎ119873+2

minus[119885

3]119879

119890119887

ℎ119901

119889119911

119863119905119904119901

= int

ℎ119873+3

ℎ119873+2

minus119890119904

ℎ119901

119889119911

119863119903119904119901

= int

ℎ119873+3

ℎ119873+2

minus[119885

6]119879

119890119904

ℎ119901

119889119911

(26)

8 Journal of Composites

Applying the dynamic version of the virtual work principle[32] the following open-loop equations of motion for anelement of the overall plate integrated with the ACLDtreatment are obtained

[119872119890

] 119889119890

119905

+ [119870119890

119905119905

] 119889119890

119905

+ [119870119890

119905119903

] 119889119890

119903

= 119865119890

119905119901

119881 + 119865119890

[119870119890

119905119903

]119879

119889119890

119905

+ [119870119890

119903119903

] 119889119890

119903

= 119865119890

119903119901

119881

(27)

Since the elastic constant matrix of the viscoelastic layer iscomplex the stiffness matrices of an element integrated withthe ACLD treatment are complex In case of an elementnot integrated with the ACLD treatment the electro-elasticcoupling matrices become null vectors and the elementalstiffnessmatrices are to be computed without considering thepiezoelectric and viscoelastic layers It should also be notedthat as the stiffness matrices associated with the transverseshear strains are derived separately one can employ theselective integration rule in a straight forward manner toavoid shear locking problem in case of thin plates Therestrained boundary conditions at the skew edges of theplate are such that the displacements of the points locatedat the skew edges are to be restrained along 119909

1015840- 1199101015840- and

1199111015840-directions (Figure 1(a)) Hence in order to impose theboundary conditions in a straight forward manner thegeneralized displacement vectors of a point lying on the skewedge are to be transformed as follows

119889119905 = [119871

119905] 119889

1015840

119905

119889119903 = [119871

119903] 119889

1015840

119903

(28)

where 1198891015840

119905

and 1198891015840

119903

are the generalized displacement vectorsof the point with respect to 119909

1015840

1199101015840

1199111015840 coordinate system and are

given by

1198891015840

119905

= [1199061015840

119900

V1015840119900

1199081015840

119900

]119879

1198891015840

119903

= [1205791015840

119909

1205791015840

119910

1205791015840

119911

1206011015840

119909

1206011015840

119910

1206011015840

119911

1205741015840

119911

1205741015840

119910

1205741015840

119911]119879

(29)

Also the transformation matrices [119871119905] and [119871

119903] are given by

[119871119905] = [

[

119888 119904 0

minus119904 119888 0

0 0 1

]

]

[119871119903] =

[[[[[[[[[[[[

[

119888 119904 0 0 0 0 0 0 0

minus119904 119888 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 119888 119904 0 0 0 0

0 0 0 minus119904 119888 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 119888 119904 0

0 0 0 0 0 0 minus119904 119888 0

0 0 0 0 0 0 0 0 1

]]]]]]]]]]]]

]

(30)

in which 119888 = cos120572 and 119904 = sin120572Thus the elementalmatricesof the element containing the nodes lying on the skew edgeare to be augmented as follows

[119870119890

119905119905

] = [1198791]119879

[119870119890

119905119905

] [1198791] [119870

119890

119905119903

] = [1198791]119879

[119870119890

119905119903

] [1198792]

[119870119890

119903119903

] = [1198792]119879

[119870119890

119903119903

] [1198792] [119872

119890

] = [1198791]119879

[119872119890

] [1198791]

(31)

The forms of the transformation matrices [1198791] and [119879

2] are

given by

[1198791] =

[[[[[[[[[[[

[

[119871119905] 119900 119900 119900 119900 119900 119900 119900

119900 [119871119905] 119900 119900 119900 119900 119900 119900

119900 119900 [119871119905] 119900 119900 119900 119900 119900

119900 119900 119900 [119871119905] 119900 119900 119900 119900

119900 119900 119900 119900 [119871119905] 119900 119900 119900

119900 119900 119900 119900 119900 [119871119905] 119900 119900

119900 119900 119900 119900 119900 119900 [119871119905] 119900

119900 119900 119900 119900 119900 119900 119900 [119871119905]

]]]]]]]]]]]

]

[1198792] =

[[[[[[[[[[[

[

[119871119903] 119900 119900 119900 119900 119900 119900 119900

119900 [119871119903] 119900 119900 119900 119900 119900 119900

119900 119900 [119871119903] 119900 119900 119900 119900 119900

119900 119900 119900 [119871119903] 119900 119900 119900 119900

119900 119900 119900 119900 [119871119903] 119900 119900 119900

119900 119900 119900 119900 119900 [119871119903] 119900 119900

119900 119900 119900 119900 119900 119900 [119871119903] 119900

119900 119900 119900 119900 119900 119900 119900 [119871119903]

]]]]]]]]]]]

]

(32)

in which 119900 and 119900 are (3 times 3) and (9 times 9) null matricesrespectively It may be noted that in case of a node ofsuch element which does not lie on the skew edge thegeneralized displacement vectors of this node with respect to119909119910119911 coordinate system are not required to be transformedHence thematrices [119871

119905] and [119871

119903] corresponding to suchnode

turn out to be (3times3) and (9times9) identitymatrices respectivelyThe elemental equations of motion are now assembled

to obtain the open-loop global equations of motion of theoverall skew plate as follows

[119872] + [119870119905119905] 119883 + [119870

119905119903] 119883

119903 =

119902

sum

119895=1

119865119895

119905119901

119881119895

+ 119865

[119870119905119903]119879

119883 + [119870119903119903] 119883

119903 =

119902

sum

119895=1

119865119895

119903119901

119881119895

(33)

where [119870119905119905] [119870

119905119903] and [119870

119903119903] are the global stiffness matrices

of the overall skew laminated composite plate [119872] is theglobal mass matrix 119865

119905119901 and 119865

119903119901 are the global electro-

elastic coupling vectors 119883 and 119883119903 are the global nodal

generalized displacement vectors 119865 is the global nodalforce vector 119902 is the number of patches and 119881

119895 is thevoltage applied to the 119895th patch Since the elemental stiffnessmatrices of an element augmented with the ACLD treatmentare complex the global stiffness matrices become complexand the energy dissipation characteristics of the overallplate are attributed to the imaginary part of these matricesHence the global equations of motion as derived previouslyalso represent the modeling of the passive (uncontrolled)constrained layer damping of the substrate plate when theconstraining layer is not subjected to any control voltagefollowing a derivative control law

Journal of Composites 9

3 Closed-Loop Model

In the active control approach the active constraining layerof each patch is activated with a control voltage negativelyproportional to the transverse velocity of a point Thus thecontrol voltage for each patch can be expressed in terms of thederivatives of the global nodal degrees of freedom as follows

119881119895

= minus119896119895

119889

= minus119896119895

119889

[119880119895

119905

] minus 119896119895

119889

(ℎ

2) [119880

119895

119903

] 119903 (34)

where [119880119895

119905

] and [119880119895

119903

] are the unit vectors describing thelocation of the velocity sensor for the 119895th patch and 119896

119895

119889

isthe control gain for the 119895th patch Substituting (34) into(33) the final equations of motion governing the closed-loop dynamics of the overall skew laminated compositeplateACLD system can be obtained as follows

[119872] + [119870119905119905] 119883 + [119870

119905119903] 119883

119903 +

119898

sum

119895=1

119896119895

119889

119865119895

119905119901

[119880119895

119905

]

+

119898

sum

119895=1

119896119895

119889

(ℎ

2) 119865

119895

119905119901

[119880119895

119903

] 119903 = 119865

[119870119905119903]119879

119883 + [119870119903119903] 119883

119903 +

119898

sum

119895=1

119896119895

119889

119865119895

119905119901

[119880119895

119905

]

+

119898

sum

119895=1

119896119895

119889

(ℎ

2) 119865

119895

119905119901

[119880119895

119903

] 119903 = 0

(35)

4 Results and Discussion

In this section the numerical results obtained by the finiteelement model derived in the previous section have beenpresented Symmetric as well as antisymmetric cross-plyand antisymmetric angle-ply laminated skew substrate platesintegrated with the two patches of ACLD treatment areconsidered for presenting the numerical results As shown inFigure 1 the locations of the patches have been selected insuch a way that the energy dissipation corresponding to thefirst fewmodes becomesmaximum PZT-5Hspur compositewith 60 fiber volume fraction has been considered for thematerial of the constraining layer of the ACLD treatmentUsing themicromechanicsmodel and thematerial propertiesof the constituent phases [35] the effective elastic andpiezoelectric material properties of the 1ndash3 PZC with respectto the principle material coordinate system are computed asfollows

11986211

= 929GPa 11986212

= 618GPa 11986213

= 605GPa

11986233

= 3544GPa 11986223

= 11986213 119862

44= 158GPa

11986266

= 154GPa 11986255

= 11986244

11989031

= minus01902Cm2

11989032

= 11989031

11989033

= 184107Cm2

11989024

= 0004Cm2

11989015

= 11989024

(36)

The material of the orthotropic layers of the substrate platesis a graphiteepoxy composite and its material properties are[42]

119864119871= 172GPa 119864

119871

119864119879

= 25 119866119871119879

= 05119864119879

119866119879119879

= 02119864119879 ]

119871119879= ]

119879119879= 025

(37)

in which the symbols have the usual meaning Unless oth-erwise mentioned the aspect ratio (119886ℎ) and the thicknessof the skew substrate plates are considered as 100 and0003m respectively while the orthotropic layers of theskew substrate plates are of equal thickness Also the lengthand the skew sides of the substrate are equal (ie 119886 =

119887) The loss factor of the viscoelastic layer considered fornumerical results remains invariant [20] within a broadband (0ndash400Hz) of frequency range and the values of thecomplex shear modulus Poisson ratio and the density ofthe viscoelastic layer are considered as 20(1 + 119894)MNmminus2049 and 1140 kgmminus3 respectively [20] The thicknesses ofthe viscoelastic layer and the 1ndash3 PZC layer are consideredas 508120583m and 250 120583m respectively The value of the shearcorrection factor is used as 56 Three points of Gaussianintegration rule are considered for computing the elementmatrices corresponding to the bending deformations whiletwo points of that are used for computing the elementmatrices corresponding to transverse shear deformationsThe simply supported boundary conditions at the edges of theoverall plate are invoked as follows

at 119909 = 119910 tan120572 119909 = 119886 + 119910 tan120572

V1015840119900

= 1199081015840

119900

= 1205791015840

119910

= 1206011015840

119910

= 1205741015840

119910

= 1205791015840

119911

= 1205741015840

119911

= 0

at 119910 = 0 119910 = 119887 cos120572

119906119900= 119908

119900= 120579

119909= 120601

119909= 120574

119909= 120579

119911= 120574

119911= 0

(38)

To verify the validity of the present finite elementmodel (FEM) the natural frequencies of the skew laminatedcomposite plates integrated with the inactivated patches ofnegligible thickness are first computed and subsequentlycompared with the existing analytical results [41] of theidentical plates without being integrated with the patchesSuch comparisons for the simply supported and clamped-clamped skew laminated plates are presented in Tables 1 and2 respectively It may be observed from these tables that theresults are in excellent agreement validating the model of theplate derived here

In order to assess the performance of the verti-callyobliquely reinforced 1ndash3 PZC as the material of thedistributed actuators of smart skew laminated compositeplates frequency responses for ACLD of the substrate skewlaminated composite plates are investigated To compute the

10 Journal of Composites

Table 1 Comparison of fundamental natural frequencies (120596) of simply supported skew laminated composite plates integratedwith the patchesof negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 15699 28917 37325 14829 24656 32522 17974 33351 33351Present FEM 15635 24383 35033 15076 24380 32254 18493 33359 33370

15∘ Reference [41] 16874 30458 39600 15741 25351 30270 18313 32490 36724Present FEM 16571 29840 36505 15796 25775 29892 18675 32075 35810

30∘ Reference [41] 20840 34023 46997 18871 29372 34489 20270 34431 42361Present FEM 19596 31690 46796 18226 29585 32357 19894 32365 43208

45∘ Reference [41] 28925 41906 54149 25609 33126 40617 25609 33131 42772Present FEM 24811 44875 53289 22996 34773 44889 23194 34870 45009

lowast

120596 = (120596119886

2

120587

2

ℎ)radic120588119864119879 120596 is the circular natural frequency

Table 2 Comparison of fundamental natural frequencies (120596) of clamped-clamped skew laminated composite plates integrated with thepatches of negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 23687 35399 41122 22990 37880 37880 22119 37339 37339Present FEM 23201 34769 44102 23315 36531 36545 22433 36000 36012

15∘ Reference [41] 24663 36255 43418 23809 37516 40785 23099 36997 40438Present FEM 23699 34821 44049 23741 35856 38401 23049 35346 38092

30∘ Reference [41] 27921 39557 50220 26666 39851 47227 26325 39549 52107Present FEM 25366 35696 44734 25240 41943 45373 24945 36113 50932

45∘ Reference [41] 34739 47129 58789 33015 46290 58423 33015 46290 58423Present FEM 28665 47074 54562 28377 47614 56620 28377 47614 55162

frequency response functions the skew laminated compositeplates are harmonically excited by a force of amplitude 1Newton applied to a point (1198862 (1198864) cos120572 ℎ2) on the skewsubstrate platesThe control voltage supplied to the constrain-ing layer of the first patch is negatively proportional to thevelocity of the point (1198862 (1198864) cos120572 ℎ2) and that applied tothe constraining layer of the second patch is negatively pro-portional to the velocity of the point (1198862 (31198864) cos120572 ℎ2)

Figures 3 4 and 5 illustrate the frequency responsefunctions for the transverse displacement of a point of asimply supported symmetric cross-ply (0∘90∘0∘) plate with15∘ 30∘ and 45∘ skew angles respectively while the fiberorientation angle (120595) in the constraining layer is 0∘ Thesefigures display both uncontrolled and controlled responsesand clearly show that the active constraining layer made ofthe vertically reinforced 1ndash3 PZC material being studied heresignificantly attenuates the amplitude of vibrations enhanc-ing the damping characteristics of the overall laminatedskew composite plates over the passive damping (uncon-trolled) Similar responses are also obtained for antisym-metric cross-ply (0∘90∘0∘90∘) and antisymmetric angle-ply (45∘minus45∘45∘minus45∘) skew laminated composite plates

However for brevity these results are not presented hereThemaximum values of the control voltages required to computethe controlled responses presented in Figures 3ndash5 have beenfound to be quite low and are illustrated in Figure 6 in caseof the symmetric cross-ply (0∘90∘0∘) plate with skew angle15∘ only In order to examine the contribution of the verticalactuation in improving the damping characteristics of theskew laminated composite plates as demonstrated in Figures3ndash5 active control responses of the symmetric cross-plylaminated composite plate with 15∘ skew angle are plotted inFigure 7with andwithout considering the value of 119890

33and 119890

31

It may be noted that when the values of 11989031is zero and that of

11989033is nonzero the vertical actuation of the active constraining

layer of the ACLD treatment is responsible for increasing thetransverse shear deformations of the viscoelastic constrainedlayer over the passive counterpart resulting in improveddamping of the smart plate over its passive damping Onthe other hand if 119890

33is zero and 119890

31is nonzero the in-

plane actuation of the active constraining layer causes thetransverse shear deformation of the viscoelastic core of theACLD treatment leading to the active damping of the smartplate It is evident from Figure 7 that the contribution of the

Journal of Composites 11

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 3 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 4 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 30∘)

vertical actuation by the constraining vertically reinforced 1ndash3 PZC layer is significantly larger than that of the in-planeactuation by the same for controlling the modes displayedin Figure 7 Though not shown here similar results are alsoobtained for antisymmetric cross-ply and angle-ply skewplates

Figures 8 9 and 10 demonstrate the effect of variationof piezoelectric fiber orientation angle on the performanceof the patches for improving the damping characteristicsof simply supported symmetric cross-ply (0∘90∘0∘)

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 5 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 45∘)

0

5

10

15

20

25

30

35

40

45

50

Con

trol v

olta

ge (V

olt)

0 100 200 300 400 500 600Frequency (Hz)

119870119889 = 600

119870119889 = 1200

Figure 6 Frequency response functions for the control voltages foractive damping of the simply supported skew symmetric cross-ply(0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

antisymmetric cross-ply (0∘90∘0∘90∘) and angle-ply(45∘minus45∘45∘minus45∘) substrate skew composite plates with30∘ skew angle respectively while the orientation angle (120595)

of the piezoelectric fibers is varied in the vertical 119909119911-planeThese figures illustrate that the attenuating capability of thepatches becomesmaximumwhen the value of the orientationangle (120595) of the fibers in the 119909119911-plane is 0∘ Similar effectis also obtained when the fiber orientation angle (120595) in theconstraining layer of the ACLD treatment is varied in thevertical 119910119911-plane as illustrated in Figures 11 12 and 13 forthe symmetric cross-ply (0∘90∘0∘) antisymmetric cross-ply(0∘90∘0∘90∘) and angle-ply (45∘minus45∘45∘minus45∘) substrate

12 Journal of Composites

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

11989031 = minus01902 11989033 = 184107

11989031 = minus01902 11989033 = 011989031 = 0 11989033 = 184107

w(a2(b4)co

s120572h2

) (m

)

Figure 7 Frequency response functions demonstrating the con-tribution of vertical actuation on the controlled responses forthe transverse displacement 119908 (1198862 (1198864) cos120572 ℎ2) of the skewsymmetric cross-ply (0∘90∘0∘) plate (119870

119889

= 600 119886ℎ = 100 120595 = 0∘

120572 = 15∘)

0 100 200 300 400 500 600Frequency (Hz)

0

02

04

06

08

1

12times10minus4

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 8 Effect of variation of piezoelectric fiber orientation angle(120595) on the controlled responses of a simply supported symmetricskew cross-ply (0∘90∘0∘) plate when the piezoelectric fibers of theconstraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

skew composite plates with 30∘ skew angle respectivelyIn this regard it should be noted that for investigating theeffect of variation of piezoelectric fiber orientation angle(120595) on the performance of the ACLD patches the valueof 120595 has been smoothly varied to compute the controlledresponses However for the sake of clarity in the plots theresponses corresponding to the four specific values of 120595

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 9 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 10 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

have been presented such that the optimum performanceof the patches can be evaluated The significant effect of thevariation of the piezoelectric fiber orientation angle in thevertical 119909119911- and 119910119911-planes has been noticed when all theboundaries of the overall plates are clamped as shown inFigures 14ndash19 When the piezoelectric fibers are coplanarwith the vertical 119909119911-plane the frequency response functionsplotted in Figures 14 15 and 16 reveal that the control

Journal of Composites 13

0

02

04

06

08

1

12

14

16times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 11 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 12 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=600 119886ℎ = 100 120572 = 30

∘)

authority of the patches becomes maximum for improvingthe damping characteristics of the clamped-clamped skewcross-ply and angle-ply substrate plates if the value of thepiezoelectric fiber orientation angle (120595) is 30∘ The same isalso found to be true if the piezoelectric fibers are coplanarwith the vertical 119910119911-plane as shown in Figures 17 18 and 19for the previously mentioned skew plates Similar responsesare also found for the effect of piezoelectric fiber orientation

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 13 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers of the constraining layer are coplanar with the119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 14 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600119886ℎ = 100120572 = 30

∘)

angle on the active damping of other skew laminated plateswith different skew angles However for the sake of brevitythey are not presented here

5 Conclusions

In this paper a study has been carried out to investi-gate the performance of the ACLD treatment for active

14 Journal of Composites

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 15 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 16 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

damping of smart skew laminated composite plates whenverticallyobliquely reinforced 1ndash3 PZC materials are usedas the materials for the constraining layer of the ACLDtreatment A layerwise FSDT-based finite element modelhas been developed to describe the dynamics of the skewlaminated composite plates integrated with the patches ofACLD treatment Unlike the existing finite element modelsof smart structures integrated with ACLD treatment the

200 250 300 350 400 450 500 550 600 650 7000

02

04

06

08

1

12

14

16

18

2times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 17 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

200 300 400 500 600 700 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 18 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

derivation of the present finite element model includes thetransverse deformations of the substrate skew laminatedcomposite plate the constrained viscoelastic layer and theconstraining 1ndash3 PZC layer of the ACLD treatment along thethickness (ie 119911) direction such that both vertical and in-plane actuations by the constraining layer of the patches canbe utilized for active damping of the plates The frequency

Journal of Composites 15

0

05

1

15

2

25

3

35

4times10minus5

200 300 400 500 600 700 800Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 19 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

responses of the symmetric cross-ply antisymmetric cross-ply and angle-ply skew composite plates indicate that theactive constraining layer of the ACLD treatment being madeof the vertically reinforced 1ndash3 PZC material significantlyenhances the damping characteristics of the plates over thepassive damping The analysis revealed that if the verticallyreinforced 1ndash3 PZCmaterial is used for the constraining layerof the ACLD treatment then the contribution of the verticalactuation by the constraining layer alone for improving theactive damping characteristics of the smart skew laminatedcomposite plate is significantly larger than that due to thein-plane actuation by the constraining layer alone It isimportant to note from the present investigation that thevariation of the orientation angle of the piezoelectric fibersof the constraining layer in the vertical 119909119911- and 119910119911-planesas well as the skew angle and boundary conditions of theoverall plates significantly affects the performance of thepatches For the simply supported laminated substrate platesthe performance of the patches becomes maximumwhen thepiezoelectric fiber orientation angle (120595) is 0∘ while in caseof the clamped-clamped laminated skew substrate plates themaximum control authority of the patches is achieved whenthe value of 120595 is 30∘ irrespective of the cases in which thefibers are oriented in 119909119911- and 119910119911-planes

Appendix

In (6) and (7) the matrices [1198851] [119885

2] [119885

3] [119885

4] [119885

5] and

[1198856] are given by

[1198851] = [[119885

1] 0 0]

[1198852] = [(

ℎ

2) 119868 [119885

2] 0]

[1198853] = [(

ℎ

2) 119868 ℎV119868 [119885

3]]

[1198854] = [119868 0 0 119911119868 0 0]

[1198855] = [0 119868 0 (

ℎ

2) 119868 (119911 minus

ℎ

2) 119868 0]

[1198856] = [0 0 119868 (

ℎ

2) 119868 ℎV119868 (119911 minus ℎ

119873+2) 119868]

(A1)

in which

[1198851] =

[[[

[

119911 0 0 0

0 119911 0 0

0 0 119911 0

0 0 0 1

]]]

]

[1198852] =

[[[[[[[[

[

(119911 minusℎ

2) 0 0 0

0 (119911 minusℎ

2) 0 0

0 0 (119911 minusℎ

2) 0

0 0 0 1

]]]]]]]]

]

[1198853] =

[[[

[

(119911 minus ℎ119873+2

) 0 0 0

0 (119911 minus ℎ119873+2

) 0 0

0 0 (119911 minus ℎ119873+2

) 0

0 0 0 1

]]]

]

[119868] =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

[119868] = [1 0

0 1]

[0] = [0 0

0 0] [0] = [

0 0

0 0]

(A2)

The various submatrices 119861119905119887119894 119861

119903119887119894 and 119861

119903119904119894 appearing in (21)

are given by

119861119905119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 0

]]]]]]]]]]]

]

119861119905119904119894

=

[[[[

[

0 0120597120578

119894

120597119909

0 0120597120578

119894

120597119910

]]]]

]

16 Journal of Composites

119861119903119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 1

]]]]]]]]]]]

]

119861119903119887119894

=

[[[[[[

[

119861119903119887119894

0

0

0 119861119903119887119894

0

0

0 119861119903119887119894

]]]]]]

]

119861119903119904119894

=

[[[[[[[[[[[[[[[[[[

[

119868 0 0

0

119868 0

0 0

119868

119861119903119904119894

0 0

0 119861119903119904119894

0

0 0 119861119903119904119894

]]]]]]]]]]]]]]]]]]

]

119868= [1 0 0

0 1 0]

(A3)

wherein

0 and 0 are the (3 times 3) and (2 times 3) null matricesrespectively

References

[1] T Bailey and J E Hubbard ldquoDistributedpiezoelectric polymeractive vibration control of a cantilever beamrdquo Journal of Guid-ance Control and Dynamics vol 8 no 5 pp 605ndash611 1985

[2] E F Crawley and J de Luis ldquoUse of piezoelectric actuators aselements of intelligent structuresrdquo AIAA Journal vol 25 no 10pp 1373ndash1385 1987

[3] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988

[4] C K Lee W W Chiang and O Sulivan ldquoPiezoelectric modalsensoractuator pairs for critical active damping vibrationcontrolrdquo Journal of the Acoustical Society of America vol 90 no1 pp 374ndash384 1991

[5] S Hanagud M W Obal and A J Calise ldquoOptimal vibrationcontrol by the use of piezoceramic sensors and actuatorsrdquoJournal of Guidance Control and Dynamics vol 15 no 5 pp1199ndash1206 1992

[6] Y Gu R L Clark C R Fuller and A C Zander ldquoExperimentson active control of plate vibration using piezoelectric actuatorsand polyvinylidene fluoride (PVDF) modal sensorsrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 116 no3 pp 303ndash308 1994

[7] A Baz and S Poh ldquoOptimal vibration control with modalpositive position feedbackrdquo Optimal Control Applications andMethods vol 17 no 2 pp 141ndash149 1996

[8] A Kugi K Schlacher and H Irschik ldquoInfinite-dimensionalcontrol of nonlinear beam vibrations by piezoelectric actuatorand sensor layersrdquo Nonlinear Dynamics vol 19 no 1 pp 71ndash911999

[9] D Sun L Tong and D Wang ldquoVibration control of platesusing discretely distributed piezoelectric quasi-modal actua-torssensorsrdquo AIAA Journal vol 39 no 9 pp 1766ndash1772 2001

[10] G Caruso S Galeani and L Menini ldquoActive vibration controlof an elastic plate using multiple piezoelectric sensors andactuatorsrdquo SimulationModelling Practice andTheory vol 11 no5-6 pp 403ndash419 2003

[11] X Zhang and A G Erdman ldquoOptimal placement of piezo-electric sensors and actuators for controlled flexible linkagemechanismsrdquo Journal of Vibration and Acoustics Transactionsof the ASME vol 128 no 2 pp 256ndash260 2006

[12] G Gatti M J Brennan and P Gardonio ldquoActive dampingof a beam using a physically collocated accelerometer andpiezoelectric patch actuatorrdquo Journal of Sound and Vibrationvol 303 no 3ndash5 pp 798ndash813 2007

[13] M K Kwak S Heo and M Jeong ldquoDynamic modellingand active vibration controller design for a cylindrical shellequipped with piezoelectric sensors and actuatorsrdquo Journal ofSound and Vibration vol 321 no 3ndash5 pp 510ndash524 2009

[14] I Bruant L Gallimard and S Nikoukar ldquoOptimal piezoelectricactuator and sensor location for active vibration control usinggenetic algorithmrdquo Journal of Sound and Vibration vol 329 no10 pp 1615ndash1635 2010

[15] Y Fu J Wang and Y Mao ldquoNonlinear vibration and activecontrol of functionally graded beams with piezoelectric sensorsand actuatorsrdquo Journal of Intelligent Material Systems andStructures vol 22 no 18 pp 2093ndash2102 2011

[16] F Chen M Hong M Song and H Cui ldquoOptimal control ofa beam with discontinuously distributed piezoelectric sensorsand actuatorsrdquo Journal of Marine Science and Application vol11 no 1 pp 44ndash51 2012

[17] B Azvine G R Tomlinson and R J Wynne ldquoUse of activeconstrained-layer damping for controlling resonant vibrationrdquoSmart Materials and Structures vol 4 no 1 pp 1ndash6 1995

[18] A Baz and J Ro ldquoVibration control of plates with activeconstrained layer dampingrdquo SmartMaterials and Structures vol5 no 3 pp 272ndash280 1996

[19] C Chantalakhana and R Stanway ldquoActive constrained layerdamping of clamped-clamped plate vibrationsrdquo Journal ofSound and Vibration vol 241 no 5 pp 755ndash777 2001

[20] J Ro and A Baz ldquoOptimum placement and control ofactive constrained layer damping using modal strain energyapproachrdquo JVCJournal of Vibration and Control vol 8 no 6pp 861ndash876 2002

[21] M C Ray and J N Reddy ldquoOptimal control of thin circularcylindrical laminated composite shells using active constrainedlayer damping treatmentrdquo Smart Materials and Structures vol13 no 1 pp 64ndash72 2004

[22] C M A Vasques B R Mace P Gardonio and J D RodriguesldquoArbitrary active constrained layer damping treatments onbeams finite element modelling and experimental validationrdquoComputers and Structures vol 84 no 22-23 pp 1384ndash14012006

[23] F M Li K Kishimoto K Wang Z B Chen and W HHuang ldquoVibration control of beams with active constrainedlayer dampingrdquo Smart Materials and Structures vol 17 no 6Article ID 065036 2008

[24] L Yuan Y Xiang Y Huang and J Lu ldquoA semi-analyticalmethod and the circumferential dominant modal control ofcircular cylindrical shells with active constrained layer dampingtreatmentrdquo SmartMaterials and Structures vol 19 no 2 ArticleID 025010 2010

[25] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

Journal of Composites 17

[26] W A Smith and B A Auld ldquoModeling 1ndash3 composite piezo-electrics thickness-mode oscillationsrdquo IEEE Transactions onUltrasonics Ferroelectrics and Frequency Control vol 38 no 1pp 40ndash47 1991

[27] PiezocompositesMaterial Systems Inc 543 Great Road Little-ton MA 01460

[28] Y Benveniste and G J Dvorak ldquoUniform fields and universalrelations in piezoelectric compositesrdquo Journal of the Mechanicsand Physics of Solids vol 40 no 6 pp 1295ndash1312 1992

[29] M L Dunn and M Taya ldquoMicromechanics predictions ofthe effective electroelastic moduli of piezoelectric compositesrdquoInternational Journal of Solids and Structures vol 30 no 2 pp161ndash175 1993

[30] J Aboudi ldquoMicromechanical prediction of the effective coeffi-cients of thermo-piezoelectric multiphase compositesrdquo Journalof Intelligent Material Systems and Structures vol 9 no 9 pp713ndash722 1999

[31] X Ruan T W Chou A Safari and S C Danforth ldquoA 3-Dconnectivity model for effective piezoelectric properties of yarncompositesrdquo Journal of Composite Materials vol 36 no 14 pp1693ndash1708 2002

[32] A Baz and A Tempia ldquoActive piezoelectric damping compos-itesrdquo Sensors and Actuators A vol 112 no 2-3 pp 340ndash3502004

[33] H Berger S Kari U Gabbert et al ldquoUnit cell models ofpiezoelectric fiber composites for numerical and analytical cal-culation of effective propertiesrdquo SmartMaterials and Structuresvol 15 no 2 pp 451ndash458 2006

[34] M Sakthivel and A Arockiarajan ldquoAn analytical model forpredicting thermo-electro-mechanical response of 1ndash3 piezo-electric compositesrdquo Computational Materials Science vol 48no 4 pp 759ndash767 2010

[35] M C Ray and A K Pradhan ldquoOn the use of verticallyreinforced 1ndash3 piezoelectric composites for hybrid damping oflaminated composite platesrdquo Mechanics of Advanced Materialsand Structures vol 14 no 4 pp 245ndash261 2007

[36] M C Ray and A K Pradhan ldquoPerformance of vertically andobliquely reinforced 1ndash3 piezoelectric composites for activedamping of laminated composite shellsrdquo Journal of Sound andVibration vol 315 no 4-5 pp 816ndash835 2008

[37] S K Sarangi and M C Ray ldquoSmart damping of geometricallynonlinear vibrations of laminated composite beams using verti-cally reinforced 1ndash3 piezoelectric compositesrdquo Smart Materialsand Structures vol 19 no 7 Article ID 075020 2010

[38] A Krishnan and J V Deshpande ldquoVibration of skew laminatesrdquoJournal of Sound and Vibration vol 153 no 2 pp 351ndash358 1992

[39] A R Krishna Reddy and R Palaninathan ldquoFree vibration ofskew laminatesrdquo Computers and Structures vol 70 no 4 pp415ndash423 1999

[40] C S Babu and T Kant ldquoTwo shear deformable finite elementmodels for buckling analysis of skew fibre-reinforced compositeand sandwich panelsrdquo Composite Structures vol 46 no 2 pp115ndash124 1999

[41] A K Garg R K Khare and T Kant ldquoFree vibration of skewfiber-reinforced composite and sandwich laminates using ashear deformable finite element modelrdquo Journal of SandwichStructures and Materials vol 8 no 1 pp 33ndash53 2006

[42] J N Reddy Mechanics of Laminated Composites Plates Theoryand Analysis CRC press Boca Raton Fla USA 1996

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

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Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

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BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 4: Research Article Active Constrained Layer Damping of Smart …downloads.hindawi.com/archive/2013/824163.pdf · 2019-07-31 · Research Article Active Constrained Layer Damping of

4 Journal of Composites

the 1ndash3 PZC layer are assumed to be linearly varying acrosstheir thicknesses Thus similar to the axial displacement thetransverse displacements at any point in the overall plate canbe expressed as

119908 (119909 119910 119911 119905) = 1199080(119909 119910 119905) + (119911 minus ⟨119911 minus

ℎ

2⟩)120579

119911(119909 119910 119905)

+ (⟨119911 minusℎ

2⟩ minus ⟨119911 minus ℎ

119873+2⟩) 120601

119911(119909 119910 119905)

+ ⟨119911 minus ℎ119873+2

⟩ 120574119911(119909 119910 119905)

(3)

in which1199080refers to the transverse displacement at any point

on the reference plane and 120579119911 120601

119911 and 120574

119911are the generalized

displacements representing the gradients of the transversedisplacement in the substrate plate the viscoelastic layer andthe 1ndash3 PZC layer respectively with respect to the thicknesscoordinate (119911)

For the ease of analysis the generalized displacementvariables are grouped into the following two vectors

119889119905 = [119906

0V0

1199080]119879

119889119903 = [120579

119909120579119910

120579119911

120601119909

120601119910

120601119911

120574119909

120574119910

120574119911]119879

(4)

In order to implement the selective integration rule forcomputing the element stiffness matrices corresponding tothe transverse shear deformations the state of strain at any

point in the overall plate is divided into the following twostrain vectors 120576

119887 and 120576

119904

120576119887 = [120576

119909120576119910

120576119909119910

120576119911]119879

120576119904 = [120576

119909119911120576119910119911

]119879 (5)

in which 120576119909 120576

119910 and 120576

119911are the normal strains along 119909-

119910- and 119911-directions respectively 120576119909119910

is the in-plane shearstrain 120576

119909119911and 120576

119910119911are the transverse shear strains Using the

displacement fields given by (1)ndash(3) and the linear strain-displacement relations the vectors 120576

119887119888

120576119887V and 120576

119887119901

defining the state of in-plane and transverse normal strainsat any point in the substrate skew composite plate the vis-coelastic layer and the active constraining layer respectivelycan be expressed as

120576119887119888

= 120576119887119905 + [119885

1] 120576

119887119903 120576

119887V = 120576

119887119905 + [119885

2] 120576

119887119903

120576119887119901

= 120576119887119905 + [119885

3] 120576

119887119903

(6)

Similarly the vectors 120576119904119888

120576119904V and 120576

119904119901

defining the stateof transverse shear strains at any point in the substrate com-posite plate the viscoelastic layer and the active constraininglayer respectively can be expressed as

120576119904119888

= 120576119904119905 + [119885

4] 120576

119904119903 120576

119904V = 120576

119904119905 + [119885

5] 120576

119904119903

120576119904119901

= 120576119904119905 + [119885

6] 120576

119904119903

(7)

The various matrices appearing in (6) and (7) have beendefined in the Appendix while the generalized strain vectorsare given by

120576119887119905 = [

1205971199060

120597119909

120597V0

120597119910

1205971199060

120597119910+

120597V0

1205971199090]

119879

120576119904119905 = [

1205971199080

120597119909

1205971199080

120597119910]

119879

120576119887119903 = [

120597120579119909

120597119909

120597120579119910

120597119910

120597120579119909

120597119910+

120597120579119910

120597119909120579119911

120597120601119909

120597119909

120597120601119910

120597119910

120597120601119909

120597119910+

120597120601119910

120597119909120601119911

120597120574119909

120597119909

120597120574119910

120597119910

120597120574119909

120597119910+

120597120574119910

120597119909120574119911

]

119879

120576119904119903 = [120579

119909120579119910

120601119909

120601119910

120574119909

120574119910

120597120579119911

120597119909

120597120579119911

120597119910

120597120601119911

120597119909

120597120601119911

120597119910

120597120574119911

120597119909

120597120574119911

120597119910]

119879

(8)

Similar to the strain vectors given by (5) the state of stressesat any point in the overall plat is described by the followingstress vectors

120590119887 = [120590

119909120590119910

120590119909119910

120590119911]119879

120590119904 = [120590

119909119911120590119910119911

]119879

(9)

where 120590119909 120590

119910 and 120590

119911are the normal stresses along 119909- 119910-

and 119911-directions respectively 120590119909119910is the in-plane shear stress

120590119909119911

and 120590119910119911

are the transverse shear stresses The constitutive

relations for the material of any orthotropic layer of thesubstrate plate are given by

120590119896

119887

= [119862119896

119887

] 120576119896

119887

120590119896

119904

= [119862119896

119904

] 120576119896

119904

(119896 = 1 2 3 119873)

(10)

Journal of Composites 5

where the elastic coefficient matrices are

[119862119896

119887

] =

[[[[[[[[[[[[[[

[

119862119896

11

119862119896

12

119862119896

16

119862119896

13

119862119896

12

119862119896

22

119862119896

26

119862119896

23

119862119896

16

119862119896

26

119862119896

66

119862119896

36

119862119896

13

119862119896

23

119862119896

36

119862119896

33

]]]]]]]]]]]]]]

]

[119862119896

119904

] =[[

[

119862119896

55

119862119896

45

119862119896

45

119862119896

44

]]

]

(11)

and 119862119896

119894119895

(119894 119895 = 1 2 3 6) are the transformed elasticcoefficients with respect to the reference coordinate systemThematerial of the viscoelastic layer is assumed to be linearlyviscoelastic and homogenous isotropic and is modeled byusing the complex modulus approach Thus the shear mod-ulus119866 and Youngrsquos modulus 119864 of the viscoelastic material aregiven by

119866 = 1198661015840

(1 + 119894120578) 119864 = 2119866 (1 + ]) (12)

in which 1198661015840 is the storage modulus ] is Poissonrsquos ratio and

120578 is the loss factor at a particular operating temperature andfrequency Employing the complex modulus approach theconstitutive relations for the material of the viscoelastic layer(119896 = 119873 + 1) can also be represented by (10) with 119862

119896

119894119895

(119894 119895 =

1 2 3 6) being the complex elastic constants [29 30]The constraining PZC layer will be subjected to the appliedelectric field (119864

119911) acting across its thickness (ie along the

119911-direction) only Accordingly the constitutive relations forthe 1ndash3 PZC material with respect to the laminate coordinatesystem (119909119910119911) can be expressed as

120590119896

119887

= [119862119896

119887

] 120576119896

119887

+ [119862119896

119887119904

] 120576119896

119904

minus 119890119887 119864

119911

120590119896

119904

= [119862119896

119887119904

] 120576119896

119887

+ [119862119896

119904

] 120576119896

119904

minus 119890119904 119864

119911

119863119911= 119890

119887119879

120576119896

119887

+ 119890119904119879

120576119896

119904

+ 12057633119864119911

119896 = 119873 + 2

(13)

Here 119863119911represents the electric displacement along the 119911-

direction and 12057633

is the dielectric constant It may be notedfrom the previous form of the constitutive relations that thetransverse shear strains are coupled with the in-plane normalstrains due to the orientation of piezoelectric fibers in the

vertical 119909119911- or 119910119911-plane and the corresponding couplingelastic constant matrices [119862

119873+2

119887119904

] are given by

[119862119873+2

119887119904

] =

[[[[[[[[[[

[

119862119873+2

15

0

119862119873+2

25

0

0 119862119873+2

46

119862119873+2

35

0

]]]]]]]]]]

]

or

[119862119873+2

119887119904

] =

[[[[[[[[[[

[

0 119862119873+2

14

0 119862119873+2

24

119862119873+2

56

0

0 119862119873+2

34

]]]]]]]]]]

]

(14)

as the piezoelectric fibers are coplanar with the vertical 119909119911-or 119910119911-plane respectively Note that if the fibers are coplanarwith both 119909119911- and 119910119911-planes (ie 120595 = 0

∘) this couplingmatrix becomes a nullmatrix Also the piezoelectric constantmatrices 119890

119887 and 119890

119904 appearing in (13) contain the following

transformed effective piezoelectric coefficients of the 1ndash3PZC

119890119887 = [119890

3111989032

11989036

11989033]119879

119890119904 = [119890

3511989034]119879

(15)

The total potential energy 119879119901and the kinetic energy 119879

119896of the

overall plateACLD system are given by [35]

119879119901

=1

2[

119873+2

sum

119896=1

intΩ

(120576119896

119887

119879

120590119896

119887

+ 120576119903

119904

119879

120590119896

119904

) 119889Ω

minusintΩ

119863119911119864119911119889Ω] minus int

119860

119889119879

119891 119889119860

119879119896=

1

2

119873+2

sum

119896=1

intΩ

120588119896

(2

+ V2 + 2

) 119889Ω

(16)

in which 120588119896 is the mass density of the 119896th layer 119891 is the

externally applied surface traction acting over a surface area119860 andΩ represents the volume of the concerned layer Sincethe plates under study are considered to be thin the rotaryinertia of the overall plate has been neglected in estimatingthe kinetic energy The overall plate is discretized by eight-noded isoparametric quadrilateral elements Following (4)the generalized displacement vectors associated with the 119894th(119894 = 1 2 3 8) node of the element can be written as

119889119905119894 = [119906

0119894V0119894

1199080119894]119879

119889119903119894 = [120579

119909119894120579119910119894

120579119911119894

120601119909119894

120601119910119894

120601119911119894

120574119909119894

120574119910119894

120574119911119894]119879

(17)

6 Journal of Composites

Thus the generalized displacement vectors at any pointwithin the element can be expressed in terms of the nodalgeneralized displacement vectors 119889

119890

119905

and 119889119890

119903

as follows

119889119890

119905

= [119873119905] 119889

119890

119905

119889119890

119903

= [119873119903] 119889

119890

119903

(18)

in which

119889119890

119905

= [119889119890

1199051

119879

119889119890

1199052

119879

sdot sdot sdot 119889119890

1199058

119879

]119879

119889119890

119903

= [119889119890

1199031

119879

119889119890

1199032

119879

sdot sdot sdot 119889119890

1199038

119879

]119879

[119873119905] = [119873

11990511198731199052

sdot sdot sdot 1198731199058]119879

[119873119903] = [119873

11990311198731199032

sdot sdot sdot 1198731199038]119879

119873119905119894

= 119899119894119868119905 119873

119903119894= 119899

119894119868119903

(19)

while 119868119905and 119868

119903are (3times3) and (9times 9) identitymatrices respec-

tively and 119899119894is the shape function of natural coordinates

associatedwith the 119894th nodeMaking use of the relations givenby (6)ndash(8) and (18) the strain vectors at any point within theelement can be expressed in terms of the nodal generalizeddisplacement vectors as follows

120576119887119888

= [119861119905119887] 119889

119890

119905

+ [1198851] [119861

119903119887] 119889

119890

119903

120576119887V = [119861

119905119887] 119889

119890

119905

+ [1198852] [119861

119903119887] 119889

119890

119903

120576119887119901

= [119861119905119887] 119889

119890

119905

+ [1198853] [119861

119903119887] 119889

119890

119903

120576119904119888

= [119861119905119904] 119889

119890

119905

+ [1198854] [119861

119903119904] 119889

119890

119903

120576119904V = [119861

119905119904] 119889

119890

119905

+ [1198855] [119861

119903119904] 119889

119890

119903

120576119904119901

= [119861119905119904] 119889

119890

119905

+ [1198856] [119861

119903119904] 119889

119890

119903

(20)

in which the nodal strain-displacement matrices [119861119905119887] [119861

119903119887]

[119861119905119904] and [119861

119903119904] are given by

[119861119905119887] = [119861

11990511988711198611199051198872

sdot sdot sdot 1198611199051198878

]

[119861119903119887] = [119861

11990311988711198611199031198872

sdot sdot sdot 1198611199031198878

]

[119861119905119904] = [119861

11990511990411198611199051199042

sdot sdot sdot 1198611199051199048

]

[119861119903119904] = [119861

11990311990411198611199031199042

sdot sdot sdot 1198611199031199048

]

(21)

The submatrices of [119861119905119887] [119861

119903119887] [119861

119905119904] and [119861

119903119904] have been

explicitly presented in the Appendix On substitution of (9)ndash(13) and (20) into (16) the total potential energy 119879

119890

119901

and the

kinetic energy 119879119890

119896

of a typical element augmented with theACLD treatment can be expressed as

119879119890

119901

=1

2[119889

119890

119905

119879

[119870119890

119905119905

] 119889119890

119905

+ 119889119890

119905

119879

[119870119890

119905119903

] 119889119890

119903

+ 119889119890

119903

119879

[119870119890

119905119903

]119879

times 119889119890

119905

+ 119889119890

119903

119879

[119870119890

119903119903

] 119889119890

119903

minus 2119889119890

119905

119879

119865119890

119905119901

119881

minus 2119889119890

119903

119879

119865119890

119903119901

119881 minus 2119889119890

119905

119879

119865119890

minus 12057633

1198812

ℎ119901

]

(22)

119879119890

119896

=1

2 119889

119890

119905

119879

[119872119890

] 119889119890

119905

(23)

In (22)119881 represents the voltage difference applied across thethickness of the PZC layer The elemental stiffness matrices[119870

119890

119905119905

] [119870119890

119905119903

] and [119870119890

119903119903

] the elemental electroelastic couplingvectors 119865

119890

119905119901

and 119865119890

119903119901

the elemental load vector 119865119890

andthe elemental mass matrix [119872

119890

] appearing in (22) and (23)are given by

[119870119890

119905119905

] = [119870119890

119905119887

] + [119870119890

119905119904

] + [119870119890

119905119887119904

]119901119887

+ [119870119890

119905119887119904

]119901119904

[119870119890

119905119903

] = [119870119890

119905119903119887

] + [119870119890

119905119903119904

]

+1

2([119870

119890

119905119903119887119904

]119901119887

+ [119870119890

119903119905119887119904

]119879

119901119887

+ [119870119890

119905119903119887119904

]119901119904

+ [119870119890

119903119905119887119904

]119879

119901119904

)

[119870119890

119903119905

] = [119870119890

119905119903

]119879

[119870119890

119903119903

] = [119870119890

119903119903119887

] + [119870119890

119903119903119904

] + [119870119890

119903119903119887119904

]119901119887

+ [119870119890

119903119903119887119904

]119901119904

119865119890

119905119901

= 119865119890

119905119887

119901

+ 119865119890

119905119904

119901

119865119890

119903119901

= 119865119890

119903119887

119901

+ 119865119890

119903119904

119901

119865119890

= int

119886119890

0

int

119887119890

0

[119873119905]119879

119891 119889119909 119889119910

[119872119890

] = int

119886119890

0

int

119887119890

0

119898[119873]119879

[119873] 119889119909 119889119910 119898 =

119873+2

sum

119896=1

120588119896

(ℎ119896+1

minus ℎ119896)

(24)

where

[119870119890

119905119887

] = int119860

[119861119905119887]119879

([119863119905119887] + [119863

119905119887]V + [119863

119905119887]119901

) [119861119905119887] 119889119909 119889119910

[119870119890

119905119904

] = int119860

[119861119905119904]119879

([119863119905119904] + [119863

119905119904]V + [119863

119905119904]119901

) [119861119905119904] 119889119909 119889119910

[119870119890

119905119887119904

]119901119887

= int119860

[119861119905119887]119879

[119863119905119887119904

]119901

[119861119905119904] 119889119909 119889119910

[119870119890

119905119887119904

]119901119904

= int119860

[119861119905119904]119879

[119863119905119887119904

]119901

[119861119905119887] 119889119909 119889119910

[119870119890

119905119903119887

] = int119860

[119861119905119903119887

]119879

([119863119905119903119887

] + [119863119905119903119887

]V + [119863119905119903119887

]119901

)

times [119861119905119903119887

] 119889119909 119889119910

Journal of Composites 7

[119870119890

119905119903119887119904

]119901119887

= int119860

[119861119905119887]119879

[119863119905119903119887119904

]119901

[119861119903119904] 119889119909 119889119910

[119870119890

119903119905119887119904

]119901119887

= int119860

[119861119903119887]119879

[119863119903119905119887119904

]119901

[119861119905119904] 119889119909 119889119910

[119870119890

119905119903119887119904

]119901119904

= int119860

[119861119905119904]119879

[119863119903119905119887119904

]119901

[119861119903119887] 119889119909 119889119910

[119870119890

119903119905119887119904

]119901119904

= int119860

[119861119903119904]119879

[119863119905119903119887119904

]119901

[119861119905119887] 119889119909 119889119910

[119870119890

119905119903119904

] = int119860

[119861119905119903119904

]119879

([119863119905119903119904

] + [119863119905119903119904

]V + [119863119905119903119904

]119901

)

times [119861119905119903119904

] 119889119909 119889119910

[119870119890

119903119903119887

] = int119860

[119861119903119903119887

]119879

([119863119903119903119887

] + [119863119903119903119887

]V + [119863119903119903119887

]119901

)

times [119861119903119903119887

] 119889119909 119889119910

[119870119890

119903119903119904

] = int119860

[119861119903119903119904

]119879

([119863119903119903119904

] + [119863119903119903119904

]V + [119863119903119903119904

]119901

)

times [119861119903119903119904

] 119889119909 119889119910

[119870119890

119903119903119887119904

]119901119887

= int119860

[119861119903119887]119879

[119863119903119903119887119904

]119901

[119861119903119904] 119889119909 119889119910

[119870119890

119903119903119887119904

]119901119904

= int119860

[119861119903119904]119879

[119863119903119903119887119904

]119901

[119861119903119887] 119889119909 119889119910

119865119890

119905119887

119901

= int119860

[119861119905119887]119879

119863119905119887119901

119889119909 119889119910

119865119890

119905119904

119901

= int119860

[119861119905119904]119879

119863119905119904119901

119889119909 119889119910

119865119890

119903119904

119901

= int119860

[119861119903119904]119879

119863119903119904119901

119889119909 119889119910

(25)

The various rigidity matrices and the rigidity vectors forelectro-elastic coupling appearing in the previous elementalmatrices are given by

[119863119905119887] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] 119889119911

[119863119905119903119887

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] [1198851] 119889119911

[119863119903119903119887

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[1198851]119879

[119862119896

119887

] [1198851] 119889119911

[119863119905119904] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119904

] 119889119911

[119863119905119903119904

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] [1198854] 119889119911

[119863119903119903119904

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[1198854]119879

[119862119896

119887

] [1198854] 119889119911

[119863119905119887]V = ℎV [119862

119873+1

119887

]

[119863119905119903119887

]V = int

ℎ119873+2

ℎ119873+1

[119862119873+1

119887

] [1198852] 119889119911

[119863119903119903119887

]V = int

ℎ119873+2

ℎ119873+1

[1198852]119879

[119862119873+1

119887

] [1198852] 119889119911

[119863119905119904]V = ℎV [119862

119873+1

119904

]

[119863119905119903119904

]V = int

ℎ119873+2

ℎ119873+1

[119862119873+1

119904

] [1198855] 119889119911

[119863119903119903119904

]V = int

ℎ119873+2

ℎ119873+1

[1198855]119879

[119862119873+1

119887

] [1198855] 119889119911

[119863119905119887]119901

= ℎ119901[119862

119873+2

119887

]

[119863119905119903119887

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887

] [1198853] 119889119911

[119863119903119903119887

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+2

119887

] [1198853] 119889119911

[119863119905119904]119901

= ℎ119901[119862

119873+2

119904

]

[119863119905119903119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119904

] [1198856] 119889119911

[119863119903119903119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198856]119879

[119862119873+2

119887

] [1198856] 119889119911

[119863119905119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887119904

] 119889119911

[119863119905119903119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887119904

] [1198856] 119889119911

[119863119903119905119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+2

119887119904

] 119889119911

[119863119903119903119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+3

119887119904

] [1198856] 119889119911

119863119905119887119901

= int

ℎ119873+3

ℎ119873+2

minus119890119887

ℎ119901

119889119911

119863119903119887119901

= int

ℎ119873+3

ℎ119873+2

minus[119885

3]119879

119890119887

ℎ119901

119889119911

119863119905119904119901

= int

ℎ119873+3

ℎ119873+2

minus119890119904

ℎ119901

119889119911

119863119903119904119901

= int

ℎ119873+3

ℎ119873+2

minus[119885

6]119879

119890119904

ℎ119901

119889119911

(26)

8 Journal of Composites

Applying the dynamic version of the virtual work principle[32] the following open-loop equations of motion for anelement of the overall plate integrated with the ACLDtreatment are obtained

[119872119890

] 119889119890

119905

+ [119870119890

119905119905

] 119889119890

119905

+ [119870119890

119905119903

] 119889119890

119903

= 119865119890

119905119901

119881 + 119865119890

[119870119890

119905119903

]119879

119889119890

119905

+ [119870119890

119903119903

] 119889119890

119903

= 119865119890

119903119901

119881

(27)

Since the elastic constant matrix of the viscoelastic layer iscomplex the stiffness matrices of an element integrated withthe ACLD treatment are complex In case of an elementnot integrated with the ACLD treatment the electro-elasticcoupling matrices become null vectors and the elementalstiffnessmatrices are to be computed without considering thepiezoelectric and viscoelastic layers It should also be notedthat as the stiffness matrices associated with the transverseshear strains are derived separately one can employ theselective integration rule in a straight forward manner toavoid shear locking problem in case of thin plates Therestrained boundary conditions at the skew edges of theplate are such that the displacements of the points locatedat the skew edges are to be restrained along 119909

1015840- 1199101015840- and

1199111015840-directions (Figure 1(a)) Hence in order to impose theboundary conditions in a straight forward manner thegeneralized displacement vectors of a point lying on the skewedge are to be transformed as follows

119889119905 = [119871

119905] 119889

1015840

119905

119889119903 = [119871

119903] 119889

1015840

119903

(28)

where 1198891015840

119905

and 1198891015840

119903

are the generalized displacement vectorsof the point with respect to 119909

1015840

1199101015840

1199111015840 coordinate system and are

given by

1198891015840

119905

= [1199061015840

119900

V1015840119900

1199081015840

119900

]119879

1198891015840

119903

= [1205791015840

119909

1205791015840

119910

1205791015840

119911

1206011015840

119909

1206011015840

119910

1206011015840

119911

1205741015840

119911

1205741015840

119910

1205741015840

119911]119879

(29)

Also the transformation matrices [119871119905] and [119871

119903] are given by

[119871119905] = [

[

119888 119904 0

minus119904 119888 0

0 0 1

]

]

[119871119903] =

[[[[[[[[[[[[

[

119888 119904 0 0 0 0 0 0 0

minus119904 119888 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 119888 119904 0 0 0 0

0 0 0 minus119904 119888 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 119888 119904 0

0 0 0 0 0 0 minus119904 119888 0

0 0 0 0 0 0 0 0 1

]]]]]]]]]]]]

]

(30)

in which 119888 = cos120572 and 119904 = sin120572Thus the elementalmatricesof the element containing the nodes lying on the skew edgeare to be augmented as follows

[119870119890

119905119905

] = [1198791]119879

[119870119890

119905119905

] [1198791] [119870

119890

119905119903

] = [1198791]119879

[119870119890

119905119903

] [1198792]

[119870119890

119903119903

] = [1198792]119879

[119870119890

119903119903

] [1198792] [119872

119890

] = [1198791]119879

[119872119890

] [1198791]

(31)

The forms of the transformation matrices [1198791] and [119879

2] are

given by

[1198791] =

[[[[[[[[[[[

[

[119871119905] 119900 119900 119900 119900 119900 119900 119900

119900 [119871119905] 119900 119900 119900 119900 119900 119900

119900 119900 [119871119905] 119900 119900 119900 119900 119900

119900 119900 119900 [119871119905] 119900 119900 119900 119900

119900 119900 119900 119900 [119871119905] 119900 119900 119900

119900 119900 119900 119900 119900 [119871119905] 119900 119900

119900 119900 119900 119900 119900 119900 [119871119905] 119900

119900 119900 119900 119900 119900 119900 119900 [119871119905]

]]]]]]]]]]]

]

[1198792] =

[[[[[[[[[[[

[

[119871119903] 119900 119900 119900 119900 119900 119900 119900

119900 [119871119903] 119900 119900 119900 119900 119900 119900

119900 119900 [119871119903] 119900 119900 119900 119900 119900

119900 119900 119900 [119871119903] 119900 119900 119900 119900

119900 119900 119900 119900 [119871119903] 119900 119900 119900

119900 119900 119900 119900 119900 [119871119903] 119900 119900

119900 119900 119900 119900 119900 119900 [119871119903] 119900

119900 119900 119900 119900 119900 119900 119900 [119871119903]

]]]]]]]]]]]

]

(32)

in which 119900 and 119900 are (3 times 3) and (9 times 9) null matricesrespectively It may be noted that in case of a node ofsuch element which does not lie on the skew edge thegeneralized displacement vectors of this node with respect to119909119910119911 coordinate system are not required to be transformedHence thematrices [119871

119905] and [119871

119903] corresponding to suchnode

turn out to be (3times3) and (9times9) identitymatrices respectivelyThe elemental equations of motion are now assembled

to obtain the open-loop global equations of motion of theoverall skew plate as follows

[119872] + [119870119905119905] 119883 + [119870

119905119903] 119883

119903 =

119902

sum

119895=1

119865119895

119905119901

119881119895

+ 119865

[119870119905119903]119879

119883 + [119870119903119903] 119883

119903 =

119902

sum

119895=1

119865119895

119903119901

119881119895

(33)

where [119870119905119905] [119870

119905119903] and [119870

119903119903] are the global stiffness matrices

of the overall skew laminated composite plate [119872] is theglobal mass matrix 119865

119905119901 and 119865

119903119901 are the global electro-

elastic coupling vectors 119883 and 119883119903 are the global nodal

generalized displacement vectors 119865 is the global nodalforce vector 119902 is the number of patches and 119881

119895 is thevoltage applied to the 119895th patch Since the elemental stiffnessmatrices of an element augmented with the ACLD treatmentare complex the global stiffness matrices become complexand the energy dissipation characteristics of the overallplate are attributed to the imaginary part of these matricesHence the global equations of motion as derived previouslyalso represent the modeling of the passive (uncontrolled)constrained layer damping of the substrate plate when theconstraining layer is not subjected to any control voltagefollowing a derivative control law

Journal of Composites 9

3 Closed-Loop Model

In the active control approach the active constraining layerof each patch is activated with a control voltage negativelyproportional to the transverse velocity of a point Thus thecontrol voltage for each patch can be expressed in terms of thederivatives of the global nodal degrees of freedom as follows

119881119895

= minus119896119895

119889

= minus119896119895

119889

[119880119895

119905

] minus 119896119895

119889

(ℎ

2) [119880

119895

119903

] 119903 (34)

where [119880119895

119905

] and [119880119895

119903

] are the unit vectors describing thelocation of the velocity sensor for the 119895th patch and 119896

119895

119889

isthe control gain for the 119895th patch Substituting (34) into(33) the final equations of motion governing the closed-loop dynamics of the overall skew laminated compositeplateACLD system can be obtained as follows

[119872] + [119870119905119905] 119883 + [119870

119905119903] 119883

119903 +

119898

sum

119895=1

119896119895

119889

119865119895

119905119901

[119880119895

119905

]

+

119898

sum

119895=1

119896119895

119889

(ℎ

2) 119865

119895

119905119901

[119880119895

119903

] 119903 = 119865

[119870119905119903]119879

119883 + [119870119903119903] 119883

119903 +

119898

sum

119895=1

119896119895

119889

119865119895

119905119901

[119880119895

119905

]

+

119898

sum

119895=1

119896119895

119889

(ℎ

2) 119865

119895

119905119901

[119880119895

119903

] 119903 = 0

(35)

4 Results and Discussion

In this section the numerical results obtained by the finiteelement model derived in the previous section have beenpresented Symmetric as well as antisymmetric cross-plyand antisymmetric angle-ply laminated skew substrate platesintegrated with the two patches of ACLD treatment areconsidered for presenting the numerical results As shown inFigure 1 the locations of the patches have been selected insuch a way that the energy dissipation corresponding to thefirst fewmodes becomesmaximum PZT-5Hspur compositewith 60 fiber volume fraction has been considered for thematerial of the constraining layer of the ACLD treatmentUsing themicromechanicsmodel and thematerial propertiesof the constituent phases [35] the effective elastic andpiezoelectric material properties of the 1ndash3 PZC with respectto the principle material coordinate system are computed asfollows

11986211

= 929GPa 11986212

= 618GPa 11986213

= 605GPa

11986233

= 3544GPa 11986223

= 11986213 119862

44= 158GPa

11986266

= 154GPa 11986255

= 11986244

11989031

= minus01902Cm2

11989032

= 11989031

11989033

= 184107Cm2

11989024

= 0004Cm2

11989015

= 11989024

(36)

The material of the orthotropic layers of the substrate platesis a graphiteepoxy composite and its material properties are[42]

119864119871= 172GPa 119864

119871

119864119879

= 25 119866119871119879

= 05119864119879

119866119879119879

= 02119864119879 ]

119871119879= ]

119879119879= 025

(37)

in which the symbols have the usual meaning Unless oth-erwise mentioned the aspect ratio (119886ℎ) and the thicknessof the skew substrate plates are considered as 100 and0003m respectively while the orthotropic layers of theskew substrate plates are of equal thickness Also the lengthand the skew sides of the substrate are equal (ie 119886 =

119887) The loss factor of the viscoelastic layer considered fornumerical results remains invariant [20] within a broadband (0ndash400Hz) of frequency range and the values of thecomplex shear modulus Poisson ratio and the density ofthe viscoelastic layer are considered as 20(1 + 119894)MNmminus2049 and 1140 kgmminus3 respectively [20] The thicknesses ofthe viscoelastic layer and the 1ndash3 PZC layer are consideredas 508120583m and 250 120583m respectively The value of the shearcorrection factor is used as 56 Three points of Gaussianintegration rule are considered for computing the elementmatrices corresponding to the bending deformations whiletwo points of that are used for computing the elementmatrices corresponding to transverse shear deformationsThe simply supported boundary conditions at the edges of theoverall plate are invoked as follows

at 119909 = 119910 tan120572 119909 = 119886 + 119910 tan120572

V1015840119900

= 1199081015840

119900

= 1205791015840

119910

= 1206011015840

119910

= 1205741015840

119910

= 1205791015840

119911

= 1205741015840

119911

= 0

at 119910 = 0 119910 = 119887 cos120572

119906119900= 119908

119900= 120579

119909= 120601

119909= 120574

119909= 120579

119911= 120574

119911= 0

(38)

To verify the validity of the present finite elementmodel (FEM) the natural frequencies of the skew laminatedcomposite plates integrated with the inactivated patches ofnegligible thickness are first computed and subsequentlycompared with the existing analytical results [41] of theidentical plates without being integrated with the patchesSuch comparisons for the simply supported and clamped-clamped skew laminated plates are presented in Tables 1 and2 respectively It may be observed from these tables that theresults are in excellent agreement validating the model of theplate derived here

In order to assess the performance of the verti-callyobliquely reinforced 1ndash3 PZC as the material of thedistributed actuators of smart skew laminated compositeplates frequency responses for ACLD of the substrate skewlaminated composite plates are investigated To compute the

10 Journal of Composites

Table 1 Comparison of fundamental natural frequencies (120596) of simply supported skew laminated composite plates integratedwith the patchesof negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 15699 28917 37325 14829 24656 32522 17974 33351 33351Present FEM 15635 24383 35033 15076 24380 32254 18493 33359 33370

15∘ Reference [41] 16874 30458 39600 15741 25351 30270 18313 32490 36724Present FEM 16571 29840 36505 15796 25775 29892 18675 32075 35810

30∘ Reference [41] 20840 34023 46997 18871 29372 34489 20270 34431 42361Present FEM 19596 31690 46796 18226 29585 32357 19894 32365 43208

45∘ Reference [41] 28925 41906 54149 25609 33126 40617 25609 33131 42772Present FEM 24811 44875 53289 22996 34773 44889 23194 34870 45009

lowast

120596 = (120596119886

2

120587

2

ℎ)radic120588119864119879 120596 is the circular natural frequency

Table 2 Comparison of fundamental natural frequencies (120596) of clamped-clamped skew laminated composite plates integrated with thepatches of negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 23687 35399 41122 22990 37880 37880 22119 37339 37339Present FEM 23201 34769 44102 23315 36531 36545 22433 36000 36012

15∘ Reference [41] 24663 36255 43418 23809 37516 40785 23099 36997 40438Present FEM 23699 34821 44049 23741 35856 38401 23049 35346 38092

30∘ Reference [41] 27921 39557 50220 26666 39851 47227 26325 39549 52107Present FEM 25366 35696 44734 25240 41943 45373 24945 36113 50932

45∘ Reference [41] 34739 47129 58789 33015 46290 58423 33015 46290 58423Present FEM 28665 47074 54562 28377 47614 56620 28377 47614 55162

frequency response functions the skew laminated compositeplates are harmonically excited by a force of amplitude 1Newton applied to a point (1198862 (1198864) cos120572 ℎ2) on the skewsubstrate platesThe control voltage supplied to the constrain-ing layer of the first patch is negatively proportional to thevelocity of the point (1198862 (1198864) cos120572 ℎ2) and that applied tothe constraining layer of the second patch is negatively pro-portional to the velocity of the point (1198862 (31198864) cos120572 ℎ2)

Figures 3 4 and 5 illustrate the frequency responsefunctions for the transverse displacement of a point of asimply supported symmetric cross-ply (0∘90∘0∘) plate with15∘ 30∘ and 45∘ skew angles respectively while the fiberorientation angle (120595) in the constraining layer is 0∘ Thesefigures display both uncontrolled and controlled responsesand clearly show that the active constraining layer made ofthe vertically reinforced 1ndash3 PZC material being studied heresignificantly attenuates the amplitude of vibrations enhanc-ing the damping characteristics of the overall laminatedskew composite plates over the passive damping (uncon-trolled) Similar responses are also obtained for antisym-metric cross-ply (0∘90∘0∘90∘) and antisymmetric angle-ply (45∘minus45∘45∘minus45∘) skew laminated composite plates

However for brevity these results are not presented hereThemaximum values of the control voltages required to computethe controlled responses presented in Figures 3ndash5 have beenfound to be quite low and are illustrated in Figure 6 in caseof the symmetric cross-ply (0∘90∘0∘) plate with skew angle15∘ only In order to examine the contribution of the verticalactuation in improving the damping characteristics of theskew laminated composite plates as demonstrated in Figures3ndash5 active control responses of the symmetric cross-plylaminated composite plate with 15∘ skew angle are plotted inFigure 7with andwithout considering the value of 119890

33and 119890

31

It may be noted that when the values of 11989031is zero and that of

11989033is nonzero the vertical actuation of the active constraining

layer of the ACLD treatment is responsible for increasing thetransverse shear deformations of the viscoelastic constrainedlayer over the passive counterpart resulting in improveddamping of the smart plate over its passive damping Onthe other hand if 119890

33is zero and 119890

31is nonzero the in-

plane actuation of the active constraining layer causes thetransverse shear deformation of the viscoelastic core of theACLD treatment leading to the active damping of the smartplate It is evident from Figure 7 that the contribution of the

Journal of Composites 11

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 3 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 4 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 30∘)

vertical actuation by the constraining vertically reinforced 1ndash3 PZC layer is significantly larger than that of the in-planeactuation by the same for controlling the modes displayedin Figure 7 Though not shown here similar results are alsoobtained for antisymmetric cross-ply and angle-ply skewplates

Figures 8 9 and 10 demonstrate the effect of variationof piezoelectric fiber orientation angle on the performanceof the patches for improving the damping characteristicsof simply supported symmetric cross-ply (0∘90∘0∘)

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 5 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 45∘)

0

5

10

15

20

25

30

35

40

45

50

Con

trol v

olta

ge (V

olt)

0 100 200 300 400 500 600Frequency (Hz)

119870119889 = 600

119870119889 = 1200

Figure 6 Frequency response functions for the control voltages foractive damping of the simply supported skew symmetric cross-ply(0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

antisymmetric cross-ply (0∘90∘0∘90∘) and angle-ply(45∘minus45∘45∘minus45∘) substrate skew composite plates with30∘ skew angle respectively while the orientation angle (120595)

of the piezoelectric fibers is varied in the vertical 119909119911-planeThese figures illustrate that the attenuating capability of thepatches becomesmaximumwhen the value of the orientationangle (120595) of the fibers in the 119909119911-plane is 0∘ Similar effectis also obtained when the fiber orientation angle (120595) in theconstraining layer of the ACLD treatment is varied in thevertical 119910119911-plane as illustrated in Figures 11 12 and 13 forthe symmetric cross-ply (0∘90∘0∘) antisymmetric cross-ply(0∘90∘0∘90∘) and angle-ply (45∘minus45∘45∘minus45∘) substrate

12 Journal of Composites

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

11989031 = minus01902 11989033 = 184107

11989031 = minus01902 11989033 = 011989031 = 0 11989033 = 184107

w(a2(b4)co

s120572h2

) (m

)

Figure 7 Frequency response functions demonstrating the con-tribution of vertical actuation on the controlled responses forthe transverse displacement 119908 (1198862 (1198864) cos120572 ℎ2) of the skewsymmetric cross-ply (0∘90∘0∘) plate (119870

119889

= 600 119886ℎ = 100 120595 = 0∘

120572 = 15∘)

0 100 200 300 400 500 600Frequency (Hz)

0

02

04

06

08

1

12times10minus4

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 8 Effect of variation of piezoelectric fiber orientation angle(120595) on the controlled responses of a simply supported symmetricskew cross-ply (0∘90∘0∘) plate when the piezoelectric fibers of theconstraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

skew composite plates with 30∘ skew angle respectivelyIn this regard it should be noted that for investigating theeffect of variation of piezoelectric fiber orientation angle(120595) on the performance of the ACLD patches the valueof 120595 has been smoothly varied to compute the controlledresponses However for the sake of clarity in the plots theresponses corresponding to the four specific values of 120595

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 9 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 10 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

have been presented such that the optimum performanceof the patches can be evaluated The significant effect of thevariation of the piezoelectric fiber orientation angle in thevertical 119909119911- and 119910119911-planes has been noticed when all theboundaries of the overall plates are clamped as shown inFigures 14ndash19 When the piezoelectric fibers are coplanarwith the vertical 119909119911-plane the frequency response functionsplotted in Figures 14 15 and 16 reveal that the control

Journal of Composites 13

0

02

04

06

08

1

12

14

16times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 11 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 12 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=600 119886ℎ = 100 120572 = 30

∘)

authority of the patches becomes maximum for improvingthe damping characteristics of the clamped-clamped skewcross-ply and angle-ply substrate plates if the value of thepiezoelectric fiber orientation angle (120595) is 30∘ The same isalso found to be true if the piezoelectric fibers are coplanarwith the vertical 119910119911-plane as shown in Figures 17 18 and 19for the previously mentioned skew plates Similar responsesare also found for the effect of piezoelectric fiber orientation

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 13 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers of the constraining layer are coplanar with the119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 14 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600119886ℎ = 100120572 = 30

∘)

angle on the active damping of other skew laminated plateswith different skew angles However for the sake of brevitythey are not presented here

5 Conclusions

In this paper a study has been carried out to investi-gate the performance of the ACLD treatment for active

14 Journal of Composites

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 15 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 16 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

damping of smart skew laminated composite plates whenverticallyobliquely reinforced 1ndash3 PZC materials are usedas the materials for the constraining layer of the ACLDtreatment A layerwise FSDT-based finite element modelhas been developed to describe the dynamics of the skewlaminated composite plates integrated with the patches ofACLD treatment Unlike the existing finite element modelsof smart structures integrated with ACLD treatment the

200 250 300 350 400 450 500 550 600 650 7000

02

04

06

08

1

12

14

16

18

2times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 17 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

200 300 400 500 600 700 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 18 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

derivation of the present finite element model includes thetransverse deformations of the substrate skew laminatedcomposite plate the constrained viscoelastic layer and theconstraining 1ndash3 PZC layer of the ACLD treatment along thethickness (ie 119911) direction such that both vertical and in-plane actuations by the constraining layer of the patches canbe utilized for active damping of the plates The frequency

Journal of Composites 15

0

05

1

15

2

25

3

35

4times10minus5

200 300 400 500 600 700 800Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 19 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

responses of the symmetric cross-ply antisymmetric cross-ply and angle-ply skew composite plates indicate that theactive constraining layer of the ACLD treatment being madeof the vertically reinforced 1ndash3 PZC material significantlyenhances the damping characteristics of the plates over thepassive damping The analysis revealed that if the verticallyreinforced 1ndash3 PZCmaterial is used for the constraining layerof the ACLD treatment then the contribution of the verticalactuation by the constraining layer alone for improving theactive damping characteristics of the smart skew laminatedcomposite plate is significantly larger than that due to thein-plane actuation by the constraining layer alone It isimportant to note from the present investigation that thevariation of the orientation angle of the piezoelectric fibersof the constraining layer in the vertical 119909119911- and 119910119911-planesas well as the skew angle and boundary conditions of theoverall plates significantly affects the performance of thepatches For the simply supported laminated substrate platesthe performance of the patches becomes maximumwhen thepiezoelectric fiber orientation angle (120595) is 0∘ while in caseof the clamped-clamped laminated skew substrate plates themaximum control authority of the patches is achieved whenthe value of 120595 is 30∘ irrespective of the cases in which thefibers are oriented in 119909119911- and 119910119911-planes

Appendix

In (6) and (7) the matrices [1198851] [119885

2] [119885

3] [119885

4] [119885

5] and

[1198856] are given by

[1198851] = [[119885

1] 0 0]

[1198852] = [(

ℎ

2) 119868 [119885

2] 0]

[1198853] = [(

ℎ

2) 119868 ℎV119868 [119885

3]]

[1198854] = [119868 0 0 119911119868 0 0]

[1198855] = [0 119868 0 (

ℎ

2) 119868 (119911 minus

ℎ

2) 119868 0]

[1198856] = [0 0 119868 (

ℎ

2) 119868 ℎV119868 (119911 minus ℎ

119873+2) 119868]

(A1)

in which

[1198851] =

[[[

[

119911 0 0 0

0 119911 0 0

0 0 119911 0

0 0 0 1

]]]

]

[1198852] =

[[[[[[[[

[

(119911 minusℎ

2) 0 0 0

0 (119911 minusℎ

2) 0 0

0 0 (119911 minusℎ

2) 0

0 0 0 1

]]]]]]]]

]

[1198853] =

[[[

[

(119911 minus ℎ119873+2

) 0 0 0

0 (119911 minus ℎ119873+2

) 0 0

0 0 (119911 minus ℎ119873+2

) 0

0 0 0 1

]]]

]

[119868] =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

[119868] = [1 0

0 1]

[0] = [0 0

0 0] [0] = [

0 0

0 0]

(A2)

The various submatrices 119861119905119887119894 119861

119903119887119894 and 119861

119903119904119894 appearing in (21)

are given by

119861119905119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 0

]]]]]]]]]]]

]

119861119905119904119894

=

[[[[

[

0 0120597120578

119894

120597119909

0 0120597120578

119894

120597119910

]]]]

]

16 Journal of Composites

119861119903119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 1

]]]]]]]]]]]

]

119861119903119887119894

=

[[[[[[

[

119861119903119887119894

0

0

0 119861119903119887119894

0

0

0 119861119903119887119894

]]]]]]

]

119861119903119904119894

=

[[[[[[[[[[[[[[[[[[

[

119868 0 0

0

119868 0

0 0

119868

119861119903119904119894

0 0

0 119861119903119904119894

0

0 0 119861119903119904119894

]]]]]]]]]]]]]]]]]]

]

119868= [1 0 0

0 1 0]

(A3)

wherein

0 and 0 are the (3 times 3) and (2 times 3) null matricesrespectively

References

[1] T Bailey and J E Hubbard ldquoDistributedpiezoelectric polymeractive vibration control of a cantilever beamrdquo Journal of Guid-ance Control and Dynamics vol 8 no 5 pp 605ndash611 1985

[2] E F Crawley and J de Luis ldquoUse of piezoelectric actuators aselements of intelligent structuresrdquo AIAA Journal vol 25 no 10pp 1373ndash1385 1987

[3] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988

[4] C K Lee W W Chiang and O Sulivan ldquoPiezoelectric modalsensoractuator pairs for critical active damping vibrationcontrolrdquo Journal of the Acoustical Society of America vol 90 no1 pp 374ndash384 1991

[5] S Hanagud M W Obal and A J Calise ldquoOptimal vibrationcontrol by the use of piezoceramic sensors and actuatorsrdquoJournal of Guidance Control and Dynamics vol 15 no 5 pp1199ndash1206 1992

[6] Y Gu R L Clark C R Fuller and A C Zander ldquoExperimentson active control of plate vibration using piezoelectric actuatorsand polyvinylidene fluoride (PVDF) modal sensorsrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 116 no3 pp 303ndash308 1994

[7] A Baz and S Poh ldquoOptimal vibration control with modalpositive position feedbackrdquo Optimal Control Applications andMethods vol 17 no 2 pp 141ndash149 1996

[8] A Kugi K Schlacher and H Irschik ldquoInfinite-dimensionalcontrol of nonlinear beam vibrations by piezoelectric actuatorand sensor layersrdquo Nonlinear Dynamics vol 19 no 1 pp 71ndash911999

[9] D Sun L Tong and D Wang ldquoVibration control of platesusing discretely distributed piezoelectric quasi-modal actua-torssensorsrdquo AIAA Journal vol 39 no 9 pp 1766ndash1772 2001

[10] G Caruso S Galeani and L Menini ldquoActive vibration controlof an elastic plate using multiple piezoelectric sensors andactuatorsrdquo SimulationModelling Practice andTheory vol 11 no5-6 pp 403ndash419 2003

[11] X Zhang and A G Erdman ldquoOptimal placement of piezo-electric sensors and actuators for controlled flexible linkagemechanismsrdquo Journal of Vibration and Acoustics Transactionsof the ASME vol 128 no 2 pp 256ndash260 2006

[12] G Gatti M J Brennan and P Gardonio ldquoActive dampingof a beam using a physically collocated accelerometer andpiezoelectric patch actuatorrdquo Journal of Sound and Vibrationvol 303 no 3ndash5 pp 798ndash813 2007

[13] M K Kwak S Heo and M Jeong ldquoDynamic modellingand active vibration controller design for a cylindrical shellequipped with piezoelectric sensors and actuatorsrdquo Journal ofSound and Vibration vol 321 no 3ndash5 pp 510ndash524 2009

[14] I Bruant L Gallimard and S Nikoukar ldquoOptimal piezoelectricactuator and sensor location for active vibration control usinggenetic algorithmrdquo Journal of Sound and Vibration vol 329 no10 pp 1615ndash1635 2010

[15] Y Fu J Wang and Y Mao ldquoNonlinear vibration and activecontrol of functionally graded beams with piezoelectric sensorsand actuatorsrdquo Journal of Intelligent Material Systems andStructures vol 22 no 18 pp 2093ndash2102 2011

[16] F Chen M Hong M Song and H Cui ldquoOptimal control ofa beam with discontinuously distributed piezoelectric sensorsand actuatorsrdquo Journal of Marine Science and Application vol11 no 1 pp 44ndash51 2012

[17] B Azvine G R Tomlinson and R J Wynne ldquoUse of activeconstrained-layer damping for controlling resonant vibrationrdquoSmart Materials and Structures vol 4 no 1 pp 1ndash6 1995

[18] A Baz and J Ro ldquoVibration control of plates with activeconstrained layer dampingrdquo SmartMaterials and Structures vol5 no 3 pp 272ndash280 1996

[19] C Chantalakhana and R Stanway ldquoActive constrained layerdamping of clamped-clamped plate vibrationsrdquo Journal ofSound and Vibration vol 241 no 5 pp 755ndash777 2001

[20] J Ro and A Baz ldquoOptimum placement and control ofactive constrained layer damping using modal strain energyapproachrdquo JVCJournal of Vibration and Control vol 8 no 6pp 861ndash876 2002

[21] M C Ray and J N Reddy ldquoOptimal control of thin circularcylindrical laminated composite shells using active constrainedlayer damping treatmentrdquo Smart Materials and Structures vol13 no 1 pp 64ndash72 2004

[22] C M A Vasques B R Mace P Gardonio and J D RodriguesldquoArbitrary active constrained layer damping treatments onbeams finite element modelling and experimental validationrdquoComputers and Structures vol 84 no 22-23 pp 1384ndash14012006

[23] F M Li K Kishimoto K Wang Z B Chen and W HHuang ldquoVibration control of beams with active constrainedlayer dampingrdquo Smart Materials and Structures vol 17 no 6Article ID 065036 2008

[24] L Yuan Y Xiang Y Huang and J Lu ldquoA semi-analyticalmethod and the circumferential dominant modal control ofcircular cylindrical shells with active constrained layer dampingtreatmentrdquo SmartMaterials and Structures vol 19 no 2 ArticleID 025010 2010

[25] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

Journal of Composites 17

[26] W A Smith and B A Auld ldquoModeling 1ndash3 composite piezo-electrics thickness-mode oscillationsrdquo IEEE Transactions onUltrasonics Ferroelectrics and Frequency Control vol 38 no 1pp 40ndash47 1991

[27] PiezocompositesMaterial Systems Inc 543 Great Road Little-ton MA 01460

[28] Y Benveniste and G J Dvorak ldquoUniform fields and universalrelations in piezoelectric compositesrdquo Journal of the Mechanicsand Physics of Solids vol 40 no 6 pp 1295ndash1312 1992

[29] M L Dunn and M Taya ldquoMicromechanics predictions ofthe effective electroelastic moduli of piezoelectric compositesrdquoInternational Journal of Solids and Structures vol 30 no 2 pp161ndash175 1993

[30] J Aboudi ldquoMicromechanical prediction of the effective coeffi-cients of thermo-piezoelectric multiphase compositesrdquo Journalof Intelligent Material Systems and Structures vol 9 no 9 pp713ndash722 1999

[31] X Ruan T W Chou A Safari and S C Danforth ldquoA 3-Dconnectivity model for effective piezoelectric properties of yarncompositesrdquo Journal of Composite Materials vol 36 no 14 pp1693ndash1708 2002

[32] A Baz and A Tempia ldquoActive piezoelectric damping compos-itesrdquo Sensors and Actuators A vol 112 no 2-3 pp 340ndash3502004

[33] H Berger S Kari U Gabbert et al ldquoUnit cell models ofpiezoelectric fiber composites for numerical and analytical cal-culation of effective propertiesrdquo SmartMaterials and Structuresvol 15 no 2 pp 451ndash458 2006

[34] M Sakthivel and A Arockiarajan ldquoAn analytical model forpredicting thermo-electro-mechanical response of 1ndash3 piezo-electric compositesrdquo Computational Materials Science vol 48no 4 pp 759ndash767 2010

[35] M C Ray and A K Pradhan ldquoOn the use of verticallyreinforced 1ndash3 piezoelectric composites for hybrid damping oflaminated composite platesrdquo Mechanics of Advanced Materialsand Structures vol 14 no 4 pp 245ndash261 2007

[36] M C Ray and A K Pradhan ldquoPerformance of vertically andobliquely reinforced 1ndash3 piezoelectric composites for activedamping of laminated composite shellsrdquo Journal of Sound andVibration vol 315 no 4-5 pp 816ndash835 2008

[37] S K Sarangi and M C Ray ldquoSmart damping of geometricallynonlinear vibrations of laminated composite beams using verti-cally reinforced 1ndash3 piezoelectric compositesrdquo Smart Materialsand Structures vol 19 no 7 Article ID 075020 2010

[38] A Krishnan and J V Deshpande ldquoVibration of skew laminatesrdquoJournal of Sound and Vibration vol 153 no 2 pp 351ndash358 1992

[39] A R Krishna Reddy and R Palaninathan ldquoFree vibration ofskew laminatesrdquo Computers and Structures vol 70 no 4 pp415ndash423 1999

[40] C S Babu and T Kant ldquoTwo shear deformable finite elementmodels for buckling analysis of skew fibre-reinforced compositeand sandwich panelsrdquo Composite Structures vol 46 no 2 pp115ndash124 1999

[41] A K Garg R K Khare and T Kant ldquoFree vibration of skewfiber-reinforced composite and sandwich laminates using ashear deformable finite element modelrdquo Journal of SandwichStructures and Materials vol 8 no 1 pp 33ndash53 2006

[42] J N Reddy Mechanics of Laminated Composites Plates Theoryand Analysis CRC press Boca Raton Fla USA 1996

Submit your manuscripts athttpwwwhindawicom

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materials

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Journal ofNanomaterials

Page 5: Research Article Active Constrained Layer Damping of Smart …downloads.hindawi.com/archive/2013/824163.pdf · 2019-07-31 · Research Article Active Constrained Layer Damping of

Journal of Composites 5

where the elastic coefficient matrices are

[119862119896

119887

] =

[[[[[[[[[[[[[[

[

119862119896

11

119862119896

12

119862119896

16

119862119896

13

119862119896

12

119862119896

22

119862119896

26

119862119896

23

119862119896

16

119862119896

26

119862119896

66

119862119896

36

119862119896

13

119862119896

23

119862119896

36

119862119896

33

]]]]]]]]]]]]]]

]

[119862119896

119904

] =[[

[

119862119896

55

119862119896

45

119862119896

45

119862119896

44

]]

]

(11)

and 119862119896

119894119895

(119894 119895 = 1 2 3 6) are the transformed elasticcoefficients with respect to the reference coordinate systemThematerial of the viscoelastic layer is assumed to be linearlyviscoelastic and homogenous isotropic and is modeled byusing the complex modulus approach Thus the shear mod-ulus119866 and Youngrsquos modulus 119864 of the viscoelastic material aregiven by

119866 = 1198661015840

(1 + 119894120578) 119864 = 2119866 (1 + ]) (12)

in which 1198661015840 is the storage modulus ] is Poissonrsquos ratio and

120578 is the loss factor at a particular operating temperature andfrequency Employing the complex modulus approach theconstitutive relations for the material of the viscoelastic layer(119896 = 119873 + 1) can also be represented by (10) with 119862

119896

119894119895

(119894 119895 =

1 2 3 6) being the complex elastic constants [29 30]The constraining PZC layer will be subjected to the appliedelectric field (119864

119911) acting across its thickness (ie along the

119911-direction) only Accordingly the constitutive relations forthe 1ndash3 PZC material with respect to the laminate coordinatesystem (119909119910119911) can be expressed as

120590119896

119887

= [119862119896

119887

] 120576119896

119887

+ [119862119896

119887119904

] 120576119896

119904

minus 119890119887 119864

119911

120590119896

119904

= [119862119896

119887119904

] 120576119896

119887

+ [119862119896

119904

] 120576119896

119904

minus 119890119904 119864

119911

119863119911= 119890

119887119879

120576119896

119887

+ 119890119904119879

120576119896

119904

+ 12057633119864119911

119896 = 119873 + 2

(13)

Here 119863119911represents the electric displacement along the 119911-

direction and 12057633

is the dielectric constant It may be notedfrom the previous form of the constitutive relations that thetransverse shear strains are coupled with the in-plane normalstrains due to the orientation of piezoelectric fibers in the

vertical 119909119911- or 119910119911-plane and the corresponding couplingelastic constant matrices [119862

119873+2

119887119904

] are given by

[119862119873+2

119887119904

] =

[[[[[[[[[[

[

119862119873+2

15

0

119862119873+2

25

0

0 119862119873+2

46

119862119873+2

35

0

]]]]]]]]]]

]

or

[119862119873+2

119887119904

] =

[[[[[[[[[[

[

0 119862119873+2

14

0 119862119873+2

24

119862119873+2

56

0

0 119862119873+2

34

]]]]]]]]]]

]

(14)

as the piezoelectric fibers are coplanar with the vertical 119909119911-or 119910119911-plane respectively Note that if the fibers are coplanarwith both 119909119911- and 119910119911-planes (ie 120595 = 0

∘) this couplingmatrix becomes a nullmatrix Also the piezoelectric constantmatrices 119890

119887 and 119890

119904 appearing in (13) contain the following

transformed effective piezoelectric coefficients of the 1ndash3PZC

119890119887 = [119890

3111989032

11989036

11989033]119879

119890119904 = [119890

3511989034]119879

(15)

The total potential energy 119879119901and the kinetic energy 119879

119896of the

overall plateACLD system are given by [35]

119879119901

=1

2[

119873+2

sum

119896=1

intΩ

(120576119896

119887

119879

120590119896

119887

+ 120576119903

119904

119879

120590119896

119904

) 119889Ω

minusintΩ

119863119911119864119911119889Ω] minus int

119860

119889119879

119891 119889119860

119879119896=

1

2

119873+2

sum

119896=1

intΩ

120588119896

(2

+ V2 + 2

) 119889Ω

(16)

in which 120588119896 is the mass density of the 119896th layer 119891 is the

externally applied surface traction acting over a surface area119860 andΩ represents the volume of the concerned layer Sincethe plates under study are considered to be thin the rotaryinertia of the overall plate has been neglected in estimatingthe kinetic energy The overall plate is discretized by eight-noded isoparametric quadrilateral elements Following (4)the generalized displacement vectors associated with the 119894th(119894 = 1 2 3 8) node of the element can be written as

119889119905119894 = [119906

0119894V0119894

1199080119894]119879

119889119903119894 = [120579

119909119894120579119910119894

120579119911119894

120601119909119894

120601119910119894

120601119911119894

120574119909119894

120574119910119894

120574119911119894]119879

(17)

6 Journal of Composites

Thus the generalized displacement vectors at any pointwithin the element can be expressed in terms of the nodalgeneralized displacement vectors 119889

119890

119905

and 119889119890

119903

as follows

119889119890

119905

= [119873119905] 119889

119890

119905

119889119890

119903

= [119873119903] 119889

119890

119903

(18)

in which

119889119890

119905

= [119889119890

1199051

119879

119889119890

1199052

119879

sdot sdot sdot 119889119890

1199058

119879

]119879

119889119890

119903

= [119889119890

1199031

119879

119889119890

1199032

119879

sdot sdot sdot 119889119890

1199038

119879

]119879

[119873119905] = [119873

11990511198731199052

sdot sdot sdot 1198731199058]119879

[119873119903] = [119873

11990311198731199032

sdot sdot sdot 1198731199038]119879

119873119905119894

= 119899119894119868119905 119873

119903119894= 119899

119894119868119903

(19)

while 119868119905and 119868

119903are (3times3) and (9times 9) identitymatrices respec-

tively and 119899119894is the shape function of natural coordinates

associatedwith the 119894th nodeMaking use of the relations givenby (6)ndash(8) and (18) the strain vectors at any point within theelement can be expressed in terms of the nodal generalizeddisplacement vectors as follows

120576119887119888

= [119861119905119887] 119889

119890

119905

+ [1198851] [119861

119903119887] 119889

119890

119903

120576119887V = [119861

119905119887] 119889

119890

119905

+ [1198852] [119861

119903119887] 119889

119890

119903

120576119887119901

= [119861119905119887] 119889

119890

119905

+ [1198853] [119861

119903119887] 119889

119890

119903

120576119904119888

= [119861119905119904] 119889

119890

119905

+ [1198854] [119861

119903119904] 119889

119890

119903

120576119904V = [119861

119905119904] 119889

119890

119905

+ [1198855] [119861

119903119904] 119889

119890

119903

120576119904119901

= [119861119905119904] 119889

119890

119905

+ [1198856] [119861

119903119904] 119889

119890

119903

(20)

in which the nodal strain-displacement matrices [119861119905119887] [119861

119903119887]

[119861119905119904] and [119861

119903119904] are given by

[119861119905119887] = [119861

11990511988711198611199051198872

sdot sdot sdot 1198611199051198878

]

[119861119903119887] = [119861

11990311988711198611199031198872

sdot sdot sdot 1198611199031198878

]

[119861119905119904] = [119861

11990511990411198611199051199042

sdot sdot sdot 1198611199051199048

]

[119861119903119904] = [119861

11990311990411198611199031199042

sdot sdot sdot 1198611199031199048

]

(21)

The submatrices of [119861119905119887] [119861

119903119887] [119861

119905119904] and [119861

119903119904] have been

explicitly presented in the Appendix On substitution of (9)ndash(13) and (20) into (16) the total potential energy 119879

119890

119901

and the

kinetic energy 119879119890

119896

of a typical element augmented with theACLD treatment can be expressed as

119879119890

119901

=1

2[119889

119890

119905

119879

[119870119890

119905119905

] 119889119890

119905

+ 119889119890

119905

119879

[119870119890

119905119903

] 119889119890

119903

+ 119889119890

119903

119879

[119870119890

119905119903

]119879

times 119889119890

119905

+ 119889119890

119903

119879

[119870119890

119903119903

] 119889119890

119903

minus 2119889119890

119905

119879

119865119890

119905119901

119881

minus 2119889119890

119903

119879

119865119890

119903119901

119881 minus 2119889119890

119905

119879

119865119890

minus 12057633

1198812

ℎ119901

]

(22)

119879119890

119896

=1

2 119889

119890

119905

119879

[119872119890

] 119889119890

119905

(23)

In (22)119881 represents the voltage difference applied across thethickness of the PZC layer The elemental stiffness matrices[119870

119890

119905119905

] [119870119890

119905119903

] and [119870119890

119903119903

] the elemental electroelastic couplingvectors 119865

119890

119905119901

and 119865119890

119903119901

the elemental load vector 119865119890

andthe elemental mass matrix [119872

119890

] appearing in (22) and (23)are given by

[119870119890

119905119905

] = [119870119890

119905119887

] + [119870119890

119905119904

] + [119870119890

119905119887119904

]119901119887

+ [119870119890

119905119887119904

]119901119904

[119870119890

119905119903

] = [119870119890

119905119903119887

] + [119870119890

119905119903119904

]

+1

2([119870

119890

119905119903119887119904

]119901119887

+ [119870119890

119903119905119887119904

]119879

119901119887

+ [119870119890

119905119903119887119904

]119901119904

+ [119870119890

119903119905119887119904

]119879

119901119904

)

[119870119890

119903119905

] = [119870119890

119905119903

]119879

[119870119890

119903119903

] = [119870119890

119903119903119887

] + [119870119890

119903119903119904

] + [119870119890

119903119903119887119904

]119901119887

+ [119870119890

119903119903119887119904

]119901119904

119865119890

119905119901

= 119865119890

119905119887

119901

+ 119865119890

119905119904

119901

119865119890

119903119901

= 119865119890

119903119887

119901

+ 119865119890

119903119904

119901

119865119890

= int

119886119890

0

int

119887119890

0

[119873119905]119879

119891 119889119909 119889119910

[119872119890

] = int

119886119890

0

int

119887119890

0

119898[119873]119879

[119873] 119889119909 119889119910 119898 =

119873+2

sum

119896=1

120588119896

(ℎ119896+1

minus ℎ119896)

(24)

where

[119870119890

119905119887

] = int119860

[119861119905119887]119879

([119863119905119887] + [119863

119905119887]V + [119863

119905119887]119901

) [119861119905119887] 119889119909 119889119910

[119870119890

119905119904

] = int119860

[119861119905119904]119879

([119863119905119904] + [119863

119905119904]V + [119863

119905119904]119901

) [119861119905119904] 119889119909 119889119910

[119870119890

119905119887119904

]119901119887

= int119860

[119861119905119887]119879

[119863119905119887119904

]119901

[119861119905119904] 119889119909 119889119910

[119870119890

119905119887119904

]119901119904

= int119860

[119861119905119904]119879

[119863119905119887119904

]119901

[119861119905119887] 119889119909 119889119910

[119870119890

119905119903119887

] = int119860

[119861119905119903119887

]119879

([119863119905119903119887

] + [119863119905119903119887

]V + [119863119905119903119887

]119901

)

times [119861119905119903119887

] 119889119909 119889119910

Journal of Composites 7

[119870119890

119905119903119887119904

]119901119887

= int119860

[119861119905119887]119879

[119863119905119903119887119904

]119901

[119861119903119904] 119889119909 119889119910

[119870119890

119903119905119887119904

]119901119887

= int119860

[119861119903119887]119879

[119863119903119905119887119904

]119901

[119861119905119904] 119889119909 119889119910

[119870119890

119905119903119887119904

]119901119904

= int119860

[119861119905119904]119879

[119863119903119905119887119904

]119901

[119861119903119887] 119889119909 119889119910

[119870119890

119903119905119887119904

]119901119904

= int119860

[119861119903119904]119879

[119863119905119903119887119904

]119901

[119861119905119887] 119889119909 119889119910

[119870119890

119905119903119904

] = int119860

[119861119905119903119904

]119879

([119863119905119903119904

] + [119863119905119903119904

]V + [119863119905119903119904

]119901

)

times [119861119905119903119904

] 119889119909 119889119910

[119870119890

119903119903119887

] = int119860

[119861119903119903119887

]119879

([119863119903119903119887

] + [119863119903119903119887

]V + [119863119903119903119887

]119901

)

times [119861119903119903119887

] 119889119909 119889119910

[119870119890

119903119903119904

] = int119860

[119861119903119903119904

]119879

([119863119903119903119904

] + [119863119903119903119904

]V + [119863119903119903119904

]119901

)

times [119861119903119903119904

] 119889119909 119889119910

[119870119890

119903119903119887119904

]119901119887

= int119860

[119861119903119887]119879

[119863119903119903119887119904

]119901

[119861119903119904] 119889119909 119889119910

[119870119890

119903119903119887119904

]119901119904

= int119860

[119861119903119904]119879

[119863119903119903119887119904

]119901

[119861119903119887] 119889119909 119889119910

119865119890

119905119887

119901

= int119860

[119861119905119887]119879

119863119905119887119901

119889119909 119889119910

119865119890

119905119904

119901

= int119860

[119861119905119904]119879

119863119905119904119901

119889119909 119889119910

119865119890

119903119904

119901

= int119860

[119861119903119904]119879

119863119903119904119901

119889119909 119889119910

(25)

The various rigidity matrices and the rigidity vectors forelectro-elastic coupling appearing in the previous elementalmatrices are given by

[119863119905119887] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] 119889119911

[119863119905119903119887

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] [1198851] 119889119911

[119863119903119903119887

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[1198851]119879

[119862119896

119887

] [1198851] 119889119911

[119863119905119904] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119904

] 119889119911

[119863119905119903119904

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] [1198854] 119889119911

[119863119903119903119904

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[1198854]119879

[119862119896

119887

] [1198854] 119889119911

[119863119905119887]V = ℎV [119862

119873+1

119887

]

[119863119905119903119887

]V = int

ℎ119873+2

ℎ119873+1

[119862119873+1

119887

] [1198852] 119889119911

[119863119903119903119887

]V = int

ℎ119873+2

ℎ119873+1

[1198852]119879

[119862119873+1

119887

] [1198852] 119889119911

[119863119905119904]V = ℎV [119862

119873+1

119904

]

[119863119905119903119904

]V = int

ℎ119873+2

ℎ119873+1

[119862119873+1

119904

] [1198855] 119889119911

[119863119903119903119904

]V = int

ℎ119873+2

ℎ119873+1

[1198855]119879

[119862119873+1

119887

] [1198855] 119889119911

[119863119905119887]119901

= ℎ119901[119862

119873+2

119887

]

[119863119905119903119887

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887

] [1198853] 119889119911

[119863119903119903119887

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+2

119887

] [1198853] 119889119911

[119863119905119904]119901

= ℎ119901[119862

119873+2

119904

]

[119863119905119903119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119904

] [1198856] 119889119911

[119863119903119903119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198856]119879

[119862119873+2

119887

] [1198856] 119889119911

[119863119905119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887119904

] 119889119911

[119863119905119903119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887119904

] [1198856] 119889119911

[119863119903119905119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+2

119887119904

] 119889119911

[119863119903119903119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+3

119887119904

] [1198856] 119889119911

119863119905119887119901

= int

ℎ119873+3

ℎ119873+2

minus119890119887

ℎ119901

119889119911

119863119903119887119901

= int

ℎ119873+3

ℎ119873+2

minus[119885

3]119879

119890119887

ℎ119901

119889119911

119863119905119904119901

= int

ℎ119873+3

ℎ119873+2

minus119890119904

ℎ119901

119889119911

119863119903119904119901

= int

ℎ119873+3

ℎ119873+2

minus[119885

6]119879

119890119904

ℎ119901

119889119911

(26)

8 Journal of Composites

Applying the dynamic version of the virtual work principle[32] the following open-loop equations of motion for anelement of the overall plate integrated with the ACLDtreatment are obtained

[119872119890

] 119889119890

119905

+ [119870119890

119905119905

] 119889119890

119905

+ [119870119890

119905119903

] 119889119890

119903

= 119865119890

119905119901

119881 + 119865119890

[119870119890

119905119903

]119879

119889119890

119905

+ [119870119890

119903119903

] 119889119890

119903

= 119865119890

119903119901

119881

(27)

Since the elastic constant matrix of the viscoelastic layer iscomplex the stiffness matrices of an element integrated withthe ACLD treatment are complex In case of an elementnot integrated with the ACLD treatment the electro-elasticcoupling matrices become null vectors and the elementalstiffnessmatrices are to be computed without considering thepiezoelectric and viscoelastic layers It should also be notedthat as the stiffness matrices associated with the transverseshear strains are derived separately one can employ theselective integration rule in a straight forward manner toavoid shear locking problem in case of thin plates Therestrained boundary conditions at the skew edges of theplate are such that the displacements of the points locatedat the skew edges are to be restrained along 119909

1015840- 1199101015840- and

1199111015840-directions (Figure 1(a)) Hence in order to impose theboundary conditions in a straight forward manner thegeneralized displacement vectors of a point lying on the skewedge are to be transformed as follows

119889119905 = [119871

119905] 119889

1015840

119905

119889119903 = [119871

119903] 119889

1015840

119903

(28)

where 1198891015840

119905

and 1198891015840

119903

are the generalized displacement vectorsof the point with respect to 119909

1015840

1199101015840

1199111015840 coordinate system and are

given by

1198891015840

119905

= [1199061015840

119900

V1015840119900

1199081015840

119900

]119879

1198891015840

119903

= [1205791015840

119909

1205791015840

119910

1205791015840

119911

1206011015840

119909

1206011015840

119910

1206011015840

119911

1205741015840

119911

1205741015840

119910

1205741015840

119911]119879

(29)

Also the transformation matrices [119871119905] and [119871

119903] are given by

[119871119905] = [

[

119888 119904 0

minus119904 119888 0

0 0 1

]

]

[119871119903] =

[[[[[[[[[[[[

[

119888 119904 0 0 0 0 0 0 0

minus119904 119888 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 119888 119904 0 0 0 0

0 0 0 minus119904 119888 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 119888 119904 0

0 0 0 0 0 0 minus119904 119888 0

0 0 0 0 0 0 0 0 1

]]]]]]]]]]]]

]

(30)

in which 119888 = cos120572 and 119904 = sin120572Thus the elementalmatricesof the element containing the nodes lying on the skew edgeare to be augmented as follows

[119870119890

119905119905

] = [1198791]119879

[119870119890

119905119905

] [1198791] [119870

119890

119905119903

] = [1198791]119879

[119870119890

119905119903

] [1198792]

[119870119890

119903119903

] = [1198792]119879

[119870119890

119903119903

] [1198792] [119872

119890

] = [1198791]119879

[119872119890

] [1198791]

(31)

The forms of the transformation matrices [1198791] and [119879

2] are

given by

[1198791] =

[[[[[[[[[[[

[

[119871119905] 119900 119900 119900 119900 119900 119900 119900

119900 [119871119905] 119900 119900 119900 119900 119900 119900

119900 119900 [119871119905] 119900 119900 119900 119900 119900

119900 119900 119900 [119871119905] 119900 119900 119900 119900

119900 119900 119900 119900 [119871119905] 119900 119900 119900

119900 119900 119900 119900 119900 [119871119905] 119900 119900

119900 119900 119900 119900 119900 119900 [119871119905] 119900

119900 119900 119900 119900 119900 119900 119900 [119871119905]

]]]]]]]]]]]

]

[1198792] =

[[[[[[[[[[[

[

[119871119903] 119900 119900 119900 119900 119900 119900 119900

119900 [119871119903] 119900 119900 119900 119900 119900 119900

119900 119900 [119871119903] 119900 119900 119900 119900 119900

119900 119900 119900 [119871119903] 119900 119900 119900 119900

119900 119900 119900 119900 [119871119903] 119900 119900 119900

119900 119900 119900 119900 119900 [119871119903] 119900 119900

119900 119900 119900 119900 119900 119900 [119871119903] 119900

119900 119900 119900 119900 119900 119900 119900 [119871119903]

]]]]]]]]]]]

]

(32)

in which 119900 and 119900 are (3 times 3) and (9 times 9) null matricesrespectively It may be noted that in case of a node ofsuch element which does not lie on the skew edge thegeneralized displacement vectors of this node with respect to119909119910119911 coordinate system are not required to be transformedHence thematrices [119871

119905] and [119871

119903] corresponding to suchnode

turn out to be (3times3) and (9times9) identitymatrices respectivelyThe elemental equations of motion are now assembled

to obtain the open-loop global equations of motion of theoverall skew plate as follows

[119872] + [119870119905119905] 119883 + [119870

119905119903] 119883

119903 =

119902

sum

119895=1

119865119895

119905119901

119881119895

+ 119865

[119870119905119903]119879

119883 + [119870119903119903] 119883

119903 =

119902

sum

119895=1

119865119895

119903119901

119881119895

(33)

where [119870119905119905] [119870

119905119903] and [119870

119903119903] are the global stiffness matrices

of the overall skew laminated composite plate [119872] is theglobal mass matrix 119865

119905119901 and 119865

119903119901 are the global electro-

elastic coupling vectors 119883 and 119883119903 are the global nodal

generalized displacement vectors 119865 is the global nodalforce vector 119902 is the number of patches and 119881

119895 is thevoltage applied to the 119895th patch Since the elemental stiffnessmatrices of an element augmented with the ACLD treatmentare complex the global stiffness matrices become complexand the energy dissipation characteristics of the overallplate are attributed to the imaginary part of these matricesHence the global equations of motion as derived previouslyalso represent the modeling of the passive (uncontrolled)constrained layer damping of the substrate plate when theconstraining layer is not subjected to any control voltagefollowing a derivative control law

Journal of Composites 9

3 Closed-Loop Model

In the active control approach the active constraining layerof each patch is activated with a control voltage negativelyproportional to the transverse velocity of a point Thus thecontrol voltage for each patch can be expressed in terms of thederivatives of the global nodal degrees of freedom as follows

119881119895

= minus119896119895

119889

= minus119896119895

119889

[119880119895

119905

] minus 119896119895

119889

(ℎ

2) [119880

119895

119903

] 119903 (34)

where [119880119895

119905

] and [119880119895

119903

] are the unit vectors describing thelocation of the velocity sensor for the 119895th patch and 119896

119895

119889

isthe control gain for the 119895th patch Substituting (34) into(33) the final equations of motion governing the closed-loop dynamics of the overall skew laminated compositeplateACLD system can be obtained as follows

[119872] + [119870119905119905] 119883 + [119870

119905119903] 119883

119903 +

119898

sum

119895=1

119896119895

119889

119865119895

119905119901

[119880119895

119905

]

+

119898

sum

119895=1

119896119895

119889

(ℎ

2) 119865

119895

119905119901

[119880119895

119903

] 119903 = 119865

[119870119905119903]119879

119883 + [119870119903119903] 119883

119903 +

119898

sum

119895=1

119896119895

119889

119865119895

119905119901

[119880119895

119905

]

+

119898

sum

119895=1

119896119895

119889

(ℎ

2) 119865

119895

119905119901

[119880119895

119903

] 119903 = 0

(35)

4 Results and Discussion

In this section the numerical results obtained by the finiteelement model derived in the previous section have beenpresented Symmetric as well as antisymmetric cross-plyand antisymmetric angle-ply laminated skew substrate platesintegrated with the two patches of ACLD treatment areconsidered for presenting the numerical results As shown inFigure 1 the locations of the patches have been selected insuch a way that the energy dissipation corresponding to thefirst fewmodes becomesmaximum PZT-5Hspur compositewith 60 fiber volume fraction has been considered for thematerial of the constraining layer of the ACLD treatmentUsing themicromechanicsmodel and thematerial propertiesof the constituent phases [35] the effective elastic andpiezoelectric material properties of the 1ndash3 PZC with respectto the principle material coordinate system are computed asfollows

11986211

= 929GPa 11986212

= 618GPa 11986213

= 605GPa

11986233

= 3544GPa 11986223

= 11986213 119862

44= 158GPa

11986266

= 154GPa 11986255

= 11986244

11989031

= minus01902Cm2

11989032

= 11989031

11989033

= 184107Cm2

11989024

= 0004Cm2

11989015

= 11989024

(36)

The material of the orthotropic layers of the substrate platesis a graphiteepoxy composite and its material properties are[42]

119864119871= 172GPa 119864

119871

119864119879

= 25 119866119871119879

= 05119864119879

119866119879119879

= 02119864119879 ]

119871119879= ]

119879119879= 025

(37)

in which the symbols have the usual meaning Unless oth-erwise mentioned the aspect ratio (119886ℎ) and the thicknessof the skew substrate plates are considered as 100 and0003m respectively while the orthotropic layers of theskew substrate plates are of equal thickness Also the lengthand the skew sides of the substrate are equal (ie 119886 =

119887) The loss factor of the viscoelastic layer considered fornumerical results remains invariant [20] within a broadband (0ndash400Hz) of frequency range and the values of thecomplex shear modulus Poisson ratio and the density ofthe viscoelastic layer are considered as 20(1 + 119894)MNmminus2049 and 1140 kgmminus3 respectively [20] The thicknesses ofthe viscoelastic layer and the 1ndash3 PZC layer are consideredas 508120583m and 250 120583m respectively The value of the shearcorrection factor is used as 56 Three points of Gaussianintegration rule are considered for computing the elementmatrices corresponding to the bending deformations whiletwo points of that are used for computing the elementmatrices corresponding to transverse shear deformationsThe simply supported boundary conditions at the edges of theoverall plate are invoked as follows

at 119909 = 119910 tan120572 119909 = 119886 + 119910 tan120572

V1015840119900

= 1199081015840

119900

= 1205791015840

119910

= 1206011015840

119910

= 1205741015840

119910

= 1205791015840

119911

= 1205741015840

119911

= 0

at 119910 = 0 119910 = 119887 cos120572

119906119900= 119908

119900= 120579

119909= 120601

119909= 120574

119909= 120579

119911= 120574

119911= 0

(38)

To verify the validity of the present finite elementmodel (FEM) the natural frequencies of the skew laminatedcomposite plates integrated with the inactivated patches ofnegligible thickness are first computed and subsequentlycompared with the existing analytical results [41] of theidentical plates without being integrated with the patchesSuch comparisons for the simply supported and clamped-clamped skew laminated plates are presented in Tables 1 and2 respectively It may be observed from these tables that theresults are in excellent agreement validating the model of theplate derived here

In order to assess the performance of the verti-callyobliquely reinforced 1ndash3 PZC as the material of thedistributed actuators of smart skew laminated compositeplates frequency responses for ACLD of the substrate skewlaminated composite plates are investigated To compute the

10 Journal of Composites

Table 1 Comparison of fundamental natural frequencies (120596) of simply supported skew laminated composite plates integratedwith the patchesof negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 15699 28917 37325 14829 24656 32522 17974 33351 33351Present FEM 15635 24383 35033 15076 24380 32254 18493 33359 33370

15∘ Reference [41] 16874 30458 39600 15741 25351 30270 18313 32490 36724Present FEM 16571 29840 36505 15796 25775 29892 18675 32075 35810

30∘ Reference [41] 20840 34023 46997 18871 29372 34489 20270 34431 42361Present FEM 19596 31690 46796 18226 29585 32357 19894 32365 43208

45∘ Reference [41] 28925 41906 54149 25609 33126 40617 25609 33131 42772Present FEM 24811 44875 53289 22996 34773 44889 23194 34870 45009

lowast

120596 = (120596119886

2

120587

2

ℎ)radic120588119864119879 120596 is the circular natural frequency

Table 2 Comparison of fundamental natural frequencies (120596) of clamped-clamped skew laminated composite plates integrated with thepatches of negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 23687 35399 41122 22990 37880 37880 22119 37339 37339Present FEM 23201 34769 44102 23315 36531 36545 22433 36000 36012

15∘ Reference [41] 24663 36255 43418 23809 37516 40785 23099 36997 40438Present FEM 23699 34821 44049 23741 35856 38401 23049 35346 38092

30∘ Reference [41] 27921 39557 50220 26666 39851 47227 26325 39549 52107Present FEM 25366 35696 44734 25240 41943 45373 24945 36113 50932

45∘ Reference [41] 34739 47129 58789 33015 46290 58423 33015 46290 58423Present FEM 28665 47074 54562 28377 47614 56620 28377 47614 55162

frequency response functions the skew laminated compositeplates are harmonically excited by a force of amplitude 1Newton applied to a point (1198862 (1198864) cos120572 ℎ2) on the skewsubstrate platesThe control voltage supplied to the constrain-ing layer of the first patch is negatively proportional to thevelocity of the point (1198862 (1198864) cos120572 ℎ2) and that applied tothe constraining layer of the second patch is negatively pro-portional to the velocity of the point (1198862 (31198864) cos120572 ℎ2)

Figures 3 4 and 5 illustrate the frequency responsefunctions for the transverse displacement of a point of asimply supported symmetric cross-ply (0∘90∘0∘) plate with15∘ 30∘ and 45∘ skew angles respectively while the fiberorientation angle (120595) in the constraining layer is 0∘ Thesefigures display both uncontrolled and controlled responsesand clearly show that the active constraining layer made ofthe vertically reinforced 1ndash3 PZC material being studied heresignificantly attenuates the amplitude of vibrations enhanc-ing the damping characteristics of the overall laminatedskew composite plates over the passive damping (uncon-trolled) Similar responses are also obtained for antisym-metric cross-ply (0∘90∘0∘90∘) and antisymmetric angle-ply (45∘minus45∘45∘minus45∘) skew laminated composite plates

However for brevity these results are not presented hereThemaximum values of the control voltages required to computethe controlled responses presented in Figures 3ndash5 have beenfound to be quite low and are illustrated in Figure 6 in caseof the symmetric cross-ply (0∘90∘0∘) plate with skew angle15∘ only In order to examine the contribution of the verticalactuation in improving the damping characteristics of theskew laminated composite plates as demonstrated in Figures3ndash5 active control responses of the symmetric cross-plylaminated composite plate with 15∘ skew angle are plotted inFigure 7with andwithout considering the value of 119890

33and 119890

31

It may be noted that when the values of 11989031is zero and that of

11989033is nonzero the vertical actuation of the active constraining

layer of the ACLD treatment is responsible for increasing thetransverse shear deformations of the viscoelastic constrainedlayer over the passive counterpart resulting in improveddamping of the smart plate over its passive damping Onthe other hand if 119890

33is zero and 119890

31is nonzero the in-

plane actuation of the active constraining layer causes thetransverse shear deformation of the viscoelastic core of theACLD treatment leading to the active damping of the smartplate It is evident from Figure 7 that the contribution of the

Journal of Composites 11

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 3 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 4 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 30∘)

vertical actuation by the constraining vertically reinforced 1ndash3 PZC layer is significantly larger than that of the in-planeactuation by the same for controlling the modes displayedin Figure 7 Though not shown here similar results are alsoobtained for antisymmetric cross-ply and angle-ply skewplates

Figures 8 9 and 10 demonstrate the effect of variationof piezoelectric fiber orientation angle on the performanceof the patches for improving the damping characteristicsof simply supported symmetric cross-ply (0∘90∘0∘)

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 5 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 45∘)

0

5

10

15

20

25

30

35

40

45

50

Con

trol v

olta

ge (V

olt)

0 100 200 300 400 500 600Frequency (Hz)

119870119889 = 600

119870119889 = 1200

Figure 6 Frequency response functions for the control voltages foractive damping of the simply supported skew symmetric cross-ply(0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

antisymmetric cross-ply (0∘90∘0∘90∘) and angle-ply(45∘minus45∘45∘minus45∘) substrate skew composite plates with30∘ skew angle respectively while the orientation angle (120595)

of the piezoelectric fibers is varied in the vertical 119909119911-planeThese figures illustrate that the attenuating capability of thepatches becomesmaximumwhen the value of the orientationangle (120595) of the fibers in the 119909119911-plane is 0∘ Similar effectis also obtained when the fiber orientation angle (120595) in theconstraining layer of the ACLD treatment is varied in thevertical 119910119911-plane as illustrated in Figures 11 12 and 13 forthe symmetric cross-ply (0∘90∘0∘) antisymmetric cross-ply(0∘90∘0∘90∘) and angle-ply (45∘minus45∘45∘minus45∘) substrate

12 Journal of Composites

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

11989031 = minus01902 11989033 = 184107

11989031 = minus01902 11989033 = 011989031 = 0 11989033 = 184107

w(a2(b4)co

s120572h2

) (m

)

Figure 7 Frequency response functions demonstrating the con-tribution of vertical actuation on the controlled responses forthe transverse displacement 119908 (1198862 (1198864) cos120572 ℎ2) of the skewsymmetric cross-ply (0∘90∘0∘) plate (119870

119889

= 600 119886ℎ = 100 120595 = 0∘

120572 = 15∘)

0 100 200 300 400 500 600Frequency (Hz)

0

02

04

06

08

1

12times10minus4

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 8 Effect of variation of piezoelectric fiber orientation angle(120595) on the controlled responses of a simply supported symmetricskew cross-ply (0∘90∘0∘) plate when the piezoelectric fibers of theconstraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

skew composite plates with 30∘ skew angle respectivelyIn this regard it should be noted that for investigating theeffect of variation of piezoelectric fiber orientation angle(120595) on the performance of the ACLD patches the valueof 120595 has been smoothly varied to compute the controlledresponses However for the sake of clarity in the plots theresponses corresponding to the four specific values of 120595

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 9 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 10 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

have been presented such that the optimum performanceof the patches can be evaluated The significant effect of thevariation of the piezoelectric fiber orientation angle in thevertical 119909119911- and 119910119911-planes has been noticed when all theboundaries of the overall plates are clamped as shown inFigures 14ndash19 When the piezoelectric fibers are coplanarwith the vertical 119909119911-plane the frequency response functionsplotted in Figures 14 15 and 16 reveal that the control

Journal of Composites 13

0

02

04

06

08

1

12

14

16times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 11 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 12 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=600 119886ℎ = 100 120572 = 30

∘)

authority of the patches becomes maximum for improvingthe damping characteristics of the clamped-clamped skewcross-ply and angle-ply substrate plates if the value of thepiezoelectric fiber orientation angle (120595) is 30∘ The same isalso found to be true if the piezoelectric fibers are coplanarwith the vertical 119910119911-plane as shown in Figures 17 18 and 19for the previously mentioned skew plates Similar responsesare also found for the effect of piezoelectric fiber orientation

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 13 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers of the constraining layer are coplanar with the119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 14 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600119886ℎ = 100120572 = 30

∘)

angle on the active damping of other skew laminated plateswith different skew angles However for the sake of brevitythey are not presented here

5 Conclusions

In this paper a study has been carried out to investi-gate the performance of the ACLD treatment for active

14 Journal of Composites

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 15 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 16 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

damping of smart skew laminated composite plates whenverticallyobliquely reinforced 1ndash3 PZC materials are usedas the materials for the constraining layer of the ACLDtreatment A layerwise FSDT-based finite element modelhas been developed to describe the dynamics of the skewlaminated composite plates integrated with the patches ofACLD treatment Unlike the existing finite element modelsof smart structures integrated with ACLD treatment the

200 250 300 350 400 450 500 550 600 650 7000

02

04

06

08

1

12

14

16

18

2times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 17 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

200 300 400 500 600 700 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 18 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

derivation of the present finite element model includes thetransverse deformations of the substrate skew laminatedcomposite plate the constrained viscoelastic layer and theconstraining 1ndash3 PZC layer of the ACLD treatment along thethickness (ie 119911) direction such that both vertical and in-plane actuations by the constraining layer of the patches canbe utilized for active damping of the plates The frequency

Journal of Composites 15

0

05

1

15

2

25

3

35

4times10minus5

200 300 400 500 600 700 800Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 19 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

responses of the symmetric cross-ply antisymmetric cross-ply and angle-ply skew composite plates indicate that theactive constraining layer of the ACLD treatment being madeof the vertically reinforced 1ndash3 PZC material significantlyenhances the damping characteristics of the plates over thepassive damping The analysis revealed that if the verticallyreinforced 1ndash3 PZCmaterial is used for the constraining layerof the ACLD treatment then the contribution of the verticalactuation by the constraining layer alone for improving theactive damping characteristics of the smart skew laminatedcomposite plate is significantly larger than that due to thein-plane actuation by the constraining layer alone It isimportant to note from the present investigation that thevariation of the orientation angle of the piezoelectric fibersof the constraining layer in the vertical 119909119911- and 119910119911-planesas well as the skew angle and boundary conditions of theoverall plates significantly affects the performance of thepatches For the simply supported laminated substrate platesthe performance of the patches becomes maximumwhen thepiezoelectric fiber orientation angle (120595) is 0∘ while in caseof the clamped-clamped laminated skew substrate plates themaximum control authority of the patches is achieved whenthe value of 120595 is 30∘ irrespective of the cases in which thefibers are oriented in 119909119911- and 119910119911-planes

Appendix

In (6) and (7) the matrices [1198851] [119885

2] [119885

3] [119885

4] [119885

5] and

[1198856] are given by

[1198851] = [[119885

1] 0 0]

[1198852] = [(

ℎ

2) 119868 [119885

2] 0]

[1198853] = [(

ℎ

2) 119868 ℎV119868 [119885

3]]

[1198854] = [119868 0 0 119911119868 0 0]

[1198855] = [0 119868 0 (

ℎ

2) 119868 (119911 minus

ℎ

2) 119868 0]

[1198856] = [0 0 119868 (

ℎ

2) 119868 ℎV119868 (119911 minus ℎ

119873+2) 119868]

(A1)

in which

[1198851] =

[[[

[

119911 0 0 0

0 119911 0 0

0 0 119911 0

0 0 0 1

]]]

]

[1198852] =

[[[[[[[[

[

(119911 minusℎ

2) 0 0 0

0 (119911 minusℎ

2) 0 0

0 0 (119911 minusℎ

2) 0

0 0 0 1

]]]]]]]]

]

[1198853] =

[[[

[

(119911 minus ℎ119873+2

) 0 0 0

0 (119911 minus ℎ119873+2

) 0 0

0 0 (119911 minus ℎ119873+2

) 0

0 0 0 1

]]]

]

[119868] =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

[119868] = [1 0

0 1]

[0] = [0 0

0 0] [0] = [

0 0

0 0]

(A2)

The various submatrices 119861119905119887119894 119861

119903119887119894 and 119861

119903119904119894 appearing in (21)

are given by

119861119905119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 0

]]]]]]]]]]]

]

119861119905119904119894

=

[[[[

[

0 0120597120578

119894

120597119909

0 0120597120578

119894

120597119910

]]]]

]

16 Journal of Composites

119861119903119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 1

]]]]]]]]]]]

]

119861119903119887119894

=

[[[[[[

[

119861119903119887119894

0

0

0 119861119903119887119894

0

0

0 119861119903119887119894

]]]]]]

]

119861119903119904119894

=

[[[[[[[[[[[[[[[[[[

[

119868 0 0

0

119868 0

0 0

119868

119861119903119904119894

0 0

0 119861119903119904119894

0

0 0 119861119903119904119894

]]]]]]]]]]]]]]]]]]

]

119868= [1 0 0

0 1 0]

(A3)

wherein

0 and 0 are the (3 times 3) and (2 times 3) null matricesrespectively

References

[1] T Bailey and J E Hubbard ldquoDistributedpiezoelectric polymeractive vibration control of a cantilever beamrdquo Journal of Guid-ance Control and Dynamics vol 8 no 5 pp 605ndash611 1985

[2] E F Crawley and J de Luis ldquoUse of piezoelectric actuators aselements of intelligent structuresrdquo AIAA Journal vol 25 no 10pp 1373ndash1385 1987

[3] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988

[4] C K Lee W W Chiang and O Sulivan ldquoPiezoelectric modalsensoractuator pairs for critical active damping vibrationcontrolrdquo Journal of the Acoustical Society of America vol 90 no1 pp 374ndash384 1991

[5] S Hanagud M W Obal and A J Calise ldquoOptimal vibrationcontrol by the use of piezoceramic sensors and actuatorsrdquoJournal of Guidance Control and Dynamics vol 15 no 5 pp1199ndash1206 1992

[6] Y Gu R L Clark C R Fuller and A C Zander ldquoExperimentson active control of plate vibration using piezoelectric actuatorsand polyvinylidene fluoride (PVDF) modal sensorsrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 116 no3 pp 303ndash308 1994

[7] A Baz and S Poh ldquoOptimal vibration control with modalpositive position feedbackrdquo Optimal Control Applications andMethods vol 17 no 2 pp 141ndash149 1996

[8] A Kugi K Schlacher and H Irschik ldquoInfinite-dimensionalcontrol of nonlinear beam vibrations by piezoelectric actuatorand sensor layersrdquo Nonlinear Dynamics vol 19 no 1 pp 71ndash911999

[9] D Sun L Tong and D Wang ldquoVibration control of platesusing discretely distributed piezoelectric quasi-modal actua-torssensorsrdquo AIAA Journal vol 39 no 9 pp 1766ndash1772 2001

[10] G Caruso S Galeani and L Menini ldquoActive vibration controlof an elastic plate using multiple piezoelectric sensors andactuatorsrdquo SimulationModelling Practice andTheory vol 11 no5-6 pp 403ndash419 2003

[11] X Zhang and A G Erdman ldquoOptimal placement of piezo-electric sensors and actuators for controlled flexible linkagemechanismsrdquo Journal of Vibration and Acoustics Transactionsof the ASME vol 128 no 2 pp 256ndash260 2006

[12] G Gatti M J Brennan and P Gardonio ldquoActive dampingof a beam using a physically collocated accelerometer andpiezoelectric patch actuatorrdquo Journal of Sound and Vibrationvol 303 no 3ndash5 pp 798ndash813 2007

[13] M K Kwak S Heo and M Jeong ldquoDynamic modellingand active vibration controller design for a cylindrical shellequipped with piezoelectric sensors and actuatorsrdquo Journal ofSound and Vibration vol 321 no 3ndash5 pp 510ndash524 2009

[14] I Bruant L Gallimard and S Nikoukar ldquoOptimal piezoelectricactuator and sensor location for active vibration control usinggenetic algorithmrdquo Journal of Sound and Vibration vol 329 no10 pp 1615ndash1635 2010

[15] Y Fu J Wang and Y Mao ldquoNonlinear vibration and activecontrol of functionally graded beams with piezoelectric sensorsand actuatorsrdquo Journal of Intelligent Material Systems andStructures vol 22 no 18 pp 2093ndash2102 2011

[16] F Chen M Hong M Song and H Cui ldquoOptimal control ofa beam with discontinuously distributed piezoelectric sensorsand actuatorsrdquo Journal of Marine Science and Application vol11 no 1 pp 44ndash51 2012

[17] B Azvine G R Tomlinson and R J Wynne ldquoUse of activeconstrained-layer damping for controlling resonant vibrationrdquoSmart Materials and Structures vol 4 no 1 pp 1ndash6 1995

[18] A Baz and J Ro ldquoVibration control of plates with activeconstrained layer dampingrdquo SmartMaterials and Structures vol5 no 3 pp 272ndash280 1996

[19] C Chantalakhana and R Stanway ldquoActive constrained layerdamping of clamped-clamped plate vibrationsrdquo Journal ofSound and Vibration vol 241 no 5 pp 755ndash777 2001

[20] J Ro and A Baz ldquoOptimum placement and control ofactive constrained layer damping using modal strain energyapproachrdquo JVCJournal of Vibration and Control vol 8 no 6pp 861ndash876 2002

[21] M C Ray and J N Reddy ldquoOptimal control of thin circularcylindrical laminated composite shells using active constrainedlayer damping treatmentrdquo Smart Materials and Structures vol13 no 1 pp 64ndash72 2004

[22] C M A Vasques B R Mace P Gardonio and J D RodriguesldquoArbitrary active constrained layer damping treatments onbeams finite element modelling and experimental validationrdquoComputers and Structures vol 84 no 22-23 pp 1384ndash14012006

[23] F M Li K Kishimoto K Wang Z B Chen and W HHuang ldquoVibration control of beams with active constrainedlayer dampingrdquo Smart Materials and Structures vol 17 no 6Article ID 065036 2008

[24] L Yuan Y Xiang Y Huang and J Lu ldquoA semi-analyticalmethod and the circumferential dominant modal control ofcircular cylindrical shells with active constrained layer dampingtreatmentrdquo SmartMaterials and Structures vol 19 no 2 ArticleID 025010 2010

[25] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

Journal of Composites 17

[26] W A Smith and B A Auld ldquoModeling 1ndash3 composite piezo-electrics thickness-mode oscillationsrdquo IEEE Transactions onUltrasonics Ferroelectrics and Frequency Control vol 38 no 1pp 40ndash47 1991

[27] PiezocompositesMaterial Systems Inc 543 Great Road Little-ton MA 01460

[28] Y Benveniste and G J Dvorak ldquoUniform fields and universalrelations in piezoelectric compositesrdquo Journal of the Mechanicsand Physics of Solids vol 40 no 6 pp 1295ndash1312 1992

[29] M L Dunn and M Taya ldquoMicromechanics predictions ofthe effective electroelastic moduli of piezoelectric compositesrdquoInternational Journal of Solids and Structures vol 30 no 2 pp161ndash175 1993

[30] J Aboudi ldquoMicromechanical prediction of the effective coeffi-cients of thermo-piezoelectric multiphase compositesrdquo Journalof Intelligent Material Systems and Structures vol 9 no 9 pp713ndash722 1999

[31] X Ruan T W Chou A Safari and S C Danforth ldquoA 3-Dconnectivity model for effective piezoelectric properties of yarncompositesrdquo Journal of Composite Materials vol 36 no 14 pp1693ndash1708 2002

[32] A Baz and A Tempia ldquoActive piezoelectric damping compos-itesrdquo Sensors and Actuators A vol 112 no 2-3 pp 340ndash3502004

[33] H Berger S Kari U Gabbert et al ldquoUnit cell models ofpiezoelectric fiber composites for numerical and analytical cal-culation of effective propertiesrdquo SmartMaterials and Structuresvol 15 no 2 pp 451ndash458 2006

[34] M Sakthivel and A Arockiarajan ldquoAn analytical model forpredicting thermo-electro-mechanical response of 1ndash3 piezo-electric compositesrdquo Computational Materials Science vol 48no 4 pp 759ndash767 2010

[35] M C Ray and A K Pradhan ldquoOn the use of verticallyreinforced 1ndash3 piezoelectric composites for hybrid damping oflaminated composite platesrdquo Mechanics of Advanced Materialsand Structures vol 14 no 4 pp 245ndash261 2007

[36] M C Ray and A K Pradhan ldquoPerformance of vertically andobliquely reinforced 1ndash3 piezoelectric composites for activedamping of laminated composite shellsrdquo Journal of Sound andVibration vol 315 no 4-5 pp 816ndash835 2008

[37] S K Sarangi and M C Ray ldquoSmart damping of geometricallynonlinear vibrations of laminated composite beams using verti-cally reinforced 1ndash3 piezoelectric compositesrdquo Smart Materialsand Structures vol 19 no 7 Article ID 075020 2010

[38] A Krishnan and J V Deshpande ldquoVibration of skew laminatesrdquoJournal of Sound and Vibration vol 153 no 2 pp 351ndash358 1992

[39] A R Krishna Reddy and R Palaninathan ldquoFree vibration ofskew laminatesrdquo Computers and Structures vol 70 no 4 pp415ndash423 1999

[40] C S Babu and T Kant ldquoTwo shear deformable finite elementmodels for buckling analysis of skew fibre-reinforced compositeand sandwich panelsrdquo Composite Structures vol 46 no 2 pp115ndash124 1999

[41] A K Garg R K Khare and T Kant ldquoFree vibration of skewfiber-reinforced composite and sandwich laminates using ashear deformable finite element modelrdquo Journal of SandwichStructures and Materials vol 8 no 1 pp 33ndash53 2006

[42] J N Reddy Mechanics of Laminated Composites Plates Theoryand Analysis CRC press Boca Raton Fla USA 1996

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 6: Research Article Active Constrained Layer Damping of Smart …downloads.hindawi.com/archive/2013/824163.pdf · 2019-07-31 · Research Article Active Constrained Layer Damping of

6 Journal of Composites

Thus the generalized displacement vectors at any pointwithin the element can be expressed in terms of the nodalgeneralized displacement vectors 119889

119890

119905

and 119889119890

119903

as follows

119889119890

119905

= [119873119905] 119889

119890

119905

119889119890

119903

= [119873119903] 119889

119890

119903

(18)

in which

119889119890

119905

= [119889119890

1199051

119879

119889119890

1199052

119879

sdot sdot sdot 119889119890

1199058

119879

]119879

119889119890

119903

= [119889119890

1199031

119879

119889119890

1199032

119879

sdot sdot sdot 119889119890

1199038

119879

]119879

[119873119905] = [119873

11990511198731199052

sdot sdot sdot 1198731199058]119879

[119873119903] = [119873

11990311198731199032

sdot sdot sdot 1198731199038]119879

119873119905119894

= 119899119894119868119905 119873

119903119894= 119899

119894119868119903

(19)

while 119868119905and 119868

119903are (3times3) and (9times 9) identitymatrices respec-

tively and 119899119894is the shape function of natural coordinates

associatedwith the 119894th nodeMaking use of the relations givenby (6)ndash(8) and (18) the strain vectors at any point within theelement can be expressed in terms of the nodal generalizeddisplacement vectors as follows

120576119887119888

= [119861119905119887] 119889

119890

119905

+ [1198851] [119861

119903119887] 119889

119890

119903

120576119887V = [119861

119905119887] 119889

119890

119905

+ [1198852] [119861

119903119887] 119889

119890

119903

120576119887119901

= [119861119905119887] 119889

119890

119905

+ [1198853] [119861

119903119887] 119889

119890

119903

120576119904119888

= [119861119905119904] 119889

119890

119905

+ [1198854] [119861

119903119904] 119889

119890

119903

120576119904V = [119861

119905119904] 119889

119890

119905

+ [1198855] [119861

119903119904] 119889

119890

119903

120576119904119901

= [119861119905119904] 119889

119890

119905

+ [1198856] [119861

119903119904] 119889

119890

119903

(20)

in which the nodal strain-displacement matrices [119861119905119887] [119861

119903119887]

[119861119905119904] and [119861

119903119904] are given by

[119861119905119887] = [119861

11990511988711198611199051198872

sdot sdot sdot 1198611199051198878

]

[119861119903119887] = [119861

11990311988711198611199031198872

sdot sdot sdot 1198611199031198878

]

[119861119905119904] = [119861

11990511990411198611199051199042

sdot sdot sdot 1198611199051199048

]

[119861119903119904] = [119861

11990311990411198611199031199042

sdot sdot sdot 1198611199031199048

]

(21)

The submatrices of [119861119905119887] [119861

119903119887] [119861

119905119904] and [119861

119903119904] have been

explicitly presented in the Appendix On substitution of (9)ndash(13) and (20) into (16) the total potential energy 119879

119890

119901

and the

kinetic energy 119879119890

119896

of a typical element augmented with theACLD treatment can be expressed as

119879119890

119901

=1

2[119889

119890

119905

119879

[119870119890

119905119905

] 119889119890

119905

+ 119889119890

119905

119879

[119870119890

119905119903

] 119889119890

119903

+ 119889119890

119903

119879

[119870119890

119905119903

]119879

times 119889119890

119905

+ 119889119890

119903

119879

[119870119890

119903119903

] 119889119890

119903

minus 2119889119890

119905

119879

119865119890

119905119901

119881

minus 2119889119890

119903

119879

119865119890

119903119901

119881 minus 2119889119890

119905

119879

119865119890

minus 12057633

1198812

ℎ119901

]

(22)

119879119890

119896

=1

2 119889

119890

119905

119879

[119872119890

] 119889119890

119905

(23)

In (22)119881 represents the voltage difference applied across thethickness of the PZC layer The elemental stiffness matrices[119870

119890

119905119905

] [119870119890

119905119903

] and [119870119890

119903119903

] the elemental electroelastic couplingvectors 119865

119890

119905119901

and 119865119890

119903119901

the elemental load vector 119865119890

andthe elemental mass matrix [119872

119890

] appearing in (22) and (23)are given by

[119870119890

119905119905

] = [119870119890

119905119887

] + [119870119890

119905119904

] + [119870119890

119905119887119904

]119901119887

+ [119870119890

119905119887119904

]119901119904

[119870119890

119905119903

] = [119870119890

119905119903119887

] + [119870119890

119905119903119904

]

+1

2([119870

119890

119905119903119887119904

]119901119887

+ [119870119890

119903119905119887119904

]119879

119901119887

+ [119870119890

119905119903119887119904

]119901119904

+ [119870119890

119903119905119887119904

]119879

119901119904

)

[119870119890

119903119905

] = [119870119890

119905119903

]119879

[119870119890

119903119903

] = [119870119890

119903119903119887

] + [119870119890

119903119903119904

] + [119870119890

119903119903119887119904

]119901119887

+ [119870119890

119903119903119887119904

]119901119904

119865119890

119905119901

= 119865119890

119905119887

119901

+ 119865119890

119905119904

119901

119865119890

119903119901

= 119865119890

119903119887

119901

+ 119865119890

119903119904

119901

119865119890

= int

119886119890

0

int

119887119890

0

[119873119905]119879

119891 119889119909 119889119910

[119872119890

] = int

119886119890

0

int

119887119890

0

119898[119873]119879

[119873] 119889119909 119889119910 119898 =

119873+2

sum

119896=1

120588119896

(ℎ119896+1

minus ℎ119896)

(24)

where

[119870119890

119905119887

] = int119860

[119861119905119887]119879

([119863119905119887] + [119863

119905119887]V + [119863

119905119887]119901

) [119861119905119887] 119889119909 119889119910

[119870119890

119905119904

] = int119860

[119861119905119904]119879

([119863119905119904] + [119863

119905119904]V + [119863

119905119904]119901

) [119861119905119904] 119889119909 119889119910

[119870119890

119905119887119904

]119901119887

= int119860

[119861119905119887]119879

[119863119905119887119904

]119901

[119861119905119904] 119889119909 119889119910

[119870119890

119905119887119904

]119901119904

= int119860

[119861119905119904]119879

[119863119905119887119904

]119901

[119861119905119887] 119889119909 119889119910

[119870119890

119905119903119887

] = int119860

[119861119905119903119887

]119879

([119863119905119903119887

] + [119863119905119903119887

]V + [119863119905119903119887

]119901

)

times [119861119905119903119887

] 119889119909 119889119910

Journal of Composites 7

[119870119890

119905119903119887119904

]119901119887

= int119860

[119861119905119887]119879

[119863119905119903119887119904

]119901

[119861119903119904] 119889119909 119889119910

[119870119890

119903119905119887119904

]119901119887

= int119860

[119861119903119887]119879

[119863119903119905119887119904

]119901

[119861119905119904] 119889119909 119889119910

[119870119890

119905119903119887119904

]119901119904

= int119860

[119861119905119904]119879

[119863119903119905119887119904

]119901

[119861119903119887] 119889119909 119889119910

[119870119890

119903119905119887119904

]119901119904

= int119860

[119861119903119904]119879

[119863119905119903119887119904

]119901

[119861119905119887] 119889119909 119889119910

[119870119890

119905119903119904

] = int119860

[119861119905119903119904

]119879

([119863119905119903119904

] + [119863119905119903119904

]V + [119863119905119903119904

]119901

)

times [119861119905119903119904

] 119889119909 119889119910

[119870119890

119903119903119887

] = int119860

[119861119903119903119887

]119879

([119863119903119903119887

] + [119863119903119903119887

]V + [119863119903119903119887

]119901

)

times [119861119903119903119887

] 119889119909 119889119910

[119870119890

119903119903119904

] = int119860

[119861119903119903119904

]119879

([119863119903119903119904

] + [119863119903119903119904

]V + [119863119903119903119904

]119901

)

times [119861119903119903119904

] 119889119909 119889119910

[119870119890

119903119903119887119904

]119901119887

= int119860

[119861119903119887]119879

[119863119903119903119887119904

]119901

[119861119903119904] 119889119909 119889119910

[119870119890

119903119903119887119904

]119901119904

= int119860

[119861119903119904]119879

[119863119903119903119887119904

]119901

[119861119903119887] 119889119909 119889119910

119865119890

119905119887

119901

= int119860

[119861119905119887]119879

119863119905119887119901

119889119909 119889119910

119865119890

119905119904

119901

= int119860

[119861119905119904]119879

119863119905119904119901

119889119909 119889119910

119865119890

119903119904

119901

= int119860

[119861119903119904]119879

119863119903119904119901

119889119909 119889119910

(25)

The various rigidity matrices and the rigidity vectors forelectro-elastic coupling appearing in the previous elementalmatrices are given by

[119863119905119887] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] 119889119911

[119863119905119903119887

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] [1198851] 119889119911

[119863119903119903119887

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[1198851]119879

[119862119896

119887

] [1198851] 119889119911

[119863119905119904] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119904

] 119889119911

[119863119905119903119904

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] [1198854] 119889119911

[119863119903119903119904

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[1198854]119879

[119862119896

119887

] [1198854] 119889119911

[119863119905119887]V = ℎV [119862

119873+1

119887

]

[119863119905119903119887

]V = int

ℎ119873+2

ℎ119873+1

[119862119873+1

119887

] [1198852] 119889119911

[119863119903119903119887

]V = int

ℎ119873+2

ℎ119873+1

[1198852]119879

[119862119873+1

119887

] [1198852] 119889119911

[119863119905119904]V = ℎV [119862

119873+1

119904

]

[119863119905119903119904

]V = int

ℎ119873+2

ℎ119873+1

[119862119873+1

119904

] [1198855] 119889119911

[119863119903119903119904

]V = int

ℎ119873+2

ℎ119873+1

[1198855]119879

[119862119873+1

119887

] [1198855] 119889119911

[119863119905119887]119901

= ℎ119901[119862

119873+2

119887

]

[119863119905119903119887

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887

] [1198853] 119889119911

[119863119903119903119887

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+2

119887

] [1198853] 119889119911

[119863119905119904]119901

= ℎ119901[119862

119873+2

119904

]

[119863119905119903119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119904

] [1198856] 119889119911

[119863119903119903119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198856]119879

[119862119873+2

119887

] [1198856] 119889119911

[119863119905119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887119904

] 119889119911

[119863119905119903119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887119904

] [1198856] 119889119911

[119863119903119905119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+2

119887119904

] 119889119911

[119863119903119903119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+3

119887119904

] [1198856] 119889119911

119863119905119887119901

= int

ℎ119873+3

ℎ119873+2

minus119890119887

ℎ119901

119889119911

119863119903119887119901

= int

ℎ119873+3

ℎ119873+2

minus[119885

3]119879

119890119887

ℎ119901

119889119911

119863119905119904119901

= int

ℎ119873+3

ℎ119873+2

minus119890119904

ℎ119901

119889119911

119863119903119904119901

= int

ℎ119873+3

ℎ119873+2

minus[119885

6]119879

119890119904

ℎ119901

119889119911

(26)

8 Journal of Composites

Applying the dynamic version of the virtual work principle[32] the following open-loop equations of motion for anelement of the overall plate integrated with the ACLDtreatment are obtained

[119872119890

] 119889119890

119905

+ [119870119890

119905119905

] 119889119890

119905

+ [119870119890

119905119903

] 119889119890

119903

= 119865119890

119905119901

119881 + 119865119890

[119870119890

119905119903

]119879

119889119890

119905

+ [119870119890

119903119903

] 119889119890

119903

= 119865119890

119903119901

119881

(27)

Since the elastic constant matrix of the viscoelastic layer iscomplex the stiffness matrices of an element integrated withthe ACLD treatment are complex In case of an elementnot integrated with the ACLD treatment the electro-elasticcoupling matrices become null vectors and the elementalstiffnessmatrices are to be computed without considering thepiezoelectric and viscoelastic layers It should also be notedthat as the stiffness matrices associated with the transverseshear strains are derived separately one can employ theselective integration rule in a straight forward manner toavoid shear locking problem in case of thin plates Therestrained boundary conditions at the skew edges of theplate are such that the displacements of the points locatedat the skew edges are to be restrained along 119909

1015840- 1199101015840- and

1199111015840-directions (Figure 1(a)) Hence in order to impose theboundary conditions in a straight forward manner thegeneralized displacement vectors of a point lying on the skewedge are to be transformed as follows

119889119905 = [119871

119905] 119889

1015840

119905

119889119903 = [119871

119903] 119889

1015840

119903

(28)

where 1198891015840

119905

and 1198891015840

119903

are the generalized displacement vectorsof the point with respect to 119909

1015840

1199101015840

1199111015840 coordinate system and are

given by

1198891015840

119905

= [1199061015840

119900

V1015840119900

1199081015840

119900

]119879

1198891015840

119903

= [1205791015840

119909

1205791015840

119910

1205791015840

119911

1206011015840

119909

1206011015840

119910

1206011015840

119911

1205741015840

119911

1205741015840

119910

1205741015840

119911]119879

(29)

Also the transformation matrices [119871119905] and [119871

119903] are given by

[119871119905] = [

[

119888 119904 0

minus119904 119888 0

0 0 1

]

]

[119871119903] =

[[[[[[[[[[[[

[

119888 119904 0 0 0 0 0 0 0

minus119904 119888 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 119888 119904 0 0 0 0

0 0 0 minus119904 119888 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 119888 119904 0

0 0 0 0 0 0 minus119904 119888 0

0 0 0 0 0 0 0 0 1

]]]]]]]]]]]]

]

(30)

in which 119888 = cos120572 and 119904 = sin120572Thus the elementalmatricesof the element containing the nodes lying on the skew edgeare to be augmented as follows

[119870119890

119905119905

] = [1198791]119879

[119870119890

119905119905

] [1198791] [119870

119890

119905119903

] = [1198791]119879

[119870119890

119905119903

] [1198792]

[119870119890

119903119903

] = [1198792]119879

[119870119890

119903119903

] [1198792] [119872

119890

] = [1198791]119879

[119872119890

] [1198791]

(31)

The forms of the transformation matrices [1198791] and [119879

2] are

given by

[1198791] =

[[[[[[[[[[[

[

[119871119905] 119900 119900 119900 119900 119900 119900 119900

119900 [119871119905] 119900 119900 119900 119900 119900 119900

119900 119900 [119871119905] 119900 119900 119900 119900 119900

119900 119900 119900 [119871119905] 119900 119900 119900 119900

119900 119900 119900 119900 [119871119905] 119900 119900 119900

119900 119900 119900 119900 119900 [119871119905] 119900 119900

119900 119900 119900 119900 119900 119900 [119871119905] 119900

119900 119900 119900 119900 119900 119900 119900 [119871119905]

]]]]]]]]]]]

]

[1198792] =

[[[[[[[[[[[

[

[119871119903] 119900 119900 119900 119900 119900 119900 119900

119900 [119871119903] 119900 119900 119900 119900 119900 119900

119900 119900 [119871119903] 119900 119900 119900 119900 119900

119900 119900 119900 [119871119903] 119900 119900 119900 119900

119900 119900 119900 119900 [119871119903] 119900 119900 119900

119900 119900 119900 119900 119900 [119871119903] 119900 119900

119900 119900 119900 119900 119900 119900 [119871119903] 119900

119900 119900 119900 119900 119900 119900 119900 [119871119903]

]]]]]]]]]]]

]

(32)

in which 119900 and 119900 are (3 times 3) and (9 times 9) null matricesrespectively It may be noted that in case of a node ofsuch element which does not lie on the skew edge thegeneralized displacement vectors of this node with respect to119909119910119911 coordinate system are not required to be transformedHence thematrices [119871

119905] and [119871

119903] corresponding to suchnode

turn out to be (3times3) and (9times9) identitymatrices respectivelyThe elemental equations of motion are now assembled

to obtain the open-loop global equations of motion of theoverall skew plate as follows

[119872] + [119870119905119905] 119883 + [119870

119905119903] 119883

119903 =

119902

sum

119895=1

119865119895

119905119901

119881119895

+ 119865

[119870119905119903]119879

119883 + [119870119903119903] 119883

119903 =

119902

sum

119895=1

119865119895

119903119901

119881119895

(33)

where [119870119905119905] [119870

119905119903] and [119870

119903119903] are the global stiffness matrices

of the overall skew laminated composite plate [119872] is theglobal mass matrix 119865

119905119901 and 119865

119903119901 are the global electro-

elastic coupling vectors 119883 and 119883119903 are the global nodal

generalized displacement vectors 119865 is the global nodalforce vector 119902 is the number of patches and 119881

119895 is thevoltage applied to the 119895th patch Since the elemental stiffnessmatrices of an element augmented with the ACLD treatmentare complex the global stiffness matrices become complexand the energy dissipation characteristics of the overallplate are attributed to the imaginary part of these matricesHence the global equations of motion as derived previouslyalso represent the modeling of the passive (uncontrolled)constrained layer damping of the substrate plate when theconstraining layer is not subjected to any control voltagefollowing a derivative control law

Journal of Composites 9

3 Closed-Loop Model

In the active control approach the active constraining layerof each patch is activated with a control voltage negativelyproportional to the transverse velocity of a point Thus thecontrol voltage for each patch can be expressed in terms of thederivatives of the global nodal degrees of freedom as follows

119881119895

= minus119896119895

119889

= minus119896119895

119889

[119880119895

119905

] minus 119896119895

119889

(ℎ

2) [119880

119895

119903

] 119903 (34)

where [119880119895

119905

] and [119880119895

119903

] are the unit vectors describing thelocation of the velocity sensor for the 119895th patch and 119896

119895

119889

isthe control gain for the 119895th patch Substituting (34) into(33) the final equations of motion governing the closed-loop dynamics of the overall skew laminated compositeplateACLD system can be obtained as follows

[119872] + [119870119905119905] 119883 + [119870

119905119903] 119883

119903 +

119898

sum

119895=1

119896119895

119889

119865119895

119905119901

[119880119895

119905

]

+

119898

sum

119895=1

119896119895

119889

(ℎ

2) 119865

119895

119905119901

[119880119895

119903

] 119903 = 119865

[119870119905119903]119879

119883 + [119870119903119903] 119883

119903 +

119898

sum

119895=1

119896119895

119889

119865119895

119905119901

[119880119895

119905

]

+

119898

sum

119895=1

119896119895

119889

(ℎ

2) 119865

119895

119905119901

[119880119895

119903

] 119903 = 0

(35)

4 Results and Discussion

In this section the numerical results obtained by the finiteelement model derived in the previous section have beenpresented Symmetric as well as antisymmetric cross-plyand antisymmetric angle-ply laminated skew substrate platesintegrated with the two patches of ACLD treatment areconsidered for presenting the numerical results As shown inFigure 1 the locations of the patches have been selected insuch a way that the energy dissipation corresponding to thefirst fewmodes becomesmaximum PZT-5Hspur compositewith 60 fiber volume fraction has been considered for thematerial of the constraining layer of the ACLD treatmentUsing themicromechanicsmodel and thematerial propertiesof the constituent phases [35] the effective elastic andpiezoelectric material properties of the 1ndash3 PZC with respectto the principle material coordinate system are computed asfollows

11986211

= 929GPa 11986212

= 618GPa 11986213

= 605GPa

11986233

= 3544GPa 11986223

= 11986213 119862

44= 158GPa

11986266

= 154GPa 11986255

= 11986244

11989031

= minus01902Cm2

11989032

= 11989031

11989033

= 184107Cm2

11989024

= 0004Cm2

11989015

= 11989024

(36)

The material of the orthotropic layers of the substrate platesis a graphiteepoxy composite and its material properties are[42]

119864119871= 172GPa 119864

119871

119864119879

= 25 119866119871119879

= 05119864119879

119866119879119879

= 02119864119879 ]

119871119879= ]

119879119879= 025

(37)

in which the symbols have the usual meaning Unless oth-erwise mentioned the aspect ratio (119886ℎ) and the thicknessof the skew substrate plates are considered as 100 and0003m respectively while the orthotropic layers of theskew substrate plates are of equal thickness Also the lengthand the skew sides of the substrate are equal (ie 119886 =

119887) The loss factor of the viscoelastic layer considered fornumerical results remains invariant [20] within a broadband (0ndash400Hz) of frequency range and the values of thecomplex shear modulus Poisson ratio and the density ofthe viscoelastic layer are considered as 20(1 + 119894)MNmminus2049 and 1140 kgmminus3 respectively [20] The thicknesses ofthe viscoelastic layer and the 1ndash3 PZC layer are consideredas 508120583m and 250 120583m respectively The value of the shearcorrection factor is used as 56 Three points of Gaussianintegration rule are considered for computing the elementmatrices corresponding to the bending deformations whiletwo points of that are used for computing the elementmatrices corresponding to transverse shear deformationsThe simply supported boundary conditions at the edges of theoverall plate are invoked as follows

at 119909 = 119910 tan120572 119909 = 119886 + 119910 tan120572

V1015840119900

= 1199081015840

119900

= 1205791015840

119910

= 1206011015840

119910

= 1205741015840

119910

= 1205791015840

119911

= 1205741015840

119911

= 0

at 119910 = 0 119910 = 119887 cos120572

119906119900= 119908

119900= 120579

119909= 120601

119909= 120574

119909= 120579

119911= 120574

119911= 0

(38)

To verify the validity of the present finite elementmodel (FEM) the natural frequencies of the skew laminatedcomposite plates integrated with the inactivated patches ofnegligible thickness are first computed and subsequentlycompared with the existing analytical results [41] of theidentical plates without being integrated with the patchesSuch comparisons for the simply supported and clamped-clamped skew laminated plates are presented in Tables 1 and2 respectively It may be observed from these tables that theresults are in excellent agreement validating the model of theplate derived here

In order to assess the performance of the verti-callyobliquely reinforced 1ndash3 PZC as the material of thedistributed actuators of smart skew laminated compositeplates frequency responses for ACLD of the substrate skewlaminated composite plates are investigated To compute the

10 Journal of Composites

Table 1 Comparison of fundamental natural frequencies (120596) of simply supported skew laminated composite plates integratedwith the patchesof negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 15699 28917 37325 14829 24656 32522 17974 33351 33351Present FEM 15635 24383 35033 15076 24380 32254 18493 33359 33370

15∘ Reference [41] 16874 30458 39600 15741 25351 30270 18313 32490 36724Present FEM 16571 29840 36505 15796 25775 29892 18675 32075 35810

30∘ Reference [41] 20840 34023 46997 18871 29372 34489 20270 34431 42361Present FEM 19596 31690 46796 18226 29585 32357 19894 32365 43208

45∘ Reference [41] 28925 41906 54149 25609 33126 40617 25609 33131 42772Present FEM 24811 44875 53289 22996 34773 44889 23194 34870 45009

lowast

120596 = (120596119886

2

120587

2

ℎ)radic120588119864119879 120596 is the circular natural frequency

Table 2 Comparison of fundamental natural frequencies (120596) of clamped-clamped skew laminated composite plates integrated with thepatches of negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 23687 35399 41122 22990 37880 37880 22119 37339 37339Present FEM 23201 34769 44102 23315 36531 36545 22433 36000 36012

15∘ Reference [41] 24663 36255 43418 23809 37516 40785 23099 36997 40438Present FEM 23699 34821 44049 23741 35856 38401 23049 35346 38092

30∘ Reference [41] 27921 39557 50220 26666 39851 47227 26325 39549 52107Present FEM 25366 35696 44734 25240 41943 45373 24945 36113 50932

45∘ Reference [41] 34739 47129 58789 33015 46290 58423 33015 46290 58423Present FEM 28665 47074 54562 28377 47614 56620 28377 47614 55162

frequency response functions the skew laminated compositeplates are harmonically excited by a force of amplitude 1Newton applied to a point (1198862 (1198864) cos120572 ℎ2) on the skewsubstrate platesThe control voltage supplied to the constrain-ing layer of the first patch is negatively proportional to thevelocity of the point (1198862 (1198864) cos120572 ℎ2) and that applied tothe constraining layer of the second patch is negatively pro-portional to the velocity of the point (1198862 (31198864) cos120572 ℎ2)

Figures 3 4 and 5 illustrate the frequency responsefunctions for the transverse displacement of a point of asimply supported symmetric cross-ply (0∘90∘0∘) plate with15∘ 30∘ and 45∘ skew angles respectively while the fiberorientation angle (120595) in the constraining layer is 0∘ Thesefigures display both uncontrolled and controlled responsesand clearly show that the active constraining layer made ofthe vertically reinforced 1ndash3 PZC material being studied heresignificantly attenuates the amplitude of vibrations enhanc-ing the damping characteristics of the overall laminatedskew composite plates over the passive damping (uncon-trolled) Similar responses are also obtained for antisym-metric cross-ply (0∘90∘0∘90∘) and antisymmetric angle-ply (45∘minus45∘45∘minus45∘) skew laminated composite plates

However for brevity these results are not presented hereThemaximum values of the control voltages required to computethe controlled responses presented in Figures 3ndash5 have beenfound to be quite low and are illustrated in Figure 6 in caseof the symmetric cross-ply (0∘90∘0∘) plate with skew angle15∘ only In order to examine the contribution of the verticalactuation in improving the damping characteristics of theskew laminated composite plates as demonstrated in Figures3ndash5 active control responses of the symmetric cross-plylaminated composite plate with 15∘ skew angle are plotted inFigure 7with andwithout considering the value of 119890

33and 119890

31

It may be noted that when the values of 11989031is zero and that of

11989033is nonzero the vertical actuation of the active constraining

layer of the ACLD treatment is responsible for increasing thetransverse shear deformations of the viscoelastic constrainedlayer over the passive counterpart resulting in improveddamping of the smart plate over its passive damping Onthe other hand if 119890

33is zero and 119890

31is nonzero the in-

plane actuation of the active constraining layer causes thetransverse shear deformation of the viscoelastic core of theACLD treatment leading to the active damping of the smartplate It is evident from Figure 7 that the contribution of the

Journal of Composites 11

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 3 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 4 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 30∘)

vertical actuation by the constraining vertically reinforced 1ndash3 PZC layer is significantly larger than that of the in-planeactuation by the same for controlling the modes displayedin Figure 7 Though not shown here similar results are alsoobtained for antisymmetric cross-ply and angle-ply skewplates

Figures 8 9 and 10 demonstrate the effect of variationof piezoelectric fiber orientation angle on the performanceof the patches for improving the damping characteristicsof simply supported symmetric cross-ply (0∘90∘0∘)

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 5 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 45∘)

0

5

10

15

20

25

30

35

40

45

50

Con

trol v

olta

ge (V

olt)

0 100 200 300 400 500 600Frequency (Hz)

119870119889 = 600

119870119889 = 1200

Figure 6 Frequency response functions for the control voltages foractive damping of the simply supported skew symmetric cross-ply(0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

antisymmetric cross-ply (0∘90∘0∘90∘) and angle-ply(45∘minus45∘45∘minus45∘) substrate skew composite plates with30∘ skew angle respectively while the orientation angle (120595)

of the piezoelectric fibers is varied in the vertical 119909119911-planeThese figures illustrate that the attenuating capability of thepatches becomesmaximumwhen the value of the orientationangle (120595) of the fibers in the 119909119911-plane is 0∘ Similar effectis also obtained when the fiber orientation angle (120595) in theconstraining layer of the ACLD treatment is varied in thevertical 119910119911-plane as illustrated in Figures 11 12 and 13 forthe symmetric cross-ply (0∘90∘0∘) antisymmetric cross-ply(0∘90∘0∘90∘) and angle-ply (45∘minus45∘45∘minus45∘) substrate

12 Journal of Composites

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

11989031 = minus01902 11989033 = 184107

11989031 = minus01902 11989033 = 011989031 = 0 11989033 = 184107

w(a2(b4)co

s120572h2

) (m

)

Figure 7 Frequency response functions demonstrating the con-tribution of vertical actuation on the controlled responses forthe transverse displacement 119908 (1198862 (1198864) cos120572 ℎ2) of the skewsymmetric cross-ply (0∘90∘0∘) plate (119870

119889

= 600 119886ℎ = 100 120595 = 0∘

120572 = 15∘)

0 100 200 300 400 500 600Frequency (Hz)

0

02

04

06

08

1

12times10minus4

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 8 Effect of variation of piezoelectric fiber orientation angle(120595) on the controlled responses of a simply supported symmetricskew cross-ply (0∘90∘0∘) plate when the piezoelectric fibers of theconstraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

skew composite plates with 30∘ skew angle respectivelyIn this regard it should be noted that for investigating theeffect of variation of piezoelectric fiber orientation angle(120595) on the performance of the ACLD patches the valueof 120595 has been smoothly varied to compute the controlledresponses However for the sake of clarity in the plots theresponses corresponding to the four specific values of 120595

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 9 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 10 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

have been presented such that the optimum performanceof the patches can be evaluated The significant effect of thevariation of the piezoelectric fiber orientation angle in thevertical 119909119911- and 119910119911-planes has been noticed when all theboundaries of the overall plates are clamped as shown inFigures 14ndash19 When the piezoelectric fibers are coplanarwith the vertical 119909119911-plane the frequency response functionsplotted in Figures 14 15 and 16 reveal that the control

Journal of Composites 13

0

02

04

06

08

1

12

14

16times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 11 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 12 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=600 119886ℎ = 100 120572 = 30

∘)

authority of the patches becomes maximum for improvingthe damping characteristics of the clamped-clamped skewcross-ply and angle-ply substrate plates if the value of thepiezoelectric fiber orientation angle (120595) is 30∘ The same isalso found to be true if the piezoelectric fibers are coplanarwith the vertical 119910119911-plane as shown in Figures 17 18 and 19for the previously mentioned skew plates Similar responsesare also found for the effect of piezoelectric fiber orientation

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 13 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers of the constraining layer are coplanar with the119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 14 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600119886ℎ = 100120572 = 30

∘)

angle on the active damping of other skew laminated plateswith different skew angles However for the sake of brevitythey are not presented here

5 Conclusions

In this paper a study has been carried out to investi-gate the performance of the ACLD treatment for active

14 Journal of Composites

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 15 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 16 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

damping of smart skew laminated composite plates whenverticallyobliquely reinforced 1ndash3 PZC materials are usedas the materials for the constraining layer of the ACLDtreatment A layerwise FSDT-based finite element modelhas been developed to describe the dynamics of the skewlaminated composite plates integrated with the patches ofACLD treatment Unlike the existing finite element modelsof smart structures integrated with ACLD treatment the

200 250 300 350 400 450 500 550 600 650 7000

02

04

06

08

1

12

14

16

18

2times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 17 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

200 300 400 500 600 700 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 18 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

derivation of the present finite element model includes thetransverse deformations of the substrate skew laminatedcomposite plate the constrained viscoelastic layer and theconstraining 1ndash3 PZC layer of the ACLD treatment along thethickness (ie 119911) direction such that both vertical and in-plane actuations by the constraining layer of the patches canbe utilized for active damping of the plates The frequency

Journal of Composites 15

0

05

1

15

2

25

3

35

4times10minus5

200 300 400 500 600 700 800Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 19 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

responses of the symmetric cross-ply antisymmetric cross-ply and angle-ply skew composite plates indicate that theactive constraining layer of the ACLD treatment being madeof the vertically reinforced 1ndash3 PZC material significantlyenhances the damping characteristics of the plates over thepassive damping The analysis revealed that if the verticallyreinforced 1ndash3 PZCmaterial is used for the constraining layerof the ACLD treatment then the contribution of the verticalactuation by the constraining layer alone for improving theactive damping characteristics of the smart skew laminatedcomposite plate is significantly larger than that due to thein-plane actuation by the constraining layer alone It isimportant to note from the present investigation that thevariation of the orientation angle of the piezoelectric fibersof the constraining layer in the vertical 119909119911- and 119910119911-planesas well as the skew angle and boundary conditions of theoverall plates significantly affects the performance of thepatches For the simply supported laminated substrate platesthe performance of the patches becomes maximumwhen thepiezoelectric fiber orientation angle (120595) is 0∘ while in caseof the clamped-clamped laminated skew substrate plates themaximum control authority of the patches is achieved whenthe value of 120595 is 30∘ irrespective of the cases in which thefibers are oriented in 119909119911- and 119910119911-planes

Appendix

In (6) and (7) the matrices [1198851] [119885

2] [119885

3] [119885

4] [119885

5] and

[1198856] are given by

[1198851] = [[119885

1] 0 0]

[1198852] = [(

ℎ

2) 119868 [119885

2] 0]

[1198853] = [(

ℎ

2) 119868 ℎV119868 [119885

3]]

[1198854] = [119868 0 0 119911119868 0 0]

[1198855] = [0 119868 0 (

ℎ

2) 119868 (119911 minus

ℎ

2) 119868 0]

[1198856] = [0 0 119868 (

ℎ

2) 119868 ℎV119868 (119911 minus ℎ

119873+2) 119868]

(A1)

in which

[1198851] =

[[[

[

119911 0 0 0

0 119911 0 0

0 0 119911 0

0 0 0 1

]]]

]

[1198852] =

[[[[[[[[

[

(119911 minusℎ

2) 0 0 0

0 (119911 minusℎ

2) 0 0

0 0 (119911 minusℎ

2) 0

0 0 0 1

]]]]]]]]

]

[1198853] =

[[[

[

(119911 minus ℎ119873+2

) 0 0 0

0 (119911 minus ℎ119873+2

) 0 0

0 0 (119911 minus ℎ119873+2

) 0

0 0 0 1

]]]

]

[119868] =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

[119868] = [1 0

0 1]

[0] = [0 0

0 0] [0] = [

0 0

0 0]

(A2)

The various submatrices 119861119905119887119894 119861

119903119887119894 and 119861

119903119904119894 appearing in (21)

are given by

119861119905119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 0

]]]]]]]]]]]

]

119861119905119904119894

=

[[[[

[

0 0120597120578

119894

120597119909

0 0120597120578

119894

120597119910

]]]]

]

16 Journal of Composites

119861119903119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 1

]]]]]]]]]]]

]

119861119903119887119894

=

[[[[[[

[

119861119903119887119894

0

0

0 119861119903119887119894

0

0

0 119861119903119887119894

]]]]]]

]

119861119903119904119894

=

[[[[[[[[[[[[[[[[[[

[

119868 0 0

0

119868 0

0 0

119868

119861119903119904119894

0 0

0 119861119903119904119894

0

0 0 119861119903119904119894

]]]]]]]]]]]]]]]]]]

]

119868= [1 0 0

0 1 0]

(A3)

wherein

0 and 0 are the (3 times 3) and (2 times 3) null matricesrespectively

References

[1] T Bailey and J E Hubbard ldquoDistributedpiezoelectric polymeractive vibration control of a cantilever beamrdquo Journal of Guid-ance Control and Dynamics vol 8 no 5 pp 605ndash611 1985

[2] E F Crawley and J de Luis ldquoUse of piezoelectric actuators aselements of intelligent structuresrdquo AIAA Journal vol 25 no 10pp 1373ndash1385 1987

[3] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988

[4] C K Lee W W Chiang and O Sulivan ldquoPiezoelectric modalsensoractuator pairs for critical active damping vibrationcontrolrdquo Journal of the Acoustical Society of America vol 90 no1 pp 374ndash384 1991

[5] S Hanagud M W Obal and A J Calise ldquoOptimal vibrationcontrol by the use of piezoceramic sensors and actuatorsrdquoJournal of Guidance Control and Dynamics vol 15 no 5 pp1199ndash1206 1992

[6] Y Gu R L Clark C R Fuller and A C Zander ldquoExperimentson active control of plate vibration using piezoelectric actuatorsand polyvinylidene fluoride (PVDF) modal sensorsrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 116 no3 pp 303ndash308 1994

[7] A Baz and S Poh ldquoOptimal vibration control with modalpositive position feedbackrdquo Optimal Control Applications andMethods vol 17 no 2 pp 141ndash149 1996

[8] A Kugi K Schlacher and H Irschik ldquoInfinite-dimensionalcontrol of nonlinear beam vibrations by piezoelectric actuatorand sensor layersrdquo Nonlinear Dynamics vol 19 no 1 pp 71ndash911999

[9] D Sun L Tong and D Wang ldquoVibration control of platesusing discretely distributed piezoelectric quasi-modal actua-torssensorsrdquo AIAA Journal vol 39 no 9 pp 1766ndash1772 2001

[10] G Caruso S Galeani and L Menini ldquoActive vibration controlof an elastic plate using multiple piezoelectric sensors andactuatorsrdquo SimulationModelling Practice andTheory vol 11 no5-6 pp 403ndash419 2003

[11] X Zhang and A G Erdman ldquoOptimal placement of piezo-electric sensors and actuators for controlled flexible linkagemechanismsrdquo Journal of Vibration and Acoustics Transactionsof the ASME vol 128 no 2 pp 256ndash260 2006

[12] G Gatti M J Brennan and P Gardonio ldquoActive dampingof a beam using a physically collocated accelerometer andpiezoelectric patch actuatorrdquo Journal of Sound and Vibrationvol 303 no 3ndash5 pp 798ndash813 2007

[13] M K Kwak S Heo and M Jeong ldquoDynamic modellingand active vibration controller design for a cylindrical shellequipped with piezoelectric sensors and actuatorsrdquo Journal ofSound and Vibration vol 321 no 3ndash5 pp 510ndash524 2009

[14] I Bruant L Gallimard and S Nikoukar ldquoOptimal piezoelectricactuator and sensor location for active vibration control usinggenetic algorithmrdquo Journal of Sound and Vibration vol 329 no10 pp 1615ndash1635 2010

[15] Y Fu J Wang and Y Mao ldquoNonlinear vibration and activecontrol of functionally graded beams with piezoelectric sensorsand actuatorsrdquo Journal of Intelligent Material Systems andStructures vol 22 no 18 pp 2093ndash2102 2011

[16] F Chen M Hong M Song and H Cui ldquoOptimal control ofa beam with discontinuously distributed piezoelectric sensorsand actuatorsrdquo Journal of Marine Science and Application vol11 no 1 pp 44ndash51 2012

[17] B Azvine G R Tomlinson and R J Wynne ldquoUse of activeconstrained-layer damping for controlling resonant vibrationrdquoSmart Materials and Structures vol 4 no 1 pp 1ndash6 1995

[18] A Baz and J Ro ldquoVibration control of plates with activeconstrained layer dampingrdquo SmartMaterials and Structures vol5 no 3 pp 272ndash280 1996

[19] C Chantalakhana and R Stanway ldquoActive constrained layerdamping of clamped-clamped plate vibrationsrdquo Journal ofSound and Vibration vol 241 no 5 pp 755ndash777 2001

[20] J Ro and A Baz ldquoOptimum placement and control ofactive constrained layer damping using modal strain energyapproachrdquo JVCJournal of Vibration and Control vol 8 no 6pp 861ndash876 2002

[21] M C Ray and J N Reddy ldquoOptimal control of thin circularcylindrical laminated composite shells using active constrainedlayer damping treatmentrdquo Smart Materials and Structures vol13 no 1 pp 64ndash72 2004

[22] C M A Vasques B R Mace P Gardonio and J D RodriguesldquoArbitrary active constrained layer damping treatments onbeams finite element modelling and experimental validationrdquoComputers and Structures vol 84 no 22-23 pp 1384ndash14012006

[23] F M Li K Kishimoto K Wang Z B Chen and W HHuang ldquoVibration control of beams with active constrainedlayer dampingrdquo Smart Materials and Structures vol 17 no 6Article ID 065036 2008

[24] L Yuan Y Xiang Y Huang and J Lu ldquoA semi-analyticalmethod and the circumferential dominant modal control ofcircular cylindrical shells with active constrained layer dampingtreatmentrdquo SmartMaterials and Structures vol 19 no 2 ArticleID 025010 2010

[25] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

Journal of Composites 17

[26] W A Smith and B A Auld ldquoModeling 1ndash3 composite piezo-electrics thickness-mode oscillationsrdquo IEEE Transactions onUltrasonics Ferroelectrics and Frequency Control vol 38 no 1pp 40ndash47 1991

[27] PiezocompositesMaterial Systems Inc 543 Great Road Little-ton MA 01460

[28] Y Benveniste and G J Dvorak ldquoUniform fields and universalrelations in piezoelectric compositesrdquo Journal of the Mechanicsand Physics of Solids vol 40 no 6 pp 1295ndash1312 1992

[29] M L Dunn and M Taya ldquoMicromechanics predictions ofthe effective electroelastic moduli of piezoelectric compositesrdquoInternational Journal of Solids and Structures vol 30 no 2 pp161ndash175 1993

[30] J Aboudi ldquoMicromechanical prediction of the effective coeffi-cients of thermo-piezoelectric multiphase compositesrdquo Journalof Intelligent Material Systems and Structures vol 9 no 9 pp713ndash722 1999

[31] X Ruan T W Chou A Safari and S C Danforth ldquoA 3-Dconnectivity model for effective piezoelectric properties of yarncompositesrdquo Journal of Composite Materials vol 36 no 14 pp1693ndash1708 2002

[32] A Baz and A Tempia ldquoActive piezoelectric damping compos-itesrdquo Sensors and Actuators A vol 112 no 2-3 pp 340ndash3502004

[33] H Berger S Kari U Gabbert et al ldquoUnit cell models ofpiezoelectric fiber composites for numerical and analytical cal-culation of effective propertiesrdquo SmartMaterials and Structuresvol 15 no 2 pp 451ndash458 2006

[34] M Sakthivel and A Arockiarajan ldquoAn analytical model forpredicting thermo-electro-mechanical response of 1ndash3 piezo-electric compositesrdquo Computational Materials Science vol 48no 4 pp 759ndash767 2010

[35] M C Ray and A K Pradhan ldquoOn the use of verticallyreinforced 1ndash3 piezoelectric composites for hybrid damping oflaminated composite platesrdquo Mechanics of Advanced Materialsand Structures vol 14 no 4 pp 245ndash261 2007

[36] M C Ray and A K Pradhan ldquoPerformance of vertically andobliquely reinforced 1ndash3 piezoelectric composites for activedamping of laminated composite shellsrdquo Journal of Sound andVibration vol 315 no 4-5 pp 816ndash835 2008

[37] S K Sarangi and M C Ray ldquoSmart damping of geometricallynonlinear vibrations of laminated composite beams using verti-cally reinforced 1ndash3 piezoelectric compositesrdquo Smart Materialsand Structures vol 19 no 7 Article ID 075020 2010

[38] A Krishnan and J V Deshpande ldquoVibration of skew laminatesrdquoJournal of Sound and Vibration vol 153 no 2 pp 351ndash358 1992

[39] A R Krishna Reddy and R Palaninathan ldquoFree vibration ofskew laminatesrdquo Computers and Structures vol 70 no 4 pp415ndash423 1999

[40] C S Babu and T Kant ldquoTwo shear deformable finite elementmodels for buckling analysis of skew fibre-reinforced compositeand sandwich panelsrdquo Composite Structures vol 46 no 2 pp115ndash124 1999

[41] A K Garg R K Khare and T Kant ldquoFree vibration of skewfiber-reinforced composite and sandwich laminates using ashear deformable finite element modelrdquo Journal of SandwichStructures and Materials vol 8 no 1 pp 33ndash53 2006

[42] J N Reddy Mechanics of Laminated Composites Plates Theoryand Analysis CRC press Boca Raton Fla USA 1996

Submit your manuscripts athttpwwwhindawicom

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materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 7: Research Article Active Constrained Layer Damping of Smart …downloads.hindawi.com/archive/2013/824163.pdf · 2019-07-31 · Research Article Active Constrained Layer Damping of

Journal of Composites 7

[119870119890

119905119903119887119904

]119901119887

= int119860

[119861119905119887]119879

[119863119905119903119887119904

]119901

[119861119903119904] 119889119909 119889119910

[119870119890

119903119905119887119904

]119901119887

= int119860

[119861119903119887]119879

[119863119903119905119887119904

]119901

[119861119905119904] 119889119909 119889119910

[119870119890

119905119903119887119904

]119901119904

= int119860

[119861119905119904]119879

[119863119903119905119887119904

]119901

[119861119903119887] 119889119909 119889119910

[119870119890

119903119905119887119904

]119901119904

= int119860

[119861119903119904]119879

[119863119905119903119887119904

]119901

[119861119905119887] 119889119909 119889119910

[119870119890

119905119903119904

] = int119860

[119861119905119903119904

]119879

([119863119905119903119904

] + [119863119905119903119904

]V + [119863119905119903119904

]119901

)

times [119861119905119903119904

] 119889119909 119889119910

[119870119890

119903119903119887

] = int119860

[119861119903119903119887

]119879

([119863119903119903119887

] + [119863119903119903119887

]V + [119863119903119903119887

]119901

)

times [119861119903119903119887

] 119889119909 119889119910

[119870119890

119903119903119904

] = int119860

[119861119903119903119904

]119879

([119863119903119903119904

] + [119863119903119903119904

]V + [119863119903119903119904

]119901

)

times [119861119903119903119904

] 119889119909 119889119910

[119870119890

119903119903119887119904

]119901119887

= int119860

[119861119903119887]119879

[119863119903119903119887119904

]119901

[119861119903119904] 119889119909 119889119910

[119870119890

119903119903119887119904

]119901119904

= int119860

[119861119903119904]119879

[119863119903119903119887119904

]119901

[119861119903119887] 119889119909 119889119910

119865119890

119905119887

119901

= int119860

[119861119905119887]119879

119863119905119887119901

119889119909 119889119910

119865119890

119905119904

119901

= int119860

[119861119905119904]119879

119863119905119904119901

119889119909 119889119910

119865119890

119903119904

119901

= int119860

[119861119903119904]119879

119863119903119904119901

119889119909 119889119910

(25)

The various rigidity matrices and the rigidity vectors forelectro-elastic coupling appearing in the previous elementalmatrices are given by

[119863119905119887] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] 119889119911

[119863119905119903119887

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] [1198851] 119889119911

[119863119903119903119887

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[1198851]119879

[119862119896

119887

] [1198851] 119889119911

[119863119905119904] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119904

] 119889119911

[119863119905119903119904

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[119862119896

119887

] [1198854] 119889119911

[119863119903119903119904

] =

119873

sum

119896=1

int

ℎ119896+1

ℎ119896

[1198854]119879

[119862119896

119887

] [1198854] 119889119911

[119863119905119887]V = ℎV [119862

119873+1

119887

]

[119863119905119903119887

]V = int

ℎ119873+2

ℎ119873+1

[119862119873+1

119887

] [1198852] 119889119911

[119863119903119903119887

]V = int

ℎ119873+2

ℎ119873+1

[1198852]119879

[119862119873+1

119887

] [1198852] 119889119911

[119863119905119904]V = ℎV [119862

119873+1

119904

]

[119863119905119903119904

]V = int

ℎ119873+2

ℎ119873+1

[119862119873+1

119904

] [1198855] 119889119911

[119863119903119903119904

]V = int

ℎ119873+2

ℎ119873+1

[1198855]119879

[119862119873+1

119887

] [1198855] 119889119911

[119863119905119887]119901

= ℎ119901[119862

119873+2

119887

]

[119863119905119903119887

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887

] [1198853] 119889119911

[119863119903119903119887

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+2

119887

] [1198853] 119889119911

[119863119905119904]119901

= ℎ119901[119862

119873+2

119904

]

[119863119905119903119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119904

] [1198856] 119889119911

[119863119903119903119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198856]119879

[119862119873+2

119887

] [1198856] 119889119911

[119863119905119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887119904

] 119889119911

[119863119905119903119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[119862119873+2

119887119904

] [1198856] 119889119911

[119863119903119905119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+2

119887119904

] 119889119911

[119863119903119903119887119904

]119901

= int

ℎ119873+3

ℎ119873+2

[1198853]119879

[119862119873+3

119887119904

] [1198856] 119889119911

119863119905119887119901

= int

ℎ119873+3

ℎ119873+2

minus119890119887

ℎ119901

119889119911

119863119903119887119901

= int

ℎ119873+3

ℎ119873+2

minus[119885

3]119879

119890119887

ℎ119901

119889119911

119863119905119904119901

= int

ℎ119873+3

ℎ119873+2

minus119890119904

ℎ119901

119889119911

119863119903119904119901

= int

ℎ119873+3

ℎ119873+2

minus[119885

6]119879

119890119904

ℎ119901

119889119911

(26)

8 Journal of Composites

Applying the dynamic version of the virtual work principle[32] the following open-loop equations of motion for anelement of the overall plate integrated with the ACLDtreatment are obtained

[119872119890

] 119889119890

119905

+ [119870119890

119905119905

] 119889119890

119905

+ [119870119890

119905119903

] 119889119890

119903

= 119865119890

119905119901

119881 + 119865119890

[119870119890

119905119903

]119879

119889119890

119905

+ [119870119890

119903119903

] 119889119890

119903

= 119865119890

119903119901

119881

(27)

Since the elastic constant matrix of the viscoelastic layer iscomplex the stiffness matrices of an element integrated withthe ACLD treatment are complex In case of an elementnot integrated with the ACLD treatment the electro-elasticcoupling matrices become null vectors and the elementalstiffnessmatrices are to be computed without considering thepiezoelectric and viscoelastic layers It should also be notedthat as the stiffness matrices associated with the transverseshear strains are derived separately one can employ theselective integration rule in a straight forward manner toavoid shear locking problem in case of thin plates Therestrained boundary conditions at the skew edges of theplate are such that the displacements of the points locatedat the skew edges are to be restrained along 119909

1015840- 1199101015840- and

1199111015840-directions (Figure 1(a)) Hence in order to impose theboundary conditions in a straight forward manner thegeneralized displacement vectors of a point lying on the skewedge are to be transformed as follows

119889119905 = [119871

119905] 119889

1015840

119905

119889119903 = [119871

119903] 119889

1015840

119903

(28)

where 1198891015840

119905

and 1198891015840

119903

are the generalized displacement vectorsof the point with respect to 119909

1015840

1199101015840

1199111015840 coordinate system and are

given by

1198891015840

119905

= [1199061015840

119900

V1015840119900

1199081015840

119900

]119879

1198891015840

119903

= [1205791015840

119909

1205791015840

119910

1205791015840

119911

1206011015840

119909

1206011015840

119910

1206011015840

119911

1205741015840

119911

1205741015840

119910

1205741015840

119911]119879

(29)

Also the transformation matrices [119871119905] and [119871

119903] are given by

[119871119905] = [

[

119888 119904 0

minus119904 119888 0

0 0 1

]

]

[119871119903] =

[[[[[[[[[[[[

[

119888 119904 0 0 0 0 0 0 0

minus119904 119888 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 119888 119904 0 0 0 0

0 0 0 minus119904 119888 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 119888 119904 0

0 0 0 0 0 0 minus119904 119888 0

0 0 0 0 0 0 0 0 1

]]]]]]]]]]]]

]

(30)

in which 119888 = cos120572 and 119904 = sin120572Thus the elementalmatricesof the element containing the nodes lying on the skew edgeare to be augmented as follows

[119870119890

119905119905

] = [1198791]119879

[119870119890

119905119905

] [1198791] [119870

119890

119905119903

] = [1198791]119879

[119870119890

119905119903

] [1198792]

[119870119890

119903119903

] = [1198792]119879

[119870119890

119903119903

] [1198792] [119872

119890

] = [1198791]119879

[119872119890

] [1198791]

(31)

The forms of the transformation matrices [1198791] and [119879

2] are

given by

[1198791] =

[[[[[[[[[[[

[

[119871119905] 119900 119900 119900 119900 119900 119900 119900

119900 [119871119905] 119900 119900 119900 119900 119900 119900

119900 119900 [119871119905] 119900 119900 119900 119900 119900

119900 119900 119900 [119871119905] 119900 119900 119900 119900

119900 119900 119900 119900 [119871119905] 119900 119900 119900

119900 119900 119900 119900 119900 [119871119905] 119900 119900

119900 119900 119900 119900 119900 119900 [119871119905] 119900

119900 119900 119900 119900 119900 119900 119900 [119871119905]

]]]]]]]]]]]

]

[1198792] =

[[[[[[[[[[[

[

[119871119903] 119900 119900 119900 119900 119900 119900 119900

119900 [119871119903] 119900 119900 119900 119900 119900 119900

119900 119900 [119871119903] 119900 119900 119900 119900 119900

119900 119900 119900 [119871119903] 119900 119900 119900 119900

119900 119900 119900 119900 [119871119903] 119900 119900 119900

119900 119900 119900 119900 119900 [119871119903] 119900 119900

119900 119900 119900 119900 119900 119900 [119871119903] 119900

119900 119900 119900 119900 119900 119900 119900 [119871119903]

]]]]]]]]]]]

]

(32)

in which 119900 and 119900 are (3 times 3) and (9 times 9) null matricesrespectively It may be noted that in case of a node ofsuch element which does not lie on the skew edge thegeneralized displacement vectors of this node with respect to119909119910119911 coordinate system are not required to be transformedHence thematrices [119871

119905] and [119871

119903] corresponding to suchnode

turn out to be (3times3) and (9times9) identitymatrices respectivelyThe elemental equations of motion are now assembled

to obtain the open-loop global equations of motion of theoverall skew plate as follows

[119872] + [119870119905119905] 119883 + [119870

119905119903] 119883

119903 =

119902

sum

119895=1

119865119895

119905119901

119881119895

+ 119865

[119870119905119903]119879

119883 + [119870119903119903] 119883

119903 =

119902

sum

119895=1

119865119895

119903119901

119881119895

(33)

where [119870119905119905] [119870

119905119903] and [119870

119903119903] are the global stiffness matrices

of the overall skew laminated composite plate [119872] is theglobal mass matrix 119865

119905119901 and 119865

119903119901 are the global electro-

elastic coupling vectors 119883 and 119883119903 are the global nodal

generalized displacement vectors 119865 is the global nodalforce vector 119902 is the number of patches and 119881

119895 is thevoltage applied to the 119895th patch Since the elemental stiffnessmatrices of an element augmented with the ACLD treatmentare complex the global stiffness matrices become complexand the energy dissipation characteristics of the overallplate are attributed to the imaginary part of these matricesHence the global equations of motion as derived previouslyalso represent the modeling of the passive (uncontrolled)constrained layer damping of the substrate plate when theconstraining layer is not subjected to any control voltagefollowing a derivative control law

Journal of Composites 9

3 Closed-Loop Model

In the active control approach the active constraining layerof each patch is activated with a control voltage negativelyproportional to the transverse velocity of a point Thus thecontrol voltage for each patch can be expressed in terms of thederivatives of the global nodal degrees of freedom as follows

119881119895

= minus119896119895

119889

= minus119896119895

119889

[119880119895

119905

] minus 119896119895

119889

(ℎ

2) [119880

119895

119903

] 119903 (34)

where [119880119895

119905

] and [119880119895

119903

] are the unit vectors describing thelocation of the velocity sensor for the 119895th patch and 119896

119895

119889

isthe control gain for the 119895th patch Substituting (34) into(33) the final equations of motion governing the closed-loop dynamics of the overall skew laminated compositeplateACLD system can be obtained as follows

[119872] + [119870119905119905] 119883 + [119870

119905119903] 119883

119903 +

119898

sum

119895=1

119896119895

119889

119865119895

119905119901

[119880119895

119905

]

+

119898

sum

119895=1

119896119895

119889

(ℎ

2) 119865

119895

119905119901

[119880119895

119903

] 119903 = 119865

[119870119905119903]119879

119883 + [119870119903119903] 119883

119903 +

119898

sum

119895=1

119896119895

119889

119865119895

119905119901

[119880119895

119905

]

+

119898

sum

119895=1

119896119895

119889

(ℎ

2) 119865

119895

119905119901

[119880119895

119903

] 119903 = 0

(35)

4 Results and Discussion

In this section the numerical results obtained by the finiteelement model derived in the previous section have beenpresented Symmetric as well as antisymmetric cross-plyand antisymmetric angle-ply laminated skew substrate platesintegrated with the two patches of ACLD treatment areconsidered for presenting the numerical results As shown inFigure 1 the locations of the patches have been selected insuch a way that the energy dissipation corresponding to thefirst fewmodes becomesmaximum PZT-5Hspur compositewith 60 fiber volume fraction has been considered for thematerial of the constraining layer of the ACLD treatmentUsing themicromechanicsmodel and thematerial propertiesof the constituent phases [35] the effective elastic andpiezoelectric material properties of the 1ndash3 PZC with respectto the principle material coordinate system are computed asfollows

11986211

= 929GPa 11986212

= 618GPa 11986213

= 605GPa

11986233

= 3544GPa 11986223

= 11986213 119862

44= 158GPa

11986266

= 154GPa 11986255

= 11986244

11989031

= minus01902Cm2

11989032

= 11989031

11989033

= 184107Cm2

11989024

= 0004Cm2

11989015

= 11989024

(36)

The material of the orthotropic layers of the substrate platesis a graphiteepoxy composite and its material properties are[42]

119864119871= 172GPa 119864

119871

119864119879

= 25 119866119871119879

= 05119864119879

119866119879119879

= 02119864119879 ]

119871119879= ]

119879119879= 025

(37)

in which the symbols have the usual meaning Unless oth-erwise mentioned the aspect ratio (119886ℎ) and the thicknessof the skew substrate plates are considered as 100 and0003m respectively while the orthotropic layers of theskew substrate plates are of equal thickness Also the lengthand the skew sides of the substrate are equal (ie 119886 =

119887) The loss factor of the viscoelastic layer considered fornumerical results remains invariant [20] within a broadband (0ndash400Hz) of frequency range and the values of thecomplex shear modulus Poisson ratio and the density ofthe viscoelastic layer are considered as 20(1 + 119894)MNmminus2049 and 1140 kgmminus3 respectively [20] The thicknesses ofthe viscoelastic layer and the 1ndash3 PZC layer are consideredas 508120583m and 250 120583m respectively The value of the shearcorrection factor is used as 56 Three points of Gaussianintegration rule are considered for computing the elementmatrices corresponding to the bending deformations whiletwo points of that are used for computing the elementmatrices corresponding to transverse shear deformationsThe simply supported boundary conditions at the edges of theoverall plate are invoked as follows

at 119909 = 119910 tan120572 119909 = 119886 + 119910 tan120572

V1015840119900

= 1199081015840

119900

= 1205791015840

119910

= 1206011015840

119910

= 1205741015840

119910

= 1205791015840

119911

= 1205741015840

119911

= 0

at 119910 = 0 119910 = 119887 cos120572

119906119900= 119908

119900= 120579

119909= 120601

119909= 120574

119909= 120579

119911= 120574

119911= 0

(38)

To verify the validity of the present finite elementmodel (FEM) the natural frequencies of the skew laminatedcomposite plates integrated with the inactivated patches ofnegligible thickness are first computed and subsequentlycompared with the existing analytical results [41] of theidentical plates without being integrated with the patchesSuch comparisons for the simply supported and clamped-clamped skew laminated plates are presented in Tables 1 and2 respectively It may be observed from these tables that theresults are in excellent agreement validating the model of theplate derived here

In order to assess the performance of the verti-callyobliquely reinforced 1ndash3 PZC as the material of thedistributed actuators of smart skew laminated compositeplates frequency responses for ACLD of the substrate skewlaminated composite plates are investigated To compute the

10 Journal of Composites

Table 1 Comparison of fundamental natural frequencies (120596) of simply supported skew laminated composite plates integratedwith the patchesof negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 15699 28917 37325 14829 24656 32522 17974 33351 33351Present FEM 15635 24383 35033 15076 24380 32254 18493 33359 33370

15∘ Reference [41] 16874 30458 39600 15741 25351 30270 18313 32490 36724Present FEM 16571 29840 36505 15796 25775 29892 18675 32075 35810

30∘ Reference [41] 20840 34023 46997 18871 29372 34489 20270 34431 42361Present FEM 19596 31690 46796 18226 29585 32357 19894 32365 43208

45∘ Reference [41] 28925 41906 54149 25609 33126 40617 25609 33131 42772Present FEM 24811 44875 53289 22996 34773 44889 23194 34870 45009

lowast

120596 = (120596119886

2

120587

2

ℎ)radic120588119864119879 120596 is the circular natural frequency

Table 2 Comparison of fundamental natural frequencies (120596) of clamped-clamped skew laminated composite plates integrated with thepatches of negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 23687 35399 41122 22990 37880 37880 22119 37339 37339Present FEM 23201 34769 44102 23315 36531 36545 22433 36000 36012

15∘ Reference [41] 24663 36255 43418 23809 37516 40785 23099 36997 40438Present FEM 23699 34821 44049 23741 35856 38401 23049 35346 38092

30∘ Reference [41] 27921 39557 50220 26666 39851 47227 26325 39549 52107Present FEM 25366 35696 44734 25240 41943 45373 24945 36113 50932

45∘ Reference [41] 34739 47129 58789 33015 46290 58423 33015 46290 58423Present FEM 28665 47074 54562 28377 47614 56620 28377 47614 55162

frequency response functions the skew laminated compositeplates are harmonically excited by a force of amplitude 1Newton applied to a point (1198862 (1198864) cos120572 ℎ2) on the skewsubstrate platesThe control voltage supplied to the constrain-ing layer of the first patch is negatively proportional to thevelocity of the point (1198862 (1198864) cos120572 ℎ2) and that applied tothe constraining layer of the second patch is negatively pro-portional to the velocity of the point (1198862 (31198864) cos120572 ℎ2)

Figures 3 4 and 5 illustrate the frequency responsefunctions for the transverse displacement of a point of asimply supported symmetric cross-ply (0∘90∘0∘) plate with15∘ 30∘ and 45∘ skew angles respectively while the fiberorientation angle (120595) in the constraining layer is 0∘ Thesefigures display both uncontrolled and controlled responsesand clearly show that the active constraining layer made ofthe vertically reinforced 1ndash3 PZC material being studied heresignificantly attenuates the amplitude of vibrations enhanc-ing the damping characteristics of the overall laminatedskew composite plates over the passive damping (uncon-trolled) Similar responses are also obtained for antisym-metric cross-ply (0∘90∘0∘90∘) and antisymmetric angle-ply (45∘minus45∘45∘minus45∘) skew laminated composite plates

However for brevity these results are not presented hereThemaximum values of the control voltages required to computethe controlled responses presented in Figures 3ndash5 have beenfound to be quite low and are illustrated in Figure 6 in caseof the symmetric cross-ply (0∘90∘0∘) plate with skew angle15∘ only In order to examine the contribution of the verticalactuation in improving the damping characteristics of theskew laminated composite plates as demonstrated in Figures3ndash5 active control responses of the symmetric cross-plylaminated composite plate with 15∘ skew angle are plotted inFigure 7with andwithout considering the value of 119890

33and 119890

31

It may be noted that when the values of 11989031is zero and that of

11989033is nonzero the vertical actuation of the active constraining

layer of the ACLD treatment is responsible for increasing thetransverse shear deformations of the viscoelastic constrainedlayer over the passive counterpart resulting in improveddamping of the smart plate over its passive damping Onthe other hand if 119890

33is zero and 119890

31is nonzero the in-

plane actuation of the active constraining layer causes thetransverse shear deformation of the viscoelastic core of theACLD treatment leading to the active damping of the smartplate It is evident from Figure 7 that the contribution of the

Journal of Composites 11

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 3 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 4 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 30∘)

vertical actuation by the constraining vertically reinforced 1ndash3 PZC layer is significantly larger than that of the in-planeactuation by the same for controlling the modes displayedin Figure 7 Though not shown here similar results are alsoobtained for antisymmetric cross-ply and angle-ply skewplates

Figures 8 9 and 10 demonstrate the effect of variationof piezoelectric fiber orientation angle on the performanceof the patches for improving the damping characteristicsof simply supported symmetric cross-ply (0∘90∘0∘)

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 5 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 45∘)

0

5

10

15

20

25

30

35

40

45

50

Con

trol v

olta

ge (V

olt)

0 100 200 300 400 500 600Frequency (Hz)

119870119889 = 600

119870119889 = 1200

Figure 6 Frequency response functions for the control voltages foractive damping of the simply supported skew symmetric cross-ply(0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

antisymmetric cross-ply (0∘90∘0∘90∘) and angle-ply(45∘minus45∘45∘minus45∘) substrate skew composite plates with30∘ skew angle respectively while the orientation angle (120595)

of the piezoelectric fibers is varied in the vertical 119909119911-planeThese figures illustrate that the attenuating capability of thepatches becomesmaximumwhen the value of the orientationangle (120595) of the fibers in the 119909119911-plane is 0∘ Similar effectis also obtained when the fiber orientation angle (120595) in theconstraining layer of the ACLD treatment is varied in thevertical 119910119911-plane as illustrated in Figures 11 12 and 13 forthe symmetric cross-ply (0∘90∘0∘) antisymmetric cross-ply(0∘90∘0∘90∘) and angle-ply (45∘minus45∘45∘minus45∘) substrate

12 Journal of Composites

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

11989031 = minus01902 11989033 = 184107

11989031 = minus01902 11989033 = 011989031 = 0 11989033 = 184107

w(a2(b4)co

s120572h2

) (m

)

Figure 7 Frequency response functions demonstrating the con-tribution of vertical actuation on the controlled responses forthe transverse displacement 119908 (1198862 (1198864) cos120572 ℎ2) of the skewsymmetric cross-ply (0∘90∘0∘) plate (119870

119889

= 600 119886ℎ = 100 120595 = 0∘

120572 = 15∘)

0 100 200 300 400 500 600Frequency (Hz)

0

02

04

06

08

1

12times10minus4

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 8 Effect of variation of piezoelectric fiber orientation angle(120595) on the controlled responses of a simply supported symmetricskew cross-ply (0∘90∘0∘) plate when the piezoelectric fibers of theconstraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

skew composite plates with 30∘ skew angle respectivelyIn this regard it should be noted that for investigating theeffect of variation of piezoelectric fiber orientation angle(120595) on the performance of the ACLD patches the valueof 120595 has been smoothly varied to compute the controlledresponses However for the sake of clarity in the plots theresponses corresponding to the four specific values of 120595

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 9 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 10 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

have been presented such that the optimum performanceof the patches can be evaluated The significant effect of thevariation of the piezoelectric fiber orientation angle in thevertical 119909119911- and 119910119911-planes has been noticed when all theboundaries of the overall plates are clamped as shown inFigures 14ndash19 When the piezoelectric fibers are coplanarwith the vertical 119909119911-plane the frequency response functionsplotted in Figures 14 15 and 16 reveal that the control

Journal of Composites 13

0

02

04

06

08

1

12

14

16times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 11 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 12 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=600 119886ℎ = 100 120572 = 30

∘)

authority of the patches becomes maximum for improvingthe damping characteristics of the clamped-clamped skewcross-ply and angle-ply substrate plates if the value of thepiezoelectric fiber orientation angle (120595) is 30∘ The same isalso found to be true if the piezoelectric fibers are coplanarwith the vertical 119910119911-plane as shown in Figures 17 18 and 19for the previously mentioned skew plates Similar responsesare also found for the effect of piezoelectric fiber orientation

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 13 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers of the constraining layer are coplanar with the119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 14 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600119886ℎ = 100120572 = 30

∘)

angle on the active damping of other skew laminated plateswith different skew angles However for the sake of brevitythey are not presented here

5 Conclusions

In this paper a study has been carried out to investi-gate the performance of the ACLD treatment for active

14 Journal of Composites

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 15 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 16 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

damping of smart skew laminated composite plates whenverticallyobliquely reinforced 1ndash3 PZC materials are usedas the materials for the constraining layer of the ACLDtreatment A layerwise FSDT-based finite element modelhas been developed to describe the dynamics of the skewlaminated composite plates integrated with the patches ofACLD treatment Unlike the existing finite element modelsof smart structures integrated with ACLD treatment the

200 250 300 350 400 450 500 550 600 650 7000

02

04

06

08

1

12

14

16

18

2times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 17 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

200 300 400 500 600 700 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 18 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

derivation of the present finite element model includes thetransverse deformations of the substrate skew laminatedcomposite plate the constrained viscoelastic layer and theconstraining 1ndash3 PZC layer of the ACLD treatment along thethickness (ie 119911) direction such that both vertical and in-plane actuations by the constraining layer of the patches canbe utilized for active damping of the plates The frequency

Journal of Composites 15

0

05

1

15

2

25

3

35

4times10minus5

200 300 400 500 600 700 800Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 19 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

responses of the symmetric cross-ply antisymmetric cross-ply and angle-ply skew composite plates indicate that theactive constraining layer of the ACLD treatment being madeof the vertically reinforced 1ndash3 PZC material significantlyenhances the damping characteristics of the plates over thepassive damping The analysis revealed that if the verticallyreinforced 1ndash3 PZCmaterial is used for the constraining layerof the ACLD treatment then the contribution of the verticalactuation by the constraining layer alone for improving theactive damping characteristics of the smart skew laminatedcomposite plate is significantly larger than that due to thein-plane actuation by the constraining layer alone It isimportant to note from the present investigation that thevariation of the orientation angle of the piezoelectric fibersof the constraining layer in the vertical 119909119911- and 119910119911-planesas well as the skew angle and boundary conditions of theoverall plates significantly affects the performance of thepatches For the simply supported laminated substrate platesthe performance of the patches becomes maximumwhen thepiezoelectric fiber orientation angle (120595) is 0∘ while in caseof the clamped-clamped laminated skew substrate plates themaximum control authority of the patches is achieved whenthe value of 120595 is 30∘ irrespective of the cases in which thefibers are oriented in 119909119911- and 119910119911-planes

Appendix

In (6) and (7) the matrices [1198851] [119885

2] [119885

3] [119885

4] [119885

5] and

[1198856] are given by

[1198851] = [[119885

1] 0 0]

[1198852] = [(

ℎ

2) 119868 [119885

2] 0]

[1198853] = [(

ℎ

2) 119868 ℎV119868 [119885

3]]

[1198854] = [119868 0 0 119911119868 0 0]

[1198855] = [0 119868 0 (

ℎ

2) 119868 (119911 minus

ℎ

2) 119868 0]

[1198856] = [0 0 119868 (

ℎ

2) 119868 ℎV119868 (119911 minus ℎ

119873+2) 119868]

(A1)

in which

[1198851] =

[[[

[

119911 0 0 0

0 119911 0 0

0 0 119911 0

0 0 0 1

]]]

]

[1198852] =

[[[[[[[[

[

(119911 minusℎ

2) 0 0 0

0 (119911 minusℎ

2) 0 0

0 0 (119911 minusℎ

2) 0

0 0 0 1

]]]]]]]]

]

[1198853] =

[[[

[

(119911 minus ℎ119873+2

) 0 0 0

0 (119911 minus ℎ119873+2

) 0 0

0 0 (119911 minus ℎ119873+2

) 0

0 0 0 1

]]]

]

[119868] =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

[119868] = [1 0

0 1]

[0] = [0 0

0 0] [0] = [

0 0

0 0]

(A2)

The various submatrices 119861119905119887119894 119861

119903119887119894 and 119861

119903119904119894 appearing in (21)

are given by

119861119905119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 0

]]]]]]]]]]]

]

119861119905119904119894

=

[[[[

[

0 0120597120578

119894

120597119909

0 0120597120578

119894

120597119910

]]]]

]

16 Journal of Composites

119861119903119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 1

]]]]]]]]]]]

]

119861119903119887119894

=

[[[[[[

[

119861119903119887119894

0

0

0 119861119903119887119894

0

0

0 119861119903119887119894

]]]]]]

]

119861119903119904119894

=

[[[[[[[[[[[[[[[[[[

[

119868 0 0

0

119868 0

0 0

119868

119861119903119904119894

0 0

0 119861119903119904119894

0

0 0 119861119903119904119894

]]]]]]]]]]]]]]]]]]

]

119868= [1 0 0

0 1 0]

(A3)

wherein

0 and 0 are the (3 times 3) and (2 times 3) null matricesrespectively

References

[1] T Bailey and J E Hubbard ldquoDistributedpiezoelectric polymeractive vibration control of a cantilever beamrdquo Journal of Guid-ance Control and Dynamics vol 8 no 5 pp 605ndash611 1985

[2] E F Crawley and J de Luis ldquoUse of piezoelectric actuators aselements of intelligent structuresrdquo AIAA Journal vol 25 no 10pp 1373ndash1385 1987

[3] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988

[4] C K Lee W W Chiang and O Sulivan ldquoPiezoelectric modalsensoractuator pairs for critical active damping vibrationcontrolrdquo Journal of the Acoustical Society of America vol 90 no1 pp 374ndash384 1991

[5] S Hanagud M W Obal and A J Calise ldquoOptimal vibrationcontrol by the use of piezoceramic sensors and actuatorsrdquoJournal of Guidance Control and Dynamics vol 15 no 5 pp1199ndash1206 1992

[6] Y Gu R L Clark C R Fuller and A C Zander ldquoExperimentson active control of plate vibration using piezoelectric actuatorsand polyvinylidene fluoride (PVDF) modal sensorsrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 116 no3 pp 303ndash308 1994

[7] A Baz and S Poh ldquoOptimal vibration control with modalpositive position feedbackrdquo Optimal Control Applications andMethods vol 17 no 2 pp 141ndash149 1996

[8] A Kugi K Schlacher and H Irschik ldquoInfinite-dimensionalcontrol of nonlinear beam vibrations by piezoelectric actuatorand sensor layersrdquo Nonlinear Dynamics vol 19 no 1 pp 71ndash911999

[9] D Sun L Tong and D Wang ldquoVibration control of platesusing discretely distributed piezoelectric quasi-modal actua-torssensorsrdquo AIAA Journal vol 39 no 9 pp 1766ndash1772 2001

[10] G Caruso S Galeani and L Menini ldquoActive vibration controlof an elastic plate using multiple piezoelectric sensors andactuatorsrdquo SimulationModelling Practice andTheory vol 11 no5-6 pp 403ndash419 2003

[11] X Zhang and A G Erdman ldquoOptimal placement of piezo-electric sensors and actuators for controlled flexible linkagemechanismsrdquo Journal of Vibration and Acoustics Transactionsof the ASME vol 128 no 2 pp 256ndash260 2006

[12] G Gatti M J Brennan and P Gardonio ldquoActive dampingof a beam using a physically collocated accelerometer andpiezoelectric patch actuatorrdquo Journal of Sound and Vibrationvol 303 no 3ndash5 pp 798ndash813 2007

[13] M K Kwak S Heo and M Jeong ldquoDynamic modellingand active vibration controller design for a cylindrical shellequipped with piezoelectric sensors and actuatorsrdquo Journal ofSound and Vibration vol 321 no 3ndash5 pp 510ndash524 2009

[14] I Bruant L Gallimard and S Nikoukar ldquoOptimal piezoelectricactuator and sensor location for active vibration control usinggenetic algorithmrdquo Journal of Sound and Vibration vol 329 no10 pp 1615ndash1635 2010

[15] Y Fu J Wang and Y Mao ldquoNonlinear vibration and activecontrol of functionally graded beams with piezoelectric sensorsand actuatorsrdquo Journal of Intelligent Material Systems andStructures vol 22 no 18 pp 2093ndash2102 2011

[16] F Chen M Hong M Song and H Cui ldquoOptimal control ofa beam with discontinuously distributed piezoelectric sensorsand actuatorsrdquo Journal of Marine Science and Application vol11 no 1 pp 44ndash51 2012

[17] B Azvine G R Tomlinson and R J Wynne ldquoUse of activeconstrained-layer damping for controlling resonant vibrationrdquoSmart Materials and Structures vol 4 no 1 pp 1ndash6 1995

[18] A Baz and J Ro ldquoVibration control of plates with activeconstrained layer dampingrdquo SmartMaterials and Structures vol5 no 3 pp 272ndash280 1996

[19] C Chantalakhana and R Stanway ldquoActive constrained layerdamping of clamped-clamped plate vibrationsrdquo Journal ofSound and Vibration vol 241 no 5 pp 755ndash777 2001

[20] J Ro and A Baz ldquoOptimum placement and control ofactive constrained layer damping using modal strain energyapproachrdquo JVCJournal of Vibration and Control vol 8 no 6pp 861ndash876 2002

[21] M C Ray and J N Reddy ldquoOptimal control of thin circularcylindrical laminated composite shells using active constrainedlayer damping treatmentrdquo Smart Materials and Structures vol13 no 1 pp 64ndash72 2004

[22] C M A Vasques B R Mace P Gardonio and J D RodriguesldquoArbitrary active constrained layer damping treatments onbeams finite element modelling and experimental validationrdquoComputers and Structures vol 84 no 22-23 pp 1384ndash14012006

[23] F M Li K Kishimoto K Wang Z B Chen and W HHuang ldquoVibration control of beams with active constrainedlayer dampingrdquo Smart Materials and Structures vol 17 no 6Article ID 065036 2008

[24] L Yuan Y Xiang Y Huang and J Lu ldquoA semi-analyticalmethod and the circumferential dominant modal control ofcircular cylindrical shells with active constrained layer dampingtreatmentrdquo SmartMaterials and Structures vol 19 no 2 ArticleID 025010 2010

[25] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

Journal of Composites 17

[26] W A Smith and B A Auld ldquoModeling 1ndash3 composite piezo-electrics thickness-mode oscillationsrdquo IEEE Transactions onUltrasonics Ferroelectrics and Frequency Control vol 38 no 1pp 40ndash47 1991

[27] PiezocompositesMaterial Systems Inc 543 Great Road Little-ton MA 01460

[28] Y Benveniste and G J Dvorak ldquoUniform fields and universalrelations in piezoelectric compositesrdquo Journal of the Mechanicsand Physics of Solids vol 40 no 6 pp 1295ndash1312 1992

[29] M L Dunn and M Taya ldquoMicromechanics predictions ofthe effective electroelastic moduli of piezoelectric compositesrdquoInternational Journal of Solids and Structures vol 30 no 2 pp161ndash175 1993

[30] J Aboudi ldquoMicromechanical prediction of the effective coeffi-cients of thermo-piezoelectric multiphase compositesrdquo Journalof Intelligent Material Systems and Structures vol 9 no 9 pp713ndash722 1999

[31] X Ruan T W Chou A Safari and S C Danforth ldquoA 3-Dconnectivity model for effective piezoelectric properties of yarncompositesrdquo Journal of Composite Materials vol 36 no 14 pp1693ndash1708 2002

[32] A Baz and A Tempia ldquoActive piezoelectric damping compos-itesrdquo Sensors and Actuators A vol 112 no 2-3 pp 340ndash3502004

[33] H Berger S Kari U Gabbert et al ldquoUnit cell models ofpiezoelectric fiber composites for numerical and analytical cal-culation of effective propertiesrdquo SmartMaterials and Structuresvol 15 no 2 pp 451ndash458 2006

[34] M Sakthivel and A Arockiarajan ldquoAn analytical model forpredicting thermo-electro-mechanical response of 1ndash3 piezo-electric compositesrdquo Computational Materials Science vol 48no 4 pp 759ndash767 2010

[35] M C Ray and A K Pradhan ldquoOn the use of verticallyreinforced 1ndash3 piezoelectric composites for hybrid damping oflaminated composite platesrdquo Mechanics of Advanced Materialsand Structures vol 14 no 4 pp 245ndash261 2007

[36] M C Ray and A K Pradhan ldquoPerformance of vertically andobliquely reinforced 1ndash3 piezoelectric composites for activedamping of laminated composite shellsrdquo Journal of Sound andVibration vol 315 no 4-5 pp 816ndash835 2008

[37] S K Sarangi and M C Ray ldquoSmart damping of geometricallynonlinear vibrations of laminated composite beams using verti-cally reinforced 1ndash3 piezoelectric compositesrdquo Smart Materialsand Structures vol 19 no 7 Article ID 075020 2010

[38] A Krishnan and J V Deshpande ldquoVibration of skew laminatesrdquoJournal of Sound and Vibration vol 153 no 2 pp 351ndash358 1992

[39] A R Krishna Reddy and R Palaninathan ldquoFree vibration ofskew laminatesrdquo Computers and Structures vol 70 no 4 pp415ndash423 1999

[40] C S Babu and T Kant ldquoTwo shear deformable finite elementmodels for buckling analysis of skew fibre-reinforced compositeand sandwich panelsrdquo Composite Structures vol 46 no 2 pp115ndash124 1999

[41] A K Garg R K Khare and T Kant ldquoFree vibration of skewfiber-reinforced composite and sandwich laminates using ashear deformable finite element modelrdquo Journal of SandwichStructures and Materials vol 8 no 1 pp 33ndash53 2006

[42] J N Reddy Mechanics of Laminated Composites Plates Theoryand Analysis CRC press Boca Raton Fla USA 1996

Submit your manuscripts athttpwwwhindawicom

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materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 8: Research Article Active Constrained Layer Damping of Smart …downloads.hindawi.com/archive/2013/824163.pdf · 2019-07-31 · Research Article Active Constrained Layer Damping of

8 Journal of Composites

Applying the dynamic version of the virtual work principle[32] the following open-loop equations of motion for anelement of the overall plate integrated with the ACLDtreatment are obtained

[119872119890

] 119889119890

119905

+ [119870119890

119905119905

] 119889119890

119905

+ [119870119890

119905119903

] 119889119890

119903

= 119865119890

119905119901

119881 + 119865119890

[119870119890

119905119903

]119879

119889119890

119905

+ [119870119890

119903119903

] 119889119890

119903

= 119865119890

119903119901

119881

(27)

Since the elastic constant matrix of the viscoelastic layer iscomplex the stiffness matrices of an element integrated withthe ACLD treatment are complex In case of an elementnot integrated with the ACLD treatment the electro-elasticcoupling matrices become null vectors and the elementalstiffnessmatrices are to be computed without considering thepiezoelectric and viscoelastic layers It should also be notedthat as the stiffness matrices associated with the transverseshear strains are derived separately one can employ theselective integration rule in a straight forward manner toavoid shear locking problem in case of thin plates Therestrained boundary conditions at the skew edges of theplate are such that the displacements of the points locatedat the skew edges are to be restrained along 119909

1015840- 1199101015840- and

1199111015840-directions (Figure 1(a)) Hence in order to impose theboundary conditions in a straight forward manner thegeneralized displacement vectors of a point lying on the skewedge are to be transformed as follows

119889119905 = [119871

119905] 119889

1015840

119905

119889119903 = [119871

119903] 119889

1015840

119903

(28)

where 1198891015840

119905

and 1198891015840

119903

are the generalized displacement vectorsof the point with respect to 119909

1015840

1199101015840

1199111015840 coordinate system and are

given by

1198891015840

119905

= [1199061015840

119900

V1015840119900

1199081015840

119900

]119879

1198891015840

119903

= [1205791015840

119909

1205791015840

119910

1205791015840

119911

1206011015840

119909

1206011015840

119910

1206011015840

119911

1205741015840

119911

1205741015840

119910

1205741015840

119911]119879

(29)

Also the transformation matrices [119871119905] and [119871

119903] are given by

[119871119905] = [

[

119888 119904 0

minus119904 119888 0

0 0 1

]

]

[119871119903] =

[[[[[[[[[[[[

[

119888 119904 0 0 0 0 0 0 0

minus119904 119888 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 119888 119904 0 0 0 0

0 0 0 minus119904 119888 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 119888 119904 0

0 0 0 0 0 0 minus119904 119888 0

0 0 0 0 0 0 0 0 1

]]]]]]]]]]]]

]

(30)

in which 119888 = cos120572 and 119904 = sin120572Thus the elementalmatricesof the element containing the nodes lying on the skew edgeare to be augmented as follows

[119870119890

119905119905

] = [1198791]119879

[119870119890

119905119905

] [1198791] [119870

119890

119905119903

] = [1198791]119879

[119870119890

119905119903

] [1198792]

[119870119890

119903119903

] = [1198792]119879

[119870119890

119903119903

] [1198792] [119872

119890

] = [1198791]119879

[119872119890

] [1198791]

(31)

The forms of the transformation matrices [1198791] and [119879

2] are

given by

[1198791] =

[[[[[[[[[[[

[

[119871119905] 119900 119900 119900 119900 119900 119900 119900

119900 [119871119905] 119900 119900 119900 119900 119900 119900

119900 119900 [119871119905] 119900 119900 119900 119900 119900

119900 119900 119900 [119871119905] 119900 119900 119900 119900

119900 119900 119900 119900 [119871119905] 119900 119900 119900

119900 119900 119900 119900 119900 [119871119905] 119900 119900

119900 119900 119900 119900 119900 119900 [119871119905] 119900

119900 119900 119900 119900 119900 119900 119900 [119871119905]

]]]]]]]]]]]

]

[1198792] =

[[[[[[[[[[[

[

[119871119903] 119900 119900 119900 119900 119900 119900 119900

119900 [119871119903] 119900 119900 119900 119900 119900 119900

119900 119900 [119871119903] 119900 119900 119900 119900 119900

119900 119900 119900 [119871119903] 119900 119900 119900 119900

119900 119900 119900 119900 [119871119903] 119900 119900 119900

119900 119900 119900 119900 119900 [119871119903] 119900 119900

119900 119900 119900 119900 119900 119900 [119871119903] 119900

119900 119900 119900 119900 119900 119900 119900 [119871119903]

]]]]]]]]]]]

]

(32)

in which 119900 and 119900 are (3 times 3) and (9 times 9) null matricesrespectively It may be noted that in case of a node ofsuch element which does not lie on the skew edge thegeneralized displacement vectors of this node with respect to119909119910119911 coordinate system are not required to be transformedHence thematrices [119871

119905] and [119871

119903] corresponding to suchnode

turn out to be (3times3) and (9times9) identitymatrices respectivelyThe elemental equations of motion are now assembled

to obtain the open-loop global equations of motion of theoverall skew plate as follows

[119872] + [119870119905119905] 119883 + [119870

119905119903] 119883

119903 =

119902

sum

119895=1

119865119895

119905119901

119881119895

+ 119865

[119870119905119903]119879

119883 + [119870119903119903] 119883

119903 =

119902

sum

119895=1

119865119895

119903119901

119881119895

(33)

where [119870119905119905] [119870

119905119903] and [119870

119903119903] are the global stiffness matrices

of the overall skew laminated composite plate [119872] is theglobal mass matrix 119865

119905119901 and 119865

119903119901 are the global electro-

elastic coupling vectors 119883 and 119883119903 are the global nodal

generalized displacement vectors 119865 is the global nodalforce vector 119902 is the number of patches and 119881

119895 is thevoltage applied to the 119895th patch Since the elemental stiffnessmatrices of an element augmented with the ACLD treatmentare complex the global stiffness matrices become complexand the energy dissipation characteristics of the overallplate are attributed to the imaginary part of these matricesHence the global equations of motion as derived previouslyalso represent the modeling of the passive (uncontrolled)constrained layer damping of the substrate plate when theconstraining layer is not subjected to any control voltagefollowing a derivative control law

Journal of Composites 9

3 Closed-Loop Model

In the active control approach the active constraining layerof each patch is activated with a control voltage negativelyproportional to the transverse velocity of a point Thus thecontrol voltage for each patch can be expressed in terms of thederivatives of the global nodal degrees of freedom as follows

119881119895

= minus119896119895

119889

= minus119896119895

119889

[119880119895

119905

] minus 119896119895

119889

(ℎ

2) [119880

119895

119903

] 119903 (34)

where [119880119895

119905

] and [119880119895

119903

] are the unit vectors describing thelocation of the velocity sensor for the 119895th patch and 119896

119895

119889

isthe control gain for the 119895th patch Substituting (34) into(33) the final equations of motion governing the closed-loop dynamics of the overall skew laminated compositeplateACLD system can be obtained as follows

[119872] + [119870119905119905] 119883 + [119870

119905119903] 119883

119903 +

119898

sum

119895=1

119896119895

119889

119865119895

119905119901

[119880119895

119905

]

+

119898

sum

119895=1

119896119895

119889

(ℎ

2) 119865

119895

119905119901

[119880119895

119903

] 119903 = 119865

[119870119905119903]119879

119883 + [119870119903119903] 119883

119903 +

119898

sum

119895=1

119896119895

119889

119865119895

119905119901

[119880119895

119905

]

+

119898

sum

119895=1

119896119895

119889

(ℎ

2) 119865

119895

119905119901

[119880119895

119903

] 119903 = 0

(35)

4 Results and Discussion

In this section the numerical results obtained by the finiteelement model derived in the previous section have beenpresented Symmetric as well as antisymmetric cross-plyand antisymmetric angle-ply laminated skew substrate platesintegrated with the two patches of ACLD treatment areconsidered for presenting the numerical results As shown inFigure 1 the locations of the patches have been selected insuch a way that the energy dissipation corresponding to thefirst fewmodes becomesmaximum PZT-5Hspur compositewith 60 fiber volume fraction has been considered for thematerial of the constraining layer of the ACLD treatmentUsing themicromechanicsmodel and thematerial propertiesof the constituent phases [35] the effective elastic andpiezoelectric material properties of the 1ndash3 PZC with respectto the principle material coordinate system are computed asfollows

11986211

= 929GPa 11986212

= 618GPa 11986213

= 605GPa

11986233

= 3544GPa 11986223

= 11986213 119862

44= 158GPa

11986266

= 154GPa 11986255

= 11986244

11989031

= minus01902Cm2

11989032

= 11989031

11989033

= 184107Cm2

11989024

= 0004Cm2

11989015

= 11989024

(36)

The material of the orthotropic layers of the substrate platesis a graphiteepoxy composite and its material properties are[42]

119864119871= 172GPa 119864

119871

119864119879

= 25 119866119871119879

= 05119864119879

119866119879119879

= 02119864119879 ]

119871119879= ]

119879119879= 025

(37)

in which the symbols have the usual meaning Unless oth-erwise mentioned the aspect ratio (119886ℎ) and the thicknessof the skew substrate plates are considered as 100 and0003m respectively while the orthotropic layers of theskew substrate plates are of equal thickness Also the lengthand the skew sides of the substrate are equal (ie 119886 =

119887) The loss factor of the viscoelastic layer considered fornumerical results remains invariant [20] within a broadband (0ndash400Hz) of frequency range and the values of thecomplex shear modulus Poisson ratio and the density ofthe viscoelastic layer are considered as 20(1 + 119894)MNmminus2049 and 1140 kgmminus3 respectively [20] The thicknesses ofthe viscoelastic layer and the 1ndash3 PZC layer are consideredas 508120583m and 250 120583m respectively The value of the shearcorrection factor is used as 56 Three points of Gaussianintegration rule are considered for computing the elementmatrices corresponding to the bending deformations whiletwo points of that are used for computing the elementmatrices corresponding to transverse shear deformationsThe simply supported boundary conditions at the edges of theoverall plate are invoked as follows

at 119909 = 119910 tan120572 119909 = 119886 + 119910 tan120572

V1015840119900

= 1199081015840

119900

= 1205791015840

119910

= 1206011015840

119910

= 1205741015840

119910

= 1205791015840

119911

= 1205741015840

119911

= 0

at 119910 = 0 119910 = 119887 cos120572

119906119900= 119908

119900= 120579

119909= 120601

119909= 120574

119909= 120579

119911= 120574

119911= 0

(38)

To verify the validity of the present finite elementmodel (FEM) the natural frequencies of the skew laminatedcomposite plates integrated with the inactivated patches ofnegligible thickness are first computed and subsequentlycompared with the existing analytical results [41] of theidentical plates without being integrated with the patchesSuch comparisons for the simply supported and clamped-clamped skew laminated plates are presented in Tables 1 and2 respectively It may be observed from these tables that theresults are in excellent agreement validating the model of theplate derived here

In order to assess the performance of the verti-callyobliquely reinforced 1ndash3 PZC as the material of thedistributed actuators of smart skew laminated compositeplates frequency responses for ACLD of the substrate skewlaminated composite plates are investigated To compute the

10 Journal of Composites

Table 1 Comparison of fundamental natural frequencies (120596) of simply supported skew laminated composite plates integratedwith the patchesof negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 15699 28917 37325 14829 24656 32522 17974 33351 33351Present FEM 15635 24383 35033 15076 24380 32254 18493 33359 33370

15∘ Reference [41] 16874 30458 39600 15741 25351 30270 18313 32490 36724Present FEM 16571 29840 36505 15796 25775 29892 18675 32075 35810

30∘ Reference [41] 20840 34023 46997 18871 29372 34489 20270 34431 42361Present FEM 19596 31690 46796 18226 29585 32357 19894 32365 43208

45∘ Reference [41] 28925 41906 54149 25609 33126 40617 25609 33131 42772Present FEM 24811 44875 53289 22996 34773 44889 23194 34870 45009

lowast

120596 = (120596119886

2

120587

2

ℎ)radic120588119864119879 120596 is the circular natural frequency

Table 2 Comparison of fundamental natural frequencies (120596) of clamped-clamped skew laminated composite plates integrated with thepatches of negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 23687 35399 41122 22990 37880 37880 22119 37339 37339Present FEM 23201 34769 44102 23315 36531 36545 22433 36000 36012

15∘ Reference [41] 24663 36255 43418 23809 37516 40785 23099 36997 40438Present FEM 23699 34821 44049 23741 35856 38401 23049 35346 38092

30∘ Reference [41] 27921 39557 50220 26666 39851 47227 26325 39549 52107Present FEM 25366 35696 44734 25240 41943 45373 24945 36113 50932

45∘ Reference [41] 34739 47129 58789 33015 46290 58423 33015 46290 58423Present FEM 28665 47074 54562 28377 47614 56620 28377 47614 55162

frequency response functions the skew laminated compositeplates are harmonically excited by a force of amplitude 1Newton applied to a point (1198862 (1198864) cos120572 ℎ2) on the skewsubstrate platesThe control voltage supplied to the constrain-ing layer of the first patch is negatively proportional to thevelocity of the point (1198862 (1198864) cos120572 ℎ2) and that applied tothe constraining layer of the second patch is negatively pro-portional to the velocity of the point (1198862 (31198864) cos120572 ℎ2)

Figures 3 4 and 5 illustrate the frequency responsefunctions for the transverse displacement of a point of asimply supported symmetric cross-ply (0∘90∘0∘) plate with15∘ 30∘ and 45∘ skew angles respectively while the fiberorientation angle (120595) in the constraining layer is 0∘ Thesefigures display both uncontrolled and controlled responsesand clearly show that the active constraining layer made ofthe vertically reinforced 1ndash3 PZC material being studied heresignificantly attenuates the amplitude of vibrations enhanc-ing the damping characteristics of the overall laminatedskew composite plates over the passive damping (uncon-trolled) Similar responses are also obtained for antisym-metric cross-ply (0∘90∘0∘90∘) and antisymmetric angle-ply (45∘minus45∘45∘minus45∘) skew laminated composite plates

However for brevity these results are not presented hereThemaximum values of the control voltages required to computethe controlled responses presented in Figures 3ndash5 have beenfound to be quite low and are illustrated in Figure 6 in caseof the symmetric cross-ply (0∘90∘0∘) plate with skew angle15∘ only In order to examine the contribution of the verticalactuation in improving the damping characteristics of theskew laminated composite plates as demonstrated in Figures3ndash5 active control responses of the symmetric cross-plylaminated composite plate with 15∘ skew angle are plotted inFigure 7with andwithout considering the value of 119890

33and 119890

31

It may be noted that when the values of 11989031is zero and that of

11989033is nonzero the vertical actuation of the active constraining

layer of the ACLD treatment is responsible for increasing thetransverse shear deformations of the viscoelastic constrainedlayer over the passive counterpart resulting in improveddamping of the smart plate over its passive damping Onthe other hand if 119890

33is zero and 119890

31is nonzero the in-

plane actuation of the active constraining layer causes thetransverse shear deformation of the viscoelastic core of theACLD treatment leading to the active damping of the smartplate It is evident from Figure 7 that the contribution of the

Journal of Composites 11

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 3 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 4 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 30∘)

vertical actuation by the constraining vertically reinforced 1ndash3 PZC layer is significantly larger than that of the in-planeactuation by the same for controlling the modes displayedin Figure 7 Though not shown here similar results are alsoobtained for antisymmetric cross-ply and angle-ply skewplates

Figures 8 9 and 10 demonstrate the effect of variationof piezoelectric fiber orientation angle on the performanceof the patches for improving the damping characteristicsof simply supported symmetric cross-ply (0∘90∘0∘)

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 5 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 45∘)

0

5

10

15

20

25

30

35

40

45

50

Con

trol v

olta

ge (V

olt)

0 100 200 300 400 500 600Frequency (Hz)

119870119889 = 600

119870119889 = 1200

Figure 6 Frequency response functions for the control voltages foractive damping of the simply supported skew symmetric cross-ply(0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

antisymmetric cross-ply (0∘90∘0∘90∘) and angle-ply(45∘minus45∘45∘minus45∘) substrate skew composite plates with30∘ skew angle respectively while the orientation angle (120595)

of the piezoelectric fibers is varied in the vertical 119909119911-planeThese figures illustrate that the attenuating capability of thepatches becomesmaximumwhen the value of the orientationangle (120595) of the fibers in the 119909119911-plane is 0∘ Similar effectis also obtained when the fiber orientation angle (120595) in theconstraining layer of the ACLD treatment is varied in thevertical 119910119911-plane as illustrated in Figures 11 12 and 13 forthe symmetric cross-ply (0∘90∘0∘) antisymmetric cross-ply(0∘90∘0∘90∘) and angle-ply (45∘minus45∘45∘minus45∘) substrate

12 Journal of Composites

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

11989031 = minus01902 11989033 = 184107

11989031 = minus01902 11989033 = 011989031 = 0 11989033 = 184107

w(a2(b4)co

s120572h2

) (m

)

Figure 7 Frequency response functions demonstrating the con-tribution of vertical actuation on the controlled responses forthe transverse displacement 119908 (1198862 (1198864) cos120572 ℎ2) of the skewsymmetric cross-ply (0∘90∘0∘) plate (119870

119889

= 600 119886ℎ = 100 120595 = 0∘

120572 = 15∘)

0 100 200 300 400 500 600Frequency (Hz)

0

02

04

06

08

1

12times10minus4

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 8 Effect of variation of piezoelectric fiber orientation angle(120595) on the controlled responses of a simply supported symmetricskew cross-ply (0∘90∘0∘) plate when the piezoelectric fibers of theconstraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

skew composite plates with 30∘ skew angle respectivelyIn this regard it should be noted that for investigating theeffect of variation of piezoelectric fiber orientation angle(120595) on the performance of the ACLD patches the valueof 120595 has been smoothly varied to compute the controlledresponses However for the sake of clarity in the plots theresponses corresponding to the four specific values of 120595

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 9 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 10 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

have been presented such that the optimum performanceof the patches can be evaluated The significant effect of thevariation of the piezoelectric fiber orientation angle in thevertical 119909119911- and 119910119911-planes has been noticed when all theboundaries of the overall plates are clamped as shown inFigures 14ndash19 When the piezoelectric fibers are coplanarwith the vertical 119909119911-plane the frequency response functionsplotted in Figures 14 15 and 16 reveal that the control

Journal of Composites 13

0

02

04

06

08

1

12

14

16times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 11 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 12 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=600 119886ℎ = 100 120572 = 30

∘)

authority of the patches becomes maximum for improvingthe damping characteristics of the clamped-clamped skewcross-ply and angle-ply substrate plates if the value of thepiezoelectric fiber orientation angle (120595) is 30∘ The same isalso found to be true if the piezoelectric fibers are coplanarwith the vertical 119910119911-plane as shown in Figures 17 18 and 19for the previously mentioned skew plates Similar responsesare also found for the effect of piezoelectric fiber orientation

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 13 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers of the constraining layer are coplanar with the119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 14 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600119886ℎ = 100120572 = 30

∘)

angle on the active damping of other skew laminated plateswith different skew angles However for the sake of brevitythey are not presented here

5 Conclusions

In this paper a study has been carried out to investi-gate the performance of the ACLD treatment for active

14 Journal of Composites

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 15 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 16 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

damping of smart skew laminated composite plates whenverticallyobliquely reinforced 1ndash3 PZC materials are usedas the materials for the constraining layer of the ACLDtreatment A layerwise FSDT-based finite element modelhas been developed to describe the dynamics of the skewlaminated composite plates integrated with the patches ofACLD treatment Unlike the existing finite element modelsof smart structures integrated with ACLD treatment the

200 250 300 350 400 450 500 550 600 650 7000

02

04

06

08

1

12

14

16

18

2times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 17 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

200 300 400 500 600 700 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 18 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

derivation of the present finite element model includes thetransverse deformations of the substrate skew laminatedcomposite plate the constrained viscoelastic layer and theconstraining 1ndash3 PZC layer of the ACLD treatment along thethickness (ie 119911) direction such that both vertical and in-plane actuations by the constraining layer of the patches canbe utilized for active damping of the plates The frequency

Journal of Composites 15

0

05

1

15

2

25

3

35

4times10minus5

200 300 400 500 600 700 800Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 19 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

responses of the symmetric cross-ply antisymmetric cross-ply and angle-ply skew composite plates indicate that theactive constraining layer of the ACLD treatment being madeof the vertically reinforced 1ndash3 PZC material significantlyenhances the damping characteristics of the plates over thepassive damping The analysis revealed that if the verticallyreinforced 1ndash3 PZCmaterial is used for the constraining layerof the ACLD treatment then the contribution of the verticalactuation by the constraining layer alone for improving theactive damping characteristics of the smart skew laminatedcomposite plate is significantly larger than that due to thein-plane actuation by the constraining layer alone It isimportant to note from the present investigation that thevariation of the orientation angle of the piezoelectric fibersof the constraining layer in the vertical 119909119911- and 119910119911-planesas well as the skew angle and boundary conditions of theoverall plates significantly affects the performance of thepatches For the simply supported laminated substrate platesthe performance of the patches becomes maximumwhen thepiezoelectric fiber orientation angle (120595) is 0∘ while in caseof the clamped-clamped laminated skew substrate plates themaximum control authority of the patches is achieved whenthe value of 120595 is 30∘ irrespective of the cases in which thefibers are oriented in 119909119911- and 119910119911-planes

Appendix

In (6) and (7) the matrices [1198851] [119885

2] [119885

3] [119885

4] [119885

5] and

[1198856] are given by

[1198851] = [[119885

1] 0 0]

[1198852] = [(

ℎ

2) 119868 [119885

2] 0]

[1198853] = [(

ℎ

2) 119868 ℎV119868 [119885

3]]

[1198854] = [119868 0 0 119911119868 0 0]

[1198855] = [0 119868 0 (

ℎ

2) 119868 (119911 minus

ℎ

2) 119868 0]

[1198856] = [0 0 119868 (

ℎ

2) 119868 ℎV119868 (119911 minus ℎ

119873+2) 119868]

(A1)

in which

[1198851] =

[[[

[

119911 0 0 0

0 119911 0 0

0 0 119911 0

0 0 0 1

]]]

]

[1198852] =

[[[[[[[[

[

(119911 minusℎ

2) 0 0 0

0 (119911 minusℎ

2) 0 0

0 0 (119911 minusℎ

2) 0

0 0 0 1

]]]]]]]]

]

[1198853] =

[[[

[

(119911 minus ℎ119873+2

) 0 0 0

0 (119911 minus ℎ119873+2

) 0 0

0 0 (119911 minus ℎ119873+2

) 0

0 0 0 1

]]]

]

[119868] =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

[119868] = [1 0

0 1]

[0] = [0 0

0 0] [0] = [

0 0

0 0]

(A2)

The various submatrices 119861119905119887119894 119861

119903119887119894 and 119861

119903119904119894 appearing in (21)

are given by

119861119905119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 0

]]]]]]]]]]]

]

119861119905119904119894

=

[[[[

[

0 0120597120578

119894

120597119909

0 0120597120578

119894

120597119910

]]]]

]

16 Journal of Composites

119861119903119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 1

]]]]]]]]]]]

]

119861119903119887119894

=

[[[[[[

[

119861119903119887119894

0

0

0 119861119903119887119894

0

0

0 119861119903119887119894

]]]]]]

]

119861119903119904119894

=

[[[[[[[[[[[[[[[[[[

[

119868 0 0

0

119868 0

0 0

119868

119861119903119904119894

0 0

0 119861119903119904119894

0

0 0 119861119903119904119894

]]]]]]]]]]]]]]]]]]

]

119868= [1 0 0

0 1 0]

(A3)

wherein

0 and 0 are the (3 times 3) and (2 times 3) null matricesrespectively

References

[1] T Bailey and J E Hubbard ldquoDistributedpiezoelectric polymeractive vibration control of a cantilever beamrdquo Journal of Guid-ance Control and Dynamics vol 8 no 5 pp 605ndash611 1985

[2] E F Crawley and J de Luis ldquoUse of piezoelectric actuators aselements of intelligent structuresrdquo AIAA Journal vol 25 no 10pp 1373ndash1385 1987

[3] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988

[4] C K Lee W W Chiang and O Sulivan ldquoPiezoelectric modalsensoractuator pairs for critical active damping vibrationcontrolrdquo Journal of the Acoustical Society of America vol 90 no1 pp 374ndash384 1991

[5] S Hanagud M W Obal and A J Calise ldquoOptimal vibrationcontrol by the use of piezoceramic sensors and actuatorsrdquoJournal of Guidance Control and Dynamics vol 15 no 5 pp1199ndash1206 1992

[6] Y Gu R L Clark C R Fuller and A C Zander ldquoExperimentson active control of plate vibration using piezoelectric actuatorsand polyvinylidene fluoride (PVDF) modal sensorsrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 116 no3 pp 303ndash308 1994

[7] A Baz and S Poh ldquoOptimal vibration control with modalpositive position feedbackrdquo Optimal Control Applications andMethods vol 17 no 2 pp 141ndash149 1996

[8] A Kugi K Schlacher and H Irschik ldquoInfinite-dimensionalcontrol of nonlinear beam vibrations by piezoelectric actuatorand sensor layersrdquo Nonlinear Dynamics vol 19 no 1 pp 71ndash911999

[9] D Sun L Tong and D Wang ldquoVibration control of platesusing discretely distributed piezoelectric quasi-modal actua-torssensorsrdquo AIAA Journal vol 39 no 9 pp 1766ndash1772 2001

[10] G Caruso S Galeani and L Menini ldquoActive vibration controlof an elastic plate using multiple piezoelectric sensors andactuatorsrdquo SimulationModelling Practice andTheory vol 11 no5-6 pp 403ndash419 2003

[11] X Zhang and A G Erdman ldquoOptimal placement of piezo-electric sensors and actuators for controlled flexible linkagemechanismsrdquo Journal of Vibration and Acoustics Transactionsof the ASME vol 128 no 2 pp 256ndash260 2006

[12] G Gatti M J Brennan and P Gardonio ldquoActive dampingof a beam using a physically collocated accelerometer andpiezoelectric patch actuatorrdquo Journal of Sound and Vibrationvol 303 no 3ndash5 pp 798ndash813 2007

[13] M K Kwak S Heo and M Jeong ldquoDynamic modellingand active vibration controller design for a cylindrical shellequipped with piezoelectric sensors and actuatorsrdquo Journal ofSound and Vibration vol 321 no 3ndash5 pp 510ndash524 2009

[14] I Bruant L Gallimard and S Nikoukar ldquoOptimal piezoelectricactuator and sensor location for active vibration control usinggenetic algorithmrdquo Journal of Sound and Vibration vol 329 no10 pp 1615ndash1635 2010

[15] Y Fu J Wang and Y Mao ldquoNonlinear vibration and activecontrol of functionally graded beams with piezoelectric sensorsand actuatorsrdquo Journal of Intelligent Material Systems andStructures vol 22 no 18 pp 2093ndash2102 2011

[16] F Chen M Hong M Song and H Cui ldquoOptimal control ofa beam with discontinuously distributed piezoelectric sensorsand actuatorsrdquo Journal of Marine Science and Application vol11 no 1 pp 44ndash51 2012

[17] B Azvine G R Tomlinson and R J Wynne ldquoUse of activeconstrained-layer damping for controlling resonant vibrationrdquoSmart Materials and Structures vol 4 no 1 pp 1ndash6 1995

[18] A Baz and J Ro ldquoVibration control of plates with activeconstrained layer dampingrdquo SmartMaterials and Structures vol5 no 3 pp 272ndash280 1996

[19] C Chantalakhana and R Stanway ldquoActive constrained layerdamping of clamped-clamped plate vibrationsrdquo Journal ofSound and Vibration vol 241 no 5 pp 755ndash777 2001

[20] J Ro and A Baz ldquoOptimum placement and control ofactive constrained layer damping using modal strain energyapproachrdquo JVCJournal of Vibration and Control vol 8 no 6pp 861ndash876 2002

[21] M C Ray and J N Reddy ldquoOptimal control of thin circularcylindrical laminated composite shells using active constrainedlayer damping treatmentrdquo Smart Materials and Structures vol13 no 1 pp 64ndash72 2004

[22] C M A Vasques B R Mace P Gardonio and J D RodriguesldquoArbitrary active constrained layer damping treatments onbeams finite element modelling and experimental validationrdquoComputers and Structures vol 84 no 22-23 pp 1384ndash14012006

[23] F M Li K Kishimoto K Wang Z B Chen and W HHuang ldquoVibration control of beams with active constrainedlayer dampingrdquo Smart Materials and Structures vol 17 no 6Article ID 065036 2008

[24] L Yuan Y Xiang Y Huang and J Lu ldquoA semi-analyticalmethod and the circumferential dominant modal control ofcircular cylindrical shells with active constrained layer dampingtreatmentrdquo SmartMaterials and Structures vol 19 no 2 ArticleID 025010 2010

[25] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

Journal of Composites 17

[26] W A Smith and B A Auld ldquoModeling 1ndash3 composite piezo-electrics thickness-mode oscillationsrdquo IEEE Transactions onUltrasonics Ferroelectrics and Frequency Control vol 38 no 1pp 40ndash47 1991

[27] PiezocompositesMaterial Systems Inc 543 Great Road Little-ton MA 01460

[28] Y Benveniste and G J Dvorak ldquoUniform fields and universalrelations in piezoelectric compositesrdquo Journal of the Mechanicsand Physics of Solids vol 40 no 6 pp 1295ndash1312 1992

[29] M L Dunn and M Taya ldquoMicromechanics predictions ofthe effective electroelastic moduli of piezoelectric compositesrdquoInternational Journal of Solids and Structures vol 30 no 2 pp161ndash175 1993

[30] J Aboudi ldquoMicromechanical prediction of the effective coeffi-cients of thermo-piezoelectric multiphase compositesrdquo Journalof Intelligent Material Systems and Structures vol 9 no 9 pp713ndash722 1999

[31] X Ruan T W Chou A Safari and S C Danforth ldquoA 3-Dconnectivity model for effective piezoelectric properties of yarncompositesrdquo Journal of Composite Materials vol 36 no 14 pp1693ndash1708 2002

[32] A Baz and A Tempia ldquoActive piezoelectric damping compos-itesrdquo Sensors and Actuators A vol 112 no 2-3 pp 340ndash3502004

[33] H Berger S Kari U Gabbert et al ldquoUnit cell models ofpiezoelectric fiber composites for numerical and analytical cal-culation of effective propertiesrdquo SmartMaterials and Structuresvol 15 no 2 pp 451ndash458 2006

[34] M Sakthivel and A Arockiarajan ldquoAn analytical model forpredicting thermo-electro-mechanical response of 1ndash3 piezo-electric compositesrdquo Computational Materials Science vol 48no 4 pp 759ndash767 2010

[35] M C Ray and A K Pradhan ldquoOn the use of verticallyreinforced 1ndash3 piezoelectric composites for hybrid damping oflaminated composite platesrdquo Mechanics of Advanced Materialsand Structures vol 14 no 4 pp 245ndash261 2007

[36] M C Ray and A K Pradhan ldquoPerformance of vertically andobliquely reinforced 1ndash3 piezoelectric composites for activedamping of laminated composite shellsrdquo Journal of Sound andVibration vol 315 no 4-5 pp 816ndash835 2008

[37] S K Sarangi and M C Ray ldquoSmart damping of geometricallynonlinear vibrations of laminated composite beams using verti-cally reinforced 1ndash3 piezoelectric compositesrdquo Smart Materialsand Structures vol 19 no 7 Article ID 075020 2010

[38] A Krishnan and J V Deshpande ldquoVibration of skew laminatesrdquoJournal of Sound and Vibration vol 153 no 2 pp 351ndash358 1992

[39] A R Krishna Reddy and R Palaninathan ldquoFree vibration ofskew laminatesrdquo Computers and Structures vol 70 no 4 pp415ndash423 1999

[40] C S Babu and T Kant ldquoTwo shear deformable finite elementmodels for buckling analysis of skew fibre-reinforced compositeand sandwich panelsrdquo Composite Structures vol 46 no 2 pp115ndash124 1999

[41] A K Garg R K Khare and T Kant ldquoFree vibration of skewfiber-reinforced composite and sandwich laminates using ashear deformable finite element modelrdquo Journal of SandwichStructures and Materials vol 8 no 1 pp 33ndash53 2006

[42] J N Reddy Mechanics of Laminated Composites Plates Theoryand Analysis CRC press Boca Raton Fla USA 1996

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

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Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 9: Research Article Active Constrained Layer Damping of Smart …downloads.hindawi.com/archive/2013/824163.pdf · 2019-07-31 · Research Article Active Constrained Layer Damping of

Journal of Composites 9

3 Closed-Loop Model

In the active control approach the active constraining layerof each patch is activated with a control voltage negativelyproportional to the transverse velocity of a point Thus thecontrol voltage for each patch can be expressed in terms of thederivatives of the global nodal degrees of freedom as follows

119881119895

= minus119896119895

119889

= minus119896119895

119889

[119880119895

119905

] minus 119896119895

119889

(ℎ

2) [119880

119895

119903

] 119903 (34)

where [119880119895

119905

] and [119880119895

119903

] are the unit vectors describing thelocation of the velocity sensor for the 119895th patch and 119896

119895

119889

isthe control gain for the 119895th patch Substituting (34) into(33) the final equations of motion governing the closed-loop dynamics of the overall skew laminated compositeplateACLD system can be obtained as follows

[119872] + [119870119905119905] 119883 + [119870

119905119903] 119883

119903 +

119898

sum

119895=1

119896119895

119889

119865119895

119905119901

[119880119895

119905

]

+

119898

sum

119895=1

119896119895

119889

(ℎ

2) 119865

119895

119905119901

[119880119895

119903

] 119903 = 119865

[119870119905119903]119879

119883 + [119870119903119903] 119883

119903 +

119898

sum

119895=1

119896119895

119889

119865119895

119905119901

[119880119895

119905

]

+

119898

sum

119895=1

119896119895

119889

(ℎ

2) 119865

119895

119905119901

[119880119895

119903

] 119903 = 0

(35)

4 Results and Discussion

In this section the numerical results obtained by the finiteelement model derived in the previous section have beenpresented Symmetric as well as antisymmetric cross-plyand antisymmetric angle-ply laminated skew substrate platesintegrated with the two patches of ACLD treatment areconsidered for presenting the numerical results As shown inFigure 1 the locations of the patches have been selected insuch a way that the energy dissipation corresponding to thefirst fewmodes becomesmaximum PZT-5Hspur compositewith 60 fiber volume fraction has been considered for thematerial of the constraining layer of the ACLD treatmentUsing themicromechanicsmodel and thematerial propertiesof the constituent phases [35] the effective elastic andpiezoelectric material properties of the 1ndash3 PZC with respectto the principle material coordinate system are computed asfollows

11986211

= 929GPa 11986212

= 618GPa 11986213

= 605GPa

11986233

= 3544GPa 11986223

= 11986213 119862

44= 158GPa

11986266

= 154GPa 11986255

= 11986244

11989031

= minus01902Cm2

11989032

= 11989031

11989033

= 184107Cm2

11989024

= 0004Cm2

11989015

= 11989024

(36)

The material of the orthotropic layers of the substrate platesis a graphiteepoxy composite and its material properties are[42]

119864119871= 172GPa 119864

119871

119864119879

= 25 119866119871119879

= 05119864119879

119866119879119879

= 02119864119879 ]

119871119879= ]

119879119879= 025

(37)

in which the symbols have the usual meaning Unless oth-erwise mentioned the aspect ratio (119886ℎ) and the thicknessof the skew substrate plates are considered as 100 and0003m respectively while the orthotropic layers of theskew substrate plates are of equal thickness Also the lengthand the skew sides of the substrate are equal (ie 119886 =

119887) The loss factor of the viscoelastic layer considered fornumerical results remains invariant [20] within a broadband (0ndash400Hz) of frequency range and the values of thecomplex shear modulus Poisson ratio and the density ofthe viscoelastic layer are considered as 20(1 + 119894)MNmminus2049 and 1140 kgmminus3 respectively [20] The thicknesses ofthe viscoelastic layer and the 1ndash3 PZC layer are consideredas 508120583m and 250 120583m respectively The value of the shearcorrection factor is used as 56 Three points of Gaussianintegration rule are considered for computing the elementmatrices corresponding to the bending deformations whiletwo points of that are used for computing the elementmatrices corresponding to transverse shear deformationsThe simply supported boundary conditions at the edges of theoverall plate are invoked as follows

at 119909 = 119910 tan120572 119909 = 119886 + 119910 tan120572

V1015840119900

= 1199081015840

119900

= 1205791015840

119910

= 1206011015840

119910

= 1205741015840

119910

= 1205791015840

119911

= 1205741015840

119911

= 0

at 119910 = 0 119910 = 119887 cos120572

119906119900= 119908

119900= 120579

119909= 120601

119909= 120574

119909= 120579

119911= 120574

119911= 0

(38)

To verify the validity of the present finite elementmodel (FEM) the natural frequencies of the skew laminatedcomposite plates integrated with the inactivated patches ofnegligible thickness are first computed and subsequentlycompared with the existing analytical results [41] of theidentical plates without being integrated with the patchesSuch comparisons for the simply supported and clamped-clamped skew laminated plates are presented in Tables 1 and2 respectively It may be observed from these tables that theresults are in excellent agreement validating the model of theplate derived here

In order to assess the performance of the verti-callyobliquely reinforced 1ndash3 PZC as the material of thedistributed actuators of smart skew laminated compositeplates frequency responses for ACLD of the substrate skewlaminated composite plates are investigated To compute the

10 Journal of Composites

Table 1 Comparison of fundamental natural frequencies (120596) of simply supported skew laminated composite plates integratedwith the patchesof negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 15699 28917 37325 14829 24656 32522 17974 33351 33351Present FEM 15635 24383 35033 15076 24380 32254 18493 33359 33370

15∘ Reference [41] 16874 30458 39600 15741 25351 30270 18313 32490 36724Present FEM 16571 29840 36505 15796 25775 29892 18675 32075 35810

30∘ Reference [41] 20840 34023 46997 18871 29372 34489 20270 34431 42361Present FEM 19596 31690 46796 18226 29585 32357 19894 32365 43208

45∘ Reference [41] 28925 41906 54149 25609 33126 40617 25609 33131 42772Present FEM 24811 44875 53289 22996 34773 44889 23194 34870 45009

lowast

120596 = (120596119886

2

120587

2

ℎ)radic120588119864119879 120596 is the circular natural frequency

Table 2 Comparison of fundamental natural frequencies (120596) of clamped-clamped skew laminated composite plates integrated with thepatches of negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 23687 35399 41122 22990 37880 37880 22119 37339 37339Present FEM 23201 34769 44102 23315 36531 36545 22433 36000 36012

15∘ Reference [41] 24663 36255 43418 23809 37516 40785 23099 36997 40438Present FEM 23699 34821 44049 23741 35856 38401 23049 35346 38092

30∘ Reference [41] 27921 39557 50220 26666 39851 47227 26325 39549 52107Present FEM 25366 35696 44734 25240 41943 45373 24945 36113 50932

45∘ Reference [41] 34739 47129 58789 33015 46290 58423 33015 46290 58423Present FEM 28665 47074 54562 28377 47614 56620 28377 47614 55162

frequency response functions the skew laminated compositeplates are harmonically excited by a force of amplitude 1Newton applied to a point (1198862 (1198864) cos120572 ℎ2) on the skewsubstrate platesThe control voltage supplied to the constrain-ing layer of the first patch is negatively proportional to thevelocity of the point (1198862 (1198864) cos120572 ℎ2) and that applied tothe constraining layer of the second patch is negatively pro-portional to the velocity of the point (1198862 (31198864) cos120572 ℎ2)

Figures 3 4 and 5 illustrate the frequency responsefunctions for the transverse displacement of a point of asimply supported symmetric cross-ply (0∘90∘0∘) plate with15∘ 30∘ and 45∘ skew angles respectively while the fiberorientation angle (120595) in the constraining layer is 0∘ Thesefigures display both uncontrolled and controlled responsesand clearly show that the active constraining layer made ofthe vertically reinforced 1ndash3 PZC material being studied heresignificantly attenuates the amplitude of vibrations enhanc-ing the damping characteristics of the overall laminatedskew composite plates over the passive damping (uncon-trolled) Similar responses are also obtained for antisym-metric cross-ply (0∘90∘0∘90∘) and antisymmetric angle-ply (45∘minus45∘45∘minus45∘) skew laminated composite plates

However for brevity these results are not presented hereThemaximum values of the control voltages required to computethe controlled responses presented in Figures 3ndash5 have beenfound to be quite low and are illustrated in Figure 6 in caseof the symmetric cross-ply (0∘90∘0∘) plate with skew angle15∘ only In order to examine the contribution of the verticalactuation in improving the damping characteristics of theskew laminated composite plates as demonstrated in Figures3ndash5 active control responses of the symmetric cross-plylaminated composite plate with 15∘ skew angle are plotted inFigure 7with andwithout considering the value of 119890

33and 119890

31

It may be noted that when the values of 11989031is zero and that of

11989033is nonzero the vertical actuation of the active constraining

layer of the ACLD treatment is responsible for increasing thetransverse shear deformations of the viscoelastic constrainedlayer over the passive counterpart resulting in improveddamping of the smart plate over its passive damping Onthe other hand if 119890

33is zero and 119890

31is nonzero the in-

plane actuation of the active constraining layer causes thetransverse shear deformation of the viscoelastic core of theACLD treatment leading to the active damping of the smartplate It is evident from Figure 7 that the contribution of the

Journal of Composites 11

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 3 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 4 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 30∘)

vertical actuation by the constraining vertically reinforced 1ndash3 PZC layer is significantly larger than that of the in-planeactuation by the same for controlling the modes displayedin Figure 7 Though not shown here similar results are alsoobtained for antisymmetric cross-ply and angle-ply skewplates

Figures 8 9 and 10 demonstrate the effect of variationof piezoelectric fiber orientation angle on the performanceof the patches for improving the damping characteristicsof simply supported symmetric cross-ply (0∘90∘0∘)

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 5 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 45∘)

0

5

10

15

20

25

30

35

40

45

50

Con

trol v

olta

ge (V

olt)

0 100 200 300 400 500 600Frequency (Hz)

119870119889 = 600

119870119889 = 1200

Figure 6 Frequency response functions for the control voltages foractive damping of the simply supported skew symmetric cross-ply(0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

antisymmetric cross-ply (0∘90∘0∘90∘) and angle-ply(45∘minus45∘45∘minus45∘) substrate skew composite plates with30∘ skew angle respectively while the orientation angle (120595)

of the piezoelectric fibers is varied in the vertical 119909119911-planeThese figures illustrate that the attenuating capability of thepatches becomesmaximumwhen the value of the orientationangle (120595) of the fibers in the 119909119911-plane is 0∘ Similar effectis also obtained when the fiber orientation angle (120595) in theconstraining layer of the ACLD treatment is varied in thevertical 119910119911-plane as illustrated in Figures 11 12 and 13 forthe symmetric cross-ply (0∘90∘0∘) antisymmetric cross-ply(0∘90∘0∘90∘) and angle-ply (45∘minus45∘45∘minus45∘) substrate

12 Journal of Composites

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

11989031 = minus01902 11989033 = 184107

11989031 = minus01902 11989033 = 011989031 = 0 11989033 = 184107

w(a2(b4)co

s120572h2

) (m

)

Figure 7 Frequency response functions demonstrating the con-tribution of vertical actuation on the controlled responses forthe transverse displacement 119908 (1198862 (1198864) cos120572 ℎ2) of the skewsymmetric cross-ply (0∘90∘0∘) plate (119870

119889

= 600 119886ℎ = 100 120595 = 0∘

120572 = 15∘)

0 100 200 300 400 500 600Frequency (Hz)

0

02

04

06

08

1

12times10minus4

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 8 Effect of variation of piezoelectric fiber orientation angle(120595) on the controlled responses of a simply supported symmetricskew cross-ply (0∘90∘0∘) plate when the piezoelectric fibers of theconstraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

skew composite plates with 30∘ skew angle respectivelyIn this regard it should be noted that for investigating theeffect of variation of piezoelectric fiber orientation angle(120595) on the performance of the ACLD patches the valueof 120595 has been smoothly varied to compute the controlledresponses However for the sake of clarity in the plots theresponses corresponding to the four specific values of 120595

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 9 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 10 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

have been presented such that the optimum performanceof the patches can be evaluated The significant effect of thevariation of the piezoelectric fiber orientation angle in thevertical 119909119911- and 119910119911-planes has been noticed when all theboundaries of the overall plates are clamped as shown inFigures 14ndash19 When the piezoelectric fibers are coplanarwith the vertical 119909119911-plane the frequency response functionsplotted in Figures 14 15 and 16 reveal that the control

Journal of Composites 13

0

02

04

06

08

1

12

14

16times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 11 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 12 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=600 119886ℎ = 100 120572 = 30

∘)

authority of the patches becomes maximum for improvingthe damping characteristics of the clamped-clamped skewcross-ply and angle-ply substrate plates if the value of thepiezoelectric fiber orientation angle (120595) is 30∘ The same isalso found to be true if the piezoelectric fibers are coplanarwith the vertical 119910119911-plane as shown in Figures 17 18 and 19for the previously mentioned skew plates Similar responsesare also found for the effect of piezoelectric fiber orientation

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 13 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers of the constraining layer are coplanar with the119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 14 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600119886ℎ = 100120572 = 30

∘)

angle on the active damping of other skew laminated plateswith different skew angles However for the sake of brevitythey are not presented here

5 Conclusions

In this paper a study has been carried out to investi-gate the performance of the ACLD treatment for active

14 Journal of Composites

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 15 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 16 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

damping of smart skew laminated composite plates whenverticallyobliquely reinforced 1ndash3 PZC materials are usedas the materials for the constraining layer of the ACLDtreatment A layerwise FSDT-based finite element modelhas been developed to describe the dynamics of the skewlaminated composite plates integrated with the patches ofACLD treatment Unlike the existing finite element modelsof smart structures integrated with ACLD treatment the

200 250 300 350 400 450 500 550 600 650 7000

02

04

06

08

1

12

14

16

18

2times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 17 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

200 300 400 500 600 700 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 18 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

derivation of the present finite element model includes thetransverse deformations of the substrate skew laminatedcomposite plate the constrained viscoelastic layer and theconstraining 1ndash3 PZC layer of the ACLD treatment along thethickness (ie 119911) direction such that both vertical and in-plane actuations by the constraining layer of the patches canbe utilized for active damping of the plates The frequency

Journal of Composites 15

0

05

1

15

2

25

3

35

4times10minus5

200 300 400 500 600 700 800Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 19 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

responses of the symmetric cross-ply antisymmetric cross-ply and angle-ply skew composite plates indicate that theactive constraining layer of the ACLD treatment being madeof the vertically reinforced 1ndash3 PZC material significantlyenhances the damping characteristics of the plates over thepassive damping The analysis revealed that if the verticallyreinforced 1ndash3 PZCmaterial is used for the constraining layerof the ACLD treatment then the contribution of the verticalactuation by the constraining layer alone for improving theactive damping characteristics of the smart skew laminatedcomposite plate is significantly larger than that due to thein-plane actuation by the constraining layer alone It isimportant to note from the present investigation that thevariation of the orientation angle of the piezoelectric fibersof the constraining layer in the vertical 119909119911- and 119910119911-planesas well as the skew angle and boundary conditions of theoverall plates significantly affects the performance of thepatches For the simply supported laminated substrate platesthe performance of the patches becomes maximumwhen thepiezoelectric fiber orientation angle (120595) is 0∘ while in caseof the clamped-clamped laminated skew substrate plates themaximum control authority of the patches is achieved whenthe value of 120595 is 30∘ irrespective of the cases in which thefibers are oriented in 119909119911- and 119910119911-planes

Appendix

In (6) and (7) the matrices [1198851] [119885

2] [119885

3] [119885

4] [119885

5] and

[1198856] are given by

[1198851] = [[119885

1] 0 0]

[1198852] = [(

ℎ

2) 119868 [119885

2] 0]

[1198853] = [(

ℎ

2) 119868 ℎV119868 [119885

3]]

[1198854] = [119868 0 0 119911119868 0 0]

[1198855] = [0 119868 0 (

ℎ

2) 119868 (119911 minus

ℎ

2) 119868 0]

[1198856] = [0 0 119868 (

ℎ

2) 119868 ℎV119868 (119911 minus ℎ

119873+2) 119868]

(A1)

in which

[1198851] =

[[[

[

119911 0 0 0

0 119911 0 0

0 0 119911 0

0 0 0 1

]]]

]

[1198852] =

[[[[[[[[

[

(119911 minusℎ

2) 0 0 0

0 (119911 minusℎ

2) 0 0

0 0 (119911 minusℎ

2) 0

0 0 0 1

]]]]]]]]

]

[1198853] =

[[[

[

(119911 minus ℎ119873+2

) 0 0 0

0 (119911 minus ℎ119873+2

) 0 0

0 0 (119911 minus ℎ119873+2

) 0

0 0 0 1

]]]

]

[119868] =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

[119868] = [1 0

0 1]

[0] = [0 0

0 0] [0] = [

0 0

0 0]

(A2)

The various submatrices 119861119905119887119894 119861

119903119887119894 and 119861

119903119904119894 appearing in (21)

are given by

119861119905119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 0

]]]]]]]]]]]

]

119861119905119904119894

=

[[[[

[

0 0120597120578

119894

120597119909

0 0120597120578

119894

120597119910

]]]]

]

16 Journal of Composites

119861119903119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 1

]]]]]]]]]]]

]

119861119903119887119894

=

[[[[[[

[

119861119903119887119894

0

0

0 119861119903119887119894

0

0

0 119861119903119887119894

]]]]]]

]

119861119903119904119894

=

[[[[[[[[[[[[[[[[[[

[

119868 0 0

0

119868 0

0 0

119868

119861119903119904119894

0 0

0 119861119903119904119894

0

0 0 119861119903119904119894

]]]]]]]]]]]]]]]]]]

]

119868= [1 0 0

0 1 0]

(A3)

wherein

0 and 0 are the (3 times 3) and (2 times 3) null matricesrespectively

References

[1] T Bailey and J E Hubbard ldquoDistributedpiezoelectric polymeractive vibration control of a cantilever beamrdquo Journal of Guid-ance Control and Dynamics vol 8 no 5 pp 605ndash611 1985

[2] E F Crawley and J de Luis ldquoUse of piezoelectric actuators aselements of intelligent structuresrdquo AIAA Journal vol 25 no 10pp 1373ndash1385 1987

[3] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988

[4] C K Lee W W Chiang and O Sulivan ldquoPiezoelectric modalsensoractuator pairs for critical active damping vibrationcontrolrdquo Journal of the Acoustical Society of America vol 90 no1 pp 374ndash384 1991

[5] S Hanagud M W Obal and A J Calise ldquoOptimal vibrationcontrol by the use of piezoceramic sensors and actuatorsrdquoJournal of Guidance Control and Dynamics vol 15 no 5 pp1199ndash1206 1992

[6] Y Gu R L Clark C R Fuller and A C Zander ldquoExperimentson active control of plate vibration using piezoelectric actuatorsand polyvinylidene fluoride (PVDF) modal sensorsrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 116 no3 pp 303ndash308 1994

[7] A Baz and S Poh ldquoOptimal vibration control with modalpositive position feedbackrdquo Optimal Control Applications andMethods vol 17 no 2 pp 141ndash149 1996

[8] A Kugi K Schlacher and H Irschik ldquoInfinite-dimensionalcontrol of nonlinear beam vibrations by piezoelectric actuatorand sensor layersrdquo Nonlinear Dynamics vol 19 no 1 pp 71ndash911999

[9] D Sun L Tong and D Wang ldquoVibration control of platesusing discretely distributed piezoelectric quasi-modal actua-torssensorsrdquo AIAA Journal vol 39 no 9 pp 1766ndash1772 2001

[10] G Caruso S Galeani and L Menini ldquoActive vibration controlof an elastic plate using multiple piezoelectric sensors andactuatorsrdquo SimulationModelling Practice andTheory vol 11 no5-6 pp 403ndash419 2003

[11] X Zhang and A G Erdman ldquoOptimal placement of piezo-electric sensors and actuators for controlled flexible linkagemechanismsrdquo Journal of Vibration and Acoustics Transactionsof the ASME vol 128 no 2 pp 256ndash260 2006

[12] G Gatti M J Brennan and P Gardonio ldquoActive dampingof a beam using a physically collocated accelerometer andpiezoelectric patch actuatorrdquo Journal of Sound and Vibrationvol 303 no 3ndash5 pp 798ndash813 2007

[13] M K Kwak S Heo and M Jeong ldquoDynamic modellingand active vibration controller design for a cylindrical shellequipped with piezoelectric sensors and actuatorsrdquo Journal ofSound and Vibration vol 321 no 3ndash5 pp 510ndash524 2009

[14] I Bruant L Gallimard and S Nikoukar ldquoOptimal piezoelectricactuator and sensor location for active vibration control usinggenetic algorithmrdquo Journal of Sound and Vibration vol 329 no10 pp 1615ndash1635 2010

[15] Y Fu J Wang and Y Mao ldquoNonlinear vibration and activecontrol of functionally graded beams with piezoelectric sensorsand actuatorsrdquo Journal of Intelligent Material Systems andStructures vol 22 no 18 pp 2093ndash2102 2011

[16] F Chen M Hong M Song and H Cui ldquoOptimal control ofa beam with discontinuously distributed piezoelectric sensorsand actuatorsrdquo Journal of Marine Science and Application vol11 no 1 pp 44ndash51 2012

[17] B Azvine G R Tomlinson and R J Wynne ldquoUse of activeconstrained-layer damping for controlling resonant vibrationrdquoSmart Materials and Structures vol 4 no 1 pp 1ndash6 1995

[18] A Baz and J Ro ldquoVibration control of plates with activeconstrained layer dampingrdquo SmartMaterials and Structures vol5 no 3 pp 272ndash280 1996

[19] C Chantalakhana and R Stanway ldquoActive constrained layerdamping of clamped-clamped plate vibrationsrdquo Journal ofSound and Vibration vol 241 no 5 pp 755ndash777 2001

[20] J Ro and A Baz ldquoOptimum placement and control ofactive constrained layer damping using modal strain energyapproachrdquo JVCJournal of Vibration and Control vol 8 no 6pp 861ndash876 2002

[21] M C Ray and J N Reddy ldquoOptimal control of thin circularcylindrical laminated composite shells using active constrainedlayer damping treatmentrdquo Smart Materials and Structures vol13 no 1 pp 64ndash72 2004

[22] C M A Vasques B R Mace P Gardonio and J D RodriguesldquoArbitrary active constrained layer damping treatments onbeams finite element modelling and experimental validationrdquoComputers and Structures vol 84 no 22-23 pp 1384ndash14012006

[23] F M Li K Kishimoto K Wang Z B Chen and W HHuang ldquoVibration control of beams with active constrainedlayer dampingrdquo Smart Materials and Structures vol 17 no 6Article ID 065036 2008

[24] L Yuan Y Xiang Y Huang and J Lu ldquoA semi-analyticalmethod and the circumferential dominant modal control ofcircular cylindrical shells with active constrained layer dampingtreatmentrdquo SmartMaterials and Structures vol 19 no 2 ArticleID 025010 2010

[25] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

Journal of Composites 17

[26] W A Smith and B A Auld ldquoModeling 1ndash3 composite piezo-electrics thickness-mode oscillationsrdquo IEEE Transactions onUltrasonics Ferroelectrics and Frequency Control vol 38 no 1pp 40ndash47 1991

[27] PiezocompositesMaterial Systems Inc 543 Great Road Little-ton MA 01460

[28] Y Benveniste and G J Dvorak ldquoUniform fields and universalrelations in piezoelectric compositesrdquo Journal of the Mechanicsand Physics of Solids vol 40 no 6 pp 1295ndash1312 1992

[29] M L Dunn and M Taya ldquoMicromechanics predictions ofthe effective electroelastic moduli of piezoelectric compositesrdquoInternational Journal of Solids and Structures vol 30 no 2 pp161ndash175 1993

[30] J Aboudi ldquoMicromechanical prediction of the effective coeffi-cients of thermo-piezoelectric multiphase compositesrdquo Journalof Intelligent Material Systems and Structures vol 9 no 9 pp713ndash722 1999

[31] X Ruan T W Chou A Safari and S C Danforth ldquoA 3-Dconnectivity model for effective piezoelectric properties of yarncompositesrdquo Journal of Composite Materials vol 36 no 14 pp1693ndash1708 2002

[32] A Baz and A Tempia ldquoActive piezoelectric damping compos-itesrdquo Sensors and Actuators A vol 112 no 2-3 pp 340ndash3502004

[33] H Berger S Kari U Gabbert et al ldquoUnit cell models ofpiezoelectric fiber composites for numerical and analytical cal-culation of effective propertiesrdquo SmartMaterials and Structuresvol 15 no 2 pp 451ndash458 2006

[34] M Sakthivel and A Arockiarajan ldquoAn analytical model forpredicting thermo-electro-mechanical response of 1ndash3 piezo-electric compositesrdquo Computational Materials Science vol 48no 4 pp 759ndash767 2010

[35] M C Ray and A K Pradhan ldquoOn the use of verticallyreinforced 1ndash3 piezoelectric composites for hybrid damping oflaminated composite platesrdquo Mechanics of Advanced Materialsand Structures vol 14 no 4 pp 245ndash261 2007

[36] M C Ray and A K Pradhan ldquoPerformance of vertically andobliquely reinforced 1ndash3 piezoelectric composites for activedamping of laminated composite shellsrdquo Journal of Sound andVibration vol 315 no 4-5 pp 816ndash835 2008

[37] S K Sarangi and M C Ray ldquoSmart damping of geometricallynonlinear vibrations of laminated composite beams using verti-cally reinforced 1ndash3 piezoelectric compositesrdquo Smart Materialsand Structures vol 19 no 7 Article ID 075020 2010

[38] A Krishnan and J V Deshpande ldquoVibration of skew laminatesrdquoJournal of Sound and Vibration vol 153 no 2 pp 351ndash358 1992

[39] A R Krishna Reddy and R Palaninathan ldquoFree vibration ofskew laminatesrdquo Computers and Structures vol 70 no 4 pp415ndash423 1999

[40] C S Babu and T Kant ldquoTwo shear deformable finite elementmodels for buckling analysis of skew fibre-reinforced compositeand sandwich panelsrdquo Composite Structures vol 46 no 2 pp115ndash124 1999

[41] A K Garg R K Khare and T Kant ldquoFree vibration of skewfiber-reinforced composite and sandwich laminates using ashear deformable finite element modelrdquo Journal of SandwichStructures and Materials vol 8 no 1 pp 33ndash53 2006

[42] J N Reddy Mechanics of Laminated Composites Plates Theoryand Analysis CRC press Boca Raton Fla USA 1996

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Nano

materials

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Journal ofNanomaterials

Page 10: Research Article Active Constrained Layer Damping of Smart …downloads.hindawi.com/archive/2013/824163.pdf · 2019-07-31 · Research Article Active Constrained Layer Damping of

10 Journal of Composites

Table 1 Comparison of fundamental natural frequencies (120596) of simply supported skew laminated composite plates integratedwith the patchesof negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 15699 28917 37325 14829 24656 32522 17974 33351 33351Present FEM 15635 24383 35033 15076 24380 32254 18493 33359 33370

15∘ Reference [41] 16874 30458 39600 15741 25351 30270 18313 32490 36724Present FEM 16571 29840 36505 15796 25775 29892 18675 32075 35810

30∘ Reference [41] 20840 34023 46997 18871 29372 34489 20270 34431 42361Present FEM 19596 31690 46796 18226 29585 32357 19894 32365 43208

45∘ Reference [41] 28925 41906 54149 25609 33126 40617 25609 33131 42772Present FEM 24811 44875 53289 22996 34773 44889 23194 34870 45009

lowast

120596 = (120596119886

2

120587

2

ℎ)radic120588119864119879 120596 is the circular natural frequency

Table 2 Comparison of fundamental natural frequencies (120596) of clamped-clamped skew laminated composite plates integrated with thepatches of negligible thickness

Skew angle Source

Symmetric cross-ply Antisymmetric cross-ply Antisymmetric angle-ply(90∘0∘90∘0∘90∘) (0∘90∘0∘90∘) (45∘minus45∘45∘minus45∘)

Modes Modes Modes1 2 3 1 2 3 1 2 3

0∘ Reference [41] 23687 35399 41122 22990 37880 37880 22119 37339 37339Present FEM 23201 34769 44102 23315 36531 36545 22433 36000 36012

15∘ Reference [41] 24663 36255 43418 23809 37516 40785 23099 36997 40438Present FEM 23699 34821 44049 23741 35856 38401 23049 35346 38092

30∘ Reference [41] 27921 39557 50220 26666 39851 47227 26325 39549 52107Present FEM 25366 35696 44734 25240 41943 45373 24945 36113 50932

45∘ Reference [41] 34739 47129 58789 33015 46290 58423 33015 46290 58423Present FEM 28665 47074 54562 28377 47614 56620 28377 47614 55162

frequency response functions the skew laminated compositeplates are harmonically excited by a force of amplitude 1Newton applied to a point (1198862 (1198864) cos120572 ℎ2) on the skewsubstrate platesThe control voltage supplied to the constrain-ing layer of the first patch is negatively proportional to thevelocity of the point (1198862 (1198864) cos120572 ℎ2) and that applied tothe constraining layer of the second patch is negatively pro-portional to the velocity of the point (1198862 (31198864) cos120572 ℎ2)

Figures 3 4 and 5 illustrate the frequency responsefunctions for the transverse displacement of a point of asimply supported symmetric cross-ply (0∘90∘0∘) plate with15∘ 30∘ and 45∘ skew angles respectively while the fiberorientation angle (120595) in the constraining layer is 0∘ Thesefigures display both uncontrolled and controlled responsesand clearly show that the active constraining layer made ofthe vertically reinforced 1ndash3 PZC material being studied heresignificantly attenuates the amplitude of vibrations enhanc-ing the damping characteristics of the overall laminatedskew composite plates over the passive damping (uncon-trolled) Similar responses are also obtained for antisym-metric cross-ply (0∘90∘0∘90∘) and antisymmetric angle-ply (45∘minus45∘45∘minus45∘) skew laminated composite plates

However for brevity these results are not presented hereThemaximum values of the control voltages required to computethe controlled responses presented in Figures 3ndash5 have beenfound to be quite low and are illustrated in Figure 6 in caseof the symmetric cross-ply (0∘90∘0∘) plate with skew angle15∘ only In order to examine the contribution of the verticalactuation in improving the damping characteristics of theskew laminated composite plates as demonstrated in Figures3ndash5 active control responses of the symmetric cross-plylaminated composite plate with 15∘ skew angle are plotted inFigure 7with andwithout considering the value of 119890

33and 119890

31

It may be noted that when the values of 11989031is zero and that of

11989033is nonzero the vertical actuation of the active constraining

layer of the ACLD treatment is responsible for increasing thetransverse shear deformations of the viscoelastic constrainedlayer over the passive counterpart resulting in improveddamping of the smart plate over its passive damping Onthe other hand if 119890

33is zero and 119890

31is nonzero the in-

plane actuation of the active constraining layer causes thetransverse shear deformation of the viscoelastic core of theACLD treatment leading to the active damping of the smartplate It is evident from Figure 7 that the contribution of the

Journal of Composites 11

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 3 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 4 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 30∘)

vertical actuation by the constraining vertically reinforced 1ndash3 PZC layer is significantly larger than that of the in-planeactuation by the same for controlling the modes displayedin Figure 7 Though not shown here similar results are alsoobtained for antisymmetric cross-ply and angle-ply skewplates

Figures 8 9 and 10 demonstrate the effect of variationof piezoelectric fiber orientation angle on the performanceof the patches for improving the damping characteristicsof simply supported symmetric cross-ply (0∘90∘0∘)

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 5 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 45∘)

0

5

10

15

20

25

30

35

40

45

50

Con

trol v

olta

ge (V

olt)

0 100 200 300 400 500 600Frequency (Hz)

119870119889 = 600

119870119889 = 1200

Figure 6 Frequency response functions for the control voltages foractive damping of the simply supported skew symmetric cross-ply(0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

antisymmetric cross-ply (0∘90∘0∘90∘) and angle-ply(45∘minus45∘45∘minus45∘) substrate skew composite plates with30∘ skew angle respectively while the orientation angle (120595)

of the piezoelectric fibers is varied in the vertical 119909119911-planeThese figures illustrate that the attenuating capability of thepatches becomesmaximumwhen the value of the orientationangle (120595) of the fibers in the 119909119911-plane is 0∘ Similar effectis also obtained when the fiber orientation angle (120595) in theconstraining layer of the ACLD treatment is varied in thevertical 119910119911-plane as illustrated in Figures 11 12 and 13 forthe symmetric cross-ply (0∘90∘0∘) antisymmetric cross-ply(0∘90∘0∘90∘) and angle-ply (45∘minus45∘45∘minus45∘) substrate

12 Journal of Composites

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

11989031 = minus01902 11989033 = 184107

11989031 = minus01902 11989033 = 011989031 = 0 11989033 = 184107

w(a2(b4)co

s120572h2

) (m

)

Figure 7 Frequency response functions demonstrating the con-tribution of vertical actuation on the controlled responses forthe transverse displacement 119908 (1198862 (1198864) cos120572 ℎ2) of the skewsymmetric cross-ply (0∘90∘0∘) plate (119870

119889

= 600 119886ℎ = 100 120595 = 0∘

120572 = 15∘)

0 100 200 300 400 500 600Frequency (Hz)

0

02

04

06

08

1

12times10minus4

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 8 Effect of variation of piezoelectric fiber orientation angle(120595) on the controlled responses of a simply supported symmetricskew cross-ply (0∘90∘0∘) plate when the piezoelectric fibers of theconstraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

skew composite plates with 30∘ skew angle respectivelyIn this regard it should be noted that for investigating theeffect of variation of piezoelectric fiber orientation angle(120595) on the performance of the ACLD patches the valueof 120595 has been smoothly varied to compute the controlledresponses However for the sake of clarity in the plots theresponses corresponding to the four specific values of 120595

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 9 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 10 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

have been presented such that the optimum performanceof the patches can be evaluated The significant effect of thevariation of the piezoelectric fiber orientation angle in thevertical 119909119911- and 119910119911-planes has been noticed when all theboundaries of the overall plates are clamped as shown inFigures 14ndash19 When the piezoelectric fibers are coplanarwith the vertical 119909119911-plane the frequency response functionsplotted in Figures 14 15 and 16 reveal that the control

Journal of Composites 13

0

02

04

06

08

1

12

14

16times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 11 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 12 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=600 119886ℎ = 100 120572 = 30

∘)

authority of the patches becomes maximum for improvingthe damping characteristics of the clamped-clamped skewcross-ply and angle-ply substrate plates if the value of thepiezoelectric fiber orientation angle (120595) is 30∘ The same isalso found to be true if the piezoelectric fibers are coplanarwith the vertical 119910119911-plane as shown in Figures 17 18 and 19for the previously mentioned skew plates Similar responsesare also found for the effect of piezoelectric fiber orientation

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 13 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers of the constraining layer are coplanar with the119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 14 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600119886ℎ = 100120572 = 30

∘)

angle on the active damping of other skew laminated plateswith different skew angles However for the sake of brevitythey are not presented here

5 Conclusions

In this paper a study has been carried out to investi-gate the performance of the ACLD treatment for active

14 Journal of Composites

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 15 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 16 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

damping of smart skew laminated composite plates whenverticallyobliquely reinforced 1ndash3 PZC materials are usedas the materials for the constraining layer of the ACLDtreatment A layerwise FSDT-based finite element modelhas been developed to describe the dynamics of the skewlaminated composite plates integrated with the patches ofACLD treatment Unlike the existing finite element modelsof smart structures integrated with ACLD treatment the

200 250 300 350 400 450 500 550 600 650 7000

02

04

06

08

1

12

14

16

18

2times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 17 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

200 300 400 500 600 700 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 18 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

derivation of the present finite element model includes thetransverse deformations of the substrate skew laminatedcomposite plate the constrained viscoelastic layer and theconstraining 1ndash3 PZC layer of the ACLD treatment along thethickness (ie 119911) direction such that both vertical and in-plane actuations by the constraining layer of the patches canbe utilized for active damping of the plates The frequency

Journal of Composites 15

0

05

1

15

2

25

3

35

4times10minus5

200 300 400 500 600 700 800Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 19 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

responses of the symmetric cross-ply antisymmetric cross-ply and angle-ply skew composite plates indicate that theactive constraining layer of the ACLD treatment being madeof the vertically reinforced 1ndash3 PZC material significantlyenhances the damping characteristics of the plates over thepassive damping The analysis revealed that if the verticallyreinforced 1ndash3 PZCmaterial is used for the constraining layerof the ACLD treatment then the contribution of the verticalactuation by the constraining layer alone for improving theactive damping characteristics of the smart skew laminatedcomposite plate is significantly larger than that due to thein-plane actuation by the constraining layer alone It isimportant to note from the present investigation that thevariation of the orientation angle of the piezoelectric fibersof the constraining layer in the vertical 119909119911- and 119910119911-planesas well as the skew angle and boundary conditions of theoverall plates significantly affects the performance of thepatches For the simply supported laminated substrate platesthe performance of the patches becomes maximumwhen thepiezoelectric fiber orientation angle (120595) is 0∘ while in caseof the clamped-clamped laminated skew substrate plates themaximum control authority of the patches is achieved whenthe value of 120595 is 30∘ irrespective of the cases in which thefibers are oriented in 119909119911- and 119910119911-planes

Appendix

In (6) and (7) the matrices [1198851] [119885

2] [119885

3] [119885

4] [119885

5] and

[1198856] are given by

[1198851] = [[119885

1] 0 0]

[1198852] = [(

ℎ

2) 119868 [119885

2] 0]

[1198853] = [(

ℎ

2) 119868 ℎV119868 [119885

3]]

[1198854] = [119868 0 0 119911119868 0 0]

[1198855] = [0 119868 0 (

ℎ

2) 119868 (119911 minus

ℎ

2) 119868 0]

[1198856] = [0 0 119868 (

ℎ

2) 119868 ℎV119868 (119911 minus ℎ

119873+2) 119868]

(A1)

in which

[1198851] =

[[[

[

119911 0 0 0

0 119911 0 0

0 0 119911 0

0 0 0 1

]]]

]

[1198852] =

[[[[[[[[

[

(119911 minusℎ

2) 0 0 0

0 (119911 minusℎ

2) 0 0

0 0 (119911 minusℎ

2) 0

0 0 0 1

]]]]]]]]

]

[1198853] =

[[[

[

(119911 minus ℎ119873+2

) 0 0 0

0 (119911 minus ℎ119873+2

) 0 0

0 0 (119911 minus ℎ119873+2

) 0

0 0 0 1

]]]

]

[119868] =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

[119868] = [1 0

0 1]

[0] = [0 0

0 0] [0] = [

0 0

0 0]

(A2)

The various submatrices 119861119905119887119894 119861

119903119887119894 and 119861

119903119904119894 appearing in (21)

are given by

119861119905119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 0

]]]]]]]]]]]

]

119861119905119904119894

=

[[[[

[

0 0120597120578

119894

120597119909

0 0120597120578

119894

120597119910

]]]]

]

16 Journal of Composites

119861119903119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 1

]]]]]]]]]]]

]

119861119903119887119894

=

[[[[[[

[

119861119903119887119894

0

0

0 119861119903119887119894

0

0

0 119861119903119887119894

]]]]]]

]

119861119903119904119894

=

[[[[[[[[[[[[[[[[[[

[

119868 0 0

0

119868 0

0 0

119868

119861119903119904119894

0 0

0 119861119903119904119894

0

0 0 119861119903119904119894

]]]]]]]]]]]]]]]]]]

]

119868= [1 0 0

0 1 0]

(A3)

wherein

0 and 0 are the (3 times 3) and (2 times 3) null matricesrespectively

References

[1] T Bailey and J E Hubbard ldquoDistributedpiezoelectric polymeractive vibration control of a cantilever beamrdquo Journal of Guid-ance Control and Dynamics vol 8 no 5 pp 605ndash611 1985

[2] E F Crawley and J de Luis ldquoUse of piezoelectric actuators aselements of intelligent structuresrdquo AIAA Journal vol 25 no 10pp 1373ndash1385 1987

[3] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988

[4] C K Lee W W Chiang and O Sulivan ldquoPiezoelectric modalsensoractuator pairs for critical active damping vibrationcontrolrdquo Journal of the Acoustical Society of America vol 90 no1 pp 374ndash384 1991

[5] S Hanagud M W Obal and A J Calise ldquoOptimal vibrationcontrol by the use of piezoceramic sensors and actuatorsrdquoJournal of Guidance Control and Dynamics vol 15 no 5 pp1199ndash1206 1992

[6] Y Gu R L Clark C R Fuller and A C Zander ldquoExperimentson active control of plate vibration using piezoelectric actuatorsand polyvinylidene fluoride (PVDF) modal sensorsrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 116 no3 pp 303ndash308 1994

[7] A Baz and S Poh ldquoOptimal vibration control with modalpositive position feedbackrdquo Optimal Control Applications andMethods vol 17 no 2 pp 141ndash149 1996

[8] A Kugi K Schlacher and H Irschik ldquoInfinite-dimensionalcontrol of nonlinear beam vibrations by piezoelectric actuatorand sensor layersrdquo Nonlinear Dynamics vol 19 no 1 pp 71ndash911999

[9] D Sun L Tong and D Wang ldquoVibration control of platesusing discretely distributed piezoelectric quasi-modal actua-torssensorsrdquo AIAA Journal vol 39 no 9 pp 1766ndash1772 2001

[10] G Caruso S Galeani and L Menini ldquoActive vibration controlof an elastic plate using multiple piezoelectric sensors andactuatorsrdquo SimulationModelling Practice andTheory vol 11 no5-6 pp 403ndash419 2003

[11] X Zhang and A G Erdman ldquoOptimal placement of piezo-electric sensors and actuators for controlled flexible linkagemechanismsrdquo Journal of Vibration and Acoustics Transactionsof the ASME vol 128 no 2 pp 256ndash260 2006

[12] G Gatti M J Brennan and P Gardonio ldquoActive dampingof a beam using a physically collocated accelerometer andpiezoelectric patch actuatorrdquo Journal of Sound and Vibrationvol 303 no 3ndash5 pp 798ndash813 2007

[13] M K Kwak S Heo and M Jeong ldquoDynamic modellingand active vibration controller design for a cylindrical shellequipped with piezoelectric sensors and actuatorsrdquo Journal ofSound and Vibration vol 321 no 3ndash5 pp 510ndash524 2009

[14] I Bruant L Gallimard and S Nikoukar ldquoOptimal piezoelectricactuator and sensor location for active vibration control usinggenetic algorithmrdquo Journal of Sound and Vibration vol 329 no10 pp 1615ndash1635 2010

[15] Y Fu J Wang and Y Mao ldquoNonlinear vibration and activecontrol of functionally graded beams with piezoelectric sensorsand actuatorsrdquo Journal of Intelligent Material Systems andStructures vol 22 no 18 pp 2093ndash2102 2011

[16] F Chen M Hong M Song and H Cui ldquoOptimal control ofa beam with discontinuously distributed piezoelectric sensorsand actuatorsrdquo Journal of Marine Science and Application vol11 no 1 pp 44ndash51 2012

[17] B Azvine G R Tomlinson and R J Wynne ldquoUse of activeconstrained-layer damping for controlling resonant vibrationrdquoSmart Materials and Structures vol 4 no 1 pp 1ndash6 1995

[18] A Baz and J Ro ldquoVibration control of plates with activeconstrained layer dampingrdquo SmartMaterials and Structures vol5 no 3 pp 272ndash280 1996

[19] C Chantalakhana and R Stanway ldquoActive constrained layerdamping of clamped-clamped plate vibrationsrdquo Journal ofSound and Vibration vol 241 no 5 pp 755ndash777 2001

[20] J Ro and A Baz ldquoOptimum placement and control ofactive constrained layer damping using modal strain energyapproachrdquo JVCJournal of Vibration and Control vol 8 no 6pp 861ndash876 2002

[21] M C Ray and J N Reddy ldquoOptimal control of thin circularcylindrical laminated composite shells using active constrainedlayer damping treatmentrdquo Smart Materials and Structures vol13 no 1 pp 64ndash72 2004

[22] C M A Vasques B R Mace P Gardonio and J D RodriguesldquoArbitrary active constrained layer damping treatments onbeams finite element modelling and experimental validationrdquoComputers and Structures vol 84 no 22-23 pp 1384ndash14012006

[23] F M Li K Kishimoto K Wang Z B Chen and W HHuang ldquoVibration control of beams with active constrainedlayer dampingrdquo Smart Materials and Structures vol 17 no 6Article ID 065036 2008

[24] L Yuan Y Xiang Y Huang and J Lu ldquoA semi-analyticalmethod and the circumferential dominant modal control ofcircular cylindrical shells with active constrained layer dampingtreatmentrdquo SmartMaterials and Structures vol 19 no 2 ArticleID 025010 2010

[25] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

Journal of Composites 17

[26] W A Smith and B A Auld ldquoModeling 1ndash3 composite piezo-electrics thickness-mode oscillationsrdquo IEEE Transactions onUltrasonics Ferroelectrics and Frequency Control vol 38 no 1pp 40ndash47 1991

[27] PiezocompositesMaterial Systems Inc 543 Great Road Little-ton MA 01460

[28] Y Benveniste and G J Dvorak ldquoUniform fields and universalrelations in piezoelectric compositesrdquo Journal of the Mechanicsand Physics of Solids vol 40 no 6 pp 1295ndash1312 1992

[29] M L Dunn and M Taya ldquoMicromechanics predictions ofthe effective electroelastic moduli of piezoelectric compositesrdquoInternational Journal of Solids and Structures vol 30 no 2 pp161ndash175 1993

[30] J Aboudi ldquoMicromechanical prediction of the effective coeffi-cients of thermo-piezoelectric multiphase compositesrdquo Journalof Intelligent Material Systems and Structures vol 9 no 9 pp713ndash722 1999

[31] X Ruan T W Chou A Safari and S C Danforth ldquoA 3-Dconnectivity model for effective piezoelectric properties of yarncompositesrdquo Journal of Composite Materials vol 36 no 14 pp1693ndash1708 2002

[32] A Baz and A Tempia ldquoActive piezoelectric damping compos-itesrdquo Sensors and Actuators A vol 112 no 2-3 pp 340ndash3502004

[33] H Berger S Kari U Gabbert et al ldquoUnit cell models ofpiezoelectric fiber composites for numerical and analytical cal-culation of effective propertiesrdquo SmartMaterials and Structuresvol 15 no 2 pp 451ndash458 2006

[34] M Sakthivel and A Arockiarajan ldquoAn analytical model forpredicting thermo-electro-mechanical response of 1ndash3 piezo-electric compositesrdquo Computational Materials Science vol 48no 4 pp 759ndash767 2010

[35] M C Ray and A K Pradhan ldquoOn the use of verticallyreinforced 1ndash3 piezoelectric composites for hybrid damping oflaminated composite platesrdquo Mechanics of Advanced Materialsand Structures vol 14 no 4 pp 245ndash261 2007

[36] M C Ray and A K Pradhan ldquoPerformance of vertically andobliquely reinforced 1ndash3 piezoelectric composites for activedamping of laminated composite shellsrdquo Journal of Sound andVibration vol 315 no 4-5 pp 816ndash835 2008

[37] S K Sarangi and M C Ray ldquoSmart damping of geometricallynonlinear vibrations of laminated composite beams using verti-cally reinforced 1ndash3 piezoelectric compositesrdquo Smart Materialsand Structures vol 19 no 7 Article ID 075020 2010

[38] A Krishnan and J V Deshpande ldquoVibration of skew laminatesrdquoJournal of Sound and Vibration vol 153 no 2 pp 351ndash358 1992

[39] A R Krishna Reddy and R Palaninathan ldquoFree vibration ofskew laminatesrdquo Computers and Structures vol 70 no 4 pp415ndash423 1999

[40] C S Babu and T Kant ldquoTwo shear deformable finite elementmodels for buckling analysis of skew fibre-reinforced compositeand sandwich panelsrdquo Composite Structures vol 46 no 2 pp115ndash124 1999

[41] A K Garg R K Khare and T Kant ldquoFree vibration of skewfiber-reinforced composite and sandwich laminates using ashear deformable finite element modelrdquo Journal of SandwichStructures and Materials vol 8 no 1 pp 33ndash53 2006

[42] J N Reddy Mechanics of Laminated Composites Plates Theoryand Analysis CRC press Boca Raton Fla USA 1996

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 11: Research Article Active Constrained Layer Damping of Smart …downloads.hindawi.com/archive/2013/824163.pdf · 2019-07-31 · Research Article Active Constrained Layer Damping of

Journal of Composites 11

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 3 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

12

0 100 200 300 400 500 600Frequency (Hz)

times10minus4

0

02

04

06

08

1

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 4 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 30∘)

vertical actuation by the constraining vertically reinforced 1ndash3 PZC layer is significantly larger than that of the in-planeactuation by the same for controlling the modes displayedin Figure 7 Though not shown here similar results are alsoobtained for antisymmetric cross-ply and angle-ply skewplates

Figures 8 9 and 10 demonstrate the effect of variationof piezoelectric fiber orientation angle on the performanceof the patches for improving the damping characteristicsof simply supported symmetric cross-ply (0∘90∘0∘)

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

Uncontrolled119870119889 = 600

119870119889 = 1200

w(a2(b4)co

s120572h2

) (m

)

Figure 5 Frequency response functions for the transverse displace-ment119908 (1198862 (1198864) cos120572 ℎ2) of a simply supported skew symmetriccross-ply (0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 45∘)

0

5

10

15

20

25

30

35

40

45

50

Con

trol v

olta

ge (V

olt)

0 100 200 300 400 500 600Frequency (Hz)

119870119889 = 600

119870119889 = 1200

Figure 6 Frequency response functions for the control voltages foractive damping of the simply supported skew symmetric cross-ply(0∘90∘0∘) plate (119886ℎ = 100 120595 = 0

∘ 120572 = 15∘)

antisymmetric cross-ply (0∘90∘0∘90∘) and angle-ply(45∘minus45∘45∘minus45∘) substrate skew composite plates with30∘ skew angle respectively while the orientation angle (120595)

of the piezoelectric fibers is varied in the vertical 119909119911-planeThese figures illustrate that the attenuating capability of thepatches becomesmaximumwhen the value of the orientationangle (120595) of the fibers in the 119909119911-plane is 0∘ Similar effectis also obtained when the fiber orientation angle (120595) in theconstraining layer of the ACLD treatment is varied in thevertical 119910119911-plane as illustrated in Figures 11 12 and 13 forthe symmetric cross-ply (0∘90∘0∘) antisymmetric cross-ply(0∘90∘0∘90∘) and angle-ply (45∘minus45∘45∘minus45∘) substrate

12 Journal of Composites

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

11989031 = minus01902 11989033 = 184107

11989031 = minus01902 11989033 = 011989031 = 0 11989033 = 184107

w(a2(b4)co

s120572h2

) (m

)

Figure 7 Frequency response functions demonstrating the con-tribution of vertical actuation on the controlled responses forthe transverse displacement 119908 (1198862 (1198864) cos120572 ℎ2) of the skewsymmetric cross-ply (0∘90∘0∘) plate (119870

119889

= 600 119886ℎ = 100 120595 = 0∘

120572 = 15∘)

0 100 200 300 400 500 600Frequency (Hz)

0

02

04

06

08

1

12times10minus4

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 8 Effect of variation of piezoelectric fiber orientation angle(120595) on the controlled responses of a simply supported symmetricskew cross-ply (0∘90∘0∘) plate when the piezoelectric fibers of theconstraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

skew composite plates with 30∘ skew angle respectivelyIn this regard it should be noted that for investigating theeffect of variation of piezoelectric fiber orientation angle(120595) on the performance of the ACLD patches the valueof 120595 has been smoothly varied to compute the controlledresponses However for the sake of clarity in the plots theresponses corresponding to the four specific values of 120595

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 9 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 10 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

have been presented such that the optimum performanceof the patches can be evaluated The significant effect of thevariation of the piezoelectric fiber orientation angle in thevertical 119909119911- and 119910119911-planes has been noticed when all theboundaries of the overall plates are clamped as shown inFigures 14ndash19 When the piezoelectric fibers are coplanarwith the vertical 119909119911-plane the frequency response functionsplotted in Figures 14 15 and 16 reveal that the control

Journal of Composites 13

0

02

04

06

08

1

12

14

16times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 11 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 12 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=600 119886ℎ = 100 120572 = 30

∘)

authority of the patches becomes maximum for improvingthe damping characteristics of the clamped-clamped skewcross-ply and angle-ply substrate plates if the value of thepiezoelectric fiber orientation angle (120595) is 30∘ The same isalso found to be true if the piezoelectric fibers are coplanarwith the vertical 119910119911-plane as shown in Figures 17 18 and 19for the previously mentioned skew plates Similar responsesare also found for the effect of piezoelectric fiber orientation

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 13 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers of the constraining layer are coplanar with the119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 14 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600119886ℎ = 100120572 = 30

∘)

angle on the active damping of other skew laminated plateswith different skew angles However for the sake of brevitythey are not presented here

5 Conclusions

In this paper a study has been carried out to investi-gate the performance of the ACLD treatment for active

14 Journal of Composites

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 15 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 16 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

damping of smart skew laminated composite plates whenverticallyobliquely reinforced 1ndash3 PZC materials are usedas the materials for the constraining layer of the ACLDtreatment A layerwise FSDT-based finite element modelhas been developed to describe the dynamics of the skewlaminated composite plates integrated with the patches ofACLD treatment Unlike the existing finite element modelsof smart structures integrated with ACLD treatment the

200 250 300 350 400 450 500 550 600 650 7000

02

04

06

08

1

12

14

16

18

2times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 17 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

200 300 400 500 600 700 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 18 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

derivation of the present finite element model includes thetransverse deformations of the substrate skew laminatedcomposite plate the constrained viscoelastic layer and theconstraining 1ndash3 PZC layer of the ACLD treatment along thethickness (ie 119911) direction such that both vertical and in-plane actuations by the constraining layer of the patches canbe utilized for active damping of the plates The frequency

Journal of Composites 15

0

05

1

15

2

25

3

35

4times10minus5

200 300 400 500 600 700 800Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 19 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

responses of the symmetric cross-ply antisymmetric cross-ply and angle-ply skew composite plates indicate that theactive constraining layer of the ACLD treatment being madeof the vertically reinforced 1ndash3 PZC material significantlyenhances the damping characteristics of the plates over thepassive damping The analysis revealed that if the verticallyreinforced 1ndash3 PZCmaterial is used for the constraining layerof the ACLD treatment then the contribution of the verticalactuation by the constraining layer alone for improving theactive damping characteristics of the smart skew laminatedcomposite plate is significantly larger than that due to thein-plane actuation by the constraining layer alone It isimportant to note from the present investigation that thevariation of the orientation angle of the piezoelectric fibersof the constraining layer in the vertical 119909119911- and 119910119911-planesas well as the skew angle and boundary conditions of theoverall plates significantly affects the performance of thepatches For the simply supported laminated substrate platesthe performance of the patches becomes maximumwhen thepiezoelectric fiber orientation angle (120595) is 0∘ while in caseof the clamped-clamped laminated skew substrate plates themaximum control authority of the patches is achieved whenthe value of 120595 is 30∘ irrespective of the cases in which thefibers are oriented in 119909119911- and 119910119911-planes

Appendix

In (6) and (7) the matrices [1198851] [119885

2] [119885

3] [119885

4] [119885

5] and

[1198856] are given by

[1198851] = [[119885

1] 0 0]

[1198852] = [(

ℎ

2) 119868 [119885

2] 0]

[1198853] = [(

ℎ

2) 119868 ℎV119868 [119885

3]]

[1198854] = [119868 0 0 119911119868 0 0]

[1198855] = [0 119868 0 (

ℎ

2) 119868 (119911 minus

ℎ

2) 119868 0]

[1198856] = [0 0 119868 (

ℎ

2) 119868 ℎV119868 (119911 minus ℎ

119873+2) 119868]

(A1)

in which

[1198851] =

[[[

[

119911 0 0 0

0 119911 0 0

0 0 119911 0

0 0 0 1

]]]

]

[1198852] =

[[[[[[[[

[

(119911 minusℎ

2) 0 0 0

0 (119911 minusℎ

2) 0 0

0 0 (119911 minusℎ

2) 0

0 0 0 1

]]]]]]]]

]

[1198853] =

[[[

[

(119911 minus ℎ119873+2

) 0 0 0

0 (119911 minus ℎ119873+2

) 0 0

0 0 (119911 minus ℎ119873+2

) 0

0 0 0 1

]]]

]

[119868] =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

[119868] = [1 0

0 1]

[0] = [0 0

0 0] [0] = [

0 0

0 0]

(A2)

The various submatrices 119861119905119887119894 119861

119903119887119894 and 119861

119903119904119894 appearing in (21)

are given by

119861119905119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 0

]]]]]]]]]]]

]

119861119905119904119894

=

[[[[

[

0 0120597120578

119894

120597119909

0 0120597120578

119894

120597119910

]]]]

]

16 Journal of Composites

119861119903119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 1

]]]]]]]]]]]

]

119861119903119887119894

=

[[[[[[

[

119861119903119887119894

0

0

0 119861119903119887119894

0

0

0 119861119903119887119894

]]]]]]

]

119861119903119904119894

=

[[[[[[[[[[[[[[[[[[

[

119868 0 0

0

119868 0

0 0

119868

119861119903119904119894

0 0

0 119861119903119904119894

0

0 0 119861119903119904119894

]]]]]]]]]]]]]]]]]]

]

119868= [1 0 0

0 1 0]

(A3)

wherein

0 and 0 are the (3 times 3) and (2 times 3) null matricesrespectively

References

[1] T Bailey and J E Hubbard ldquoDistributedpiezoelectric polymeractive vibration control of a cantilever beamrdquo Journal of Guid-ance Control and Dynamics vol 8 no 5 pp 605ndash611 1985

[2] E F Crawley and J de Luis ldquoUse of piezoelectric actuators aselements of intelligent structuresrdquo AIAA Journal vol 25 no 10pp 1373ndash1385 1987

[3] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988

[4] C K Lee W W Chiang and O Sulivan ldquoPiezoelectric modalsensoractuator pairs for critical active damping vibrationcontrolrdquo Journal of the Acoustical Society of America vol 90 no1 pp 374ndash384 1991

[5] S Hanagud M W Obal and A J Calise ldquoOptimal vibrationcontrol by the use of piezoceramic sensors and actuatorsrdquoJournal of Guidance Control and Dynamics vol 15 no 5 pp1199ndash1206 1992

[6] Y Gu R L Clark C R Fuller and A C Zander ldquoExperimentson active control of plate vibration using piezoelectric actuatorsand polyvinylidene fluoride (PVDF) modal sensorsrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 116 no3 pp 303ndash308 1994

[7] A Baz and S Poh ldquoOptimal vibration control with modalpositive position feedbackrdquo Optimal Control Applications andMethods vol 17 no 2 pp 141ndash149 1996

[8] A Kugi K Schlacher and H Irschik ldquoInfinite-dimensionalcontrol of nonlinear beam vibrations by piezoelectric actuatorand sensor layersrdquo Nonlinear Dynamics vol 19 no 1 pp 71ndash911999

[9] D Sun L Tong and D Wang ldquoVibration control of platesusing discretely distributed piezoelectric quasi-modal actua-torssensorsrdquo AIAA Journal vol 39 no 9 pp 1766ndash1772 2001

[10] G Caruso S Galeani and L Menini ldquoActive vibration controlof an elastic plate using multiple piezoelectric sensors andactuatorsrdquo SimulationModelling Practice andTheory vol 11 no5-6 pp 403ndash419 2003

[11] X Zhang and A G Erdman ldquoOptimal placement of piezo-electric sensors and actuators for controlled flexible linkagemechanismsrdquo Journal of Vibration and Acoustics Transactionsof the ASME vol 128 no 2 pp 256ndash260 2006

[12] G Gatti M J Brennan and P Gardonio ldquoActive dampingof a beam using a physically collocated accelerometer andpiezoelectric patch actuatorrdquo Journal of Sound and Vibrationvol 303 no 3ndash5 pp 798ndash813 2007

[13] M K Kwak S Heo and M Jeong ldquoDynamic modellingand active vibration controller design for a cylindrical shellequipped with piezoelectric sensors and actuatorsrdquo Journal ofSound and Vibration vol 321 no 3ndash5 pp 510ndash524 2009

[14] I Bruant L Gallimard and S Nikoukar ldquoOptimal piezoelectricactuator and sensor location for active vibration control usinggenetic algorithmrdquo Journal of Sound and Vibration vol 329 no10 pp 1615ndash1635 2010

[15] Y Fu J Wang and Y Mao ldquoNonlinear vibration and activecontrol of functionally graded beams with piezoelectric sensorsand actuatorsrdquo Journal of Intelligent Material Systems andStructures vol 22 no 18 pp 2093ndash2102 2011

[16] F Chen M Hong M Song and H Cui ldquoOptimal control ofa beam with discontinuously distributed piezoelectric sensorsand actuatorsrdquo Journal of Marine Science and Application vol11 no 1 pp 44ndash51 2012

[17] B Azvine G R Tomlinson and R J Wynne ldquoUse of activeconstrained-layer damping for controlling resonant vibrationrdquoSmart Materials and Structures vol 4 no 1 pp 1ndash6 1995

[18] A Baz and J Ro ldquoVibration control of plates with activeconstrained layer dampingrdquo SmartMaterials and Structures vol5 no 3 pp 272ndash280 1996

[19] C Chantalakhana and R Stanway ldquoActive constrained layerdamping of clamped-clamped plate vibrationsrdquo Journal ofSound and Vibration vol 241 no 5 pp 755ndash777 2001

[20] J Ro and A Baz ldquoOptimum placement and control ofactive constrained layer damping using modal strain energyapproachrdquo JVCJournal of Vibration and Control vol 8 no 6pp 861ndash876 2002

[21] M C Ray and J N Reddy ldquoOptimal control of thin circularcylindrical laminated composite shells using active constrainedlayer damping treatmentrdquo Smart Materials and Structures vol13 no 1 pp 64ndash72 2004

[22] C M A Vasques B R Mace P Gardonio and J D RodriguesldquoArbitrary active constrained layer damping treatments onbeams finite element modelling and experimental validationrdquoComputers and Structures vol 84 no 22-23 pp 1384ndash14012006

[23] F M Li K Kishimoto K Wang Z B Chen and W HHuang ldquoVibration control of beams with active constrainedlayer dampingrdquo Smart Materials and Structures vol 17 no 6Article ID 065036 2008

[24] L Yuan Y Xiang Y Huang and J Lu ldquoA semi-analyticalmethod and the circumferential dominant modal control ofcircular cylindrical shells with active constrained layer dampingtreatmentrdquo SmartMaterials and Structures vol 19 no 2 ArticleID 025010 2010

[25] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

Journal of Composites 17

[26] W A Smith and B A Auld ldquoModeling 1ndash3 composite piezo-electrics thickness-mode oscillationsrdquo IEEE Transactions onUltrasonics Ferroelectrics and Frequency Control vol 38 no 1pp 40ndash47 1991

[27] PiezocompositesMaterial Systems Inc 543 Great Road Little-ton MA 01460

[28] Y Benveniste and G J Dvorak ldquoUniform fields and universalrelations in piezoelectric compositesrdquo Journal of the Mechanicsand Physics of Solids vol 40 no 6 pp 1295ndash1312 1992

[29] M L Dunn and M Taya ldquoMicromechanics predictions ofthe effective electroelastic moduli of piezoelectric compositesrdquoInternational Journal of Solids and Structures vol 30 no 2 pp161ndash175 1993

[30] J Aboudi ldquoMicromechanical prediction of the effective coeffi-cients of thermo-piezoelectric multiphase compositesrdquo Journalof Intelligent Material Systems and Structures vol 9 no 9 pp713ndash722 1999

[31] X Ruan T W Chou A Safari and S C Danforth ldquoA 3-Dconnectivity model for effective piezoelectric properties of yarncompositesrdquo Journal of Composite Materials vol 36 no 14 pp1693ndash1708 2002

[32] A Baz and A Tempia ldquoActive piezoelectric damping compos-itesrdquo Sensors and Actuators A vol 112 no 2-3 pp 340ndash3502004

[33] H Berger S Kari U Gabbert et al ldquoUnit cell models ofpiezoelectric fiber composites for numerical and analytical cal-culation of effective propertiesrdquo SmartMaterials and Structuresvol 15 no 2 pp 451ndash458 2006

[34] M Sakthivel and A Arockiarajan ldquoAn analytical model forpredicting thermo-electro-mechanical response of 1ndash3 piezo-electric compositesrdquo Computational Materials Science vol 48no 4 pp 759ndash767 2010

[35] M C Ray and A K Pradhan ldquoOn the use of verticallyreinforced 1ndash3 piezoelectric composites for hybrid damping oflaminated composite platesrdquo Mechanics of Advanced Materialsand Structures vol 14 no 4 pp 245ndash261 2007

[36] M C Ray and A K Pradhan ldquoPerformance of vertically andobliquely reinforced 1ndash3 piezoelectric composites for activedamping of laminated composite shellsrdquo Journal of Sound andVibration vol 315 no 4-5 pp 816ndash835 2008

[37] S K Sarangi and M C Ray ldquoSmart damping of geometricallynonlinear vibrations of laminated composite beams using verti-cally reinforced 1ndash3 piezoelectric compositesrdquo Smart Materialsand Structures vol 19 no 7 Article ID 075020 2010

[38] A Krishnan and J V Deshpande ldquoVibration of skew laminatesrdquoJournal of Sound and Vibration vol 153 no 2 pp 351ndash358 1992

[39] A R Krishna Reddy and R Palaninathan ldquoFree vibration ofskew laminatesrdquo Computers and Structures vol 70 no 4 pp415ndash423 1999

[40] C S Babu and T Kant ldquoTwo shear deformable finite elementmodels for buckling analysis of skew fibre-reinforced compositeand sandwich panelsrdquo Composite Structures vol 46 no 2 pp115ndash124 1999

[41] A K Garg R K Khare and T Kant ldquoFree vibration of skewfiber-reinforced composite and sandwich laminates using ashear deformable finite element modelrdquo Journal of SandwichStructures and Materials vol 8 no 1 pp 33ndash53 2006

[42] J N Reddy Mechanics of Laminated Composites Plates Theoryand Analysis CRC press Boca Raton Fla USA 1996

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 12: Research Article Active Constrained Layer Damping of Smart …downloads.hindawi.com/archive/2013/824163.pdf · 2019-07-31 · Research Article Active Constrained Layer Damping of

12 Journal of Composites

12times10minus4

0

02

04

06

08

1

0 100 200 300 400 500 600Frequency (Hz)

11989031 = minus01902 11989033 = 184107

11989031 = minus01902 11989033 = 011989031 = 0 11989033 = 184107

w(a2(b4)co

s120572h2

) (m

)

Figure 7 Frequency response functions demonstrating the con-tribution of vertical actuation on the controlled responses forthe transverse displacement 119908 (1198862 (1198864) cos120572 ℎ2) of the skewsymmetric cross-ply (0∘90∘0∘) plate (119870

119889

= 600 119886ℎ = 100 120595 = 0∘

120572 = 15∘)

0 100 200 300 400 500 600Frequency (Hz)

0

02

04

06

08

1

12times10minus4

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 8 Effect of variation of piezoelectric fiber orientation angle(120595) on the controlled responses of a simply supported symmetricskew cross-ply (0∘90∘0∘) plate when the piezoelectric fibers of theconstraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

skew composite plates with 30∘ skew angle respectivelyIn this regard it should be noted that for investigating theeffect of variation of piezoelectric fiber orientation angle(120595) on the performance of the ACLD patches the valueof 120595 has been smoothly varied to compute the controlledresponses However for the sake of clarity in the plots theresponses corresponding to the four specific values of 120595

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 9 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 10 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ =

100 120572 = 30∘)

have been presented such that the optimum performanceof the patches can be evaluated The significant effect of thevariation of the piezoelectric fiber orientation angle in thevertical 119909119911- and 119910119911-planes has been noticed when all theboundaries of the overall plates are clamped as shown inFigures 14ndash19 When the piezoelectric fibers are coplanarwith the vertical 119909119911-plane the frequency response functionsplotted in Figures 14 15 and 16 reveal that the control

Journal of Composites 13

0

02

04

06

08

1

12

14

16times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 11 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 12 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=600 119886ℎ = 100 120572 = 30

∘)

authority of the patches becomes maximum for improvingthe damping characteristics of the clamped-clamped skewcross-ply and angle-ply substrate plates if the value of thepiezoelectric fiber orientation angle (120595) is 30∘ The same isalso found to be true if the piezoelectric fibers are coplanarwith the vertical 119910119911-plane as shown in Figures 17 18 and 19for the previously mentioned skew plates Similar responsesare also found for the effect of piezoelectric fiber orientation

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 13 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers of the constraining layer are coplanar with the119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 14 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600119886ℎ = 100120572 = 30

∘)

angle on the active damping of other skew laminated plateswith different skew angles However for the sake of brevitythey are not presented here

5 Conclusions

In this paper a study has been carried out to investi-gate the performance of the ACLD treatment for active

14 Journal of Composites

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 15 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 16 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

damping of smart skew laminated composite plates whenverticallyobliquely reinforced 1ndash3 PZC materials are usedas the materials for the constraining layer of the ACLDtreatment A layerwise FSDT-based finite element modelhas been developed to describe the dynamics of the skewlaminated composite plates integrated with the patches ofACLD treatment Unlike the existing finite element modelsof smart structures integrated with ACLD treatment the

200 250 300 350 400 450 500 550 600 650 7000

02

04

06

08

1

12

14

16

18

2times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 17 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

200 300 400 500 600 700 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 18 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

derivation of the present finite element model includes thetransverse deformations of the substrate skew laminatedcomposite plate the constrained viscoelastic layer and theconstraining 1ndash3 PZC layer of the ACLD treatment along thethickness (ie 119911) direction such that both vertical and in-plane actuations by the constraining layer of the patches canbe utilized for active damping of the plates The frequency

Journal of Composites 15

0

05

1

15

2

25

3

35

4times10minus5

200 300 400 500 600 700 800Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 19 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

responses of the symmetric cross-ply antisymmetric cross-ply and angle-ply skew composite plates indicate that theactive constraining layer of the ACLD treatment being madeof the vertically reinforced 1ndash3 PZC material significantlyenhances the damping characteristics of the plates over thepassive damping The analysis revealed that if the verticallyreinforced 1ndash3 PZCmaterial is used for the constraining layerof the ACLD treatment then the contribution of the verticalactuation by the constraining layer alone for improving theactive damping characteristics of the smart skew laminatedcomposite plate is significantly larger than that due to thein-plane actuation by the constraining layer alone It isimportant to note from the present investigation that thevariation of the orientation angle of the piezoelectric fibersof the constraining layer in the vertical 119909119911- and 119910119911-planesas well as the skew angle and boundary conditions of theoverall plates significantly affects the performance of thepatches For the simply supported laminated substrate platesthe performance of the patches becomes maximumwhen thepiezoelectric fiber orientation angle (120595) is 0∘ while in caseof the clamped-clamped laminated skew substrate plates themaximum control authority of the patches is achieved whenthe value of 120595 is 30∘ irrespective of the cases in which thefibers are oriented in 119909119911- and 119910119911-planes

Appendix

In (6) and (7) the matrices [1198851] [119885

2] [119885

3] [119885

4] [119885

5] and

[1198856] are given by

[1198851] = [[119885

1] 0 0]

[1198852] = [(

ℎ

2) 119868 [119885

2] 0]

[1198853] = [(

ℎ

2) 119868 ℎV119868 [119885

3]]

[1198854] = [119868 0 0 119911119868 0 0]

[1198855] = [0 119868 0 (

ℎ

2) 119868 (119911 minus

ℎ

2) 119868 0]

[1198856] = [0 0 119868 (

ℎ

2) 119868 ℎV119868 (119911 minus ℎ

119873+2) 119868]

(A1)

in which

[1198851] =

[[[

[

119911 0 0 0

0 119911 0 0

0 0 119911 0

0 0 0 1

]]]

]

[1198852] =

[[[[[[[[

[

(119911 minusℎ

2) 0 0 0

0 (119911 minusℎ

2) 0 0

0 0 (119911 minusℎ

2) 0

0 0 0 1

]]]]]]]]

]

[1198853] =

[[[

[

(119911 minus ℎ119873+2

) 0 0 0

0 (119911 minus ℎ119873+2

) 0 0

0 0 (119911 minus ℎ119873+2

) 0

0 0 0 1

]]]

]

[119868] =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

[119868] = [1 0

0 1]

[0] = [0 0

0 0] [0] = [

0 0

0 0]

(A2)

The various submatrices 119861119905119887119894 119861

119903119887119894 and 119861

119903119904119894 appearing in (21)

are given by

119861119905119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 0

]]]]]]]]]]]

]

119861119905119904119894

=

[[[[

[

0 0120597120578

119894

120597119909

0 0120597120578

119894

120597119910

]]]]

]

16 Journal of Composites

119861119903119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 1

]]]]]]]]]]]

]

119861119903119887119894

=

[[[[[[

[

119861119903119887119894

0

0

0 119861119903119887119894

0

0

0 119861119903119887119894

]]]]]]

]

119861119903119904119894

=

[[[[[[[[[[[[[[[[[[

[

119868 0 0

0

119868 0

0 0

119868

119861119903119904119894

0 0

0 119861119903119904119894

0

0 0 119861119903119904119894

]]]]]]]]]]]]]]]]]]

]

119868= [1 0 0

0 1 0]

(A3)

wherein

0 and 0 are the (3 times 3) and (2 times 3) null matricesrespectively

References

[1] T Bailey and J E Hubbard ldquoDistributedpiezoelectric polymeractive vibration control of a cantilever beamrdquo Journal of Guid-ance Control and Dynamics vol 8 no 5 pp 605ndash611 1985

[2] E F Crawley and J de Luis ldquoUse of piezoelectric actuators aselements of intelligent structuresrdquo AIAA Journal vol 25 no 10pp 1373ndash1385 1987

[3] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988

[4] C K Lee W W Chiang and O Sulivan ldquoPiezoelectric modalsensoractuator pairs for critical active damping vibrationcontrolrdquo Journal of the Acoustical Society of America vol 90 no1 pp 374ndash384 1991

[5] S Hanagud M W Obal and A J Calise ldquoOptimal vibrationcontrol by the use of piezoceramic sensors and actuatorsrdquoJournal of Guidance Control and Dynamics vol 15 no 5 pp1199ndash1206 1992

[6] Y Gu R L Clark C R Fuller and A C Zander ldquoExperimentson active control of plate vibration using piezoelectric actuatorsand polyvinylidene fluoride (PVDF) modal sensorsrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 116 no3 pp 303ndash308 1994

[7] A Baz and S Poh ldquoOptimal vibration control with modalpositive position feedbackrdquo Optimal Control Applications andMethods vol 17 no 2 pp 141ndash149 1996

[8] A Kugi K Schlacher and H Irschik ldquoInfinite-dimensionalcontrol of nonlinear beam vibrations by piezoelectric actuatorand sensor layersrdquo Nonlinear Dynamics vol 19 no 1 pp 71ndash911999

[9] D Sun L Tong and D Wang ldquoVibration control of platesusing discretely distributed piezoelectric quasi-modal actua-torssensorsrdquo AIAA Journal vol 39 no 9 pp 1766ndash1772 2001

[10] G Caruso S Galeani and L Menini ldquoActive vibration controlof an elastic plate using multiple piezoelectric sensors andactuatorsrdquo SimulationModelling Practice andTheory vol 11 no5-6 pp 403ndash419 2003

[11] X Zhang and A G Erdman ldquoOptimal placement of piezo-electric sensors and actuators for controlled flexible linkagemechanismsrdquo Journal of Vibration and Acoustics Transactionsof the ASME vol 128 no 2 pp 256ndash260 2006

[12] G Gatti M J Brennan and P Gardonio ldquoActive dampingof a beam using a physically collocated accelerometer andpiezoelectric patch actuatorrdquo Journal of Sound and Vibrationvol 303 no 3ndash5 pp 798ndash813 2007

[13] M K Kwak S Heo and M Jeong ldquoDynamic modellingand active vibration controller design for a cylindrical shellequipped with piezoelectric sensors and actuatorsrdquo Journal ofSound and Vibration vol 321 no 3ndash5 pp 510ndash524 2009

[14] I Bruant L Gallimard and S Nikoukar ldquoOptimal piezoelectricactuator and sensor location for active vibration control usinggenetic algorithmrdquo Journal of Sound and Vibration vol 329 no10 pp 1615ndash1635 2010

[15] Y Fu J Wang and Y Mao ldquoNonlinear vibration and activecontrol of functionally graded beams with piezoelectric sensorsand actuatorsrdquo Journal of Intelligent Material Systems andStructures vol 22 no 18 pp 2093ndash2102 2011

[16] F Chen M Hong M Song and H Cui ldquoOptimal control ofa beam with discontinuously distributed piezoelectric sensorsand actuatorsrdquo Journal of Marine Science and Application vol11 no 1 pp 44ndash51 2012

[17] B Azvine G R Tomlinson and R J Wynne ldquoUse of activeconstrained-layer damping for controlling resonant vibrationrdquoSmart Materials and Structures vol 4 no 1 pp 1ndash6 1995

[18] A Baz and J Ro ldquoVibration control of plates with activeconstrained layer dampingrdquo SmartMaterials and Structures vol5 no 3 pp 272ndash280 1996

[19] C Chantalakhana and R Stanway ldquoActive constrained layerdamping of clamped-clamped plate vibrationsrdquo Journal ofSound and Vibration vol 241 no 5 pp 755ndash777 2001

[20] J Ro and A Baz ldquoOptimum placement and control ofactive constrained layer damping using modal strain energyapproachrdquo JVCJournal of Vibration and Control vol 8 no 6pp 861ndash876 2002

[21] M C Ray and J N Reddy ldquoOptimal control of thin circularcylindrical laminated composite shells using active constrainedlayer damping treatmentrdquo Smart Materials and Structures vol13 no 1 pp 64ndash72 2004

[22] C M A Vasques B R Mace P Gardonio and J D RodriguesldquoArbitrary active constrained layer damping treatments onbeams finite element modelling and experimental validationrdquoComputers and Structures vol 84 no 22-23 pp 1384ndash14012006

[23] F M Li K Kishimoto K Wang Z B Chen and W HHuang ldquoVibration control of beams with active constrainedlayer dampingrdquo Smart Materials and Structures vol 17 no 6Article ID 065036 2008

[24] L Yuan Y Xiang Y Huang and J Lu ldquoA semi-analyticalmethod and the circumferential dominant modal control ofcircular cylindrical shells with active constrained layer dampingtreatmentrdquo SmartMaterials and Structures vol 19 no 2 ArticleID 025010 2010

[25] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

Journal of Composites 17

[26] W A Smith and B A Auld ldquoModeling 1ndash3 composite piezo-electrics thickness-mode oscillationsrdquo IEEE Transactions onUltrasonics Ferroelectrics and Frequency Control vol 38 no 1pp 40ndash47 1991

[27] PiezocompositesMaterial Systems Inc 543 Great Road Little-ton MA 01460

[28] Y Benveniste and G J Dvorak ldquoUniform fields and universalrelations in piezoelectric compositesrdquo Journal of the Mechanicsand Physics of Solids vol 40 no 6 pp 1295ndash1312 1992

[29] M L Dunn and M Taya ldquoMicromechanics predictions ofthe effective electroelastic moduli of piezoelectric compositesrdquoInternational Journal of Solids and Structures vol 30 no 2 pp161ndash175 1993

[30] J Aboudi ldquoMicromechanical prediction of the effective coeffi-cients of thermo-piezoelectric multiphase compositesrdquo Journalof Intelligent Material Systems and Structures vol 9 no 9 pp713ndash722 1999

[31] X Ruan T W Chou A Safari and S C Danforth ldquoA 3-Dconnectivity model for effective piezoelectric properties of yarncompositesrdquo Journal of Composite Materials vol 36 no 14 pp1693ndash1708 2002

[32] A Baz and A Tempia ldquoActive piezoelectric damping compos-itesrdquo Sensors and Actuators A vol 112 no 2-3 pp 340ndash3502004

[33] H Berger S Kari U Gabbert et al ldquoUnit cell models ofpiezoelectric fiber composites for numerical and analytical cal-culation of effective propertiesrdquo SmartMaterials and Structuresvol 15 no 2 pp 451ndash458 2006

[34] M Sakthivel and A Arockiarajan ldquoAn analytical model forpredicting thermo-electro-mechanical response of 1ndash3 piezo-electric compositesrdquo Computational Materials Science vol 48no 4 pp 759ndash767 2010

[35] M C Ray and A K Pradhan ldquoOn the use of verticallyreinforced 1ndash3 piezoelectric composites for hybrid damping oflaminated composite platesrdquo Mechanics of Advanced Materialsand Structures vol 14 no 4 pp 245ndash261 2007

[36] M C Ray and A K Pradhan ldquoPerformance of vertically andobliquely reinforced 1ndash3 piezoelectric composites for activedamping of laminated composite shellsrdquo Journal of Sound andVibration vol 315 no 4-5 pp 816ndash835 2008

[37] S K Sarangi and M C Ray ldquoSmart damping of geometricallynonlinear vibrations of laminated composite beams using verti-cally reinforced 1ndash3 piezoelectric compositesrdquo Smart Materialsand Structures vol 19 no 7 Article ID 075020 2010

[38] A Krishnan and J V Deshpande ldquoVibration of skew laminatesrdquoJournal of Sound and Vibration vol 153 no 2 pp 351ndash358 1992

[39] A R Krishna Reddy and R Palaninathan ldquoFree vibration ofskew laminatesrdquo Computers and Structures vol 70 no 4 pp415ndash423 1999

[40] C S Babu and T Kant ldquoTwo shear deformable finite elementmodels for buckling analysis of skew fibre-reinforced compositeand sandwich panelsrdquo Composite Structures vol 46 no 2 pp115ndash124 1999

[41] A K Garg R K Khare and T Kant ldquoFree vibration of skewfiber-reinforced composite and sandwich laminates using ashear deformable finite element modelrdquo Journal of SandwichStructures and Materials vol 8 no 1 pp 33ndash53 2006

[42] J N Reddy Mechanics of Laminated Composites Plates Theoryand Analysis CRC press Boca Raton Fla USA 1996

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 13: Research Article Active Constrained Layer Damping of Smart …downloads.hindawi.com/archive/2013/824163.pdf · 2019-07-31 · Research Article Active Constrained Layer Damping of

Journal of Composites 13

0

02

04

06

08

1

12

14

16times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 11 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 12 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supported skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=600 119886ℎ = 100 120572 = 30

∘)

authority of the patches becomes maximum for improvingthe damping characteristics of the clamped-clamped skewcross-ply and angle-ply substrate plates if the value of thepiezoelectric fiber orientation angle (120595) is 30∘ The same isalso found to be true if the piezoelectric fibers are coplanarwith the vertical 119910119911-plane as shown in Figures 17 18 and 19for the previously mentioned skew plates Similar responsesare also found for the effect of piezoelectric fiber orientation

0

02

04

06

08

1

12times10minus4

0 100 200 300 400 500 600Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 13 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a simply supportedskew antisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when thepiezoelectric fibers of the constraining layer are coplanar with the119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 14 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600119886ℎ = 100120572 = 30

∘)

angle on the active damping of other skew laminated plateswith different skew angles However for the sake of brevitythey are not presented here

5 Conclusions

In this paper a study has been carried out to investi-gate the performance of the ACLD treatment for active

14 Journal of Composites

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 15 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 16 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

damping of smart skew laminated composite plates whenverticallyobliquely reinforced 1ndash3 PZC materials are usedas the materials for the constraining layer of the ACLDtreatment A layerwise FSDT-based finite element modelhas been developed to describe the dynamics of the skewlaminated composite plates integrated with the patches ofACLD treatment Unlike the existing finite element modelsof smart structures integrated with ACLD treatment the

200 250 300 350 400 450 500 550 600 650 7000

02

04

06

08

1

12

14

16

18

2times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 17 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

200 300 400 500 600 700 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 18 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

derivation of the present finite element model includes thetransverse deformations of the substrate skew laminatedcomposite plate the constrained viscoelastic layer and theconstraining 1ndash3 PZC layer of the ACLD treatment along thethickness (ie 119911) direction such that both vertical and in-plane actuations by the constraining layer of the patches canbe utilized for active damping of the plates The frequency

Journal of Composites 15

0

05

1

15

2

25

3

35

4times10minus5

200 300 400 500 600 700 800Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 19 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

responses of the symmetric cross-ply antisymmetric cross-ply and angle-ply skew composite plates indicate that theactive constraining layer of the ACLD treatment being madeof the vertically reinforced 1ndash3 PZC material significantlyenhances the damping characteristics of the plates over thepassive damping The analysis revealed that if the verticallyreinforced 1ndash3 PZCmaterial is used for the constraining layerof the ACLD treatment then the contribution of the verticalactuation by the constraining layer alone for improving theactive damping characteristics of the smart skew laminatedcomposite plate is significantly larger than that due to thein-plane actuation by the constraining layer alone It isimportant to note from the present investigation that thevariation of the orientation angle of the piezoelectric fibersof the constraining layer in the vertical 119909119911- and 119910119911-planesas well as the skew angle and boundary conditions of theoverall plates significantly affects the performance of thepatches For the simply supported laminated substrate platesthe performance of the patches becomes maximumwhen thepiezoelectric fiber orientation angle (120595) is 0∘ while in caseof the clamped-clamped laminated skew substrate plates themaximum control authority of the patches is achieved whenthe value of 120595 is 30∘ irrespective of the cases in which thefibers are oriented in 119909119911- and 119910119911-planes

Appendix

In (6) and (7) the matrices [1198851] [119885

2] [119885

3] [119885

4] [119885

5] and

[1198856] are given by

[1198851] = [[119885

1] 0 0]

[1198852] = [(

ℎ

2) 119868 [119885

2] 0]

[1198853] = [(

ℎ

2) 119868 ℎV119868 [119885

3]]

[1198854] = [119868 0 0 119911119868 0 0]

[1198855] = [0 119868 0 (

ℎ

2) 119868 (119911 minus

ℎ

2) 119868 0]

[1198856] = [0 0 119868 (

ℎ

2) 119868 ℎV119868 (119911 minus ℎ

119873+2) 119868]

(A1)

in which

[1198851] =

[[[

[

119911 0 0 0

0 119911 0 0

0 0 119911 0

0 0 0 1

]]]

]

[1198852] =

[[[[[[[[

[

(119911 minusℎ

2) 0 0 0

0 (119911 minusℎ

2) 0 0

0 0 (119911 minusℎ

2) 0

0 0 0 1

]]]]]]]]

]

[1198853] =

[[[

[

(119911 minus ℎ119873+2

) 0 0 0

0 (119911 minus ℎ119873+2

) 0 0

0 0 (119911 minus ℎ119873+2

) 0

0 0 0 1

]]]

]

[119868] =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

[119868] = [1 0

0 1]

[0] = [0 0

0 0] [0] = [

0 0

0 0]

(A2)

The various submatrices 119861119905119887119894 119861

119903119887119894 and 119861

119903119904119894 appearing in (21)

are given by

119861119905119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 0

]]]]]]]]]]]

]

119861119905119904119894

=

[[[[

[

0 0120597120578

119894

120597119909

0 0120597120578

119894

120597119910

]]]]

]

16 Journal of Composites

119861119903119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 1

]]]]]]]]]]]

]

119861119903119887119894

=

[[[[[[

[

119861119903119887119894

0

0

0 119861119903119887119894

0

0

0 119861119903119887119894

]]]]]]

]

119861119903119904119894

=

[[[[[[[[[[[[[[[[[[

[

119868 0 0

0

119868 0

0 0

119868

119861119903119904119894

0 0

0 119861119903119904119894

0

0 0 119861119903119904119894

]]]]]]]]]]]]]]]]]]

]

119868= [1 0 0

0 1 0]

(A3)

wherein

0 and 0 are the (3 times 3) and (2 times 3) null matricesrespectively

References

[1] T Bailey and J E Hubbard ldquoDistributedpiezoelectric polymeractive vibration control of a cantilever beamrdquo Journal of Guid-ance Control and Dynamics vol 8 no 5 pp 605ndash611 1985

[2] E F Crawley and J de Luis ldquoUse of piezoelectric actuators aselements of intelligent structuresrdquo AIAA Journal vol 25 no 10pp 1373ndash1385 1987

[3] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988

[4] C K Lee W W Chiang and O Sulivan ldquoPiezoelectric modalsensoractuator pairs for critical active damping vibrationcontrolrdquo Journal of the Acoustical Society of America vol 90 no1 pp 374ndash384 1991

[5] S Hanagud M W Obal and A J Calise ldquoOptimal vibrationcontrol by the use of piezoceramic sensors and actuatorsrdquoJournal of Guidance Control and Dynamics vol 15 no 5 pp1199ndash1206 1992

[6] Y Gu R L Clark C R Fuller and A C Zander ldquoExperimentson active control of plate vibration using piezoelectric actuatorsand polyvinylidene fluoride (PVDF) modal sensorsrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 116 no3 pp 303ndash308 1994

[7] A Baz and S Poh ldquoOptimal vibration control with modalpositive position feedbackrdquo Optimal Control Applications andMethods vol 17 no 2 pp 141ndash149 1996

[8] A Kugi K Schlacher and H Irschik ldquoInfinite-dimensionalcontrol of nonlinear beam vibrations by piezoelectric actuatorand sensor layersrdquo Nonlinear Dynamics vol 19 no 1 pp 71ndash911999

[9] D Sun L Tong and D Wang ldquoVibration control of platesusing discretely distributed piezoelectric quasi-modal actua-torssensorsrdquo AIAA Journal vol 39 no 9 pp 1766ndash1772 2001

[10] G Caruso S Galeani and L Menini ldquoActive vibration controlof an elastic plate using multiple piezoelectric sensors andactuatorsrdquo SimulationModelling Practice andTheory vol 11 no5-6 pp 403ndash419 2003

[11] X Zhang and A G Erdman ldquoOptimal placement of piezo-electric sensors and actuators for controlled flexible linkagemechanismsrdquo Journal of Vibration and Acoustics Transactionsof the ASME vol 128 no 2 pp 256ndash260 2006

[12] G Gatti M J Brennan and P Gardonio ldquoActive dampingof a beam using a physically collocated accelerometer andpiezoelectric patch actuatorrdquo Journal of Sound and Vibrationvol 303 no 3ndash5 pp 798ndash813 2007

[13] M K Kwak S Heo and M Jeong ldquoDynamic modellingand active vibration controller design for a cylindrical shellequipped with piezoelectric sensors and actuatorsrdquo Journal ofSound and Vibration vol 321 no 3ndash5 pp 510ndash524 2009

[14] I Bruant L Gallimard and S Nikoukar ldquoOptimal piezoelectricactuator and sensor location for active vibration control usinggenetic algorithmrdquo Journal of Sound and Vibration vol 329 no10 pp 1615ndash1635 2010

[15] Y Fu J Wang and Y Mao ldquoNonlinear vibration and activecontrol of functionally graded beams with piezoelectric sensorsand actuatorsrdquo Journal of Intelligent Material Systems andStructures vol 22 no 18 pp 2093ndash2102 2011

[16] F Chen M Hong M Song and H Cui ldquoOptimal control ofa beam with discontinuously distributed piezoelectric sensorsand actuatorsrdquo Journal of Marine Science and Application vol11 no 1 pp 44ndash51 2012

[17] B Azvine G R Tomlinson and R J Wynne ldquoUse of activeconstrained-layer damping for controlling resonant vibrationrdquoSmart Materials and Structures vol 4 no 1 pp 1ndash6 1995

[18] A Baz and J Ro ldquoVibration control of plates with activeconstrained layer dampingrdquo SmartMaterials and Structures vol5 no 3 pp 272ndash280 1996

[19] C Chantalakhana and R Stanway ldquoActive constrained layerdamping of clamped-clamped plate vibrationsrdquo Journal ofSound and Vibration vol 241 no 5 pp 755ndash777 2001

[20] J Ro and A Baz ldquoOptimum placement and control ofactive constrained layer damping using modal strain energyapproachrdquo JVCJournal of Vibration and Control vol 8 no 6pp 861ndash876 2002

[21] M C Ray and J N Reddy ldquoOptimal control of thin circularcylindrical laminated composite shells using active constrainedlayer damping treatmentrdquo Smart Materials and Structures vol13 no 1 pp 64ndash72 2004

[22] C M A Vasques B R Mace P Gardonio and J D RodriguesldquoArbitrary active constrained layer damping treatments onbeams finite element modelling and experimental validationrdquoComputers and Structures vol 84 no 22-23 pp 1384ndash14012006

[23] F M Li K Kishimoto K Wang Z B Chen and W HHuang ldquoVibration control of beams with active constrainedlayer dampingrdquo Smart Materials and Structures vol 17 no 6Article ID 065036 2008

[24] L Yuan Y Xiang Y Huang and J Lu ldquoA semi-analyticalmethod and the circumferential dominant modal control ofcircular cylindrical shells with active constrained layer dampingtreatmentrdquo SmartMaterials and Structures vol 19 no 2 ArticleID 025010 2010

[25] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

Journal of Composites 17

[26] W A Smith and B A Auld ldquoModeling 1ndash3 composite piezo-electrics thickness-mode oscillationsrdquo IEEE Transactions onUltrasonics Ferroelectrics and Frequency Control vol 38 no 1pp 40ndash47 1991

[27] PiezocompositesMaterial Systems Inc 543 Great Road Little-ton MA 01460

[28] Y Benveniste and G J Dvorak ldquoUniform fields and universalrelations in piezoelectric compositesrdquo Journal of the Mechanicsand Physics of Solids vol 40 no 6 pp 1295ndash1312 1992

[29] M L Dunn and M Taya ldquoMicromechanics predictions ofthe effective electroelastic moduli of piezoelectric compositesrdquoInternational Journal of Solids and Structures vol 30 no 2 pp161ndash175 1993

[30] J Aboudi ldquoMicromechanical prediction of the effective coeffi-cients of thermo-piezoelectric multiphase compositesrdquo Journalof Intelligent Material Systems and Structures vol 9 no 9 pp713ndash722 1999

[31] X Ruan T W Chou A Safari and S C Danforth ldquoA 3-Dconnectivity model for effective piezoelectric properties of yarncompositesrdquo Journal of Composite Materials vol 36 no 14 pp1693ndash1708 2002

[32] A Baz and A Tempia ldquoActive piezoelectric damping compos-itesrdquo Sensors and Actuators A vol 112 no 2-3 pp 340ndash3502004

[33] H Berger S Kari U Gabbert et al ldquoUnit cell models ofpiezoelectric fiber composites for numerical and analytical cal-culation of effective propertiesrdquo SmartMaterials and Structuresvol 15 no 2 pp 451ndash458 2006

[34] M Sakthivel and A Arockiarajan ldquoAn analytical model forpredicting thermo-electro-mechanical response of 1ndash3 piezo-electric compositesrdquo Computational Materials Science vol 48no 4 pp 759ndash767 2010

[35] M C Ray and A K Pradhan ldquoOn the use of verticallyreinforced 1ndash3 piezoelectric composites for hybrid damping oflaminated composite platesrdquo Mechanics of Advanced Materialsand Structures vol 14 no 4 pp 245ndash261 2007

[36] M C Ray and A K Pradhan ldquoPerformance of vertically andobliquely reinforced 1ndash3 piezoelectric composites for activedamping of laminated composite shellsrdquo Journal of Sound andVibration vol 315 no 4-5 pp 816ndash835 2008

[37] S K Sarangi and M C Ray ldquoSmart damping of geometricallynonlinear vibrations of laminated composite beams using verti-cally reinforced 1ndash3 piezoelectric compositesrdquo Smart Materialsand Structures vol 19 no 7 Article ID 075020 2010

[38] A Krishnan and J V Deshpande ldquoVibration of skew laminatesrdquoJournal of Sound and Vibration vol 153 no 2 pp 351ndash358 1992

[39] A R Krishna Reddy and R Palaninathan ldquoFree vibration ofskew laminatesrdquo Computers and Structures vol 70 no 4 pp415ndash423 1999

[40] C S Babu and T Kant ldquoTwo shear deformable finite elementmodels for buckling analysis of skew fibre-reinforced compositeand sandwich panelsrdquo Composite Structures vol 46 no 2 pp115ndash124 1999

[41] A K Garg R K Khare and T Kant ldquoFree vibration of skewfiber-reinforced composite and sandwich laminates using ashear deformable finite element modelrdquo Journal of SandwichStructures and Materials vol 8 no 1 pp 33ndash53 2006

[42] J N Reddy Mechanics of Laminated Composites Plates Theoryand Analysis CRC press Boca Raton Fla USA 1996

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 14: Research Article Active Constrained Layer Damping of Smart …downloads.hindawi.com/archive/2013/824163.pdf · 2019-07-31 · Research Article Active Constrained Layer Damping of

14 Journal of Composites

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 15 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

300 350 400 450 500 550 600 650 700 750 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 16 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119909119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

damping of smart skew laminated composite plates whenverticallyobliquely reinforced 1ndash3 PZC materials are usedas the materials for the constraining layer of the ACLDtreatment A layerwise FSDT-based finite element modelhas been developed to describe the dynamics of the skewlaminated composite plates integrated with the patches ofACLD treatment Unlike the existing finite element modelsof smart structures integrated with ACLD treatment the

200 250 300 350 400 450 500 550 600 650 7000

02

04

06

08

1

12

14

16

18

2times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 17 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewsymmetric cross-ply (0∘90∘0∘) plate when the piezoelectric fibersof the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600119886ℎ = 100 120572 = 30

∘)

200 300 400 500 600 700 8000

1

2

3

4

5times10minus5

Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 18 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric cross-ply (0∘90∘0∘90∘) plate when the piezoelectricfibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

=

600 119886ℎ = 100 120572 = 30∘)

derivation of the present finite element model includes thetransverse deformations of the substrate skew laminatedcomposite plate the constrained viscoelastic layer and theconstraining 1ndash3 PZC layer of the ACLD treatment along thethickness (ie 119911) direction such that both vertical and in-plane actuations by the constraining layer of the patches canbe utilized for active damping of the plates The frequency

Journal of Composites 15

0

05

1

15

2

25

3

35

4times10minus5

200 300 400 500 600 700 800Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 19 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

responses of the symmetric cross-ply antisymmetric cross-ply and angle-ply skew composite plates indicate that theactive constraining layer of the ACLD treatment being madeof the vertically reinforced 1ndash3 PZC material significantlyenhances the damping characteristics of the plates over thepassive damping The analysis revealed that if the verticallyreinforced 1ndash3 PZCmaterial is used for the constraining layerof the ACLD treatment then the contribution of the verticalactuation by the constraining layer alone for improving theactive damping characteristics of the smart skew laminatedcomposite plate is significantly larger than that due to thein-plane actuation by the constraining layer alone It isimportant to note from the present investigation that thevariation of the orientation angle of the piezoelectric fibersof the constraining layer in the vertical 119909119911- and 119910119911-planesas well as the skew angle and boundary conditions of theoverall plates significantly affects the performance of thepatches For the simply supported laminated substrate platesthe performance of the patches becomes maximumwhen thepiezoelectric fiber orientation angle (120595) is 0∘ while in caseof the clamped-clamped laminated skew substrate plates themaximum control authority of the patches is achieved whenthe value of 120595 is 30∘ irrespective of the cases in which thefibers are oriented in 119909119911- and 119910119911-planes

Appendix

In (6) and (7) the matrices [1198851] [119885

2] [119885

3] [119885

4] [119885

5] and

[1198856] are given by

[1198851] = [[119885

1] 0 0]

[1198852] = [(

ℎ

2) 119868 [119885

2] 0]

[1198853] = [(

ℎ

2) 119868 ℎV119868 [119885

3]]

[1198854] = [119868 0 0 119911119868 0 0]

[1198855] = [0 119868 0 (

ℎ

2) 119868 (119911 minus

ℎ

2) 119868 0]

[1198856] = [0 0 119868 (

ℎ

2) 119868 ℎV119868 (119911 minus ℎ

119873+2) 119868]

(A1)

in which

[1198851] =

[[[

[

119911 0 0 0

0 119911 0 0

0 0 119911 0

0 0 0 1

]]]

]

[1198852] =

[[[[[[[[

[

(119911 minusℎ

2) 0 0 0

0 (119911 minusℎ

2) 0 0

0 0 (119911 minusℎ

2) 0

0 0 0 1

]]]]]]]]

]

[1198853] =

[[[

[

(119911 minus ℎ119873+2

) 0 0 0

0 (119911 minus ℎ119873+2

) 0 0

0 0 (119911 minus ℎ119873+2

) 0

0 0 0 1

]]]

]

[119868] =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

[119868] = [1 0

0 1]

[0] = [0 0

0 0] [0] = [

0 0

0 0]

(A2)

The various submatrices 119861119905119887119894 119861

119903119887119894 and 119861

119903119904119894 appearing in (21)

are given by

119861119905119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 0

]]]]]]]]]]]

]

119861119905119904119894

=

[[[[

[

0 0120597120578

119894

120597119909

0 0120597120578

119894

120597119910

]]]]

]

16 Journal of Composites

119861119903119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 1

]]]]]]]]]]]

]

119861119903119887119894

=

[[[[[[

[

119861119903119887119894

0

0

0 119861119903119887119894

0

0

0 119861119903119887119894

]]]]]]

]

119861119903119904119894

=

[[[[[[[[[[[[[[[[[[

[

119868 0 0

0

119868 0

0 0

119868

119861119903119904119894

0 0

0 119861119903119904119894

0

0 0 119861119903119904119894

]]]]]]]]]]]]]]]]]]

]

119868= [1 0 0

0 1 0]

(A3)

wherein

0 and 0 are the (3 times 3) and (2 times 3) null matricesrespectively

References

[1] T Bailey and J E Hubbard ldquoDistributedpiezoelectric polymeractive vibration control of a cantilever beamrdquo Journal of Guid-ance Control and Dynamics vol 8 no 5 pp 605ndash611 1985

[2] E F Crawley and J de Luis ldquoUse of piezoelectric actuators aselements of intelligent structuresrdquo AIAA Journal vol 25 no 10pp 1373ndash1385 1987

[3] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988

[4] C K Lee W W Chiang and O Sulivan ldquoPiezoelectric modalsensoractuator pairs for critical active damping vibrationcontrolrdquo Journal of the Acoustical Society of America vol 90 no1 pp 374ndash384 1991

[5] S Hanagud M W Obal and A J Calise ldquoOptimal vibrationcontrol by the use of piezoceramic sensors and actuatorsrdquoJournal of Guidance Control and Dynamics vol 15 no 5 pp1199ndash1206 1992

[6] Y Gu R L Clark C R Fuller and A C Zander ldquoExperimentson active control of plate vibration using piezoelectric actuatorsand polyvinylidene fluoride (PVDF) modal sensorsrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 116 no3 pp 303ndash308 1994

[7] A Baz and S Poh ldquoOptimal vibration control with modalpositive position feedbackrdquo Optimal Control Applications andMethods vol 17 no 2 pp 141ndash149 1996

[8] A Kugi K Schlacher and H Irschik ldquoInfinite-dimensionalcontrol of nonlinear beam vibrations by piezoelectric actuatorand sensor layersrdquo Nonlinear Dynamics vol 19 no 1 pp 71ndash911999

[9] D Sun L Tong and D Wang ldquoVibration control of platesusing discretely distributed piezoelectric quasi-modal actua-torssensorsrdquo AIAA Journal vol 39 no 9 pp 1766ndash1772 2001

[10] G Caruso S Galeani and L Menini ldquoActive vibration controlof an elastic plate using multiple piezoelectric sensors andactuatorsrdquo SimulationModelling Practice andTheory vol 11 no5-6 pp 403ndash419 2003

[11] X Zhang and A G Erdman ldquoOptimal placement of piezo-electric sensors and actuators for controlled flexible linkagemechanismsrdquo Journal of Vibration and Acoustics Transactionsof the ASME vol 128 no 2 pp 256ndash260 2006

[12] G Gatti M J Brennan and P Gardonio ldquoActive dampingof a beam using a physically collocated accelerometer andpiezoelectric patch actuatorrdquo Journal of Sound and Vibrationvol 303 no 3ndash5 pp 798ndash813 2007

[13] M K Kwak S Heo and M Jeong ldquoDynamic modellingand active vibration controller design for a cylindrical shellequipped with piezoelectric sensors and actuatorsrdquo Journal ofSound and Vibration vol 321 no 3ndash5 pp 510ndash524 2009

[14] I Bruant L Gallimard and S Nikoukar ldquoOptimal piezoelectricactuator and sensor location for active vibration control usinggenetic algorithmrdquo Journal of Sound and Vibration vol 329 no10 pp 1615ndash1635 2010

[15] Y Fu J Wang and Y Mao ldquoNonlinear vibration and activecontrol of functionally graded beams with piezoelectric sensorsand actuatorsrdquo Journal of Intelligent Material Systems andStructures vol 22 no 18 pp 2093ndash2102 2011

[16] F Chen M Hong M Song and H Cui ldquoOptimal control ofa beam with discontinuously distributed piezoelectric sensorsand actuatorsrdquo Journal of Marine Science and Application vol11 no 1 pp 44ndash51 2012

[17] B Azvine G R Tomlinson and R J Wynne ldquoUse of activeconstrained-layer damping for controlling resonant vibrationrdquoSmart Materials and Structures vol 4 no 1 pp 1ndash6 1995

[18] A Baz and J Ro ldquoVibration control of plates with activeconstrained layer dampingrdquo SmartMaterials and Structures vol5 no 3 pp 272ndash280 1996

[19] C Chantalakhana and R Stanway ldquoActive constrained layerdamping of clamped-clamped plate vibrationsrdquo Journal ofSound and Vibration vol 241 no 5 pp 755ndash777 2001

[20] J Ro and A Baz ldquoOptimum placement and control ofactive constrained layer damping using modal strain energyapproachrdquo JVCJournal of Vibration and Control vol 8 no 6pp 861ndash876 2002

[21] M C Ray and J N Reddy ldquoOptimal control of thin circularcylindrical laminated composite shells using active constrainedlayer damping treatmentrdquo Smart Materials and Structures vol13 no 1 pp 64ndash72 2004

[22] C M A Vasques B R Mace P Gardonio and J D RodriguesldquoArbitrary active constrained layer damping treatments onbeams finite element modelling and experimental validationrdquoComputers and Structures vol 84 no 22-23 pp 1384ndash14012006

[23] F M Li K Kishimoto K Wang Z B Chen and W HHuang ldquoVibration control of beams with active constrainedlayer dampingrdquo Smart Materials and Structures vol 17 no 6Article ID 065036 2008

[24] L Yuan Y Xiang Y Huang and J Lu ldquoA semi-analyticalmethod and the circumferential dominant modal control ofcircular cylindrical shells with active constrained layer dampingtreatmentrdquo SmartMaterials and Structures vol 19 no 2 ArticleID 025010 2010

[25] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

Journal of Composites 17

[26] W A Smith and B A Auld ldquoModeling 1ndash3 composite piezo-electrics thickness-mode oscillationsrdquo IEEE Transactions onUltrasonics Ferroelectrics and Frequency Control vol 38 no 1pp 40ndash47 1991

[27] PiezocompositesMaterial Systems Inc 543 Great Road Little-ton MA 01460

[28] Y Benveniste and G J Dvorak ldquoUniform fields and universalrelations in piezoelectric compositesrdquo Journal of the Mechanicsand Physics of Solids vol 40 no 6 pp 1295ndash1312 1992

[29] M L Dunn and M Taya ldquoMicromechanics predictions ofthe effective electroelastic moduli of piezoelectric compositesrdquoInternational Journal of Solids and Structures vol 30 no 2 pp161ndash175 1993

[30] J Aboudi ldquoMicromechanical prediction of the effective coeffi-cients of thermo-piezoelectric multiphase compositesrdquo Journalof Intelligent Material Systems and Structures vol 9 no 9 pp713ndash722 1999

[31] X Ruan T W Chou A Safari and S C Danforth ldquoA 3-Dconnectivity model for effective piezoelectric properties of yarncompositesrdquo Journal of Composite Materials vol 36 no 14 pp1693ndash1708 2002

[32] A Baz and A Tempia ldquoActive piezoelectric damping compos-itesrdquo Sensors and Actuators A vol 112 no 2-3 pp 340ndash3502004

[33] H Berger S Kari U Gabbert et al ldquoUnit cell models ofpiezoelectric fiber composites for numerical and analytical cal-culation of effective propertiesrdquo SmartMaterials and Structuresvol 15 no 2 pp 451ndash458 2006

[34] M Sakthivel and A Arockiarajan ldquoAn analytical model forpredicting thermo-electro-mechanical response of 1ndash3 piezo-electric compositesrdquo Computational Materials Science vol 48no 4 pp 759ndash767 2010

[35] M C Ray and A K Pradhan ldquoOn the use of verticallyreinforced 1ndash3 piezoelectric composites for hybrid damping oflaminated composite platesrdquo Mechanics of Advanced Materialsand Structures vol 14 no 4 pp 245ndash261 2007

[36] M C Ray and A K Pradhan ldquoPerformance of vertically andobliquely reinforced 1ndash3 piezoelectric composites for activedamping of laminated composite shellsrdquo Journal of Sound andVibration vol 315 no 4-5 pp 816ndash835 2008

[37] S K Sarangi and M C Ray ldquoSmart damping of geometricallynonlinear vibrations of laminated composite beams using verti-cally reinforced 1ndash3 piezoelectric compositesrdquo Smart Materialsand Structures vol 19 no 7 Article ID 075020 2010

[38] A Krishnan and J V Deshpande ldquoVibration of skew laminatesrdquoJournal of Sound and Vibration vol 153 no 2 pp 351ndash358 1992

[39] A R Krishna Reddy and R Palaninathan ldquoFree vibration ofskew laminatesrdquo Computers and Structures vol 70 no 4 pp415ndash423 1999

[40] C S Babu and T Kant ldquoTwo shear deformable finite elementmodels for buckling analysis of skew fibre-reinforced compositeand sandwich panelsrdquo Composite Structures vol 46 no 2 pp115ndash124 1999

[41] A K Garg R K Khare and T Kant ldquoFree vibration of skewfiber-reinforced composite and sandwich laminates using ashear deformable finite element modelrdquo Journal of SandwichStructures and Materials vol 8 no 1 pp 33ndash53 2006

[42] J N Reddy Mechanics of Laminated Composites Plates Theoryand Analysis CRC press Boca Raton Fla USA 1996

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 15: Research Article Active Constrained Layer Damping of Smart …downloads.hindawi.com/archive/2013/824163.pdf · 2019-07-31 · Research Article Active Constrained Layer Damping of

Journal of Composites 15

0

05

1

15

2

25

3

35

4times10minus5

200 300 400 500 600 700 800Frequency (Hz)

0∘120595 =

15∘120595 =

30∘120595 =

45∘120595 =

w(a2(b4)co

s120572h2

) (m

)

Figure 19 Effect of variation of piezoelectric fiber orientationangle (120595) on the controlled responses of a clamped-clamped skewantisymmetric angle-ply (45∘minus45∘45∘minus45∘) plate when the piezo-electric fibers of the constraining layer are coplanar with the 119910119911-plane (119870

119889

= 600 119886ℎ = 100 120572 = 30∘)

responses of the symmetric cross-ply antisymmetric cross-ply and angle-ply skew composite plates indicate that theactive constraining layer of the ACLD treatment being madeof the vertically reinforced 1ndash3 PZC material significantlyenhances the damping characteristics of the plates over thepassive damping The analysis revealed that if the verticallyreinforced 1ndash3 PZCmaterial is used for the constraining layerof the ACLD treatment then the contribution of the verticalactuation by the constraining layer alone for improving theactive damping characteristics of the smart skew laminatedcomposite plate is significantly larger than that due to thein-plane actuation by the constraining layer alone It isimportant to note from the present investigation that thevariation of the orientation angle of the piezoelectric fibersof the constraining layer in the vertical 119909119911- and 119910119911-planesas well as the skew angle and boundary conditions of theoverall plates significantly affects the performance of thepatches For the simply supported laminated substrate platesthe performance of the patches becomes maximumwhen thepiezoelectric fiber orientation angle (120595) is 0∘ while in caseof the clamped-clamped laminated skew substrate plates themaximum control authority of the patches is achieved whenthe value of 120595 is 30∘ irrespective of the cases in which thefibers are oriented in 119909119911- and 119910119911-planes

Appendix

In (6) and (7) the matrices [1198851] [119885

2] [119885

3] [119885

4] [119885

5] and

[1198856] are given by

[1198851] = [[119885

1] 0 0]

[1198852] = [(

ℎ

2) 119868 [119885

2] 0]

[1198853] = [(

ℎ

2) 119868 ℎV119868 [119885

3]]

[1198854] = [119868 0 0 119911119868 0 0]

[1198855] = [0 119868 0 (

ℎ

2) 119868 (119911 minus

ℎ

2) 119868 0]

[1198856] = [0 0 119868 (

ℎ

2) 119868 ℎV119868 (119911 minus ℎ

119873+2) 119868]

(A1)

in which

[1198851] =

[[[

[

119911 0 0 0

0 119911 0 0

0 0 119911 0

0 0 0 1

]]]

]

[1198852] =

[[[[[[[[

[

(119911 minusℎ

2) 0 0 0

0 (119911 minusℎ

2) 0 0

0 0 (119911 minusℎ

2) 0

0 0 0 1

]]]]]]]]

]

[1198853] =

[[[

[

(119911 minus ℎ119873+2

) 0 0 0

0 (119911 minus ℎ119873+2

) 0 0

0 0 (119911 minus ℎ119873+2

) 0

0 0 0 1

]]]

]

[119868] =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

[119868] = [1 0

0 1]

[0] = [0 0

0 0] [0] = [

0 0

0 0]

(A2)

The various submatrices 119861119905119887119894 119861

119903119887119894 and 119861

119903119904119894 appearing in (21)

are given by

119861119905119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 0

]]]]]]]]]]]

]

119861119905119904119894

=

[[[[

[

0 0120597120578

119894

120597119909

0 0120597120578

119894

120597119910

]]]]

]

16 Journal of Composites

119861119903119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 1

]]]]]]]]]]]

]

119861119903119887119894

=

[[[[[[

[

119861119903119887119894

0

0

0 119861119903119887119894

0

0

0 119861119903119887119894

]]]]]]

]

119861119903119904119894

=

[[[[[[[[[[[[[[[[[[

[

119868 0 0

0

119868 0

0 0

119868

119861119903119904119894

0 0

0 119861119903119904119894

0

0 0 119861119903119904119894

]]]]]]]]]]]]]]]]]]

]

119868= [1 0 0

0 1 0]

(A3)

wherein

0 and 0 are the (3 times 3) and (2 times 3) null matricesrespectively

References

[1] T Bailey and J E Hubbard ldquoDistributedpiezoelectric polymeractive vibration control of a cantilever beamrdquo Journal of Guid-ance Control and Dynamics vol 8 no 5 pp 605ndash611 1985

[2] E F Crawley and J de Luis ldquoUse of piezoelectric actuators aselements of intelligent structuresrdquo AIAA Journal vol 25 no 10pp 1373ndash1385 1987

[3] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988

[4] C K Lee W W Chiang and O Sulivan ldquoPiezoelectric modalsensoractuator pairs for critical active damping vibrationcontrolrdquo Journal of the Acoustical Society of America vol 90 no1 pp 374ndash384 1991

[5] S Hanagud M W Obal and A J Calise ldquoOptimal vibrationcontrol by the use of piezoceramic sensors and actuatorsrdquoJournal of Guidance Control and Dynamics vol 15 no 5 pp1199ndash1206 1992

[6] Y Gu R L Clark C R Fuller and A C Zander ldquoExperimentson active control of plate vibration using piezoelectric actuatorsand polyvinylidene fluoride (PVDF) modal sensorsrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 116 no3 pp 303ndash308 1994

[7] A Baz and S Poh ldquoOptimal vibration control with modalpositive position feedbackrdquo Optimal Control Applications andMethods vol 17 no 2 pp 141ndash149 1996

[8] A Kugi K Schlacher and H Irschik ldquoInfinite-dimensionalcontrol of nonlinear beam vibrations by piezoelectric actuatorand sensor layersrdquo Nonlinear Dynamics vol 19 no 1 pp 71ndash911999

[9] D Sun L Tong and D Wang ldquoVibration control of platesusing discretely distributed piezoelectric quasi-modal actua-torssensorsrdquo AIAA Journal vol 39 no 9 pp 1766ndash1772 2001

[10] G Caruso S Galeani and L Menini ldquoActive vibration controlof an elastic plate using multiple piezoelectric sensors andactuatorsrdquo SimulationModelling Practice andTheory vol 11 no5-6 pp 403ndash419 2003

[11] X Zhang and A G Erdman ldquoOptimal placement of piezo-electric sensors and actuators for controlled flexible linkagemechanismsrdquo Journal of Vibration and Acoustics Transactionsof the ASME vol 128 no 2 pp 256ndash260 2006

[12] G Gatti M J Brennan and P Gardonio ldquoActive dampingof a beam using a physically collocated accelerometer andpiezoelectric patch actuatorrdquo Journal of Sound and Vibrationvol 303 no 3ndash5 pp 798ndash813 2007

[13] M K Kwak S Heo and M Jeong ldquoDynamic modellingand active vibration controller design for a cylindrical shellequipped with piezoelectric sensors and actuatorsrdquo Journal ofSound and Vibration vol 321 no 3ndash5 pp 510ndash524 2009

[14] I Bruant L Gallimard and S Nikoukar ldquoOptimal piezoelectricactuator and sensor location for active vibration control usinggenetic algorithmrdquo Journal of Sound and Vibration vol 329 no10 pp 1615ndash1635 2010

[15] Y Fu J Wang and Y Mao ldquoNonlinear vibration and activecontrol of functionally graded beams with piezoelectric sensorsand actuatorsrdquo Journal of Intelligent Material Systems andStructures vol 22 no 18 pp 2093ndash2102 2011

[16] F Chen M Hong M Song and H Cui ldquoOptimal control ofa beam with discontinuously distributed piezoelectric sensorsand actuatorsrdquo Journal of Marine Science and Application vol11 no 1 pp 44ndash51 2012

[17] B Azvine G R Tomlinson and R J Wynne ldquoUse of activeconstrained-layer damping for controlling resonant vibrationrdquoSmart Materials and Structures vol 4 no 1 pp 1ndash6 1995

[18] A Baz and J Ro ldquoVibration control of plates with activeconstrained layer dampingrdquo SmartMaterials and Structures vol5 no 3 pp 272ndash280 1996

[19] C Chantalakhana and R Stanway ldquoActive constrained layerdamping of clamped-clamped plate vibrationsrdquo Journal ofSound and Vibration vol 241 no 5 pp 755ndash777 2001

[20] J Ro and A Baz ldquoOptimum placement and control ofactive constrained layer damping using modal strain energyapproachrdquo JVCJournal of Vibration and Control vol 8 no 6pp 861ndash876 2002

[21] M C Ray and J N Reddy ldquoOptimal control of thin circularcylindrical laminated composite shells using active constrainedlayer damping treatmentrdquo Smart Materials and Structures vol13 no 1 pp 64ndash72 2004

[22] C M A Vasques B R Mace P Gardonio and J D RodriguesldquoArbitrary active constrained layer damping treatments onbeams finite element modelling and experimental validationrdquoComputers and Structures vol 84 no 22-23 pp 1384ndash14012006

[23] F M Li K Kishimoto K Wang Z B Chen and W HHuang ldquoVibration control of beams with active constrainedlayer dampingrdquo Smart Materials and Structures vol 17 no 6Article ID 065036 2008

[24] L Yuan Y Xiang Y Huang and J Lu ldquoA semi-analyticalmethod and the circumferential dominant modal control ofcircular cylindrical shells with active constrained layer dampingtreatmentrdquo SmartMaterials and Structures vol 19 no 2 ArticleID 025010 2010

[25] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

Journal of Composites 17

[26] W A Smith and B A Auld ldquoModeling 1ndash3 composite piezo-electrics thickness-mode oscillationsrdquo IEEE Transactions onUltrasonics Ferroelectrics and Frequency Control vol 38 no 1pp 40ndash47 1991

[27] PiezocompositesMaterial Systems Inc 543 Great Road Little-ton MA 01460

[28] Y Benveniste and G J Dvorak ldquoUniform fields and universalrelations in piezoelectric compositesrdquo Journal of the Mechanicsand Physics of Solids vol 40 no 6 pp 1295ndash1312 1992

[29] M L Dunn and M Taya ldquoMicromechanics predictions ofthe effective electroelastic moduli of piezoelectric compositesrdquoInternational Journal of Solids and Structures vol 30 no 2 pp161ndash175 1993

[30] J Aboudi ldquoMicromechanical prediction of the effective coeffi-cients of thermo-piezoelectric multiphase compositesrdquo Journalof Intelligent Material Systems and Structures vol 9 no 9 pp713ndash722 1999

[31] X Ruan T W Chou A Safari and S C Danforth ldquoA 3-Dconnectivity model for effective piezoelectric properties of yarncompositesrdquo Journal of Composite Materials vol 36 no 14 pp1693ndash1708 2002

[32] A Baz and A Tempia ldquoActive piezoelectric damping compos-itesrdquo Sensors and Actuators A vol 112 no 2-3 pp 340ndash3502004

[33] H Berger S Kari U Gabbert et al ldquoUnit cell models ofpiezoelectric fiber composites for numerical and analytical cal-culation of effective propertiesrdquo SmartMaterials and Structuresvol 15 no 2 pp 451ndash458 2006

[34] M Sakthivel and A Arockiarajan ldquoAn analytical model forpredicting thermo-electro-mechanical response of 1ndash3 piezo-electric compositesrdquo Computational Materials Science vol 48no 4 pp 759ndash767 2010

[35] M C Ray and A K Pradhan ldquoOn the use of verticallyreinforced 1ndash3 piezoelectric composites for hybrid damping oflaminated composite platesrdquo Mechanics of Advanced Materialsand Structures vol 14 no 4 pp 245ndash261 2007

[36] M C Ray and A K Pradhan ldquoPerformance of vertically andobliquely reinforced 1ndash3 piezoelectric composites for activedamping of laminated composite shellsrdquo Journal of Sound andVibration vol 315 no 4-5 pp 816ndash835 2008

[37] S K Sarangi and M C Ray ldquoSmart damping of geometricallynonlinear vibrations of laminated composite beams using verti-cally reinforced 1ndash3 piezoelectric compositesrdquo Smart Materialsand Structures vol 19 no 7 Article ID 075020 2010

[38] A Krishnan and J V Deshpande ldquoVibration of skew laminatesrdquoJournal of Sound and Vibration vol 153 no 2 pp 351ndash358 1992

[39] A R Krishna Reddy and R Palaninathan ldquoFree vibration ofskew laminatesrdquo Computers and Structures vol 70 no 4 pp415ndash423 1999

[40] C S Babu and T Kant ldquoTwo shear deformable finite elementmodels for buckling analysis of skew fibre-reinforced compositeand sandwich panelsrdquo Composite Structures vol 46 no 2 pp115ndash124 1999

[41] A K Garg R K Khare and T Kant ldquoFree vibration of skewfiber-reinforced composite and sandwich laminates using ashear deformable finite element modelrdquo Journal of SandwichStructures and Materials vol 8 no 1 pp 33ndash53 2006

[42] J N Reddy Mechanics of Laminated Composites Plates Theoryand Analysis CRC press Boca Raton Fla USA 1996

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 16: Research Article Active Constrained Layer Damping of Smart …downloads.hindawi.com/archive/2013/824163.pdf · 2019-07-31 · Research Article Active Constrained Layer Damping of

16 Journal of Composites

119861119903119887119894

=

[[[[[[[[[[[

[

120597120578119894

1205971199090 0

0120597120578

119894

1205971199100

120597120578119894

120597119910

120597120578119894

1205971199090

0 0 1

]]]]]]]]]]]

]

119861119903119887119894

=

[[[[[[

[

119861119903119887119894

0

0

0 119861119903119887119894

0

0

0 119861119903119887119894

]]]]]]

]

119861119903119904119894

=

[[[[[[[[[[[[[[[[[[

[

119868 0 0

0

119868 0

0 0

119868

119861119903119904119894

0 0

0 119861119903119904119894

0

0 0 119861119903119904119894

]]]]]]]]]]]]]]]]]]

]

119868= [1 0 0

0 1 0]

(A3)

wherein

0 and 0 are the (3 times 3) and (2 times 3) null matricesrespectively

References

[1] T Bailey and J E Hubbard ldquoDistributedpiezoelectric polymeractive vibration control of a cantilever beamrdquo Journal of Guid-ance Control and Dynamics vol 8 no 5 pp 605ndash611 1985

[2] E F Crawley and J de Luis ldquoUse of piezoelectric actuators aselements of intelligent structuresrdquo AIAA Journal vol 25 no 10pp 1373ndash1385 1987

[3] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988

[4] C K Lee W W Chiang and O Sulivan ldquoPiezoelectric modalsensoractuator pairs for critical active damping vibrationcontrolrdquo Journal of the Acoustical Society of America vol 90 no1 pp 374ndash384 1991

[5] S Hanagud M W Obal and A J Calise ldquoOptimal vibrationcontrol by the use of piezoceramic sensors and actuatorsrdquoJournal of Guidance Control and Dynamics vol 15 no 5 pp1199ndash1206 1992

[6] Y Gu R L Clark C R Fuller and A C Zander ldquoExperimentson active control of plate vibration using piezoelectric actuatorsand polyvinylidene fluoride (PVDF) modal sensorsrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 116 no3 pp 303ndash308 1994

[7] A Baz and S Poh ldquoOptimal vibration control with modalpositive position feedbackrdquo Optimal Control Applications andMethods vol 17 no 2 pp 141ndash149 1996

[8] A Kugi K Schlacher and H Irschik ldquoInfinite-dimensionalcontrol of nonlinear beam vibrations by piezoelectric actuatorand sensor layersrdquo Nonlinear Dynamics vol 19 no 1 pp 71ndash911999

[9] D Sun L Tong and D Wang ldquoVibration control of platesusing discretely distributed piezoelectric quasi-modal actua-torssensorsrdquo AIAA Journal vol 39 no 9 pp 1766ndash1772 2001

[10] G Caruso S Galeani and L Menini ldquoActive vibration controlof an elastic plate using multiple piezoelectric sensors andactuatorsrdquo SimulationModelling Practice andTheory vol 11 no5-6 pp 403ndash419 2003

[11] X Zhang and A G Erdman ldquoOptimal placement of piezo-electric sensors and actuators for controlled flexible linkagemechanismsrdquo Journal of Vibration and Acoustics Transactionsof the ASME vol 128 no 2 pp 256ndash260 2006

[12] G Gatti M J Brennan and P Gardonio ldquoActive dampingof a beam using a physically collocated accelerometer andpiezoelectric patch actuatorrdquo Journal of Sound and Vibrationvol 303 no 3ndash5 pp 798ndash813 2007

[13] M K Kwak S Heo and M Jeong ldquoDynamic modellingand active vibration controller design for a cylindrical shellequipped with piezoelectric sensors and actuatorsrdquo Journal ofSound and Vibration vol 321 no 3ndash5 pp 510ndash524 2009

[14] I Bruant L Gallimard and S Nikoukar ldquoOptimal piezoelectricactuator and sensor location for active vibration control usinggenetic algorithmrdquo Journal of Sound and Vibration vol 329 no10 pp 1615ndash1635 2010

[15] Y Fu J Wang and Y Mao ldquoNonlinear vibration and activecontrol of functionally graded beams with piezoelectric sensorsand actuatorsrdquo Journal of Intelligent Material Systems andStructures vol 22 no 18 pp 2093ndash2102 2011

[16] F Chen M Hong M Song and H Cui ldquoOptimal control ofa beam with discontinuously distributed piezoelectric sensorsand actuatorsrdquo Journal of Marine Science and Application vol11 no 1 pp 44ndash51 2012

[17] B Azvine G R Tomlinson and R J Wynne ldquoUse of activeconstrained-layer damping for controlling resonant vibrationrdquoSmart Materials and Structures vol 4 no 1 pp 1ndash6 1995

[18] A Baz and J Ro ldquoVibration control of plates with activeconstrained layer dampingrdquo SmartMaterials and Structures vol5 no 3 pp 272ndash280 1996

[19] C Chantalakhana and R Stanway ldquoActive constrained layerdamping of clamped-clamped plate vibrationsrdquo Journal ofSound and Vibration vol 241 no 5 pp 755ndash777 2001

[20] J Ro and A Baz ldquoOptimum placement and control ofactive constrained layer damping using modal strain energyapproachrdquo JVCJournal of Vibration and Control vol 8 no 6pp 861ndash876 2002

[21] M C Ray and J N Reddy ldquoOptimal control of thin circularcylindrical laminated composite shells using active constrainedlayer damping treatmentrdquo Smart Materials and Structures vol13 no 1 pp 64ndash72 2004

[22] C M A Vasques B R Mace P Gardonio and J D RodriguesldquoArbitrary active constrained layer damping treatments onbeams finite element modelling and experimental validationrdquoComputers and Structures vol 84 no 22-23 pp 1384ndash14012006

[23] F M Li K Kishimoto K Wang Z B Chen and W HHuang ldquoVibration control of beams with active constrainedlayer dampingrdquo Smart Materials and Structures vol 17 no 6Article ID 065036 2008

[24] L Yuan Y Xiang Y Huang and J Lu ldquoA semi-analyticalmethod and the circumferential dominant modal control ofcircular cylindrical shells with active constrained layer dampingtreatmentrdquo SmartMaterials and Structures vol 19 no 2 ArticleID 025010 2010

[25] N Kumar and S P Singh ldquoVibration control of curved panelusing smart dampingrdquo Mechanical Systems and Signal Process-ing vol 30 pp 232ndash247 2012

Journal of Composites 17

[26] W A Smith and B A Auld ldquoModeling 1ndash3 composite piezo-electrics thickness-mode oscillationsrdquo IEEE Transactions onUltrasonics Ferroelectrics and Frequency Control vol 38 no 1pp 40ndash47 1991

[27] PiezocompositesMaterial Systems Inc 543 Great Road Little-ton MA 01460

[28] Y Benveniste and G J Dvorak ldquoUniform fields and universalrelations in piezoelectric compositesrdquo Journal of the Mechanicsand Physics of Solids vol 40 no 6 pp 1295ndash1312 1992

[29] M L Dunn and M Taya ldquoMicromechanics predictions ofthe effective electroelastic moduli of piezoelectric compositesrdquoInternational Journal of Solids and Structures vol 30 no 2 pp161ndash175 1993

[30] J Aboudi ldquoMicromechanical prediction of the effective coeffi-cients of thermo-piezoelectric multiphase compositesrdquo Journalof Intelligent Material Systems and Structures vol 9 no 9 pp713ndash722 1999

[31] X Ruan T W Chou A Safari and S C Danforth ldquoA 3-Dconnectivity model for effective piezoelectric properties of yarncompositesrdquo Journal of Composite Materials vol 36 no 14 pp1693ndash1708 2002

[32] A Baz and A Tempia ldquoActive piezoelectric damping compos-itesrdquo Sensors and Actuators A vol 112 no 2-3 pp 340ndash3502004

[33] H Berger S Kari U Gabbert et al ldquoUnit cell models ofpiezoelectric fiber composites for numerical and analytical cal-culation of effective propertiesrdquo SmartMaterials and Structuresvol 15 no 2 pp 451ndash458 2006

[34] M Sakthivel and A Arockiarajan ldquoAn analytical model forpredicting thermo-electro-mechanical response of 1ndash3 piezo-electric compositesrdquo Computational Materials Science vol 48no 4 pp 759ndash767 2010

[35] M C Ray and A K Pradhan ldquoOn the use of verticallyreinforced 1ndash3 piezoelectric composites for hybrid damping oflaminated composite platesrdquo Mechanics of Advanced Materialsand Structures vol 14 no 4 pp 245ndash261 2007

[36] M C Ray and A K Pradhan ldquoPerformance of vertically andobliquely reinforced 1ndash3 piezoelectric composites for activedamping of laminated composite shellsrdquo Journal of Sound andVibration vol 315 no 4-5 pp 816ndash835 2008

[37] S K Sarangi and M C Ray ldquoSmart damping of geometricallynonlinear vibrations of laminated composite beams using verti-cally reinforced 1ndash3 piezoelectric compositesrdquo Smart Materialsand Structures vol 19 no 7 Article ID 075020 2010

[38] A Krishnan and J V Deshpande ldquoVibration of skew laminatesrdquoJournal of Sound and Vibration vol 153 no 2 pp 351ndash358 1992

[39] A R Krishna Reddy and R Palaninathan ldquoFree vibration ofskew laminatesrdquo Computers and Structures vol 70 no 4 pp415ndash423 1999

[40] C S Babu and T Kant ldquoTwo shear deformable finite elementmodels for buckling analysis of skew fibre-reinforced compositeand sandwich panelsrdquo Composite Structures vol 46 no 2 pp115ndash124 1999

[41] A K Garg R K Khare and T Kant ldquoFree vibration of skewfiber-reinforced composite and sandwich laminates using ashear deformable finite element modelrdquo Journal of SandwichStructures and Materials vol 8 no 1 pp 33ndash53 2006

[42] J N Reddy Mechanics of Laminated Composites Plates Theoryand Analysis CRC press Boca Raton Fla USA 1996

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 17: Research Article Active Constrained Layer Damping of Smart …downloads.hindawi.com/archive/2013/824163.pdf · 2019-07-31 · Research Article Active Constrained Layer Damping of

Journal of Composites 17

[26] W A Smith and B A Auld ldquoModeling 1ndash3 composite piezo-electrics thickness-mode oscillationsrdquo IEEE Transactions onUltrasonics Ferroelectrics and Frequency Control vol 38 no 1pp 40ndash47 1991

[27] PiezocompositesMaterial Systems Inc 543 Great Road Little-ton MA 01460

[28] Y Benveniste and G J Dvorak ldquoUniform fields and universalrelations in piezoelectric compositesrdquo Journal of the Mechanicsand Physics of Solids vol 40 no 6 pp 1295ndash1312 1992

[29] M L Dunn and M Taya ldquoMicromechanics predictions ofthe effective electroelastic moduli of piezoelectric compositesrdquoInternational Journal of Solids and Structures vol 30 no 2 pp161ndash175 1993

[30] J Aboudi ldquoMicromechanical prediction of the effective coeffi-cients of thermo-piezoelectric multiphase compositesrdquo Journalof Intelligent Material Systems and Structures vol 9 no 9 pp713ndash722 1999

[31] X Ruan T W Chou A Safari and S C Danforth ldquoA 3-Dconnectivity model for effective piezoelectric properties of yarncompositesrdquo Journal of Composite Materials vol 36 no 14 pp1693ndash1708 2002

[32] A Baz and A Tempia ldquoActive piezoelectric damping compos-itesrdquo Sensors and Actuators A vol 112 no 2-3 pp 340ndash3502004

[33] H Berger S Kari U Gabbert et al ldquoUnit cell models ofpiezoelectric fiber composites for numerical and analytical cal-culation of effective propertiesrdquo SmartMaterials and Structuresvol 15 no 2 pp 451ndash458 2006

[34] M Sakthivel and A Arockiarajan ldquoAn analytical model forpredicting thermo-electro-mechanical response of 1ndash3 piezo-electric compositesrdquo Computational Materials Science vol 48no 4 pp 759ndash767 2010

[35] M C Ray and A K Pradhan ldquoOn the use of verticallyreinforced 1ndash3 piezoelectric composites for hybrid damping oflaminated composite platesrdquo Mechanics of Advanced Materialsand Structures vol 14 no 4 pp 245ndash261 2007

[36] M C Ray and A K Pradhan ldquoPerformance of vertically andobliquely reinforced 1ndash3 piezoelectric composites for activedamping of laminated composite shellsrdquo Journal of Sound andVibration vol 315 no 4-5 pp 816ndash835 2008

[37] S K Sarangi and M C Ray ldquoSmart damping of geometricallynonlinear vibrations of laminated composite beams using verti-cally reinforced 1ndash3 piezoelectric compositesrdquo Smart Materialsand Structures vol 19 no 7 Article ID 075020 2010

[38] A Krishnan and J V Deshpande ldquoVibration of skew laminatesrdquoJournal of Sound and Vibration vol 153 no 2 pp 351ndash358 1992

[39] A R Krishna Reddy and R Palaninathan ldquoFree vibration ofskew laminatesrdquo Computers and Structures vol 70 no 4 pp415ndash423 1999

[40] C S Babu and T Kant ldquoTwo shear deformable finite elementmodels for buckling analysis of skew fibre-reinforced compositeand sandwich panelsrdquo Composite Structures vol 46 no 2 pp115ndash124 1999

[41] A K Garg R K Khare and T Kant ldquoFree vibration of skewfiber-reinforced composite and sandwich laminates using ashear deformable finite element modelrdquo Journal of SandwichStructures and Materials vol 8 no 1 pp 33ndash53 2006

[42] J N Reddy Mechanics of Laminated Composites Plates Theoryand Analysis CRC press Boca Raton Fla USA 1996

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 18: Research Article Active Constrained Layer Damping of Smart …downloads.hindawi.com/archive/2013/824163.pdf · 2019-07-31 · Research Article Active Constrained Layer Damping of

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials


Distribution Optimization of Constrained Damping Materials ... · three-bay and 3 3 grid) covered by the constrained damping materials considering first modal loss factors. The vibration

Distribution Optimization of Constrained Damping Materials ... · three-bay and 3 3 grid) covered by the constrained damping materials considering first modal loss factors. The vibration

THE EFFECT OF ISOLATED DAMPING LAYER SYSTEM ON EARTH …eprints.utm.my/id/eprint/77888/1/BehrouzGordanPFKA2014TH.pdfTHE EFFECT OF ISOLATED DAMPING LAYER SYSTEM ON EARTH DAM UNDER EARTHQUAKE

THE EFFECT OF ISOLATED DAMPING LAYER SYSTEM ON EARTH …eprints.utm.my/id/eprint/77888/1/BehrouzGordanPFKA2014TH.pdfTHE EFFECT OF ISOLATED DAMPING LAYER SYSTEM ON EARTH DAM UNDER EARTHQUAKE

Column dampers with negative stiffness: high damping at ...silver.neep.wisc.edu/~lakes/AdvDamperSmallPre.pdf · Cedolin, 1991) . If the bar is constrained laterally, it may be stabilized

Column dampers with negative stiffness: high damping at ...silver.neep.wisc.edu/~lakes/AdvDamperSmallPre.pdf · Cedolin, 1991) . If the bar is constrained laterally, it may be stabilized

Strain Dependent Damping Characteristics of a High Damping ...

Strain Dependent Damping Characteristics of a High Damping ...

Topology optimization of passive constrained layer damping ...

Topology optimization of passive constrained layer damping ...

Evaluation of Interaction Effects on the System Period and the System Damping Due to Foundation Embedment and Layer Depth - IMPORTANTE

Evaluation of Interaction Effects on the System Period and the System Damping Due to Foundation Embedment and Layer Depth - IMPORTANTE

Topology optimization of constrained layer damping on plates ...Z. Ling et al. / Topology optimization of constrained layer damping on plates using MMA approach 223 Fig. 1. A schematic

Topology optimization of constrained layer damping on plates ...Z. Ling et al. / Topology optimization of constrained layer damping on plates using MMA approach 223 Fig. 1. A schematic

Optimization of constrained layer damping for strain ...rdo.psu.ac.th/sjstweb/journal/34-2/0125-3395-volume-179-187.pdf · Original Article Optimization of constrained layer damping

Optimization of constrained layer damping for strain ...rdo.psu.ac.th/sjstweb/journal/34-2/0125-3395-volume-179-187.pdf · Original Article Optimization of constrained layer damping

Noise and Vibration Control with Constrained Layer Damping ...srehsv.com/wp-content/uploads/2013/08/RAM-6-Workshop-Roush... · Noise and Vibration Control with Constrained Layer Damping

Noise and Vibration Control with Constrained Layer Damping ...srehsv.com/wp-content/uploads/2013/08/RAM-6-Workshop-Roush... · Noise and Vibration Control with Constrained Layer Damping

Journal of Reinforced Plastics and Composites...1023 Analytical and Experimental Investigations of Prestressed Laminated Composite Beams with Constrained Viscoelastic Damping Layer

Journal of Reinforced Plastics and Composites...1023 Analytical and Experimental Investigations of Prestressed Laminated Composite Beams with Constrained Viscoelastic Damping Layer

Cross-Layer Frameworks for Constrained Power and Resources Management of Embedded Systems

Cross-Layer Frameworks for Constrained Power and Resources Management of Embedded Systems

Surface plasmon damping effects due to Ti …...Surface plasmon damping effects due to Ti adhesion layer in individual gold nanodisks DESALEGN T. DEBU, PIJUSH K. GHOSH, DAVID FRENCH,

Surface plasmon damping effects due to Ti …...Surface plasmon damping effects due to Ti adhesion layer in individual gold nanodisks DESALEGN T. DEBU, PIJUSH K. GHOSH, DAVID FRENCH,

159 Modal Analysis of Plates With Partial Constrained-Layer Damping Treatments

159 Modal Analysis of Plates With Partial Constrained-Layer Damping Treatments

CoAP over BP for a Delay-Tolerant Internet of ThingsA. Constrained Application Protocol (CoAP) CoAP [10] offers an application layer protocol that al-lows resource-constrained devices

CoAP over BP for a Delay-Tolerant Internet of ThingsA. Constrained Application Protocol (CoAP) CoAP [10] offers an application layer protocol that al-lows resource-constrained devices

Research Article Dynamic Analysis of Three-Layer Sandwich …downloads.hindawi.com/journals/sv/2015/736256.pdf · 2019-07-31 · Shock and Vibration H 1 H 2 H 3 hn b Damping layer

Research Article Dynamic Analysis of Three-Layer Sandwich …downloads.hindawi.com/journals/sv/2015/736256.pdf · 2019-07-31 · Shock and Vibration H 1 H 2 H 3 hn b Damping layer

A Polynomial Time Approximation Scheme For Timing Constrained Minimum Cost Layer Assignment

A Polynomial Time Approximation Scheme For Timing Constrained Minimum Cost Layer Assignment

Professor Mike Brennan - feis.unesp.br · KK bd o To be effective damping layers of similar thickness to the structure are required Weight penalty . Unconstrained layer damping 10-1

Professor Mike Brennan - feis.unesp.br · KK bd o To be effective damping layers of similar thickness to the structure are required Weight penalty . Unconstrained layer damping 10-1

Noise and Vibration Control with Constrained Layer Damping … · 2014-01-29 · Constrained layer damping (CLD) systems can be applied to control the resonant response of a variety

Noise and Vibration Control with Constrained Layer Damping … · 2014-01-29 · Constrained layer damping (CLD) systems can be applied to control the resonant response of a variety

SCALING VIDEO ANALYTICS ON CONSTRAINED EDGE N › static › paper_files › mlsys › 2019 › 19… · Scaling Video Analytics on Constrained Edge Nodes ... up to a given layer

SCALING VIDEO ANALYTICS ON CONSTRAINED EDGE N › static › paper_files › mlsys › 2019 › 19… · Scaling Video Analytics on Constrained Edge Nodes ... up to a given layer

Topology optimization of constrained layer damping on ...downloads.hindawi.com/journals/sv/2011/830793.pdf · Topology optimization of constrained layer damping on plates using Method

Topology optimization of constrained layer damping on ...downloads.hindawi.com/journals/sv/2011/830793.pdf · Topology optimization of constrained layer damping on plates using Method