Research Article On the Finite Element Free Vibration Analysis of Delaminated Layered...

Research ArticleOn the Finite Element Free Vibration Analysis of DelaminatedLayered Beams A New Assembly Technique

Nicholas H Erdelyi and Seyed M Hashemi

Department of Aerospace Engineering Ryerson University 350 Victoria Street Toronto Ontario Canada M5B 2K3

Correspondence should be addressed to Seyed M Hashemi smhashemryersonca

Received 20 July 2015 Revised 16 October 2015 Accepted 21 October 2015

Academic Editor Longjun Dong

Copyright copy 2016 N H Erdelyi and S M Hashemi 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

The dynamic analysis of flexible delaminated layered beams is revisited Exploiting Boolean vectors a novel assembly schemeis developed which can be used to enforce the continuity requirements at the edges of delamination region leading to adelamination stiffness term The proposed assembly technique can be used to form various beam configurations with through-width delaminations irrespective of the formulation used to model each beam segment The proposed assembly system and theGalerkin Finite Element Method (FEM) formulation are subsequently used to investigate the natural frequencies and modes of2- and 3-layer beam configurations Using the Euler-Bernoulli bending beam theory and free mode delamination the governingdifferential equations are exploited and two beam finite elements are developed The free bending vibration of three illustrativeexample problems characterized by delamination zones of variable length is investigated The intact and defective beam naturalfrequencies and modes obtained from the proposed assemblyFEM beam formulations are presented along with the analyticalresults and those available in the literature

1 Introduction

Layered structures have seen greatly increased use in civilshipbuilding mechanical and aerospace structural applica-tions in recent decades Delamination a common failuremode in layered structures may arise from loss of adhe-sion between two layers of the structure from interlaminarstresses arising from geometric or material discontinuitiesor from mechanical loadings The presence of delaminationmay significantly reduce the stiffness and strength of thestructures A reduction in the stiffness in turn will affectthe vibration characteristics of the structures Changes inthe natural frequency as a direct result of the reduction ofstiffness may lead to resonance if the reduced frequency isclose to an excitation frequency

Several experimental methods exist to predict the onsetsize and growth of delamination as a failure mode incomposite materials Using acoustic emission (AE) sensorsdifferent levels of amplitude signals emitted by the materialscan be monitored [1] The different amplitudes correspondto loading types and are assigned damage mechanisms

Using this technique continuous monitoring of damage ispossible experimentally Another research [2] has shown thatacoustic emission is a viable and effective tool for identifyingdamage and distinguishing damage types in self-reinforcedpolyethylene composites More recently further research hasbeen done using neural networks and unsupervised learningtechniques applied to the data set of acoustic emission signals[3] The signals were successfully used to classify the AEpatterns caused by different damage mechanisms in carbon-reinforced composites (delamination and matrix cracking)

The time-domain stability of vibrating delaminated sys-tems has also been an area of study Particularly the mech-anisms that cause the different laminates to separate fromeach other are typically not defined in most theoreticalapplications They are described as physically inadmissiblemode shapes whose existence in the frequency domain is aproduct of delamination tip boundary conditions The studyof the real phenomenon however has shown that time-dependent normal forces in the delaminated segments donot influence the global free vibration frequencies but maycontribute to localized buckling [4 5] Instability and critical

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 3707658 14 pageshttpdxdoiorg10115520163707658

2 Shock and Vibration

dynamic forces can be predicted allowing for study of theonset of delamination opening

The vibration modelling and analysis of delaminatedmultilayer beams has been a topic of interest for manyresearchers The earliest delamination models formulated inthe 1980s [6] addressed the vibration of two-layer beamswhere each layer was modelled using Euler-Bernoulli bend-ing beam theory The upper and lower portions of thedelaminated segment were assumed to vibrate independentof each other that is ldquofree moderdquo delamination The freemode however underpredicts natural frequencies for off-midplane delaminations due to unrestricted penetration ofthe beams into each other In 1988 [7] Mujumdar andSuryanarayan proposed the ldquoconstrained moderdquo delamina-tion model which assumes equal transverse displacementsfor the top and bottom beams and the rigid connectorassumption The rigid connector assumption states that forthe beam models presented the delamination faces whichare planar and normal to the neutral axis of the undeformedbeam remain planar and normal to the neutral axis of thedeformed beamThis assumption produces a set of kinematicand force continuity conditions at the delamination tipsIn a recent work by Szekrenyes [8] an extensive literaturesurvey of the research works related to the vibrations ofdelaminated elements was presented and based on coupledflexural-longitudinal vibration model the equality of axialforces in the top and bottom beams was derived and shownin an exact way Also the continuity of the effective bendingmoments was related to the equilibrium equations and it wasalso concluded that delamination buckling can take place ifthe normal force is compressive in one of the half periods ofthe vibration and reaches a critical value [8]

The constrained mode delamination model predictingvibration behaviour much more accurately for off-midplanedelamination is in fact simply a limiting case of the freemodedelaminationmodel However opening delaminationmodesthat is where the layers separate from each other commonlyseen in experimental analysis [9ndash11] cannot be capturedusing the constrained model Therefore in the present studythe free mode delamination model will be investigated andthe constrained mode delamination model can be derived ina similar manner

The accuracy of dynamicforced response analysis of aflexible structure depends greatly on the reliability of themodal analysis method used and the resulting natural fre-quencies and modes There are various numerical semiana-lytical and analytical methods to predict the natural frequen-cies andmode shapes of such a system Several exact solutionmethods exist for well-defined systems such as delaminatedisotropic beams with constant geometric and material prop-erties Single [9 12 13] multiple [14] and various overlappingand enveloped delamination conditions in space and onvarious elastic media such as Pasternak soil [15] have beenstudied using analytical solution methods Some work hasalso been done on delaminated sandwich structures [16]albeit with some mathematical simplification These solutionmethods generally use the same procedure as Mujumdarand Suryanarayan [7] to formulate the kinematic conti-nuity conditions across the delamination tips The power

of this type of formulation lies in the ability to be appliedto any number of different system configurations However apotential drawback to this procedure is that the system equa-tion must be reformulated after any configuration changepotentially limiting its applicability

The conventional Finite Element Method (FEM) has along well-established history and with the advent of digitalcomputers it is commonly used for structural analysis TheFEM is a general and systematic approach to formulate theelement matrices for a given system and is easily adaptableto complex systems such as nonuniform geometry oftenmodeled as a stepped piecewise-uniform configurationExploiting polynomial interpolation (shape) functions theFEM leads to constant element mass and stiffness matricesand ultimately a linear eigenvalue problem from whichthe natural frequencies and modes of the system can bereadily extracted The FEM method for a single beam can bemodified to accurately model delaminated multilayer beamsAmong others Lee [17ndash19] used the layerwise FEM theoryto investigate the free vibration of delaminated beams In therecent years layered sandwich and composite elements havebeen integrated in certain commercial software and are usedto analyze the vibration of composite structures Howevermodelling a delaminated configuration in commercial soft-ware packages such as ANSYS is not straightforward and caninvolve cumbersome complex time-consuming and error-prone processes It requiresmanualmodel creation involvingthe use of for example multipoint constraint rigid link(ANSYS element MPC184) [20] to enforce the displacementand slope continuity at the edges of delamination region [21]

Semianalytical formulations such as the so-calledDynamic Finite Element (DFE) method [22] have also beendeveloped to carry out structural modal analysis The hybridDFE formulation results in a more accurate predictionmethod than traditional and FEM modeling techniquesallowing for a reduced mesh size The DFE technique followsthe same typical procedure as the FEM by formulating theelement equations discretized to a local domain where ele-ment stiffness matrices are constructed and then assembledinto a single global matrix The application of the DFE to thepreliminary free vibration analysis of a delaminated 2-layerbeamhas been reported in an earlier work by the authors [23]

Analytical methods namely the Dynamic StiffnessMatrix (DSM) have also been used for the vibrationalanalysis of isotropic sandwich and composite structuralelements and beam-structures The DSM approach exploitsthe general closed-form solution to the governing differentialequations of motion of the system to formulate a frequency-dependent stiffness matrix The DSM produces exact resultsfor simple structural elements such as uniform beams andBanerjee and his coworkers [24 25] have developed a numberof DSM formulations for various beam configurations TheDSMmethod for a single beam can be modified to accuratelymodel delaminated multilayer beams A DSM-based analysisof a two-layer split beam has also been presented in earlierworks by the authors [26 27]

The aimof this paper is to present an FEM formulation forthe linear free vibration analysis of a delaminated two-layer

Shock and Vibration 3

H1

H2

H3

Beam 1

Beam 2

Beam 3

Beam 4

Through-width delamination

Mw998400

L1 L4

S w

z

x

L0

L2L3a

x1 x4x2x3

P uEI1

EI2

EI3

EI4

Figure 1 The coordinate system and notation for a delaminated layered beam

beam using the free mode delamination model The delami-nation is represented by two intact beam segments one foreach of the top and bottom sections of the delaminationThe delaminated region is bounded on either side by intactfull-height beams The beams transverse displacements aregoverned by the Euler-Bernoulli slender beam bending the-ory and shear deformation and rotary inertia are neglectedContinuity conditions for forces moments displacementsand slopes at the delamination tips are enforced through anovel Boolean vector assembly scheme leading to the integralFEM model of the system In fact through the presentedmethod one obtains specific matrices for an intact fullydelaminated and delaminated elements attached to an intactsegment from leftright Therefore a direct assembly methodcan be directly used to form various multiple-delaminatedbeam configurations without the need to manually createthe model and to use a constraint element (eg ANSYSelementMPC184 [20]) to enforce the displacement and slopecontinuity at the edges of delamination region Thence thedirect assembly of element matrices and the application ofsystemrsquos global boundary conditions results in the lineareigenvalue problem of the defective system In addition twoMATLAB-based computer codes based on the DynamicStiffness Matrix (DSM) method [26 27] and the analyticalsolutions reported in the literature [28ndash31] are developed andused as a benchmark for comparison Two 2-noded and 3-noded beam FEM elements are presented where cubic Her-mite and quartic interpolation functions of approximationrespectively are used to express the flexural displacementfunctions that is field variables and weighting functions[32] The FEM models are used to compute the naturalfrequencies of an illustrative defective beam example char-acterized by a single delamination zone of variable lengthThe frequency values are then compared with DSM data andthose from the literature Certain modal characteristics ofthe system are also discussed It is worth noting that whilethe model used in this study assumes isotropic materialsfurther research is underway to extend it to sandwich [33] and

fibre-reinforced laminated composite beams characterizedby an extensional response coupled with flexuraltorsionaland coupled bending-torsion vibration [34 35]

2 Mathematical Model

Figure 1 shows the general coordinate system and notationfor a single-delaminated beam with total length 119871 intactbeam segment lengths 119871

1and 119871

4 delamination length 119886 and

total height 1198671 This model incorporates a general delami-

nation which can include laminated composites or bilayeredisotropic materials with different material and geometricproperties above and below the delamination planeThus thetop layer has thickness 119867

2 Youngrsquos modulus 119864

2 density 120588

2

cross-sectional area 1198602 and second moment of area 119868

2 The

bottom layer has corresponding properties with subscript3 The delamination tips occur at stations 119909

2and 119909

3 and

torsion shear deformation axial (warping effects and axialdeformation) and out-of-plane delamination are ignoredFollowing this notation the general equation of motion forthe 119894th Euler-Bernoulli beam in free vibration is written as[26ndash28 31]

119864119868119894

1205974119908119894

1205971199094+ 120588119894119860119894

1205972119908119894

1205971199052= 0 119894 = 1 4 (1)

For harmonic oscillations the transverse displacements canbe written as

119908119894(119905) = 119882

119894sin (120596119905) (2)

where 119882119894is the amplitude of the displacement 119908

119894 subscript

ldquo119894rdquo represents the beam segment number and120596 is the circularfrequency of excitation of the system Back-substituting (2)into (1) the equations of motion reduce to

119864119868119894

1205974119882119894

1205971199094minus 1205881198941198601198941205962119882119894= 0 119894 = 1 4 (3)


The general solution to the 4th-order homogeneous differen-tial equation (3) can be written in the following form

119882119894(119909) = 119860

119894cos(120582

119894

119909119894

119871119894

) + 119861119894sin(120582

119894

119909119894

119871119894

)

+ 119862119894cosh(120582

119894

119909119894

119871119894

) + 119863119894sinh(120582

119894

119909119894

119871119894

)

(4)

which represents the bending displacement 119882119894of beam

segment ldquo119894rdquo 119871119894is the beam segment length and 120582

119894stands for

nondimensional frequency of oscillation defined as

1205824

119894=

1205962120588119894119860119894

119864119868119894

1198714

119894 (5)

Coefficients 119860119894 119861119894 119862119894 and 119863

119894(119894 = 1 4) are evaluated

to satisfy the displacement continuity requirements of thebeam segments and the system boundary conditions As alsoreported by several researchers [8 26ndash28 31] the inclusionof delamination into the beam model leads to coupled axial-transversemotion of the delaminated beamportions primar-ily associated with the continuity requirements imposed atthe delamination endpoints In order for the delamination tipcross sections to remain planar after deformation the ends ofthe top and bottom beams must have the same relative axiallocation after deformation preventing interlaminar slip Themidplanes that is the neutral axes of the beam segmentsin the delaminated region are located at a distance fromthe midplanes of the intact segments Hence they will nothave the same axial deformation unless some internal axialforce is imposed As mentioned earlier in this paper basedon coupled flexural-longitudinal vibration model [8] theequality of axial forces in the top and bottom beams hasbeen recently derived and shown in an exact way and thecontinuity of the effective bending moments was related tothe equilibrium equations However in what follows thisimposed axial force is briefly presented for completenessfollowing the method derived and discussed in [7]

Consider a delamination tip after deformationAccordingto the numbering scheme in Figure 1 and since no externalaxial load is applied the top and bottom beam segmentsmusthave equal and opposite internal axial forces that is119875

3= minus1198752

applied to prevent interlaminar slip (see Figure 2) Addition-ally the requirement that the delamination tip faces remainplanar after deformation results in at the left delaminationtip

1199062(1199092= 0) minus 119906

3(1199093= 0) =

1198671

21198821015840

1(1199091= 1198711) (6)

where 119906119894is the axial displacement of beam section 119894 and

1198821015840

1(1199091

= 1198711) = 119882

1015840

2(1199092

= 0) = 1198821015840

3(1199093

= 0) from thekinematic continuity conditions If this is combined with thesame formulation from the right delamination tip

(1199063(1199093= 1198713) minus 1199063(1199093= 0))

minus (1199062(1199092= 1198712) minus 1199062(1199092= 0))

=1198671

2(1198821015840

4(1199094= 0) minus119882

1015840

1(1199091= 1198711))

(7)

+

23

23

23

Faces remain planar after deformation

M2M2

M3M3

=

P2 P2

P3 = minusP2P3

Figure 2 The faces of the delamination remain planar afterdeformation adapted from [31]

The assumption is made by Mujumdar and Suryanarayan [7]and by other researchers (see eg [27 31]) that the axialdisplacementwill behave according to the following for smalldeformations and material and geometric properties whichremain constant along the length of the beam

119906119894(119909119894= 119871119894) minus 119906119894(119909119894= 0) = int

119871 119894

0

119875119894(119909119894)

119864119860119894(119909119894)119889119909119894=

119875119894119871119894

119864119860119894

(8)

where 119864119860119894is the axial stiffness of beam section 119894 Substituting

this into (7) yields

11987521198712

1198641198602

minus11987531198713

1198641198603

=1198671

2(1198821015840

4(1199094= 0) minus119882

1015840

1(1199091= 1198711)) (9)

Using the continuity of axial forces across the delaminationtip 1198753= minus1198752

1198753= Λlowast(1198821015840

4(1199094= 0) minus119882

1015840

1(1199091= 1198711)) (10)

where 1198821015840

119894is the slope of the 119894th beam segment where

ldquoprimerdquo represents the differentiation with respect to thebeam longitudinal axis 119909 and the parameter Λlowast is definedas

Λlowast=

1198671

21198712

(11986411986021198641198603

1198641198602+ 1198641198603

) (11)

Expression (11) can be further simplified if the cross-sectionalshape is known and continuity of bending moments at theleft and right delamination tips respectively leads to thefollowing equations

1198721(1199091= 1198711) = 119872

2(1199092= 0) +119872

3(1199093= 0) minus 119875

2

1198673

2

+ 1198753

1198672

2

(12a)

1198724(1199094= 0) = 119872

2(1199092= 1198712) + 119872

3(1199093= 1198713)

minus 1198752

1198673

2+ 1198753

1198672

2

(12b)


Exploiting (12a) and conditions ((10) (11)) for internal axialforce and noting that bending moments and shear forces inbeam segment ldquo119894rdquo are related to displacements 119882

119894 through

119872119894= minus119864119868

11989411988210158401015840

119894 and 119878

119894= 119864119868119894119882101584010158401015840

119894 respectively it can be shown

that for continuity of bending moments

119864119868111988210158401015840

1(1199091= 1198711)

= 119864119868211988210158401015840

2(1199092= 0) + 119864119868

311988210158401015840

3(1199093= 0)

+ Λ [1198821015840

4(1199094= 0) minus119882

1015840

1(1199091= 1198711)]

(13)

where the coefficient Λ is defined as

Λ =1198672

1

41198712

(11986411986021198641198603

1198641198602+ 1198641198603

) (14)

To satisfy the continuity of shear forces about the leftdelamination tip

1198641198681119882101584010158401015840

1(1199091= 1198711) = 119864119868

2119882101584010158401015840

2(1199092= 0)

+ 1198641198683119882101584010158401015840

3(1199093= 0)

(15)

Likewise using (12b) for 1199094= 0 relevant relationships

similar to (13) and (15) can be derived for the left delaminationtip Two boundary conditions at the intact beam ends conti-nuity of displacements and slopes at the delamination tipsresult in 12 equations Along with additional four equationsresulting from the continuity of bending moments and shearforces at the delamination tips the total 16 equations can beused to solve for the 16 unknowns 119860

119894minus 119863119894for each beam

119894 = 1 4 as appear in Expression (4) In other words the 16equations can be solved simultaneously using a root findingalgorithm to find the natural frequencies and mode shapesof the system Thus an analytical solution can be producedfor each set of systemrsquos imposed boundary conditions (seeeg [28ndash31]) This solution method based on finding thecoefficient matrix of the system herein referred to as theldquoCoefficient Method (CM)rdquo has been used to predict thevibration behavior of different systems of varying complexity(see eg [28ndash31]) However one of the main drawbacks ofthis method is that it remains a relatively problem-specificsolution technique which should be reformulated everytime the systemrsquos global boundary conditions change Inan earlier publication by the authors [27] an equivalentyet more conveniently applicable Dynamic Stiffness Matrix(DSM) formulation was presented The DSM however lacksgenerality and is limited to uniform beam configurationsTherefore in what follows a general FEM-based model ispresented which can be readily extended to more complexcases with variable geometric and material parameters

21 Finite Element Method (FEM) Formulation The finiteelement approach used here is based on the Galerkin methodof weighted residuals The equations of motion for eachbeam are used as the basis of this solution method Simpleharmonic motion is again assumed and the equations of

motion according to the Euler-Bernoulli beam theory takethe following form

119864119868119894

1205974119882119894

1205971199094minus 119864119868119894(120582119894

119871119894

)

4

119882119894= 0 119894 = 1 4 (16)

where119882119894is the actual transverse displacement of beam 119894 and

the same nondimensionalization used in (5) has been appliedAn approximate transverse displacement119882

119894is introduced in

place of the actual displacement such that 119882119894cong 119882119894 This

results in the following residual equation

119864119868119894

1205974119882119894

1205971199094minus 119864119868119894(120582119894

119871119894

)

4

119882119894= R 119894 = 1 4 (17)

whereR is the residual of the approximate equation Follow-ing the Galerkin method of weighted residuals the residualabove is weighted by a virtual displacement 120575119882 and theintegral is set to zero across the domain of the systemSince the system is composed of four distinct beam sectionsoccupying their own subset of the domain the followingis representative of the Galerkin method applied to thedelaminated system

4

sum

119894=1

(int

119871 119894

0

(1198641198681198941205751198821198941198821015840101584010158401015840

119894minus 119864119868119894(120582119894

119871119894

)

4

120575119882119894119882119894)119889119909119894)

= 0

(18)

with

119882119894(119909119894) = ⟨119873

119894(119909119894)⟩ 119882119899 (19)

where ⟨119873119894(119909119894)⟩ are the shape functions of the beam elements

which will be defined later Since the virtual displacement isapplied to the entire domain and the four different beamsections occupy unique subdomains 120575119882 = sum

4

119894=1120575119882119894 In

order to produce the force and displacement continuityterms a set of integrations by parts is performed on the aboveresulting in the following weak form

4

sum

119894=1

(119864119868119894[120575119882119894119882101584010158401015840

119894minus 120575119882

1015840

11989411988210158401015840

119894]119871 119894

0)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

lowast

+

4

sum

119894=1

(int

119871 119894

0

(11986411986811989412057511988210158401015840

11989411988210158401015840

119894minus 119864119868119894(120582119894

119871119894

)

4

120575119882119894119882119894)119889119909119894)

= 0

(20)

The terms in (lowast) above represent the boundary and continu-ity conditions imposed on the system Using Euler-Bernoullibeam theory the shear force and bending moment at anypoint are defined based on the transverse displacement as

119878 (119909) = 119864119868 (119909)119882 (119909)101584010158401015840

119872 (119909) = 119864119868 (119909)119882 (119909)10158401015840

(21)


For the endpoints of beam sections 1 and 4 the following istrue for free vibration

1198641198681(1205751198821119882101584010158401015840

1minus 120575119882

1015840

111988210158401015840

1)100381610038161003816100381610038161199091=0

= 120575Wext10038161003816100381610038161199091=0

1198641198684(1205751198824119882101584010158401015840

4minus 120575119882

1015840

411988210158401015840

4)100381610038161003816100381610038161199094=1198714

= 120575Wext10038161003816100381610038161199094=1198714

(22)

where 120575Wext is the external virtual work caused by appliedexternal forces on the system causing virtual displacementsFor the free vibration of this system the total external workis 120575Wext = 120575Wext|1199091=0 + 120575Wext|1199094=1198714 = 0 The remainingterms in (lowast) above can be resolved by applying the necessarycontinuity conditions for displacements (eg119882

1= 1198822= 1198823

slopes11988210158401= 1198821015840

2= 1198821015840

3 and shear forces119864119868

1119882101584010158401015840

1= 1198641198682119882101584010158401015840

2+

1198641198683119882101584010158401015840

3) leading to

4

sum

119894=1

(119864119868119894[120575119882119894119882101584010158401015840

119894minus 120575119882

1015840

11989411988210158401015840

119894]119871 119894

0) = 120575Wext + 120575119882

2(0)

sdot (1198641198681119882101584010158401015840

1(1198711) minus 119864119868

2119882101584010158401015840

2(0) minus 119864119868

3119882101584010158401015840

3(0))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

lowastlowast

minus 1205751198821015840

2(0)

sdot (119864119868111988210158401015840

1(1198711) minus 119864119868

211988210158401015840

2(0) minus 119864119868

311988210158401015840

3(0))

minus 1205751198822(1198712)

sdot (1198641198684119882101584010158401015840

4(0) minus 119864119868

2119882101584010158401015840

2(1198712) minus 119864119868

3119882101584010158401015840

3(1198713))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

lowastlowastlowast

+ 1205751198821015840

2(1198712)

sdot (119864119868411988210158401015840

4(0) minus 119864119868

211988210158401015840

2(1198712) minus 119864119868

311988210158401015840

3(1198713))

(23)

The terms (lowastlowast) and (lowast lowast lowast) in (23) as well as the externalwork term go to zero directly as a result of the shear forcecontinuity conditions However the remaining terms do notvanish since the continuity of bending moments ((13) (14))contains an additional implicit bending-axial coupling termsuch that

4

sum

119894=1

(119864119868119894[120575119882119894119882101584010158401015840

119894minus 120575119882

1015840

11989411988210158401015840

119894]119871 119894

0)

= (1205751198821015840

2(1198712) minus 120575119882

1015840

2(0)) (Λ (119882

1015840

2(1198712) minus 120575119882

1015840

2(0)))

(24)

With the boundary and continuity conditions satisfied thesystem can be discretized into elements which will each beapproximated using their own basis functions fromwhich FEshape functions can be found The system can be discretizedas follows using the result of (24)

Λ(1205751198821015840

2(1198712) minus 120575119882

1015840

2(0)) (119882

1015840

2(1198712) minus 120575119882

1015840

2(0))

+

4

sum

119894=1

elements119894sum

119898=1

(int

119909119898+1

119909119898

(11986411986811989412057511988210158401015840

11989411988210158401015840

119894minus 119864119868119894(120582119894

119871119894

)

4

120575119882119894119882119894)119889119909)

= 0

(25)

where ldquoelements119894rdquo is the number of elements in beam

section 119894It is worth noting that following the same above-

described procedure expressions (13) and (14) respectivelyfor the case of double-delaminated three-layer beam config-urations can be written as

119864119868111988210158401015840

1= 119864119868211988210158401015840

2+ 119864119868311988210158401015840

3+ 119864119868411988210158401015840

4+ Λ (119882

1015840

1(1199091= 1198711) minus 119882

1015840

5(1199095= 0)) (26a)

Λ =(1198671+ 1198673)2

11986411986021198641198604+ (1198672+ 1198673)2

11986411986021198641198603+ (1198673+ 1198674)2

11986411986031198641198604

4119886 (1198641198602+ 1198641198603+ 1198641198604)

(26b)

where subscripts 119894 = 1 5 stand for the intact beam (left andright extremity) segments and the three delaminated layersare represented by 119894 = 2 3 4 The necessary displacementsslopes and shear force continuity conditions at the delami-nation tips respectively are then written as

1198821= 1198822= 1198823= 1198824

1198821015840

1= 1198821015840

2= 1198821015840

3= 1198821015840

4

1198641198681119882101584010158401015840

1= 1198641198682119882101584010158401015840

2+ 1198641198683119882101584010158401015840

3+ 1198641198684119882101584010158401015840

4

(27)

22 2-Node Beam Element (1) Following the traditionalEuler-Bernoulli finite element development Hermite cubic

polynomials [32] were used as the basis functions of approxi-mation for each beam such that for a two-node 2-degree-of-freedom per node beam element that is transverse displace-ment and slope defined at each node (Figure 3)

119882(119909) = ⟨1 119909 1199092

1199093⟩ 119862 (28)

where 119862 is a column vector of unknown constant coeffi-cients The following represents the vector of nodal displace-ments used in further FE development

119882119899 =

1198821

1198821015840

1

1198822

1198821015840

2

=

[[[[[

[

1 0 0 0

0 1 0 0

1 119871 1198712

1198713

0 1 2119871 31198712

]]]]]

]

119862 = [119875119899] 119862 (29)


1 2

L

W1

W9984001

W2

W9984002

x

Figure 3 A 2-node 4-degree-of-freedom (4-DOF) beam element

Thus

119882(119909) = ⟨1 119909 1199092

1199093⟩ [119875119899]minus1

119882119899 = ⟨119873 (119909)⟩ 119882119899 (30)

where ⟨119873(119909)⟩ is a row vector of shape functions whichdescribe the displacements at any point along the domain ofthe element in terms of the nodal displacements and slopesat the endpoints of the element domain 119882

119899 Additionally

the shape functions may also be used to approximate thevirtual displacements 120575119882(119909) = ⟨119873(119909)⟩120575119882

119899 With the

shape functions fully defined they may be substituted for theapproximate displacements in (25)

⟨120575119882119899⟩ (Λ (119873

21015840

(1198712) minus 119873

21015840

(0)) (⟨1198732⟩1015840

(1198712) minus ⟨119873

2⟩1015840

sdot (0))) 119882119899 +

4

sum

119894=1

elements119894sum

119899=1

⟨120575119882119899⟩

sdot (int

119909119898+1

119909119898

(11986411986811989411987311989410158401015840

⟨119873119894⟩10158401015840

minus 119864119868119894(120582119894

119871119894

)

4

119873119894 ⟨119873119894⟩)119889119909)

sdot 119882119899 = 0

(31)

Frequency-dependent and non-frequency-dependent termsabove can be gathered to form the following eigenvalue prob-lem common to structural vibration analysis with FEM witha modification caused by the presence of the delamination

⟨120575119882119899⟩ ((119870 + 119870delam) minus 120596

2119872) 119882

119899 = 0 (32)

if

det ((119870 + 119870delam) minus 1205962119872) = 0 (33)

where 119870 is the structural stiffness matrix formed by assem-bling the associated beam elements as per (31) 119870delam isthe delamination stiffness matrix from the term appearingoutside the integral expression in (31) and119872 is the structuralmass matrix From this formulation the simplest solutionmethods involve eigensolutions

23 3-Node Beam Element (2) Exploiting the same conceptof a polynomial interpolation function a 3-node 5-DOFelement is also developed (see Figure 4) Making use of ahigher-order polynomial interpolation functions increasesthe accuracy of the solution Whereas for the 2-node beamelement (Figure 3) a 3rd-order polynomial was required fora higher-order interpolation of 4th-order one requires theaddition of another single degree of freedom to the systemThis was accomplished by adding a midpoint node with onlyone degree of freedom (lateral displacement) to the beam

1 2 3W9984001

W3W2W1

W9984002

x

L2 L2

Figure 4 The 3-node 5-degree-of-freedom beam element

model used previously This third node while increasing themesh fineness allows for a greater solution accuracy andpossibly faster convergence which will be investigated The3-node beam element (Figure 4) was developed in the sameway as the 2-node beam element except using the followinginterpolation function

119882(119909) = ⟨1 119909 1199092

1199093

1199094⟩ 119862 (34)

Consequently the degrees of freedom for the system weremodified as discussedThe addition of themidpoint node andits associated lateral degree of freedom are compensated forby using the following degrees of freedom

119882119899 =

1198821

1198821015840

1

1198822

1198823

1198821015840

3

=

[[[[[[[[[[

[

1 0 0 0 0

0 1 0 0 0

1119871

2

1198712

4

1198713

8

1198714

16

1 119871 1198712

1198713

1198714

0 1 2119871 31198712

41198713

]]]]]]]]]]

]

119862

= [119875119899] 119862

(35)

Thus

119882(119909) = ⟨1 119909 1199092

1199093

1199094⟩ [119875119899]minus1

119882119899 = ⟨119873⟩ V119899 (36)

3 Numerical Tests

Numerical checkswere performed to confirm the predictabil-ity accuracy and practical applicability of the proposeddelaminated FEM models Both the FEM and CoefficientMethod (CM) [31] formulations were programmed in MAT-LAB codesThe linear eigenvalue problem resulting from theconventional FEM formulations was solved using MATLABldquoeigrdquo function The use of the nondimensional frequency (5)in the calculations removed material dependencies from thesystem provided that the material was isotropic or at leastorthotropic with principal axes aligned with the Cartesiancoordinate system in Figure 1

In what follows two illustrative examples of fixed-fixedhomogeneous 2-layer and 3-layer delaminated beams areexamined In the first example the natural frequencies of thesystem with a central split about the midsection (119871

1= 1198714)

of various lengths up to 60 of the span (0 le 119886119871 le 06)occurring symmetrically along the midplane of the beamand surrounded by intact beam segments are consideredThis split beam configuration has also been presented andstudied in [6 7 27ndash31] The split FEM and DSM modelswere created and used to compute the natural frequencies


Table 1 First frequency (1205822) of the delaminated beam with a split occurring symmetrically about the midsection along the midplaneproposed FEM DSM CM and standard FEMmodels

Delaminationlength119886119871 tot

Six 2-node FEMdaggerelements

Ten 2-node FEMdaggerelements DSM [27] Wang et al [6]

reported by [31]Della andShu [31]

LayerwiseFEM [17]

Mode 1 Mode 1 Mode 1 Mode 1 Mode 1 Mode 1Intact mdash mdash 2239 2239 2237 223601 2237 2237 2237 2237 2237 223602 2236 2236 2236 2235 2236 223503 2224 2224 2224 2223 2224 222304 2184 2184 2183 2183 2183 218205 2089 2089 2089 2088 2089 208806 1929 1929 1930 1929 1930 1928daggerConventional FEM displays numerical instabilities with delamination lengths approaching 0

Table 2 2nd frequency (1205822) of the delaminated beam with a split occurring symmetrically about the midsection along the midplaneproposed FEM DSM CM and standard FEMmodels




reported by [31]Della and Shu

[31]LayerwiseFEM [17]


and mode shapes of various delamination cases As thebenchmarks for comparison and validation purposes theresults from [6 7 27] for the constrained mode as wellas an alternative formulation from [6] were used As alsosuggested in [6] the first two frequencies were computedfor a delamination length of 00002119871 to check for numericalinstability when the split length becomes extremely smallThis case showed negligible discrepancies from those of asolid intact beam The effect of the longitudinal motionof the upper and lower parts of the split region on thefrequencies examined in [6] has been neglected here forthis class of example problems As discussed earlier in thispaper the differential stretching of the top and bottom layerswas present to keep the delamination faces planar afterdeformation (ie no interlaminar slip at the delaminationfaces) The FEM formulation results in an additional stiffnessterm called ldquodelamination stiffnessrdquo which has the effect ofstiffening the system at the delamination tips

Tables 1 and 2 summarize the first two natural frequenciesobtained using the beam element 1 (4-DOF) Finite ElementMethod (FEM) with 6- and 10-element discretizations ofmidplane delaminated region (up to 60 of span)The intactbeam segments were modeled using single beam elementsAs can be seen fromTables 1 and 2 the FEM formulation pro-duced excellent agreement with analytical results and the lay-erwise FEM theory [17] and exhibits a convergence towardsthe DSM results as the number of elements is increased

While the FEM result discrepancies are generally low evenfor a coarse mesh size it is observed to be even lower for thefirst naturalmode than the second oneThis is consistent withtraditional FEM theory where more elements are required toguarantee accurate solutions for higher mode numbers

Figure 5 shows the first two natural modes of the 2-layered beam with 60 of span midplane delaminationcompared with those of an intact configuration It is worthnoting that the conventional FEM-based models are char-acterized by constant mass and stiffness matrices of limitednumber of total degrees of freedom (DOF) that is numberof nodes times number of DOF per node Accordingly thenaturalmodes obtained from the conventional FEMmodelmdashbeing the eigenmodes of the governing linear eigenvalueproblemmdashhave the same dimension as the total degrees offreedom of the FEM model Unlike the conventional FEM(eg 4-DOF Hermite beam element) the DSM matrices areformulated based on continuous element assumptions whichintroduces infinite number of degrees of freedomwithin eachelement (see eg [24 25]) Therefore as also reported in[27] usingDSM technique additionalmodes of vibration canbe found These modes are the result of the denominator ofthe global stiffness matrix going to zero and correspondinglythe determinant of the global stiffness matrix approachinginfinity |119870(120596)| rarr infin Also known as the poles of a systemthey can represent real physical mode shapes describing thestructure vibrating at zero nodal displacements [36] outside


DelaminatedIntact

0

01

02

03

04

05

06

07

08

09

1

Mod

e sha

pe

01 02 03 04 05 06 07 08 09 10x location

aL = 06 H2 = 05H1

(a)

DelaminatedIntact

0 01 02 03 04 05 06 07 08 09 1x location

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Mod

e sha

pe

aL = 06 H2 = 05H1

(b)

Figure 5 The first two natural modes for a 2-layered beam with centrally located 60 midplane delamination compared with those of theintact configuration (a) 1st modes (b) 2nd modes

aL = 05 15 elements

minus15

minus1

minus05

0

05

1

15

Mod

e sha

pe

05 1 15 2 25 3 35 4 45 50x location

Figure 6The inadmissiblemode interpenetration of equithicknesstop and bottombeamsWhilemathematically possible this situationwould not be encountered in practical applications 60 of spanmidplane delamination 15-element FEM model within the delami-nation zone

of the delaminated region Zero-nodal-displacement modeshave also been observed and reported in the literature forother structural configurations (see eg [22 36]) There arealso certain frequencies captured through the system modalanalysis whose mode shapes while mathematically possibledo not represent physically admissible displacements Thesemodes for example a second mode (120582 = 310) in the caseof present study are simply the result of the free modelassumptions [27] They correspond to interpenetration ofthe beams as illustrated in Figure 6 and would not be

DelaminatedIntact

0

01

02

03

04

05

06

07

08

09

1

Mod

e sha

pe

01 02 03 04 05 06 07 08 09 10x location

aL = 06 H2 = 04H1

Figure 7 The first opening mode for a delaminated beam with topbeam thickness equal to 40 of the height of the intact beam 60of span off-midplane delamination

present in a constrained mode analysis Similar inadmissiblepartial and complete interpenetration modes have also beenreported in the literature [37] In addition as also reportedin the literature (see eg [27ndash31]) under small vibrationamplitudes a split layered beam may exhibit a mode at afrequency corresponding to a delamination-opening modeThe first opening mode for a delaminated beam with topbeam thickness equal to 40 of the height of the intact beam60 of span off-midplane delamination obtained using a 17-element FEMmodel is depicted in Figure 7


Table 3 The first and second nondimensional frequency parameter (120582) for model 1

1198861198712-nodeFEM

3-nodeFEM Analytical (CM) [31] Ref [31] DSM [27]

02 4725 4725 4725 47 4725

Mode 103 4691 4695 4695 47 469504 4574 4572 4575 46 457505 4318 4315 4315 43 431502 7054 7046 7045 71 7045

Mode 203 6337 6334 6335 63 633504 5965 5960 5965 60 596505 5860 5846 5845 59 5845

H1

03H1

03H1

aL

(a)

H1 H13H13H13

aL

(b)

Figure 8 The first (a) and second (b) double-delamination models tested

1 2

3 6 9

1074

5 8 11

12 13

Figure 9 Mesh discretization for double-delaminated configura-tions

In order to further assess the accuracy of the proposedFEM method in what follows two different 3-layer beamconfigurations with central dual delaminations are analyzed(refer to Figure 8)

The FEM frequency results obtained using both 2- and3-node elements for both models above are validated againstvarious analytical solutions obtained from a MATLAB codewritten based on the ldquoexactanalyticalrdquo (CM) method [31]those extracted by interpolation from a graph by Della andShu [31] and the frequency data obtained from a DSM code(see an earlier work by the authors [27]) as well as thosegathered from the literature For the relatively simple modelspresented in this section the discretization (Figure 9) wasused to mesh the domains

In the first delamination model tested (see Figure 8) thetop and centre delaminated beams each have a height of30 of the intact beam height In addition the delaminationlength a was varied as a percentage of the total beam lengthfrom 20 to 50 The delamination is central meaning thatthe left and right intact segments have equal lengths Thefrequency results for these delaminated clamped-clampedbeam configurations are presented in Table 3

As it can be observed from Table 3 both 2- and 3-node beam elements perform well with respect to both theanalytical solution and those taken from the literature As

expected light deviations (026 for 2-node mode 2 withrespect to the exactanalytical solution) are present for largerdelamination sizes and for higher modes of vibration Forhigher modes the 3-node beam tends to perform slightlybetter than the 2-node beam This difference is expected toincrease with an increase in mode number However for thefirstmode and for small delamination sizes the 2- and 3-nodebeam elements perform similarly and differences between theresults for each element were negligibleThis could be used tojustify the use of a 2-node beam element in such situations tosave on processor requirements and solution times

Since identifying nonphysical or physically inadmissiblebehaviour can be essential in improving a given model thesystemrsquos natural modes are also an important considerationin free vibration modeling Therefore the first two modeshapes were compiled in Figure 10 generated from thesystemrsquos eigenvectors results the shape functions derived andpresented earlier Since the variance between mode shapesfor the 2- and 3-node beam elements was minimal onlythe mode shape for the 3-node beam will be plotted Notethat in Figure 10 the element end nodes are represented bycircular markers and the midpoint nodes are represented bytimesmarkers

In the second delamination model tested (see Figure 8)the total thickness is the same as in Model 1 however eachbeam segmentrsquos thickness is assumed to be one-third (13)of the total thickness As before the results reported from[31] were converted from chart to numerical form The finalresults of this second analysis are detailed in Table 4

As it can be observed from Table 4 for smaller delamina-tion sizes the 2-node and 3-node beam elements performedwith similar accuracy and both converged to a reasonableaccuracy (largest deviation approximately 016 from theexact solution) However for highermodes and larger delam-ination sizes the 3-node beam exhibits higher accuracy and



119886119871FEM2-node

FEM3-node Analytical (CM) [31] Ref [31] DSM [27]

02 4726 4725 4725 47 4725

Mode 103 4692 4695 4695 47 469504 4580 4578 4575 46 457505 4342 4338 4335 43 433502 7010 7002 7005 70 7005

Mode 203 6284 6281 6285 63 628504 5922 5917 5915 59 591505 5833 5820 5815 58 5815

Figure 10The first two natural modes for a 2-layered beamwith centrally located delamination obtained fromModel 1 top 1st mode shapesBottom 2nd mode shapes left 30 middle 40 and right 50 delaminations

better convergence characteristics It is also worth notingthat the FEM results presented in this section were obtainedwith only slight modifications to the single delaminationtechnique presented earlier whereas the analytical solutionhad to be completely modifiedThis lends further credence tothe advantages of presented FEM-based formulations sincegood correlation with analytical results can be achieved withless development overhead

The systemrsquos first two natural modes evaluated in thesame manner as in Model 1 are presented in Figure 11Furthermore some mode shapes emerged from the analysiswhich involved physically inadmissible mode shapes In thiscase the inadmissibility comes from the interpenetrationof different beam layers with each other (see Figure 12)That is one beam segment would vibrate laterally in onedirection and another beam segment occupying the sameaxial domain would vibrate laterally in the opposite direc-tion In the present study the physically inadmissible mode

shapes were found to occur when the difference in flexuralstiffness of the beams is nonzero and worsens with increasingdifference seen for both exact and FEM solutions It wasobserved that if the difference in beam stiffness between thethree delaminated beams was sufficiently large these modeswould appear to be slight interpenetrations

Delaminations not centrally located along the beam (ie1198711

= 1198715) were observed to generate mode shapes deviating

from standard beam mode shapes and opening of the beamsegments can be seen (see Figure 13) The delaminated beamsegment with the smallest bending stiffness deforms thelargest and will thus be forced into the other stiffer beamsegments Although not physically admissible there is noprovision within the free model to prevent this behaviourThis phenomenon was observed for all doubly delaminatedbeammodels with disparate delaminated beamheights (119867

2=

1198673

= 1198674) as well as for any noncentral delamination (119871

1=

1198715)



(a) (b)

Figure 12 Examples of physically inadmissible mode shapes for a 3-layer delaminated beam obtained using 3-node elements (double-delamination model 1) 119867

2= 03119867

1 1198673= 05119867

1 and 119886119871 = 05 (a) off-delamination level partial pole 2nd mode (ie no vibrations at

the intact segments) 120582 = 467 (b) interpenetration due to natural vibration 4th mode 120582 = 596

Finally it is also worth mentioning that by provingthe existence of the parametric excitation in delaminatedbeams [4] it was recently shown and experimentally ver-ified that the opening is amplitude dependent that is thedelamination opening takes place only at a certain criticalamplitude In reality the mode shapes are asymmetric andcan be approximated by the superposition of the global shapeof the entire beam and the local buckling eigenshape ofthe delaminated part based on a dynamic stability analysis[8] Furthermore the experimentally observed delaminationopening was reported to be significantly less than those

calculated by the free model which would rather justify theuse of the constrained model for further analysis [8]

4 Conclusion

Based on the conventional Finite Element Method (FEM)formulation and employing Boolean vectors a novel assem-bly scheme for the free vibration modelling and analysisof a delaminated layered beams was developed Based onthe free mode delamination model 2- and 3-node beamelements equipped with cubic and quartic shape functions


(a) (b)

Figure 13 The 1st (a) and the 2nd mode (b) for a noncentrally delaminated beam

respectively were used to extract the flexural natural fre-quencies of single- and double-delaminated beam configu-rations While the results showed very good agreement withthose obtained from analytical and Dynamic Stiffness Matrix(DSM) models presented in the literature the use of 3-nodebeam elements with quartic shape functions as opposed tothe more common Hermite cubic shape functions found in2-node beams did not produce significantly better resultsfor the same mesh size Systemrsquos natural modes and openingmodes for both midplane and off-midplane delaminationswere also examined and illustrated The constrained modelcan be applied to prevent the interpenetration modes at theexpense of increased system stiffness [7 31]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

The authors declare that this paper presents the results of aresearch conducted by the first author under the supervisionof the second (corresponding) author

Acknowledgments

The authors wish to acknowledge the support provided byThe Natural Sciences and Engineering Research Council ofCanada (NSERC) Ontario Graduate Scholarship (OGS) andRyerson University

References

[1] S Benmedakhene M Kenane and M L Benzeggagh ldquoIniti-ation and growth of delamination in glassepoxy compositessubjected to static and dynamic loading by acoustic emissionmonitoringrdquo Composites Science and Technology vol 59 no 2pp 201ndash208 1999

[2] X Zhuang and X Yan ldquoInvestigation of damagemechanisms inself-reinforced polyethylene composites by acoustic emissionrdquo

Composites Science and Technology vol 66 no 3-4 pp 444ndash449 2006

[3] D Crivelli M Guagliano M Eaton et al ldquoLocalisation andidentification of fatigue matrix cracking and delamination ina carbon fibre panel by acoustic emissionrdquo Composites Part BEngineering vol 74 pp 1ndash12 2015

[4] A Szekrenyes ldquoA special case of parametrically excited systemsfree vibration of delaminated composite beamsrdquo EuropeanJournal of MechanicsmdashASolids vol 49 pp 82ndash105 2015

[5] Z Juhasz and A Szekrenyes ldquoEstimation of local delaminationbuckling in orthotropic composite plates using Kirchhoff platefinite elementsrdquo Mathematical Problems in Engineering vol2015 Article ID 749607 14 pages 2015

[6] J T S Wang Y Y Liu and J A Gibby ldquoVibrations of splitbeamsrdquo Journal of Sound and Vibration vol 84 no 4 pp 491ndash502 1982

[7] P M Mujumdar and S Suryanarayan ldquoFlexural vibrations ofbeams with delaminationsrdquo Journal of Sound and Vibration vol125 no 3 pp 441ndash461 1988

[8] A Szekrenyes ldquoCoupled flexural-longitudinal vibration ofdelaminated composite beams with local stability analysisrdquoJournal of Sound and Vibration vol 333 no 20 pp 5141ndash51642014

[9] J J Tracy and G C Pardoen ldquoEffect of delamination onthe natural frequencies of composite laminatesrdquo Journal ofComposite Materials vol 23 no 12 pp 1200ndash1216 1989

[10] M-H H Shen and J E Grady ldquoFree vibrations of delaminatedbeamsrdquo AIAA Journal vol 30 no 5 pp 1361ndash1370 1992

[11] W Lestari and S Hanagud ldquoHealth monitoring of structuresmultiple delamination dynamics in composite beamsrdquo Reportfor American Institute of Aeronautics and Astronautics AIAA-99-1509 American Institute of Aeronautics and AstronauticsReston Va USA 1999

[12] L Brandinelli and R Massabo ldquoFree vibrations of delaminatedbeam-type structures with crack bridgingrdquo Composite Struc-tures vol 61 no 1-2 pp 129ndash142 2003

[13] D Shu and H Fan ldquoFree vibration of bimaterial split beamrdquoComposites Part B Engineering vol 27 no 1 pp 79ndash84 1996

[14] D Shu ldquoVibration of sandwich beams with double delamina-tionsrdquoComposites Science and Technology vol 54 no 1 pp 101ndash109 1995

[15] H Hein ldquoThe influence of delamination on free vibrations ofcomposite beams on Pasternak soilrdquo Proceedings of the Estonian


Academy of Sciences Physics Mathematics vol 55 no 4 pp220ndash234 2006

[16] SH Jian andCHwu ldquoFree vibration of delaminated compositesandwich beamsrdquo AIAA Journal vol 33 no 10 pp 1911ndash19181995

[17] J Lee ldquoFree vibration analysis of delaminated compositebeamsrdquoComputers amp Structures vol 74 no 2 pp 121ndash129 2000

[18] S Lee T Park and G Z Voyiadjis ldquoFree vibration analysis ofaxially compressed laminated composite beam-columns withmultiple delaminationsrdquoComposites Part B Engineering vol 33no 8 pp 605ndash617 2002

[19] S Lee T Park and G Z Voyiadjis ldquoVibration analysis of multi-delaminated beamsrdquoComposites Part B Engineering vol 34 no7 pp 647ndash659 2003

[20] ANSYS Academic Research Release 130 Help System ANSYSCanonsburg Pa USA 2009

[21] A R Ghorbanzad Free vibration analysis of layered beamswith delamination damagemdashanANSYS-based FEM investigation[MS thesis] Department of Aerospace Engineering RyersonUniversity Toronto Canada 2012

[22] S M Hashemi Free vibrational analysis of rotating beam-like structures a dynamic finite element approach [PhD the-sis] Department of Mechanical Engineering Laval UniversityQuebec Canada 1998

[23] N Erdelyi and S M Hashemi ldquoFree vibration analysis ofdelaminated layered beams a dynamic finite element (DFE)techniquerdquo in Proceedings of the 8th Joint Canada-Japan Work-shop on CompositeMaterials p 10 Montreal Canada July 2010

[24] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997

[25] J R Banerjee C W Cheung R Morishima M Pereraand J Njuguna ldquoFree vibration of a three-layered sandwichbeam using the dynamic stiffness method and experimentrdquoInternational Journal of Solids and Structures vol 44 no 22-23pp 7543ndash7563 2007

[26] N Erdelyi and S M Hashemi ldquoAn exact dynamic stiffnessmatrix (DSM) formulation for free vibration analysis of delam-inated beamsrdquo in Proceedings of the 8th Joint Canada-JapanWorkshop on Composite Materials p 10 Montreal Canada July2010

[27] N H Erdelyi and S M Hashemi ldquoA dynamic stiffness elementfor free vibration analysis of delaminated layered beamsrdquoModelling and Simulation in Engineering vol 2012 Article ID492415 8 pages 2012

[28] C N Della and D Shu ldquoVibration of delaminated multilayerbeamsrdquoComposites Part B Engineering vol 37 no 2-3 pp 227ndash236 2006

[29] C N Della and D Shu ldquoFree vibration analysis of compositebeams with overlapping delaminationsrdquo European Journal ofMechanicsmdashASolids vol 24 no 3 pp 491ndash503 2005

[30] C N Della and D Shu ldquoVibration of delaminated compositelaminates a reviewrdquo Applied Mechanics Reviews vol 60 no 1ndash6 pp 1ndash20 2007

[31] C N Della and D Shu ldquoFree vibration analysis of multipledelaminated beams under axial compressive loadrdquo Journal ofReinforced Plastics and Composites vol 28 no 11 pp 1365ndash13812009

[32] K-J Bathe Finite Element Procedures in Engineering AnalysisPrentice-Hall 1982

[33] S M Hashemi and E J Adique ldquoA quasi-exact dynamic finiteelement for free vibration analysis of sandwich beamsrdquo AppliedComposite Materials vol 17 no 2 pp 259ndash269 2010

[34] S M Hashemi and A Roach ldquoDynamic finite element analysisof extensional-torsional coupled vibration in nonuniform com-posite beamsrdquo Applied Composite Materials vol 18 no 6 pp521ndash538 2011

[35] S M Hashemi and S Borneman ldquoDoubly-coupled vibrationsof nonuniform composite wings a dynamic finite elementrdquo inMathematical Problems in Engineering Aerospace and SciencesS Sivasundaram Ed vol 5 pp 141ndash152 Cambridge ScientificPublishers 2011

[36] F W Williams and W H Wittrick ldquoAn automatic computa-tional procedure for calculating natural frequencies of skeletalstructuresrdquo International Journal of Mechanical Sciences vol 12no 9 pp 781ndash791 1970

[37] C H Roche and M L Accorsi ldquoA new finite element for globalmodeling of delaminations in laminated beamsrdquoFinite Elementsin Analysis and Design vol 31 no 2 pp 165ndash177 1998

International Journal of

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Control Scienceand Engineering

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Journal ofEngineeringVolume 2014

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VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



dynamic forces can be predicted allowing for study of theonset of delamination opening

The vibration modelling and analysis of delaminatedmultilayer beams has been a topic of interest for manyresearchers The earliest delamination models formulated inthe 1980s [6] addressed the vibration of two-layer beamswhere each layer was modelled using Euler-Bernoulli bend-ing beam theory The upper and lower portions of thedelaminated segment were assumed to vibrate independentof each other that is ldquofree moderdquo delamination The freemode however underpredicts natural frequencies for off-midplane delaminations due to unrestricted penetration ofthe beams into each other In 1988 [7] Mujumdar andSuryanarayan proposed the ldquoconstrained moderdquo delamina-tion model which assumes equal transverse displacementsfor the top and bottom beams and the rigid connectorassumption The rigid connector assumption states that forthe beam models presented the delamination faces whichare planar and normal to the neutral axis of the undeformedbeam remain planar and normal to the neutral axis of thedeformed beamThis assumption produces a set of kinematicand force continuity conditions at the delamination tipsIn a recent work by Szekrenyes [8] an extensive literaturesurvey of the research works related to the vibrations ofdelaminated elements was presented and based on coupledflexural-longitudinal vibration model the equality of axialforces in the top and bottom beams was derived and shownin an exact way Also the continuity of the effective bendingmoments was related to the equilibrium equations and it wasalso concluded that delamination buckling can take place ifthe normal force is compressive in one of the half periods ofthe vibration and reaches a critical value [8]

The constrained mode delamination model predictingvibration behaviour much more accurately for off-midplanedelamination is in fact simply a limiting case of the freemodedelaminationmodel However opening delaminationmodesthat is where the layers separate from each other commonlyseen in experimental analysis [9ndash11] cannot be capturedusing the constrained model Therefore in the present studythe free mode delamination model will be investigated andthe constrained mode delamination model can be derived ina similar manner

The accuracy of dynamicforced response analysis of aflexible structure depends greatly on the reliability of themodal analysis method used and the resulting natural fre-quencies and modes There are various numerical semiana-lytical and analytical methods to predict the natural frequen-cies andmode shapes of such a system Several exact solutionmethods exist for well-defined systems such as delaminatedisotropic beams with constant geometric and material prop-erties Single [9 12 13] multiple [14] and various overlappingand enveloped delamination conditions in space and onvarious elastic media such as Pasternak soil [15] have beenstudied using analytical solution methods Some work hasalso been done on delaminated sandwich structures [16]albeit with some mathematical simplification These solutionmethods generally use the same procedure as Mujumdarand Suryanarayan [7] to formulate the kinematic conti-nuity conditions across the delamination tips The power

of this type of formulation lies in the ability to be appliedto any number of different system configurations However apotential drawback to this procedure is that the system equa-tion must be reformulated after any configuration changepotentially limiting its applicability

The conventional Finite Element Method (FEM) has along well-established history and with the advent of digitalcomputers it is commonly used for structural analysis TheFEM is a general and systematic approach to formulate theelement matrices for a given system and is easily adaptableto complex systems such as nonuniform geometry oftenmodeled as a stepped piecewise-uniform configurationExploiting polynomial interpolation (shape) functions theFEM leads to constant element mass and stiffness matricesand ultimately a linear eigenvalue problem from whichthe natural frequencies and modes of the system can bereadily extracted The FEM method for a single beam can bemodified to accurately model delaminated multilayer beamsAmong others Lee [17ndash19] used the layerwise FEM theoryto investigate the free vibration of delaminated beams In therecent years layered sandwich and composite elements havebeen integrated in certain commercial software and are usedto analyze the vibration of composite structures Howevermodelling a delaminated configuration in commercial soft-ware packages such as ANSYS is not straightforward and caninvolve cumbersome complex time-consuming and error-prone processes It requiresmanualmodel creation involvingthe use of for example multipoint constraint rigid link(ANSYS element MPC184) [20] to enforce the displacementand slope continuity at the edges of delamination region [21]

Semianalytical formulations such as the so-calledDynamic Finite Element (DFE) method [22] have also beendeveloped to carry out structural modal analysis The hybridDFE formulation results in a more accurate predictionmethod than traditional and FEM modeling techniquesallowing for a reduced mesh size The DFE technique followsthe same typical procedure as the FEM by formulating theelement equations discretized to a local domain where ele-ment stiffness matrices are constructed and then assembledinto a single global matrix The application of the DFE to thepreliminary free vibration analysis of a delaminated 2-layerbeamhas been reported in an earlier work by the authors [23]

Analytical methods namely the Dynamic StiffnessMatrix (DSM) have also been used for the vibrationalanalysis of isotropic sandwich and composite structuralelements and beam-structures The DSM approach exploitsthe general closed-form solution to the governing differentialequations of motion of the system to formulate a frequency-dependent stiffness matrix The DSM produces exact resultsfor simple structural elements such as uniform beams andBanerjee and his coworkers [24 25] have developed a numberof DSM formulations for various beam configurations TheDSMmethod for a single beam can be modified to accuratelymodel delaminated multilayer beams A DSM-based analysisof a two-layer split beam has also been presented in earlierworks by the authors [26 27]

The aimof this paper is to present an FEM formulation forthe linear free vibration analysis of a delaminated two-layer


H1

H2

H3

Beam 1

Beam 2

Beam 3

Beam 4


Mw998400

L1 L4

S w

z

x

L0

L2L3a

x1 x4x2x3

P uEI1

EI2

EI3

EI4






1and 119871





2 density 120588

2


2 The


2and 119909

3 and


119864119868119894

1205974119908119894

1205971199094+ 120588119894119860119894

1205972119908119894

1205971199052= 0 119894 = 1 4 (1)


119908119894(119905) = 119882

119894sin (120596119905) (2)


119894 subscript


119864119868119894

1205974119882119894

1205971199094minus 1205881198941198601198941205962119882119894= 0 119894 = 1 4 (3)



119882119894(119909) = 119860

119894cos(120582

119894

119909119894

119871119894

) + 119861119894sin(120582

119894

119909119894

119871119894

)

+ 119862119894cosh(120582

119894

119909119894

119871119894

) + 119863119894sinh(120582

119894

119909119894

119871119894

)

(4)



119894stands for


1205824

119894=

1205962120588119894119860119894

119864119868119894

1198714

119894 (5)


119894(119894 = 1 4) are evaluated



3= minus1198752


1199062(1199092= 0) minus 119906

3(1199093= 0) =

1198671

21198821015840

1(1199091= 1198711) (6)


1198821015840

1(1199091

= 1198711) = 119882

1015840

2(1199092

= 0) = 1198821015840

3(1199093


(1199063(1199093= 1198713) minus 1199063(1199093= 0))

minus (1199062(1199092= 1198712) minus 1199062(1199092= 0))

=1198671

2(1198821015840

4(1199094= 0) minus119882

1015840

1(1199091= 1198711))

(7)

+

23

23

23


M2M2

M3M3

=

P2 P2

P3 = minusP2P3



119906119894(119909119894= 119871119894) minus 119906119894(119909119894= 0) = int

119871 119894

0

119875119894(119909119894)

119864119860119894(119909119894)119889119909119894=

119875119894119871119894

119864119860119894

(8)



11987521198712

1198641198602

minus11987531198713

1198641198603

=1198671

2(1198821015840

4(1199094= 0) minus119882

1015840

1(1199091= 1198711)) (9)


1198753= Λlowast(1198821015840

4(1199094= 0) minus119882

1015840

1(1199091= 1198711)) (10)

where 1198821015840



Λlowast=

1198671

21198712

(11986411986021198641198603

1198641198602+ 1198641198603

) (11)


1198721(1199091= 1198711) = 119872

2(1199092= 0) +119872

3(1199093= 0) minus 119875

2

1198673

2

+ 1198753

1198672

2

(12a)

1198724(1199094= 0) = 119872

2(1199092= 1198712) + 119872

3(1199093= 1198713)

minus 1198752

1198673

2+ 1198753

1198672

2

(12b)



119894 through

119872119894= minus119864119868

11989411988210158401015840

119894 and 119878

119894= 119864119868119894119882101584010158401015840



119864119868111988210158401015840

1(1199091= 1198711)

= 119864119868211988210158401015840

2(1199092= 0) + 119864119868

311988210158401015840

3(1199093= 0)

+ Λ [1198821015840

4(1199094= 0) minus119882

1015840

1(1199091= 1198711)]

(13)


Λ =1198672

1

41198712

(11986411986021198641198603

1198641198602+ 1198641198603

) (14)


1198641198681119882101584010158401015840

1(1199091= 1198711) = 119864119868

2119882101584010158401015840

2(1199092= 0)

+ 1198641198683119882101584010158401015840

3(1199093= 0)

(15)







119864119868119894

1205974119882119894

1205971199094minus 119864119868119894(120582119894

119871119894

)

4

119882119894= 0 119894 = 1 4 (16)






119864119868119894

1205974119882119894

1205971199094minus 119864119868119894(120582119894

119871119894

)

4

119882119894= R 119894 = 1 4 (17)


4

sum

119894=1

(int

119871 119894

0

(1198641198681198941205751198821198941198821015840101584010158401015840

119894minus 119864119868119894(120582119894

119871119894

)

4

120575119882119894119882119894)119889119909119894)

= 0

(18)

with

119882119894(119909119894) = ⟨119873

119894(119909119894)⟩ 119882119899 (19)



4

119894=1120575119882119894 In


4

sum

119894=1

(119864119868119894[120575119882119894119882101584010158401015840

119894minus 120575119882

1015840

11989411988210158401015840

119894]119871 119894

0)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

lowast

+

4

sum

119894=1

(int

119871 119894

0

(11986411986811989412057511988210158401015840

11989411988210158401015840

119894minus 119864119868119894(120582119894

119871119894

)

4

120575119882119894119882119894)119889119909119894)

= 0

(20)


119878 (119909) = 119864119868 (119909)119882 (119909)101584010158401015840

119872 (119909) = 119864119868 (119909)119882 (119909)10158401015840

(21)



1198641198681(1205751198821119882101584010158401015840

1minus 120575119882

1015840

111988210158401015840

1)100381610038161003816100381610038161199091=0

= 120575Wext10038161003816100381610038161199091=0

1198641198684(1205751198824119882101584010158401015840

4minus 120575119882

1015840

411988210158401015840

4)100381610038161003816100381610038161199094=1198714

= 120575Wext10038161003816100381610038161199094=1198714

(22)


1= 1198822= 1198823

slopes11988210158401= 1198821015840

2= 1198821015840


1119882101584010158401015840

1= 1198641198682119882101584010158401015840

2+

1198641198683119882101584010158401015840

3) leading to

4

sum

119894=1

(119864119868119894[120575119882119894119882101584010158401015840

119894minus 120575119882

1015840

11989411988210158401015840

119894]119871 119894

0) = 120575Wext + 120575119882

2(0)

sdot (1198641198681119882101584010158401015840

1(1198711) minus 119864119868

2119882101584010158401015840

2(0) minus 119864119868

3119882101584010158401015840

3(0))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

lowastlowast

minus 1205751198821015840

2(0)

sdot (119864119868111988210158401015840

1(1198711) minus 119864119868

211988210158401015840

2(0) minus 119864119868

311988210158401015840

3(0))

minus 1205751198822(1198712)

sdot (1198641198684119882101584010158401015840

4(0) minus 119864119868

2119882101584010158401015840

2(1198712) minus 119864119868

3119882101584010158401015840

3(1198713))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

lowastlowastlowast

+ 1205751198821015840

2(1198712)

sdot (119864119868411988210158401015840

4(0) minus 119864119868

211988210158401015840

2(1198712) minus 119864119868

311988210158401015840

3(1198713))

(23)


4

sum

119894=1

(119864119868119894[120575119882119894119882101584010158401015840

119894minus 120575119882

1015840

11989411988210158401015840

119894]119871 119894

0)

= (1205751198821015840

2(1198712) minus 120575119882

1015840

2(0)) (Λ (119882

1015840

2(1198712) minus 120575119882

1015840

2(0)))

(24)


Λ(1205751198821015840

2(1198712) minus 120575119882

1015840

2(0)) (119882

1015840

2(1198712) minus 120575119882

1015840

2(0))

+

4

sum

119894=1

elements119894sum

119898=1

(int

119909119898+1

119909119898

(11986411986811989412057511988210158401015840

11989411988210158401015840

119894minus 119864119868119894(120582119894

119871119894

)

4

120575119882119894119882119894)119889119909)

= 0

(25)




119864119868111988210158401015840

1= 119864119868211988210158401015840

2+ 119864119868311988210158401015840

3+ 119864119868411988210158401015840

4+ Λ (119882

1015840

1(1199091= 1198711) minus 119882

1015840

5(1199095= 0)) (26a)

Λ =(1198671+ 1198673)2

11986411986021198641198604+ (1198672+ 1198673)2

11986411986021198641198603+ (1198673+ 1198674)2

11986411986031198641198604

4119886 (1198641198602+ 1198641198603+ 1198641198604)

(26b)


1198821= 1198822= 1198823= 1198824

1198821015840

1= 1198821015840

2= 1198821015840

3= 1198821015840

4

1198641198681119882101584010158401015840

1= 1198641198682119882101584010158401015840

2+ 1198641198683119882101584010158401015840

3+ 1198641198684119882101584010158401015840

4

(27)



119882(119909) = ⟨1 119909 1199092

1199093⟩ 119862 (28)


119882119899 =

1198821

1198821015840

1

1198822

1198821015840

2

=

[[[[[

[

1 0 0 0

0 1 0 0

1 119871 1198712

1198713

0 1 2119871 31198712

]]]]]

]

119862 = [119875119899] 119862 (29)


1 2

L

W1

W9984001

W2

W9984002

x


Thus

119882(119909) = ⟨1 119909 1199092

1199093⟩ [119875119899]minus1

119882119899 = ⟨119873 (119909)⟩ 119882119899 (30)


119899 Additionally


119899 With the


⟨120575119882119899⟩ (Λ (119873

21015840

(1198712) minus 119873

21015840

(0)) (⟨1198732⟩1015840

(1198712) minus ⟨119873

2⟩1015840

sdot (0))) 119882119899 +

4

sum

119894=1

elements119894sum

119899=1

⟨120575119882119899⟩

sdot (int

119909119898+1

119909119898

(11986411986811989411987311989410158401015840

⟨119873119894⟩10158401015840

minus 119864119868119894(120582119894

119871119894

)

4

119873119894 ⟨119873119894⟩)119889119909)

sdot 119882119899 = 0

(31)


⟨120575119882119899⟩ ((119870 + 119870delam) minus 120596

2119872) 119882

119899 = 0 (32)

if

det ((119870 + 119870delam) minus 1205962119872) = 0 (33)



1 2 3W9984001

W3W2W1

W9984002

x

L2 L2



119882(119909) = ⟨1 119909 1199092

1199093

1199094⟩ 119862 (34)


119882119899 =

1198821

1198821015840

1

1198822

1198823

1198821015840

3

=

[[[[[[[[[[

[

1 0 0 0 0

0 1 0 0 0

1119871

2

1198712

4

1198713

8

1198714

16

1 119871 1198712

1198713

1198714

0 1 2119871 31198712

41198713

]]]]]]]]]]

]

119862

= [119875119899] 119862

(35)

Thus

119882(119909) = ⟨1 119909 1199092

1199093

1199094⟩ [119875119899]minus1

119882119899 = ⟨119873⟩ V119899 (36)

3 Numerical Tests



1= 1198714)








LayerwiseFEM [17]














DelaminatedIntact

0

01

02

03

04

05

06

07

08

09

1

Mod

e sha

pe

01 02 03 04 05 06 07 08 09 10x location

aL = 06 H2 = 05H1

(a)

DelaminatedIntact

0 01 02 03 04 05 06 07 08 09 1x location

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Mod

e sha

pe

aL = 06 H2 = 05H1

(b)


aL = 05 15 elements

minus15

minus1

minus05

0

05

1

15

Mod

e sha

pe

05 1 15 2 25 3 35 4 45 50x location



DelaminatedIntact

0

01

02

03

04

05

06

07

08

09

1

Mod

e sha

pe

01 02 03 04 05 06 07 08 09 10x location

aL = 06 H2 = 04H1





1198861198712-nodeFEM


02 4725 4725 4725 47 4725

Mode 103 4691 4695 4695 47 469504 4574 4572 4575 46 457505 4318 4315 4315 43 431502 7054 7046 7045 71 7045

Mode 203 6337 6334 6335 63 633504 5965 5960 5965 60 596505 5860 5846 5845 59 5845

H1

03H1

03H1

aL

(a)

H1 H13H13H13

aL

(b)


1 2

3 6 9

1074

5 8 11

12 13












119886119871FEM2-node


02 4726 4725 4725 47 4725

Mode 103 4692 4695 4695 47 469504 4580 4578 4575 46 457505 4342 4338 4335 43 433502 7010 7002 7005 70 7005

Mode 203 6284 6281 6285 63 628504 5922 5917 5915 59 591505 5833 5820 5815 58 5815








2=

1198673


1=

1198715)



(a) (b)


2= 03119867

1 1198673= 05119867





4 Conclusion



(a) (b)







Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










H1

H2

H3

Beam 1

Beam 2

Beam 3

Beam 4


Mw998400

L1 L4

S w

z

x

L0

L2L3a

x1 x4x2x3

P uEI1

EI2

EI3

EI4






1and 119871





2 density 120588

2


2 The


2and 119909

3 and


119864119868119894

1205974119908119894

1205971199094+ 120588119894119860119894

1205972119908119894

1205971199052= 0 119894 = 1 4 (1)


119908119894(119905) = 119882

119894sin (120596119905) (2)


119894 subscript


119864119868119894

1205974119882119894

1205971199094minus 1205881198941198601198941205962119882119894= 0 119894 = 1 4 (3)



119882119894(119909) = 119860

119894cos(120582

119894

119909119894

119871119894

) + 119861119894sin(120582

119894

119909119894

119871119894

)

+ 119862119894cosh(120582

119894

119909119894

119871119894

) + 119863119894sinh(120582

119894

119909119894

119871119894

)

(4)



119894stands for


1205824

119894=

1205962120588119894119860119894

119864119868119894

1198714

119894 (5)


119894(119894 = 1 4) are evaluated



3= minus1198752


1199062(1199092= 0) minus 119906

3(1199093= 0) =

1198671

21198821015840

1(1199091= 1198711) (6)


1198821015840

1(1199091

= 1198711) = 119882

1015840

2(1199092

= 0) = 1198821015840

3(1199093


(1199063(1199093= 1198713) minus 1199063(1199093= 0))

minus (1199062(1199092= 1198712) minus 1199062(1199092= 0))

=1198671

2(1198821015840

4(1199094= 0) minus119882

1015840

1(1199091= 1198711))

(7)

+

23

23

23


M2M2

M3M3

=

P2 P2

P3 = minusP2P3



119906119894(119909119894= 119871119894) minus 119906119894(119909119894= 0) = int

119871 119894

0

119875119894(119909119894)

119864119860119894(119909119894)119889119909119894=

119875119894119871119894

119864119860119894

(8)



11987521198712

1198641198602

minus11987531198713

1198641198603

=1198671

2(1198821015840

4(1199094= 0) minus119882

1015840

1(1199091= 1198711)) (9)


1198753= Λlowast(1198821015840

4(1199094= 0) minus119882

1015840

1(1199091= 1198711)) (10)

where 1198821015840



Λlowast=

1198671

21198712

(11986411986021198641198603

1198641198602+ 1198641198603

) (11)


1198721(1199091= 1198711) = 119872

2(1199092= 0) +119872

3(1199093= 0) minus 119875

2

1198673

2

+ 1198753

1198672

2

(12a)

1198724(1199094= 0) = 119872

2(1199092= 1198712) + 119872

3(1199093= 1198713)

minus 1198752

1198673

2+ 1198753

1198672

2

(12b)



119894 through

119872119894= minus119864119868

11989411988210158401015840

119894 and 119878

119894= 119864119868119894119882101584010158401015840



119864119868111988210158401015840

1(1199091= 1198711)

= 119864119868211988210158401015840

2(1199092= 0) + 119864119868

311988210158401015840

3(1199093= 0)

+ Λ [1198821015840

4(1199094= 0) minus119882

1015840

1(1199091= 1198711)]

(13)


Λ =1198672

1

41198712

(11986411986021198641198603

1198641198602+ 1198641198603

) (14)


1198641198681119882101584010158401015840

1(1199091= 1198711) = 119864119868

2119882101584010158401015840

2(1199092= 0)

+ 1198641198683119882101584010158401015840

3(1199093= 0)

(15)







119864119868119894

1205974119882119894

1205971199094minus 119864119868119894(120582119894

119871119894

)

4

119882119894= 0 119894 = 1 4 (16)






119864119868119894

1205974119882119894

1205971199094minus 119864119868119894(120582119894

119871119894

)

4

119882119894= R 119894 = 1 4 (17)


4

sum

119894=1

(int

119871 119894

0

(1198641198681198941205751198821198941198821015840101584010158401015840

119894minus 119864119868119894(120582119894

119871119894

)

4

120575119882119894119882119894)119889119909119894)

= 0

(18)

with

119882119894(119909119894) = ⟨119873

119894(119909119894)⟩ 119882119899 (19)



4

119894=1120575119882119894 In


4

sum

119894=1

(119864119868119894[120575119882119894119882101584010158401015840

119894minus 120575119882

1015840

11989411988210158401015840

119894]119871 119894

0)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

lowast

+

4

sum

119894=1

(int

119871 119894

0

(11986411986811989412057511988210158401015840

11989411988210158401015840

119894minus 119864119868119894(120582119894

119871119894

)

4

120575119882119894119882119894)119889119909119894)

= 0

(20)


119878 (119909) = 119864119868 (119909)119882 (119909)101584010158401015840

119872 (119909) = 119864119868 (119909)119882 (119909)10158401015840

(21)



1198641198681(1205751198821119882101584010158401015840

1minus 120575119882

1015840

111988210158401015840

1)100381610038161003816100381610038161199091=0

= 120575Wext10038161003816100381610038161199091=0

1198641198684(1205751198824119882101584010158401015840

4minus 120575119882

1015840

411988210158401015840

4)100381610038161003816100381610038161199094=1198714

= 120575Wext10038161003816100381610038161199094=1198714

(22)


1= 1198822= 1198823

slopes11988210158401= 1198821015840

2= 1198821015840


1119882101584010158401015840

1= 1198641198682119882101584010158401015840

2+

1198641198683119882101584010158401015840

3) leading to

4

sum

119894=1

(119864119868119894[120575119882119894119882101584010158401015840

119894minus 120575119882

1015840

11989411988210158401015840

119894]119871 119894

0) = 120575Wext + 120575119882

2(0)

sdot (1198641198681119882101584010158401015840

1(1198711) minus 119864119868

2119882101584010158401015840

2(0) minus 119864119868

3119882101584010158401015840

3(0))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

lowastlowast

minus 1205751198821015840

2(0)

sdot (119864119868111988210158401015840

1(1198711) minus 119864119868

211988210158401015840

2(0) minus 119864119868

311988210158401015840

3(0))

minus 1205751198822(1198712)

sdot (1198641198684119882101584010158401015840

4(0) minus 119864119868

2119882101584010158401015840

2(1198712) minus 119864119868

3119882101584010158401015840

3(1198713))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

lowastlowastlowast

+ 1205751198821015840

2(1198712)

sdot (119864119868411988210158401015840

4(0) minus 119864119868

211988210158401015840

2(1198712) minus 119864119868

311988210158401015840

3(1198713))

(23)


4

sum

119894=1

(119864119868119894[120575119882119894119882101584010158401015840

119894minus 120575119882

1015840

11989411988210158401015840

119894]119871 119894

0)

= (1205751198821015840

2(1198712) minus 120575119882

1015840

2(0)) (Λ (119882

1015840

2(1198712) minus 120575119882

1015840

2(0)))

(24)


Λ(1205751198821015840

2(1198712) minus 120575119882

1015840

2(0)) (119882

1015840

2(1198712) minus 120575119882

1015840

2(0))

+

4

sum

119894=1

elements119894sum

119898=1

(int

119909119898+1

119909119898

(11986411986811989412057511988210158401015840

11989411988210158401015840

119894minus 119864119868119894(120582119894

119871119894

)

4

120575119882119894119882119894)119889119909)

= 0

(25)




119864119868111988210158401015840

1= 119864119868211988210158401015840

2+ 119864119868311988210158401015840

3+ 119864119868411988210158401015840

4+ Λ (119882

1015840

1(1199091= 1198711) minus 119882

1015840

5(1199095= 0)) (26a)

Λ =(1198671+ 1198673)2

11986411986021198641198604+ (1198672+ 1198673)2

11986411986021198641198603+ (1198673+ 1198674)2

11986411986031198641198604

4119886 (1198641198602+ 1198641198603+ 1198641198604)

(26b)


1198821= 1198822= 1198823= 1198824

1198821015840

1= 1198821015840

2= 1198821015840

3= 1198821015840

4

1198641198681119882101584010158401015840

1= 1198641198682119882101584010158401015840

2+ 1198641198683119882101584010158401015840

3+ 1198641198684119882101584010158401015840

4

(27)



119882(119909) = ⟨1 119909 1199092

1199093⟩ 119862 (28)


119882119899 =

1198821

1198821015840

1

1198822

1198821015840

2

=

[[[[[

[

1 0 0 0

0 1 0 0

1 119871 1198712

1198713

0 1 2119871 31198712

]]]]]

]

119862 = [119875119899] 119862 (29)


1 2

L

W1

W9984001

W2

W9984002

x


Thus

119882(119909) = ⟨1 119909 1199092

1199093⟩ [119875119899]minus1

119882119899 = ⟨119873 (119909)⟩ 119882119899 (30)


119899 Additionally


119899 With the


⟨120575119882119899⟩ (Λ (119873

21015840

(1198712) minus 119873

21015840

(0)) (⟨1198732⟩1015840

(1198712) minus ⟨119873

2⟩1015840

sdot (0))) 119882119899 +

4

sum

119894=1

elements119894sum

119899=1

⟨120575119882119899⟩

sdot (int

119909119898+1

119909119898

(11986411986811989411987311989410158401015840

⟨119873119894⟩10158401015840

minus 119864119868119894(120582119894

119871119894

)

4

119873119894 ⟨119873119894⟩)119889119909)

sdot 119882119899 = 0

(31)


⟨120575119882119899⟩ ((119870 + 119870delam) minus 120596

2119872) 119882

119899 = 0 (32)

if

det ((119870 + 119870delam) minus 1205962119872) = 0 (33)



1 2 3W9984001

W3W2W1

W9984002

x

L2 L2



119882(119909) = ⟨1 119909 1199092

1199093

1199094⟩ 119862 (34)


119882119899 =

1198821

1198821015840

1

1198822

1198823

1198821015840

3

=

[[[[[[[[[[

[

1 0 0 0 0

0 1 0 0 0

1119871

2

1198712

4

1198713

8

1198714

16

1 119871 1198712

1198713

1198714

0 1 2119871 31198712

41198713

]]]]]]]]]]

]

119862

= [119875119899] 119862

(35)

Thus

119882(119909) = ⟨1 119909 1199092

1199093

1199094⟩ [119875119899]minus1

119882119899 = ⟨119873⟩ V119899 (36)

3 Numerical Tests



1= 1198714)








LayerwiseFEM [17]














DelaminatedIntact

0

01

02

03

04

05

06

07

08

09

1

Mod

e sha

pe

01 02 03 04 05 06 07 08 09 10x location

aL = 06 H2 = 05H1

(a)

DelaminatedIntact

0 01 02 03 04 05 06 07 08 09 1x location

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Mod

e sha

pe

aL = 06 H2 = 05H1

(b)


aL = 05 15 elements

minus15

minus1

minus05

0

05

1

15

Mod

e sha

pe

05 1 15 2 25 3 35 4 45 50x location



DelaminatedIntact

0

01

02

03

04

05

06

07

08

09

1

Mod

e sha

pe

01 02 03 04 05 06 07 08 09 10x location

aL = 06 H2 = 04H1





1198861198712-nodeFEM


02 4725 4725 4725 47 4725

Mode 103 4691 4695 4695 47 469504 4574 4572 4575 46 457505 4318 4315 4315 43 431502 7054 7046 7045 71 7045

Mode 203 6337 6334 6335 63 633504 5965 5960 5965 60 596505 5860 5846 5845 59 5845

H1

03H1

03H1

aL

(a)

H1 H13H13H13

aL

(b)


1 2

3 6 9

1074

5 8 11

12 13












119886119871FEM2-node


02 4726 4725 4725 47 4725

Mode 103 4692 4695 4695 47 469504 4580 4578 4575 46 457505 4342 4338 4335 43 433502 7010 7002 7005 70 7005

Mode 203 6284 6281 6285 63 628504 5922 5917 5915 59 591505 5833 5820 5815 58 5815








2=

1198673


1=

1198715)



(a) (b)


2= 03119867

1 1198673= 05119867





4 Conclusion



(a) (b)







Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119882119894(119909) = 119860

119894cos(120582

119894

119909119894

119871119894

) + 119861119894sin(120582

119894

119909119894

119871119894

)

+ 119862119894cosh(120582

119894

119909119894

119871119894

) + 119863119894sinh(120582

119894

119909119894

119871119894

)

(4)



119894stands for


1205824

119894=

1205962120588119894119860119894

119864119868119894

1198714

119894 (5)


119894(119894 = 1 4) are evaluated



3= minus1198752


1199062(1199092= 0) minus 119906

3(1199093= 0) =

1198671

21198821015840

1(1199091= 1198711) (6)


1198821015840

1(1199091

= 1198711) = 119882

1015840

2(1199092

= 0) = 1198821015840

3(1199093


(1199063(1199093= 1198713) minus 1199063(1199093= 0))

minus (1199062(1199092= 1198712) minus 1199062(1199092= 0))

=1198671

2(1198821015840

4(1199094= 0) minus119882

1015840

1(1199091= 1198711))

(7)

+

23

23

23


M2M2

M3M3

=

P2 P2

P3 = minusP2P3



119906119894(119909119894= 119871119894) minus 119906119894(119909119894= 0) = int

119871 119894

0

119875119894(119909119894)

119864119860119894(119909119894)119889119909119894=

119875119894119871119894

119864119860119894

(8)



11987521198712

1198641198602

minus11987531198713

1198641198603

=1198671

2(1198821015840

4(1199094= 0) minus119882

1015840

1(1199091= 1198711)) (9)


1198753= Λlowast(1198821015840

4(1199094= 0) minus119882

1015840

1(1199091= 1198711)) (10)

where 1198821015840



Λlowast=

1198671

21198712

(11986411986021198641198603

1198641198602+ 1198641198603

) (11)


1198721(1199091= 1198711) = 119872

2(1199092= 0) +119872

3(1199093= 0) minus 119875

2

1198673

2

+ 1198753

1198672

2

(12a)

1198724(1199094= 0) = 119872

2(1199092= 1198712) + 119872

3(1199093= 1198713)

minus 1198752

1198673

2+ 1198753

1198672

2

(12b)



119894 through

119872119894= minus119864119868

11989411988210158401015840

119894 and 119878

119894= 119864119868119894119882101584010158401015840



119864119868111988210158401015840

1(1199091= 1198711)

= 119864119868211988210158401015840

2(1199092= 0) + 119864119868

311988210158401015840

3(1199093= 0)

+ Λ [1198821015840

4(1199094= 0) minus119882

1015840

1(1199091= 1198711)]

(13)


Λ =1198672

1

41198712

(11986411986021198641198603

1198641198602+ 1198641198603

) (14)


1198641198681119882101584010158401015840

1(1199091= 1198711) = 119864119868

2119882101584010158401015840

2(1199092= 0)

+ 1198641198683119882101584010158401015840

3(1199093= 0)

(15)







119864119868119894

1205974119882119894

1205971199094minus 119864119868119894(120582119894

119871119894

)

4

119882119894= 0 119894 = 1 4 (16)






119864119868119894

1205974119882119894

1205971199094minus 119864119868119894(120582119894

119871119894

)

4

119882119894= R 119894 = 1 4 (17)


4

sum

119894=1

(int

119871 119894

0

(1198641198681198941205751198821198941198821015840101584010158401015840

119894minus 119864119868119894(120582119894

119871119894

)

4

120575119882119894119882119894)119889119909119894)

= 0

(18)

with

119882119894(119909119894) = ⟨119873

119894(119909119894)⟩ 119882119899 (19)



4

119894=1120575119882119894 In


4

sum

119894=1

(119864119868119894[120575119882119894119882101584010158401015840

119894minus 120575119882

1015840

11989411988210158401015840

119894]119871 119894

0)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

lowast

+

4

sum

119894=1

(int

119871 119894

0

(11986411986811989412057511988210158401015840

11989411988210158401015840

119894minus 119864119868119894(120582119894

119871119894

)

4

120575119882119894119882119894)119889119909119894)

= 0

(20)


119878 (119909) = 119864119868 (119909)119882 (119909)101584010158401015840

119872 (119909) = 119864119868 (119909)119882 (119909)10158401015840

(21)



1198641198681(1205751198821119882101584010158401015840

1minus 120575119882

1015840

111988210158401015840

1)100381610038161003816100381610038161199091=0

= 120575Wext10038161003816100381610038161199091=0

1198641198684(1205751198824119882101584010158401015840

4minus 120575119882

1015840

411988210158401015840

4)100381610038161003816100381610038161199094=1198714

= 120575Wext10038161003816100381610038161199094=1198714

(22)


1= 1198822= 1198823

slopes11988210158401= 1198821015840

2= 1198821015840


1119882101584010158401015840

1= 1198641198682119882101584010158401015840

2+

1198641198683119882101584010158401015840

3) leading to

4

sum

119894=1

(119864119868119894[120575119882119894119882101584010158401015840

119894minus 120575119882

1015840

11989411988210158401015840

119894]119871 119894

0) = 120575Wext + 120575119882

2(0)

sdot (1198641198681119882101584010158401015840

1(1198711) minus 119864119868

2119882101584010158401015840

2(0) minus 119864119868

3119882101584010158401015840

3(0))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

lowastlowast

minus 1205751198821015840

2(0)

sdot (119864119868111988210158401015840

1(1198711) minus 119864119868

211988210158401015840

2(0) minus 119864119868

311988210158401015840

3(0))

minus 1205751198822(1198712)

sdot (1198641198684119882101584010158401015840

4(0) minus 119864119868

2119882101584010158401015840

2(1198712) minus 119864119868

3119882101584010158401015840

3(1198713))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

lowastlowastlowast

+ 1205751198821015840

2(1198712)

sdot (119864119868411988210158401015840

4(0) minus 119864119868

211988210158401015840

2(1198712) minus 119864119868

311988210158401015840

3(1198713))

(23)


4

sum

119894=1

(119864119868119894[120575119882119894119882101584010158401015840

119894minus 120575119882

1015840

11989411988210158401015840

119894]119871 119894

0)

= (1205751198821015840

2(1198712) minus 120575119882

1015840

2(0)) (Λ (119882

1015840

2(1198712) minus 120575119882

1015840

2(0)))

(24)


Λ(1205751198821015840

2(1198712) minus 120575119882

1015840

2(0)) (119882

1015840

2(1198712) minus 120575119882

1015840

2(0))

+

4

sum

119894=1

elements119894sum

119898=1

(int

119909119898+1

119909119898

(11986411986811989412057511988210158401015840

11989411988210158401015840

119894minus 119864119868119894(120582119894

119871119894

)

4

120575119882119894119882119894)119889119909)

= 0

(25)




119864119868111988210158401015840

1= 119864119868211988210158401015840

2+ 119864119868311988210158401015840

3+ 119864119868411988210158401015840

4+ Λ (119882

1015840

1(1199091= 1198711) minus 119882

1015840

5(1199095= 0)) (26a)

Λ =(1198671+ 1198673)2

11986411986021198641198604+ (1198672+ 1198673)2

11986411986021198641198603+ (1198673+ 1198674)2

11986411986031198641198604

4119886 (1198641198602+ 1198641198603+ 1198641198604)

(26b)


1198821= 1198822= 1198823= 1198824

1198821015840

1= 1198821015840

2= 1198821015840

3= 1198821015840

4

1198641198681119882101584010158401015840

1= 1198641198682119882101584010158401015840

2+ 1198641198683119882101584010158401015840

3+ 1198641198684119882101584010158401015840

4

(27)



119882(119909) = ⟨1 119909 1199092

1199093⟩ 119862 (28)


119882119899 =

1198821

1198821015840

1

1198822

1198821015840

2

=

[[[[[

[

1 0 0 0

0 1 0 0

1 119871 1198712

1198713

0 1 2119871 31198712

]]]]]

]

119862 = [119875119899] 119862 (29)


1 2

L

W1

W9984001

W2

W9984002

x


Thus

119882(119909) = ⟨1 119909 1199092

1199093⟩ [119875119899]minus1

119882119899 = ⟨119873 (119909)⟩ 119882119899 (30)


119899 Additionally


119899 With the


⟨120575119882119899⟩ (Λ (119873

21015840

(1198712) minus 119873

21015840

(0)) (⟨1198732⟩1015840

(1198712) minus ⟨119873

2⟩1015840

sdot (0))) 119882119899 +

4

sum

119894=1

elements119894sum

119899=1

⟨120575119882119899⟩

sdot (int

119909119898+1

119909119898

(11986411986811989411987311989410158401015840

⟨119873119894⟩10158401015840

minus 119864119868119894(120582119894

119871119894

)

4

119873119894 ⟨119873119894⟩)119889119909)

sdot 119882119899 = 0

(31)


⟨120575119882119899⟩ ((119870 + 119870delam) minus 120596

2119872) 119882

119899 = 0 (32)

if

det ((119870 + 119870delam) minus 1205962119872) = 0 (33)



1 2 3W9984001

W3W2W1

W9984002

x

L2 L2



119882(119909) = ⟨1 119909 1199092

1199093

1199094⟩ 119862 (34)


119882119899 =

1198821

1198821015840

1

1198822

1198823

1198821015840

3

=

[[[[[[[[[[

[

1 0 0 0 0

0 1 0 0 0

1119871

2

1198712

4

1198713

8

1198714

16

1 119871 1198712

1198713

1198714

0 1 2119871 31198712

41198713

]]]]]]]]]]

]

119862

= [119875119899] 119862

(35)

Thus

119882(119909) = ⟨1 119909 1199092

1199093

1199094⟩ [119875119899]minus1

119882119899 = ⟨119873⟩ V119899 (36)

3 Numerical Tests



1= 1198714)








LayerwiseFEM [17]














DelaminatedIntact

0

01

02

03

04

05

06

07

08

09

1

Mod

e sha

pe

01 02 03 04 05 06 07 08 09 10x location

aL = 06 H2 = 05H1

(a)

DelaminatedIntact

0 01 02 03 04 05 06 07 08 09 1x location

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Mod

e sha

pe

aL = 06 H2 = 05H1

(b)


aL = 05 15 elements

minus15

minus1

minus05

0

05

1

15

Mod

e sha

pe

05 1 15 2 25 3 35 4 45 50x location



DelaminatedIntact

0

01

02

03

04

05

06

07

08

09

1

Mod

e sha

pe

01 02 03 04 05 06 07 08 09 10x location

aL = 06 H2 = 04H1





1198861198712-nodeFEM


02 4725 4725 4725 47 4725

Mode 103 4691 4695 4695 47 469504 4574 4572 4575 46 457505 4318 4315 4315 43 431502 7054 7046 7045 71 7045

Mode 203 6337 6334 6335 63 633504 5965 5960 5965 60 596505 5860 5846 5845 59 5845

H1

03H1

03H1

aL

(a)

H1 H13H13H13

aL

(b)


1 2

3 6 9

1074

5 8 11

12 13












119886119871FEM2-node


02 4726 4725 4725 47 4725

Mode 103 4692 4695 4695 47 469504 4580 4578 4575 46 457505 4342 4338 4335 43 433502 7010 7002 7005 70 7005

Mode 203 6284 6281 6285 63 628504 5922 5917 5915 59 591505 5833 5820 5815 58 5815








2=

1198673


1=

1198715)



(a) (b)


2= 03119867

1 1198673= 05119867





4 Conclusion



(a) (b)







Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119894 through

119872119894= minus119864119868

11989411988210158401015840

119894 and 119878

119894= 119864119868119894119882101584010158401015840



119864119868111988210158401015840

1(1199091= 1198711)

= 119864119868211988210158401015840

2(1199092= 0) + 119864119868

311988210158401015840

3(1199093= 0)

+ Λ [1198821015840

4(1199094= 0) minus119882

1015840

1(1199091= 1198711)]

(13)


Λ =1198672

1

41198712

(11986411986021198641198603

1198641198602+ 1198641198603

) (14)


1198641198681119882101584010158401015840

1(1199091= 1198711) = 119864119868

2119882101584010158401015840

2(1199092= 0)

+ 1198641198683119882101584010158401015840

3(1199093= 0)

(15)







119864119868119894

1205974119882119894

1205971199094minus 119864119868119894(120582119894

119871119894

)

4

119882119894= 0 119894 = 1 4 (16)






119864119868119894

1205974119882119894

1205971199094minus 119864119868119894(120582119894

119871119894

)

4

119882119894= R 119894 = 1 4 (17)


4

sum

119894=1

(int

119871 119894

0

(1198641198681198941205751198821198941198821015840101584010158401015840

119894minus 119864119868119894(120582119894

119871119894

)

4

120575119882119894119882119894)119889119909119894)

= 0

(18)

with

119882119894(119909119894) = ⟨119873

119894(119909119894)⟩ 119882119899 (19)



4

119894=1120575119882119894 In


4

sum

119894=1

(119864119868119894[120575119882119894119882101584010158401015840

119894minus 120575119882

1015840

11989411988210158401015840

119894]119871 119894

0)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

lowast

+

4

sum

119894=1

(int

119871 119894

0

(11986411986811989412057511988210158401015840

11989411988210158401015840

119894minus 119864119868119894(120582119894

119871119894

)

4

120575119882119894119882119894)119889119909119894)

= 0

(20)


119878 (119909) = 119864119868 (119909)119882 (119909)101584010158401015840

119872 (119909) = 119864119868 (119909)119882 (119909)10158401015840

(21)



1198641198681(1205751198821119882101584010158401015840

1minus 120575119882

1015840

111988210158401015840

1)100381610038161003816100381610038161199091=0

= 120575Wext10038161003816100381610038161199091=0

1198641198684(1205751198824119882101584010158401015840

4minus 120575119882

1015840

411988210158401015840

4)100381610038161003816100381610038161199094=1198714

= 120575Wext10038161003816100381610038161199094=1198714

(22)


1= 1198822= 1198823

slopes11988210158401= 1198821015840

2= 1198821015840


1119882101584010158401015840

1= 1198641198682119882101584010158401015840

2+

1198641198683119882101584010158401015840

3) leading to

4

sum

119894=1

(119864119868119894[120575119882119894119882101584010158401015840

119894minus 120575119882

1015840

11989411988210158401015840

119894]119871 119894

0) = 120575Wext + 120575119882

2(0)

sdot (1198641198681119882101584010158401015840

1(1198711) minus 119864119868

2119882101584010158401015840

2(0) minus 119864119868

3119882101584010158401015840

3(0))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

lowastlowast

minus 1205751198821015840

2(0)

sdot (119864119868111988210158401015840

1(1198711) minus 119864119868

211988210158401015840

2(0) minus 119864119868

311988210158401015840

3(0))

minus 1205751198822(1198712)

sdot (1198641198684119882101584010158401015840

4(0) minus 119864119868

2119882101584010158401015840

2(1198712) minus 119864119868

3119882101584010158401015840

3(1198713))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

lowastlowastlowast

+ 1205751198821015840

2(1198712)

sdot (119864119868411988210158401015840

4(0) minus 119864119868

211988210158401015840

2(1198712) minus 119864119868

311988210158401015840

3(1198713))

(23)


4

sum

119894=1

(119864119868119894[120575119882119894119882101584010158401015840

119894minus 120575119882

1015840

11989411988210158401015840

119894]119871 119894

0)

= (1205751198821015840

2(1198712) minus 120575119882

1015840

2(0)) (Λ (119882

1015840

2(1198712) minus 120575119882

1015840

2(0)))

(24)


Λ(1205751198821015840

2(1198712) minus 120575119882

1015840

2(0)) (119882

1015840

2(1198712) minus 120575119882

1015840

2(0))

+

4

sum

119894=1

elements119894sum

119898=1

(int

119909119898+1

119909119898

(11986411986811989412057511988210158401015840

11989411988210158401015840

119894minus 119864119868119894(120582119894

119871119894

)

4

120575119882119894119882119894)119889119909)

= 0

(25)




119864119868111988210158401015840

1= 119864119868211988210158401015840

2+ 119864119868311988210158401015840

3+ 119864119868411988210158401015840

4+ Λ (119882

1015840

1(1199091= 1198711) minus 119882

1015840

5(1199095= 0)) (26a)

Λ =(1198671+ 1198673)2

11986411986021198641198604+ (1198672+ 1198673)2

11986411986021198641198603+ (1198673+ 1198674)2

11986411986031198641198604

4119886 (1198641198602+ 1198641198603+ 1198641198604)

(26b)


1198821= 1198822= 1198823= 1198824

1198821015840

1= 1198821015840

2= 1198821015840

3= 1198821015840

4

1198641198681119882101584010158401015840

1= 1198641198682119882101584010158401015840

2+ 1198641198683119882101584010158401015840

3+ 1198641198684119882101584010158401015840

4

(27)



119882(119909) = ⟨1 119909 1199092

1199093⟩ 119862 (28)


119882119899 =

1198821

1198821015840

1

1198822

1198821015840

2

=

[[[[[

[

1 0 0 0

0 1 0 0

1 119871 1198712

1198713

0 1 2119871 31198712

]]]]]

]

119862 = [119875119899] 119862 (29)


1 2

L

W1

W9984001

W2

W9984002

x


Thus

119882(119909) = ⟨1 119909 1199092

1199093⟩ [119875119899]minus1

119882119899 = ⟨119873 (119909)⟩ 119882119899 (30)


119899 Additionally


119899 With the


⟨120575119882119899⟩ (Λ (119873

21015840

(1198712) minus 119873

21015840

(0)) (⟨1198732⟩1015840

(1198712) minus ⟨119873

2⟩1015840

sdot (0))) 119882119899 +

4

sum

119894=1

elements119894sum

119899=1

⟨120575119882119899⟩

sdot (int

119909119898+1

119909119898

(11986411986811989411987311989410158401015840

⟨119873119894⟩10158401015840

minus 119864119868119894(120582119894

119871119894

)

4

119873119894 ⟨119873119894⟩)119889119909)

sdot 119882119899 = 0

(31)


⟨120575119882119899⟩ ((119870 + 119870delam) minus 120596

2119872) 119882

119899 = 0 (32)

if

det ((119870 + 119870delam) minus 1205962119872) = 0 (33)



1 2 3W9984001

W3W2W1

W9984002

x

L2 L2



119882(119909) = ⟨1 119909 1199092

1199093

1199094⟩ 119862 (34)


119882119899 =

1198821

1198821015840

1

1198822

1198823

1198821015840

3

=

[[[[[[[[[[

[

1 0 0 0 0

0 1 0 0 0

1119871

2

1198712

4

1198713

8

1198714

16

1 119871 1198712

1198713

1198714

0 1 2119871 31198712

41198713

]]]]]]]]]]

]

119862

= [119875119899] 119862

(35)

Thus

119882(119909) = ⟨1 119909 1199092

1199093

1199094⟩ [119875119899]minus1

119882119899 = ⟨119873⟩ V119899 (36)

3 Numerical Tests



1= 1198714)








LayerwiseFEM [17]














DelaminatedIntact

0

01

02

03

04

05

06

07

08

09

1

Mod

e sha

pe

01 02 03 04 05 06 07 08 09 10x location

aL = 06 H2 = 05H1

(a)

DelaminatedIntact

0 01 02 03 04 05 06 07 08 09 1x location

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Mod

e sha

pe

aL = 06 H2 = 05H1

(b)


aL = 05 15 elements

minus15

minus1

minus05

0

05

1

15

Mod

e sha

pe

05 1 15 2 25 3 35 4 45 50x location



DelaminatedIntact

0

01

02

03

04

05

06

07

08

09

1

Mod

e sha

pe

01 02 03 04 05 06 07 08 09 10x location

aL = 06 H2 = 04H1





1198861198712-nodeFEM


02 4725 4725 4725 47 4725

Mode 103 4691 4695 4695 47 469504 4574 4572 4575 46 457505 4318 4315 4315 43 431502 7054 7046 7045 71 7045

Mode 203 6337 6334 6335 63 633504 5965 5960 5965 60 596505 5860 5846 5845 59 5845

H1

03H1

03H1

aL

(a)

H1 H13H13H13

aL

(b)


1 2

3 6 9

1074

5 8 11

12 13












119886119871FEM2-node


02 4726 4725 4725 47 4725

Mode 103 4692 4695 4695 47 469504 4580 4578 4575 46 457505 4342 4338 4335 43 433502 7010 7002 7005 70 7005

Mode 203 6284 6281 6285 63 628504 5922 5917 5915 59 591505 5833 5820 5815 58 5815








2=

1198673


1=

1198715)



(a) (b)


2= 03119867

1 1198673= 05119867





4 Conclusion



(a) (b)







Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1198641198681(1205751198821119882101584010158401015840

1minus 120575119882

1015840

111988210158401015840

1)100381610038161003816100381610038161199091=0

= 120575Wext10038161003816100381610038161199091=0

1198641198684(1205751198824119882101584010158401015840

4minus 120575119882

1015840

411988210158401015840

4)100381610038161003816100381610038161199094=1198714

= 120575Wext10038161003816100381610038161199094=1198714

(22)


1= 1198822= 1198823

slopes11988210158401= 1198821015840

2= 1198821015840


1119882101584010158401015840

1= 1198641198682119882101584010158401015840

2+

1198641198683119882101584010158401015840

3) leading to

4

sum

119894=1

(119864119868119894[120575119882119894119882101584010158401015840

119894minus 120575119882

1015840

11989411988210158401015840

119894]119871 119894

0) = 120575Wext + 120575119882

2(0)

sdot (1198641198681119882101584010158401015840

1(1198711) minus 119864119868

2119882101584010158401015840

2(0) minus 119864119868

3119882101584010158401015840

3(0))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

lowastlowast

minus 1205751198821015840

2(0)

sdot (119864119868111988210158401015840

1(1198711) minus 119864119868

211988210158401015840

2(0) minus 119864119868

311988210158401015840

3(0))

minus 1205751198822(1198712)

sdot (1198641198684119882101584010158401015840

4(0) minus 119864119868

2119882101584010158401015840

2(1198712) minus 119864119868

3119882101584010158401015840

3(1198713))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

lowastlowastlowast

+ 1205751198821015840

2(1198712)

sdot (119864119868411988210158401015840

4(0) minus 119864119868

211988210158401015840

2(1198712) minus 119864119868

311988210158401015840

3(1198713))

(23)


4

sum

119894=1

(119864119868119894[120575119882119894119882101584010158401015840

119894minus 120575119882

1015840

11989411988210158401015840

119894]119871 119894

0)

= (1205751198821015840

2(1198712) minus 120575119882

1015840

2(0)) (Λ (119882

1015840

2(1198712) minus 120575119882

1015840

2(0)))

(24)


Λ(1205751198821015840

2(1198712) minus 120575119882

1015840

2(0)) (119882

1015840

2(1198712) minus 120575119882

1015840

2(0))

+

4

sum

119894=1

elements119894sum

119898=1

(int

119909119898+1

119909119898

(11986411986811989412057511988210158401015840

11989411988210158401015840

119894minus 119864119868119894(120582119894

119871119894

)

4

120575119882119894119882119894)119889119909)

= 0

(25)




119864119868111988210158401015840

1= 119864119868211988210158401015840

2+ 119864119868311988210158401015840

3+ 119864119868411988210158401015840

4+ Λ (119882

1015840

1(1199091= 1198711) minus 119882

1015840

5(1199095= 0)) (26a)

Λ =(1198671+ 1198673)2

11986411986021198641198604+ (1198672+ 1198673)2

11986411986021198641198603+ (1198673+ 1198674)2

11986411986031198641198604

4119886 (1198641198602+ 1198641198603+ 1198641198604)

(26b)


1198821= 1198822= 1198823= 1198824

1198821015840

1= 1198821015840

2= 1198821015840

3= 1198821015840

4

1198641198681119882101584010158401015840

1= 1198641198682119882101584010158401015840

2+ 1198641198683119882101584010158401015840

3+ 1198641198684119882101584010158401015840

4

(27)



119882(119909) = ⟨1 119909 1199092

1199093⟩ 119862 (28)


119882119899 =

1198821

1198821015840

1

1198822

1198821015840

2

=

[[[[[

[

1 0 0 0

0 1 0 0

1 119871 1198712

1198713

0 1 2119871 31198712

]]]]]

]

119862 = [119875119899] 119862 (29)


1 2

L

W1

W9984001

W2

W9984002

x


Thus

119882(119909) = ⟨1 119909 1199092

1199093⟩ [119875119899]minus1

119882119899 = ⟨119873 (119909)⟩ 119882119899 (30)


119899 Additionally


119899 With the


⟨120575119882119899⟩ (Λ (119873

21015840

(1198712) minus 119873

21015840

(0)) (⟨1198732⟩1015840

(1198712) minus ⟨119873

2⟩1015840

sdot (0))) 119882119899 +

4

sum

119894=1

elements119894sum

119899=1

⟨120575119882119899⟩

sdot (int

119909119898+1

119909119898

(11986411986811989411987311989410158401015840

⟨119873119894⟩10158401015840

minus 119864119868119894(120582119894

119871119894

)

4

119873119894 ⟨119873119894⟩)119889119909)

sdot 119882119899 = 0

(31)


⟨120575119882119899⟩ ((119870 + 119870delam) minus 120596

2119872) 119882

119899 = 0 (32)

if

det ((119870 + 119870delam) minus 1205962119872) = 0 (33)



1 2 3W9984001

W3W2W1

W9984002

x

L2 L2



119882(119909) = ⟨1 119909 1199092

1199093

1199094⟩ 119862 (34)


119882119899 =

1198821

1198821015840

1

1198822

1198823

1198821015840

3

=

[[[[[[[[[[

[

1 0 0 0 0

0 1 0 0 0

1119871

2

1198712

4

1198713

8

1198714

16

1 119871 1198712

1198713

1198714

0 1 2119871 31198712

41198713

]]]]]]]]]]

]

119862

= [119875119899] 119862

(35)

Thus

119882(119909) = ⟨1 119909 1199092

1199093

1199094⟩ [119875119899]minus1

119882119899 = ⟨119873⟩ V119899 (36)

3 Numerical Tests



1= 1198714)








LayerwiseFEM [17]














DelaminatedIntact

0

01

02

03

04

05

06

07

08

09

1

Mod

e sha

pe

01 02 03 04 05 06 07 08 09 10x location

aL = 06 H2 = 05H1

(a)

DelaminatedIntact

0 01 02 03 04 05 06 07 08 09 1x location

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Mod

e sha

pe

aL = 06 H2 = 05H1

(b)


aL = 05 15 elements

minus15

minus1

minus05

0

05

1

15

Mod

e sha

pe

05 1 15 2 25 3 35 4 45 50x location



DelaminatedIntact

0

01

02

03

04

05

06

07

08

09

1

Mod

e sha

pe

01 02 03 04 05 06 07 08 09 10x location

aL = 06 H2 = 04H1





1198861198712-nodeFEM


02 4725 4725 4725 47 4725

Mode 103 4691 4695 4695 47 469504 4574 4572 4575 46 457505 4318 4315 4315 43 431502 7054 7046 7045 71 7045

Mode 203 6337 6334 6335 63 633504 5965 5960 5965 60 596505 5860 5846 5845 59 5845

H1

03H1

03H1

aL

(a)

H1 H13H13H13

aL

(b)


1 2

3 6 9

1074

5 8 11

12 13












119886119871FEM2-node


02 4726 4725 4725 47 4725

Mode 103 4692 4695 4695 47 469504 4580 4578 4575 46 457505 4342 4338 4335 43 433502 7010 7002 7005 70 7005

Mode 203 6284 6281 6285 63 628504 5922 5917 5915 59 591505 5833 5820 5815 58 5815








2=

1198673


1=

1198715)



(a) (b)


2= 03119867

1 1198673= 05119867





4 Conclusion



(a) (b)







Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2

L

W1

W9984001

W2

W9984002

x


Thus

119882(119909) = ⟨1 119909 1199092

1199093⟩ [119875119899]minus1

119882119899 = ⟨119873 (119909)⟩ 119882119899 (30)


119899 Additionally


119899 With the


⟨120575119882119899⟩ (Λ (119873

21015840

(1198712) minus 119873

21015840

(0)) (⟨1198732⟩1015840

(1198712) minus ⟨119873

2⟩1015840

sdot (0))) 119882119899 +

4

sum

119894=1

elements119894sum

119899=1

⟨120575119882119899⟩

sdot (int

119909119898+1

119909119898

(11986411986811989411987311989410158401015840

⟨119873119894⟩10158401015840

minus 119864119868119894(120582119894

119871119894

)

4

119873119894 ⟨119873119894⟩)119889119909)

sdot 119882119899 = 0

(31)


⟨120575119882119899⟩ ((119870 + 119870delam) minus 120596

2119872) 119882

119899 = 0 (32)

if

det ((119870 + 119870delam) minus 1205962119872) = 0 (33)



1 2 3W9984001

W3W2W1

W9984002

x

L2 L2



119882(119909) = ⟨1 119909 1199092

1199093

1199094⟩ 119862 (34)


119882119899 =

1198821

1198821015840

1

1198822

1198823

1198821015840

3

=

[[[[[[[[[[

[

1 0 0 0 0

0 1 0 0 0

1119871

2

1198712

4

1198713

8

1198714

16

1 119871 1198712

1198713

1198714

0 1 2119871 31198712

41198713

]]]]]]]]]]

]

119862

= [119875119899] 119862

(35)

Thus

119882(119909) = ⟨1 119909 1199092

1199093

1199094⟩ [119875119899]minus1

119882119899 = ⟨119873⟩ V119899 (36)

3 Numerical Tests



1= 1198714)








LayerwiseFEM [17]














DelaminatedIntact

0

01

02

03

04

05

06

07

08

09

1

Mod

e sha

pe

01 02 03 04 05 06 07 08 09 10x location

aL = 06 H2 = 05H1

(a)

DelaminatedIntact

0 01 02 03 04 05 06 07 08 09 1x location

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Mod

e sha

pe

aL = 06 H2 = 05H1

(b)


aL = 05 15 elements

minus15

minus1

minus05

0

05

1

15

Mod

e sha

pe

05 1 15 2 25 3 35 4 45 50x location



DelaminatedIntact

0

01

02

03

04

05

06

07

08

09

1

Mod

e sha

pe

01 02 03 04 05 06 07 08 09 10x location

aL = 06 H2 = 04H1





1198861198712-nodeFEM


02 4725 4725 4725 47 4725

Mode 103 4691 4695 4695 47 469504 4574 4572 4575 46 457505 4318 4315 4315 43 431502 7054 7046 7045 71 7045

Mode 203 6337 6334 6335 63 633504 5965 5960 5965 60 596505 5860 5846 5845 59 5845

H1

03H1

03H1

aL

(a)

H1 H13H13H13

aL

(b)


1 2

3 6 9

1074

5 8 11

12 13












119886119871FEM2-node


02 4726 4725 4725 47 4725

Mode 103 4692 4695 4695 47 469504 4580 4578 4575 46 457505 4342 4338 4335 43 433502 7010 7002 7005 70 7005

Mode 203 6284 6281 6285 63 628504 5922 5917 5915 59 591505 5833 5820 5815 58 5815








2=

1198673


1=

1198715)



(a) (b)


2= 03119867

1 1198673= 05119867





4 Conclusion



(a) (b)







Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















LayerwiseFEM [17]














DelaminatedIntact

0

01

02

03

04

05

06

07

08

09

1

Mod

e sha

pe

01 02 03 04 05 06 07 08 09 10x location

aL = 06 H2 = 05H1

(a)

DelaminatedIntact

0 01 02 03 04 05 06 07 08 09 1x location

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Mod

e sha

pe

aL = 06 H2 = 05H1

(b)


aL = 05 15 elements

minus15

minus1

minus05

0

05

1

15

Mod

e sha

pe

05 1 15 2 25 3 35 4 45 50x location



DelaminatedIntact

0

01

02

03

04

05

06

07

08

09

1

Mod

e sha

pe

01 02 03 04 05 06 07 08 09 10x location

aL = 06 H2 = 04H1





1198861198712-nodeFEM


02 4725 4725 4725 47 4725

Mode 103 4691 4695 4695 47 469504 4574 4572 4575 46 457505 4318 4315 4315 43 431502 7054 7046 7045 71 7045

Mode 203 6337 6334 6335 63 633504 5965 5960 5965 60 596505 5860 5846 5845 59 5845

H1

03H1

03H1

aL

(a)

H1 H13H13H13

aL

(b)


1 2

3 6 9

1074

5 8 11

12 13












119886119871FEM2-node


02 4726 4725 4725 47 4725

Mode 103 4692 4695 4695 47 469504 4580 4578 4575 46 457505 4342 4338 4335 43 433502 7010 7002 7005 70 7005

Mode 203 6284 6281 6285 63 628504 5922 5917 5915 59 591505 5833 5820 5815 58 5815








2=

1198673


1=

1198715)



(a) (b)


2= 03119867

1 1198673= 05119867





4 Conclusion



(a) (b)







Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










DelaminatedIntact

0

01

02

03

04

05

06

07

08

09

1

Mod

e sha

pe

01 02 03 04 05 06 07 08 09 10x location

aL = 06 H2 = 05H1

(a)

DelaminatedIntact

0 01 02 03 04 05 06 07 08 09 1x location

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Mod

e sha

pe

aL = 06 H2 = 05H1

(b)


aL = 05 15 elements

minus15

minus1

minus05

0

05

1

15

Mod

e sha

pe

05 1 15 2 25 3 35 4 45 50x location



DelaminatedIntact

0

01

02

03

04

05

06

07

08

09

1

Mod

e sha

pe

01 02 03 04 05 06 07 08 09 10x location

aL = 06 H2 = 04H1





1198861198712-nodeFEM


02 4725 4725 4725 47 4725

Mode 103 4691 4695 4695 47 469504 4574 4572 4575 46 457505 4318 4315 4315 43 431502 7054 7046 7045 71 7045

Mode 203 6337 6334 6335 63 633504 5965 5960 5965 60 596505 5860 5846 5845 59 5845

H1

03H1

03H1

aL

(a)

H1 H13H13H13

aL

(b)


1 2

3 6 9

1074

5 8 11

12 13












119886119871FEM2-node


02 4726 4725 4725 47 4725

Mode 103 4692 4695 4695 47 469504 4580 4578 4575 46 457505 4342 4338 4335 43 433502 7010 7002 7005 70 7005

Mode 203 6284 6281 6285 63 628504 5922 5917 5915 59 591505 5833 5820 5815 58 5815








2=

1198673


1=

1198715)



(a) (b)


2= 03119867

1 1198673= 05119867





4 Conclusion



(a) (b)







Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1198861198712-nodeFEM


02 4725 4725 4725 47 4725

Mode 103 4691 4695 4695 47 469504 4574 4572 4575 46 457505 4318 4315 4315 43 431502 7054 7046 7045 71 7045

Mode 203 6337 6334 6335 63 633504 5965 5960 5965 60 596505 5860 5846 5845 59 5845

H1

03H1

03H1

aL

(a)

H1 H13H13H13

aL

(b)


1 2

3 6 9

1074

5 8 11

12 13












119886119871FEM2-node


02 4726 4725 4725 47 4725

Mode 103 4692 4695 4695 47 469504 4580 4578 4575 46 457505 4342 4338 4335 43 433502 7010 7002 7005 70 7005

Mode 203 6284 6281 6285 63 628504 5922 5917 5915 59 591505 5833 5820 5815 58 5815








2=

1198673


1=

1198715)



(a) (b)


2= 03119867

1 1198673= 05119867





4 Conclusion



(a) (b)







Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119886119871FEM2-node


02 4726 4725 4725 47 4725

Mode 103 4692 4695 4695 47 469504 4580 4578 4575 46 457505 4342 4338 4335 43 433502 7010 7002 7005 70 7005

Mode 203 6284 6281 6285 63 628504 5922 5917 5915 59 591505 5833 5820 5815 58 5815








2=

1198673


1=

1198715)



(a) (b)


2= 03119867

1 1198673= 05119867





4 Conclusion



(a) (b)







Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(a) (b)


2= 03119867

1 1198673= 05119867





4 Conclusion



(a) (b)







Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a) (b)







Acknowledgments


References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article On the Finite Element Free Vibration Analysis of Delaminated Layered...

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Transcript of Research Article On the Finite Element Free Vibration Analysis of Delaminated Layered...