Research Article Integrated Topology Optimization of...

Research ArticleIntegrated Topology Optimization ofStructureVibration Control for Piezoelectric CylindricalShell Based on the Genetic Algorithm

Jianjun He and Xiangzi Chen

School of Automobile and Mechanical Engineering Changsha University of Science and Technology Changsha 410004 China

Correspondence should be addressed to Jianjun He hezhengde8163com

Received 26 November 2014 Revised 29 March 2015 Accepted 10 April 2015

Academic Editor Quoc Hung Nguyen

Copyright copy 2015 J He and X Chen This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A hybrid optimization strategy for integrated topological optimization design of piezoelectric cylindrical flat shell structureis proposed The method combines the genetic algorithm (GA) and linear-quadratic-regulator (LQR) theory to optimize theperformance of coupling structurecontrol system The GA is used to choose the optimal structure topology and numberand placements of actuators and control parameters meanwhile the LQR is used to design control system to suppressvibration of optimal structure under sinusoidal excitation which is based on the couple-mode space control In addition themathematical morphology operators are used for repairs of disconnected structure topology The results of numerical simulationand computations show that the proposed method is effective and feasible with good performance for the optimal and couplingpiezoelectric cylindrical shell structurecontrol system

1 Introduction

The cylindrical shell structure is widely used as main partsof the large structure design in many fields such as civilengineering and architecture marine and aerospace Forinstance the roofs of many stadiums and solid fuel tank ofrocket are this type of structure due to their excellentmechan-ical properties especially their high specific stiffness andstrength [1] However the vibrations of those structuresunder complex and extreme dynamic loads are inevitableas they are excited by strong wind and wave So it is alsonecessary to design control system to suppress the amplitudeof vibration for those structures in engineering practice [2 3]

The optimization design can effectively improve variousperformances of structure for example it can make theoptimal structure more lightweight and controllable In par-ticular the topology optimization design has huge potentialof saving material in structure design which is verifiedby many applications of this technique in aerospace andautomobile industry For example the application of topologyoptimization in the design of A380 leading edge ribs ledto the forty-five percentage weight reduction compared to

traditional design and so forth At present smart or adaptivestructures are widely used in industrial or engineering fieldsso engineers can monitor and control the normal work ofstructures in real-time However because of the complexityof the adaptive structure the structure and control system iscoupled For instance the vibration controllerrsquos coefficientsare strongly dependent on the topology of structure theplacements and number of the actuatorssensors whichdetermine the effectiveness of structural vibration controlTherefore researching the structure and control of integratedoptimization strategies for smart structures is theoreticallyand practically significant It can simultaneously realize theoptimal design of structure and control system and simplifydesign links of huge and complex structurecontrol inte-grated system So the integrated optimization design greatlyimproves the efficiency of traditional sequential optimizationdesign (namely the designer firstly optimizes the structureparameters and then optimizes the controller parameters)

Presently for the active control of truss or plate andshell structures piezoelectric patches and stacks or piezo-electric ceramic transducer (PZT) is used as one of the mainactuators For the optimization strategy the GA is the main

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 456147 10 pageshttpdxdoiorg1011552015456147

2 Shock and Vibration

optimization algorithm There have been some works pub-lished on intelligent piezoelectric structure integrated opti-mization For instance Li and Huang [4] combined theGA and the linear quadratic Gaussian (LQG) studied theintegrated optimization of actuator placement and vibrationcontrol for piezoelectric adaptive trusses and verified that theresult of experiment is consistent with numerical results Zhuet al [5] investigated simultaneous optimization with respectto the structural topology actuator locations and controlparameters of an actively controlled plate structure its resultsshowed that it can produce systems with clear structuraltopology and high control performance by the proposedapproach Kang and Tong [6] investigated the integratedoptimization of structural topology and control voltage ofpiezoelectric laminated plates Xu et al [7] discussed someissues associated with integrated optimization of structuraltopology the number and placement of actuators and controlparameters for piezoelectric smart trusses some optimalstrategies based on genetic algorithms were adopted in theirstudy Xu et al [8] studied the integrated optimizationof structural topology and number and positions of theactuators and control parameters for piezoelectric smartplates which is based on the optimal control effect in theindependentmode control and singular value decompositionof the distributed matrix of total performance index for allphysical control forces with the GA as optimization strategyHowever almost all the efforts described in the literaturesinvolve the integrated optimization design of piezoelectrictruss or plate not including shell structure In generalit is more complicated to construct the piezoelectric shellfinite element for mathematical modeling and finite elementanalysis (FEA) for shell structure Shell structure usually hasmore DOF and more complicated structural characteristicandmechanical behavior than truss and plate structure Shellstructure hasmore complicated geometric shape which leadsto its greater difficulties for modeling and analysis If the shellstructure is in vibration its vibration suppression techniqueis usually invalid andmore complicated than that of truss andplate structure Thus the main concern of this paper is toextend the integrated topology optimization design into shellstructure

In the present paper the integrated topology optimizationof structurecontrol for piezoelectric cylindrical flat shell isstudied Firstly the piezoelectric coupling finite element (FE)model was built Then the integrated optimization modelingincluding the design variables the objective function andthe constraint functions was proposed and the LQR wasused for the design of control system in coupling modalspace In addition in order to ensure the realization of inte-grated optimization the GA and mathematical morphologytreatment was applied to solution of the above problems Toverify the validity of the integrated optimization design ofshell structure and promote engineering applications of thisapproach the numerical simulations and computations wereconductedThe results show that the integrated optimizationsof piezoelectric cylindrical shell structure are successful andfeasible

t1

t2

t3

z

Actuator layer

Base layer

Sensor layer

Figure 1 Four-node piezoelectric coupling rectangular flat shell ele-ment

x

y

z

x998400

y998400

z998400

Figure 2 The coordinate system of cylindrical flat shell

2 The Piezoelectric Coupling Dynamic Modelof Cylindrical Flat Shell Structure

In general it is more complicated to construct the piezo-electric shell finite element for mathematical modeling andfinite element analysis (FEA) of shell structure Shell structureusually has more DOF and more complicated structuralcharacteristic and mechanical behavior than truss and platestructure For simplification we utilized the four-node rect-angular flat shell element with 24 displacement degrees offreedom (DOF) and 2 electrical DOF to build the piezoelec-tric coupling dynamic model of cylindrical shell structure [38] The ideal element is shown in Figure 1 The piezoelectricflat shell element is composed of middle isotropic baselayer with upper and lower surfaces of base layer that arecompletely covered by PZT patches where 119905

1and 1199052sim3

are thethickness of base layer and piezoelectric patches respectivelyThe approximate cylindrical flat shell structure is shown inFigure 2

In finite element derivation the node displacement vectorof flat shell structural element is defined as

d1015840119890119894= (1199061015840

119894V1015840119894

1199081015840

1198941205791015840

1199091198941205791015840

1199101198941205791015840

119911119894)T

119894 = 1 2 3 4 (1)

where 119894 is denoted by the four nodes of the flat shell structuralelement

Shock and Vibration 3

Assume that the element stiffness matrix correspondingto the plane stress state is expressed as

k119898119890

=

[[[[[[[

[

k11989811

k11989812

k11989813

k11989814

k11989821

k11989822

k11989823

k11989824

k11989831

k11989832

k11989833

k11989834

k11989841

k11989842

k11989843

k11989844

]]]]]]]

]

node 1

node 2

node 3

node 4

(2)

where superscript ldquo119898rdquo is corresponding to the stiffnessmatrixof plane stress k119898

119894119895is denoted by 2 times 2 submatrix

Similarly assume that the element stiffness matrix corre-sponding to the bending stress state is expressed as

k119887119890=

[[[[[[[[

[

k11988711

k11988712

k11988713

k11988714

k11988721

k11988722

k11988723

k11988724

k11988731

k11988732

k11988733

k11988734

k11988741

k11988742

k11988743

k11988744

]]]]]]]]

]

node 1

node 2

node 3

node 4

(3)

where superscript ldquo119887rdquo is corresponding to the stiffness matrixof bending stress k119887


By combining with (2) and (3) the element stiffnessmatrix of flat shell structural element in local coordinatesystem is obtained as

k1015840119890

=

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

[

k11989811

0 0 k11989812

0 0 k11989813

0 0 k11989814

0 0

0 k11988711

0 0 k11988712

0 0 k11988713

0 0 k11988714

0

0 0 0 0 0 0 0 0 0 0 0 0

k11989821

0 0 k11989822

0 0 k11989823

0 0 k11989824

0 0

0 k11988721

0 0 k11988722

0 0 k11988723

0 0 k11988724

0

0 0 0 0 0 0 0 0 0 0 0 0

k11989831

0 0 k11989832

0 0 k11989833

0 0 k11989834

0 0

0 k11988731

0 0 k11988732

0 0 k11988733

0 0 k11988734

0

0 0 0 0 0 0 0 0 0 0 0 0

k11989841

0 0 k11989842

0 0 k11989843

0 0 k11989844

0 0

0 k11988741

0 0 k11988742

0 0 k11988743

0 0 k11988744

0

0 0 0 0 0 0 0 0 0 0 0 0

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

]

(4)

where k1015840119890is derived in local coordinate system which is a 24times

24matrixBy eliminating the singularity of the whole stiffness

matrix of shell structure we should transform all the localelement stiffness matrices into global coordinate system bycoordinate transformation

The transformation matrix is defined as follows

Τ =

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

[

120593 0 0 0 0 0 0 0

0 120593 0 0 0 0 0 0

0 0 120593 0 0 0 0 0

0 0 0 120593 0 0 0 0

0 0 0 0 120593 0 0 0

0 0 0 0 0 120593 0 0

0 0 0 0 0 0 120593 0

0 0 0 0 0 0 0 120593

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

]

120593 =

[[[[

[

cos (1199091015840 119909) cos (1199091015840 119910) cos (1199091015840 119911)

cos (1199101015840 119909) cos (1199101015840 119910) cos (1199101015840 119911)

cos (1199111015840 119909) cos (1199111015840 119910) cos (1199111015840 119911)

]]]]

]

(5)

where f is denoted by direction cosine matrix of local coordi-nate system versus global coordinate system

After coordinate transformation by (6) the local elementstiffness matrix is transformed into global element stiffnessmatrix

k119890= Τ

Tk1015840119890ΤT (6)

Finally we can get the whole stiffness matrix in globalcoordinate system through elements assembly process

By using the Hamilton principle we can obtain theformula as follows

120575int

1199052

1199051

(119879119890minus 119880119890+ 119882119890) 119889119905 = 0 (7)

where119879119890119880119890 and119882119890 are the kinetic energy potential energy

and external work of the piezoelectric coupling flat shellelement shown in Figure 1 respectively

Substituting the kinetic energy potential energy andexternal work of the piezoelectric coupling flat shell elementinto (7) and calculating the variation of (7) the piezoelectriccoupling equation is obtained as follows

M119890 q119890 + (Κ119890

119901+ Κ119890

119898) q119890 minus Κ119890

119898119886119881119890

119886119911+ Κ119890

119898119904119881119890

119904119911

= 119865119890

119891

(8)

Κ119890

119886119898q119890 + Κ119890

119886119886119881119890

119886119911= minus∬

119860119890

120590119890

119886119889119909 119889119910 (9)

Κ119890

119904119898q119890 + Κ119890

119904119904119881119890

119904119911= 0 (10)

M119890 = sum

ℎ=119898119904119886

int

119881119890

ℎ

120588ℎ [N]

T[N] 119889119881 (11)

Κ119890

119898= int

119881119890

119898

1199112[B119906]T[D119898] [B119906] 119889119881 (12)


Κ119890

119901= int

119881119890

119904+119881119890

119886

1199112[B119906]T[S119864] [B

119906] 119889119881 (13)

Κ119890

119898119886= (Κ119890

119886119898)T= int

119881119890

119886

119911 [B119906]T[e]T

1199052

119889119881 (14)

Κ119890

119898119904= (Κ119890

119904119898)T= int

119881119890

119904

119911 [B119906]T[e]T

1199052

119889119881 (15)

Κ119890

119886119886= Κ119890

119904119904= int

119881119890

119886

120576119864

33

1199052

2

119889119881 (16)

F119890119891 = ∬

119860119890

NTf 119889119909 119889119910 (17)

where q119890 is denoted by node displacement vector of thepiezoelectric coupling flat shell element and the meanings ofthe other symbols are similar to the utilizations of these in theliterature [3 9]

By reducing the 119881119890

119904119911and 119881

119890

119886119911terms in (8)ndash(10) the

piezoelectric coupling finite element equation is obtained asfollows

M119890 q119890 + Κ119890 q119890 = F119890119891 minus F119890control (18)

where

Κ119890= Κ119890

119898+ Κ119890

119901+ (Κ119890

119898119886)T(Κ119890

119886119886)minus1

Κ119886119898

+ (Κ119890

119898119904)T(Κ119890

119904119904)minus1

Κ119904119898

(19)

The equivalent control force of input voltage in actuatorlayer is calculated as

F119890control = Κ119890

119898119886(Κ119890

119886119886)minus1

∬

119860119890

1205890

1199052

119881119890

119886119911119889119909 119889119910 (20)

Similarly we can finally establish the piezoelectric cou-pling dynamic FE equation for the global coupled piezo-electric structurecontrol system through elements stiffnessmatrix assembly process as follows

Mq (119905) + Kq (119905) = Fload (119905) + Fcontrol (119905) (21)

where M and K are the coupled system mass matrix andsystem stiffness matrix respectively q and q are the structurenodal displacement vector and acceleration vector respec-tively Fload(119905) and Fcontrol(119905) are the load vector and controlforce vector respectively Fcontrol(119905) can be expressed asBu(119905)where B and u(119905) are the actuator distributing matrix and thecontrol force vector respectively In (21) we did not considerthe damping term in order to emphasize the vibration sup-pression by active control This section is used for modelingand finite element analysis of piezoelectric coupling flat shellstructure and is followed by the LQR control design in theintegrated optimization procedure

3 The Mathematical Modeling forIntegrated Optimization Designof Piezoelectric Cylindrical Shell Structure

The standard form of the integrated optimization problemfor piezoelectric smart cylindrical flat shell structure can bedescribed as

find 119889119904

119894 119889119888

119896(119894 = 1 2 119873 119896 = 1 2 119872)

min 119878 =

119873

sum

119894=1

4120588119894119886119894119887119894119905119894

st 119892 (119889119904

119894 119889119888

119896)119895le 0 (119895 = 1 2 119901)

(22)

where 119889119904

119894and 119889

119888

119896are the structural design variable and

control design variable respectively to minimize the struc-turecontrol system mass 119878 is the objective function 119873 120588

119894

119886119894 119887119894 and 119905

119894are the number of elements (including ordinary

elements and piezoelectric coupling elements) the densityhalf-length of long side and short side for rectangular flat shellelement and thickness of structure element or piezoelectricelement respectively 119898 119899 are the number of structuraldesign variable and control design variable respectively119892(119889119904

119894 119889119888

119896)119895is coupled constraint functions which means it is

simultaneously in relation to the structural design variableand control design variable

31 Design Variables The topology design variable 119889119888

119894indi-

cates the presence or absence of structure material that is

119889119904

119894isin 0 1 (23)

if 119889119904119894= 1 indicates that the corresponding element exists or

else it is removed in optimization process On the contrary119889119904

119894= 0 indicates that it is a void element without materialThe elements of 119889119888

119896may be the control design parameters

and the number and positions of actuators (PZT patches)

119889119888

119896=

119899119886 119899119886isin [1 119878]

119901119899119886

119901119899119886

isin [1119873]

119877119899119886

119877119899119886

isin [119877119897 119877119906]

(24)

where 119899119886 119901119899119886

and 119877119899119886

are the number of actuators the posi-tions of actuators and weight factor of control weight matrixrespectively 119878 119873 119877119897 and 119877

119906 indicate the allowable maximalnumber of actuators the allowablemaximal element positionof actuators and the lower bound and upper bound of weightfactor respectively

32 Constraint Function Here the constraint functions con-tain the first natural frequency constraint function voltageconstraint function and weight factor constraint function

321 Natural Frequency Constraint Function In engineeringpractice structure usually suffers from dynamic load whichmay cause resonance or coupled effect between the struc-ture and external excitation These cases should be avoided


because of their huge destruction to structure In generalwe impose natural frequency constraint to the optimizationmathematical modeling especially for the first natural fre-quency So the natural frequency constraint is expressed as

119891119868

1minus 119891119877

1le 0 (25)

where 119891119868

1and 119891

119877

1are the first anticipant minimal natural

frequency of optimization structure and the first actualnatural frequency of optimal structure respectively

322 Voltage Constraint Function Presently piezoelectricpatches or PZT are used as one of themain actuators in activevibration control of structure However there is a break-over voltage for the piezoelectric patches and the appliedvoltage by the voltage amplifier is limited the active controlsystem cannot provide the unlimited and necessary controlforce beyond the break-over voltage Then the constrainton the active control force can be transformed into thecorresponding voltage constraint that is

119881119886

119896minus 119881119886

119896le 0 (26)

where 119881119886

119896and 119881

119886

119896are the actual applied voltage and its

corresponding allowed upper limit respectively

323Weight Factor Constraint Function In LQR control theselections of control weight matrixes have great impact onthe velocity of response suppression and size of control forceSo we choose weight factor as control design parameter andmake it confined to a specified range that is

119877119897

119896le 119877119896le 119877119906

119896 (27)

where 119877119897

119896and 119877

119906

119896are the corresponding lower bound and

upper bound of control weight factor respectively

4 LQR Control Modeling and Design

In modern control theory LQR is one of the mature andeffective optimal controllerrsquos design methods which is basedon state space expression and is applied in many literaturessuch as in the literature [8] In general the vibration suppres-sion of flat shell structure is more complicated than truss andplate structure For simplifications we still utilize the LQR forcontrol modeling and design here In the following the LQRapproach for design of control system is given out

By considering the first several waiting controlmodes andcombining with the initial conditions aftermodal coordinatetransformation (21) can be rewritten as in the modal space

y119888(119905)

y119903(119905)

+ [

Λ119888

0

0 Λ119903

]

y119888(119905)

y119903(119905)

=

120601T119888

120601T119903

Bu (119905) (28)

120601119888= [1206011 120601

2sdot sdot sdot 120601119899119898]

120593T119894119872120593119895= 120575119894119895

Λ119888= diag (1205962

1 1205962

2 120596

2

119899119898

)

(29)

where y119888(119905) and y

119903(119905) are denoted by the modal displacement

vector for waiting control and the rest modal displacementvector respectively Λ

119888is expressed as the eigenvalues diago-

nal matrix for waiting control modesEquation (28) can be expressed as in state space with

matrix form

Z (119905) = AZ (119905) + B1015840u (119905) (30)

Z (119905) = [

y119888(119905)

y119888(119905)

]

A = [

0119899119898times119899119898

I119899119898times119899119898

minusΛc 0119899119898times119899119898

]

B1015840 = [

0119899119898times119899119886

120593T119888B

]

(31)

To ensure that the waiting control system is stableand controllable the allocation number of actuators 119899

119898in

coupled structurecontrol system should be met with thecondition as

rank [B1015840 AB1015840 A2B1015840 sdot sdot sdot A2119899119898minus1B1015840] = 2119899119898 (32)

where rank[] is denoted as the rank computation of matrixFor obtaining the quadratic form optimal control design

u(119905) is to minimize the objective function 119869

119869 =1

2int

infin

0

(ZT(119905)QZ (119905) + uT (119905)Ru (119905)) 119889119905

997888rarr min(33)

Q = [

Λ119888

0119899119898times119899119898

0119899119898times119899119898

Ι119899119898times119899119898

]

R =

[[[

[

1199031

0

d

0 119903119899119886

]]]

]

(34)

whereR is expressed as the control design parametersmatrixTo minimize (33) we can obtain u(119905) as

u (119905) = minusRminus1 (B1015840)TPZ (119905) (35)

where P is a 2119899119898

times 2119899119898semidefinite and symmetric matrix

which satisfied the Riccati equation as follows

PA + ATP +Q minus PB1015840Rminus1 (B1015840)TP = 0 (36)

By substituting (35) into (30) we can obtain the stateequation of closed-loop system as

Z (119905) = AZ (119905)

A = A minus B1015840Rminus1 (B1015840)TP

(37)

Thus this section is used for calculating the optimalcontrol force vector u(119905) by (35) and determining the optimalcontrol design parameters for (33)


(a) Original topology (b) Final topology

Figure 3 Effect for deletions of some low-stress elements

5 Integrated OptimizationAlgorithm and Strategy

51 Optimization Algorithm Theproblem in (22) has contin-uous and discrete design variables which lead to its solutionbeing more difficult and complex than ordinary optimizationproblem with single typersquos design variable For the sakeof coupling between the structure and control system andcomplexion of constraint functions it is very difficult tofind the optimum solution for combined optimization designproblem with structural topology and control parameterGenerally there exist several equal or not equal suboptimalsolutions For the saving of computational cost and simpli-fying of optimization design we can find the suboptimalsolution instead Because of wide adaptability of the GA tothe different types of optimization problems then it may bean effective method for the solution of the above problem Sothe GA is applied for the integrated optimization design inthis paper The binary is selected for coding of chromosomewith structural design variable and control parameter designvariable [8]Meanwhile the individual fitness forminimizingthe objective function is expressed as

eval (119889119904119894 119889119888

119896) =

1

119878times (1 + penal times

sum119901

119895=1119863119895

119901) (38)

where 119901 is the number of constraints functions and penal isthe penalty factor For natural frequency constraint function1198631

= (119891119877

1minus 119891119868

1)119891119868

1 the other 119863

119895are obtained by similar

method as1198631

52 Deletions of Some Low-Stress Elements There generallyexist massive low-stress or inefficient material elements intopological optimizing structure at each iteration becauseof control requirements realization for integrated design ofcoupled structurecontrol system Those elements make themaximal lightweight potential of structure be not achievedby topology optimization Here we delete a part of low-stress

Figure 4 Morphology bridge operator

elements at each iteration to make the topology structure bemore clear and lightweight The steps of the above strategyare summarized as follows firstly according to results ofvon Mises stress computation we sort the whole elementsof structure in ascending order secondly we delete certainpercentages of low-stress elements The deletion percentagesare usually about 20 For example let us see a comparisonbetween the original topology and final topology treated bydeletions of some low-stress elements which is shown inFigure 3

53 Repair Strategy for Disconnected Topology Based on Mor-phology Operator The literature [10] points out that somedisconnected topologies created by genetic algorithm com-monly appear The disconnected topology configuration isinvalid and uncontrollable for integrated optimization designof structurecontrol system So the repair strategy based onimage mathematical morphology operators is applied to thedisconnected topology in integrated optimization design inthis paper Firstly topology of structure or topology variablematrix is considered as a gray imagewith solid and void areasThenweuse basicmorphology operators ldquobridgerdquo and ldquodilaterdquoto filter the gray image that is obtained from each iterationThe detailed morphology operators and related formulationsare expressed in the literature [11] At first let us simplyunderstand the effects of two morphology operators whichare shown in Figures 4 and 5


Figure 5 Morphology dilate operator for a piezoelectric smart plate with four simple supported corners

Begin k = 1

Generate the initial population

Binary decode

The finite element analysis (FEA)

Sort elements for deleting some low-stress elements

Morphology operations by bridge and dilateoperators for disconnected topology

Control design by LQR

Evaluate the fitness of individual

Execute genetic operators selectioncrossover and mutation

k = k + 1

Satisfy the terminationcriterion

Yes End

No

Figure 6 Flowchart for integrated optimization design of flat shell

6 The Integrated OptimizationDesign Steps of StructureControl

Thedetailed steps of integrated topology optimization designfor piezoelectric cylindrical flat shell structurecontrol areshown in Figure 6

The general genetic algorithm LQRandFEAwas utilizedfor integrated topology optimization design for piezoelectriccylindrical flat shell structurecontrol in this paper whichappeared in many literatures [7ndash9] Some steps in flowchartsuch as deletions of some low-stress elements and repair

03

02

01

0

02 0 minus02 0

02

04

06

Figure 7 The original flat shell structure and the appointed node(labeled with red color)

strategy for disconnected topology based on morphologyoperator are described in Sections 52 and 53

7 Numerical Examples

The simulation computations for two piezoelectric cylin-drical flat shell structures are carried out by the proposedmethod Here the base material for two simulated structuresis used by steel and its material properties are as followsYoungrsquos elastic modulus 119864 = 21 times 10

11Nm2 Poissonrsquosratio V = 03 and density 120588 = 7800 kgm3 Meanwhile thematerial properties of PZT patches are as follows Youngrsquoselasticmodulus119864

119901= 63times10

10Nm2 Poissonrsquos ratio V119901= 03

and density 120588119901= 7600 kgm3 Piezoelectric strain constants

11988931

= 11988932

= minus171 times 10minus12 CN and dielectric constant

120576 = 15 nfm

71 Semicylindrical Piezoelectric Shell Structure The semi-cylindrical shell structure is shown in Figure 7 The structureis discretized to 10 times 16 elements with the left and rightedges being fixed and the other edges are free The sizes of


03

02

01

0

02 0 minus02 0

02

04

06

Figure 8 The optimal number and positions of PZT patches(labeled with black color)

Table 1The results for integrated optimization design of semicylin-drical shell

1198911Hz 119881

1V 119881

2V 119881

3V 119881

4V 119878 (Kg)

1900 39208 39208 62244 62244 73553

structure are as follows the length along generatrix directionis 119871 = 06m the ring radius is 119877 = 03m and the thicknessis 119905119887

= 0004m The thickness of PZT patches is 119905119901

=

0001m The static load condition is that the load of 1000Nis imposed in the negative 119911-direction of the appointed nodewhich is shown in Figure 7The genetic algorithmparametersare as follows population size the crossover rate and themutation rate are 40 08 and 002 respectivelyThe allowablemaximal iterations and the penalty factor are 200 and 1000respectively

The vibration of the cylindrical shell structure is causedby the sinusoidal excitation 119865 = 1000 lowast sin(2 lowast 119901119894 lowast 15 lowast 119905)

that is imposed in the negative 119911-direction of the same nodeas the static load conditionThe amplitude of applied voltagesis not more than 100V the allowable maximal number ofpiezoelectric elements or actuators is not more than 6 Wehope that we can use the several PZT actuators that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 15Hz The upper and lower bounds for control weightparameters are 10 times 10

minus12 and 10 times 10minus13 respectively The

optimal number and positions of PZT patches are shown inFigure 8

In Figure 8 the black blocks are piezoelectric couplingelements as expressed in Figure 1

The results of integrated topology optimization design forsemicylindrical shell structurecontrol are shown in Table 1

The optimal structure topology is shown in Figure 9In Figure 9 the black and gray block areas are the solid

elements and void elements respectivelyThe optimal control

03

02

01

0

020 minus02

0

02

04

06

Figure 9 The optimal structure topology

14

12

10

8

60 20 40 60 80 100 120 140 160 180 200

S(k

g)

Evolutionary iterations

Figure 10 The iteration history of mass index

weight parameters are 11987711988612

= 09789 times 10minus12 and 119877

11988634

=

04155 times 10minus12 respectively The iteration history of mass

index for the coupled structurecontrol is shown in Figure 10The vibration responses comparisons between under un-

control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 11

72 Cantilever Piezoelectric Cylindrical Shell Structure Thecantilever cylindrical shell structure is shown in Figure 12The structure is discretized to 10 times 32 elements with thefront ring edges being fixed and the other edges are freeThesizes of structure are as follows the length along generatrixdirection is 119871 = 08m the ring radius is 119877 = 04mand the base thickness is 119905

119887= 0001m The thickness of

PZT patches is 119905119901

= 00015m The static load conditionis that the load of 10000N is imposed in the negative 119911-direction of the specified node which is shown in Figure 11The genetic algorithm parameters are as follows populationsize the crossover rate and the mutation rate are 40 08 and002 respectively The allowable maximal iterations and thepenalty factor are 100 and 10000 respectively

The vibration of the cantilever cylindrical shell structureis caused by the sinusoidal excitation 119865 = 10000 lowast sin(2 lowast

119901119894 lowast 175 lowast 119905) that is imposed in the negative 119911-direction ofthe same node as the static load condition The amplitude


15

1

05

0

minus05

minus1

minus150 005 01 015 02502 03 035 04 045 05

t (s)

The t

ime r

espo

nse (

m)

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol

times10minus4

Figure 11 The time response of vibration structure at the excitationnode along negative 119911-direction

0

0

02

04

06

08

Figure 12 The original flat shell structure and the specified node(labeled with red color)

of applied voltages is not more than 100V the allowablemaximal number of PZT patches is not more than 6 Wehope that we can use the several PZT patches that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 2Hz The upper and lower bounds for control weightparameters are 10 times 10


optimal number and positions of PZT patches are shown inFigure 13The four piezoelectric flat shell elements are 55 6696 and 105 respectively

The results of integrated optimization design for can-tilever cylindrical shell structurecontrol are shown inTable 2

The optimal structure topology is shown in Figure 14

0

02

04

06

08

Figure 13 The optimal number and positions of PZT patches

0

02

04

06

08


Table 2The results for integrated optimization design of cantilevercylindrical shell

1198911Hz 119881

1V 119881

2V 119881

3V 119881

4V 119878 (Kg)

2886 8842 8842 24806 24806 78205

The optimal control weight parameters are 11987711988612

=

09561times10minus8 and 119877

11988634

= 07574times10minus8 respectivelyThe iter-

ation history of mass index for the coupling structurecontrolis shown in Figure 15

The vibration responses comparisons between under un-control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 16

10 Shock and VibrationS

(kg)

9

85

8

750 20 40 60 80 100



25

2

15

1

05

0

minus05

minus1

minus15

minus2

minus250 01 02 03 04 05

t (s)

The t

ime r

espo

nse (

m)

times10minus7

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol

Figure 16 The time response comparisons at the excitation nodealong negative 119911-direction

8 Conclusions

In order to enlarge the scope of application of the integratedoptimization design of structurecontrol in engineering andpractice the integrated topology optimization design ofpiezoelectric cylindrical flat shell structurecontrol is studiedin this paper Firstly the piezoelectric flat shell elementis used for dynamic modeling of piezoelectric cylindricalshell structure Secondly based on the LQR control incoupling modal space the integrated topology optimizationdesign modeling with the GA is presented Meanwhile thestrategies with the deletions of some low-stress elements andmorphology operators are used for realization of integratedoptimization design and smooth structure topology Finallynumerical simulation computations are carried out for twocylindrical flat shell structures

The results of numerical examples show that the proposedmethod can produce clear structure topology and highcontrol performance In two examples the number and thepositions of PZT patches are reasonable and the amplitude ofapplied voltages is small which are valuable to the realizationand efficiency of integrated topology optimization design ofpiezoelectric cylindrical flat shell structure in engineeringpractice

Conflict of Interests

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

Acknowledgments

The authors would like to acknowledge the support byNatural Science Foundation of China under Grant 51305048and Scientific Research Fund of Hunan Provincial EducationDepartment under Grant 11C0045

References

[1] P Vidal L Gallimard and O Polit ldquoShell finite element basedon the Proper Generalized Decomposition for the modeling ofcylindrical composite structuresrdquo Computers amp Structures vol132 pp 1ndash11 2014

[2] Y Zhang H Niu S Xie and X Zhang ldquoNumerical and exper-imental investigation of active vibration control in a cylindricalshell partially covered by a laminated PVDF actuatorrdquo SmartMaterials and Structures vol 17 no 3 Article ID 035024 2008

[3] X T Cao L Shi X S Zhang and G Jiang ldquoActive control ofacoustic radiation from laminated cylindrical shells integratedwith a piezoelectric layerrdquo Smart Materials and Structures vol22 no 6 Article ID 065003 pp 3ndash18 2013

[4] W P Li and H Huang ldquoIntegrated optimization of actuatorplacement and vibration control for piezoelectric adaptivetrussesrdquo Journal of Sound andVibration vol 332 no 1 pp 17ndash322013

[5] Y Zhu J Qiu H Du and J Tani ldquoSimultaneous optimaldesign of structural topology actuator locations and controlparameters for a plate structurerdquoComputationalMechanics vol29 no 2 pp 89ndash97 2002

[6] Z Kang and L Tong ldquoIntegrated optimization of materiallayout and control voltage for piezoelectric laminated platesrdquoJournal of Intelligent Material Systems and Structures vol 19 no8 pp 889ndash904 2008

[7] B Xu J S Jiang and J P Ou ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart trussesusing genetic algorithmrdquo Journal of Sound and Vibration vol307 no 3ndash5 pp 393ndash427 2007

[8] B Xu J P Ou and J S Jiang ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart platebased on genetic algorithmrdquo Finite Elements in Analysis andDesign vol 64 pp 1ndash12 2013

[9] W-S Hwang and H C Park ldquoFinite element modeling ofpiezoelectric sensors and actuatorsrdquo AIAA journal vol 31 no5 pp 930ndash937 1993

[10] S Y Wang K Tai and S T Quek ldquoTopology optimization ofpiezoelectric sensorsactuators for torsional vibration control ofcomposite platesrdquo Smart Materials and Structures vol 15 no 2pp 253ndash269 2006

[11] O Sigmund ldquoMorphology-based black and white filters fortopology optimizationrdquo Structural and Multidisciplinary Opti-mization vol 33 no 4-5 pp 401ndash424 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

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



optimization algorithm There have been some works pub-lished on intelligent piezoelectric structure integrated opti-mization For instance Li and Huang [4] combined theGA and the linear quadratic Gaussian (LQG) studied theintegrated optimization of actuator placement and vibrationcontrol for piezoelectric adaptive trusses and verified that theresult of experiment is consistent with numerical results Zhuet al [5] investigated simultaneous optimization with respectto the structural topology actuator locations and controlparameters of an actively controlled plate structure its resultsshowed that it can produce systems with clear structuraltopology and high control performance by the proposedapproach Kang and Tong [6] investigated the integratedoptimization of structural topology and control voltage ofpiezoelectric laminated plates Xu et al [7] discussed someissues associated with integrated optimization of structuraltopology the number and placement of actuators and controlparameters for piezoelectric smart trusses some optimalstrategies based on genetic algorithms were adopted in theirstudy Xu et al [8] studied the integrated optimizationof structural topology and number and positions of theactuators and control parameters for piezoelectric smartplates which is based on the optimal control effect in theindependentmode control and singular value decompositionof the distributed matrix of total performance index for allphysical control forces with the GA as optimization strategyHowever almost all the efforts described in the literaturesinvolve the integrated optimization design of piezoelectrictruss or plate not including shell structure In generalit is more complicated to construct the piezoelectric shellfinite element for mathematical modeling and finite elementanalysis (FEA) for shell structure Shell structure usually hasmore DOF and more complicated structural characteristicandmechanical behavior than truss and plate structure Shellstructure hasmore complicated geometric shape which leadsto its greater difficulties for modeling and analysis If the shellstructure is in vibration its vibration suppression techniqueis usually invalid andmore complicated than that of truss andplate structure Thus the main concern of this paper is toextend the integrated topology optimization design into shellstructure

In the present paper the integrated topology optimizationof structurecontrol for piezoelectric cylindrical flat shell isstudied Firstly the piezoelectric coupling finite element (FE)model was built Then the integrated optimization modelingincluding the design variables the objective function andthe constraint functions was proposed and the LQR wasused for the design of control system in coupling modalspace In addition in order to ensure the realization of inte-grated optimization the GA and mathematical morphologytreatment was applied to solution of the above problems Toverify the validity of the integrated optimization design ofshell structure and promote engineering applications of thisapproach the numerical simulations and computations wereconductedThe results show that the integrated optimizationsof piezoelectric cylindrical shell structure are successful andfeasible

t1

t2

t3

z

Actuator layer

Base layer

Sensor layer

Figure 1 Four-node piezoelectric coupling rectangular flat shell ele-ment

x

y

z

x998400

y998400

z998400

Figure 2 The coordinate system of cylindrical flat shell

2 The Piezoelectric Coupling Dynamic Modelof Cylindrical Flat Shell Structure

In general it is more complicated to construct the piezo-electric shell finite element for mathematical modeling andfinite element analysis (FEA) of shell structure Shell structureusually has more DOF and more complicated structuralcharacteristic and mechanical behavior than truss and platestructure For simplification we utilized the four-node rect-angular flat shell element with 24 displacement degrees offreedom (DOF) and 2 electrical DOF to build the piezoelec-tric coupling dynamic model of cylindrical shell structure [38] The ideal element is shown in Figure 1 The piezoelectricflat shell element is composed of middle isotropic baselayer with upper and lower surfaces of base layer that arecompletely covered by PZT patches where 119905

1and 1199052sim3

are thethickness of base layer and piezoelectric patches respectivelyThe approximate cylindrical flat shell structure is shown inFigure 2

In finite element derivation the node displacement vectorof flat shell structural element is defined as

d1015840119890119894= (1199061015840

119894V1015840119894

1199081015840

1198941205791015840

1199091198941205791015840

1199101198941205791015840

119911119894)T

119894 = 1 2 3 4 (1)

where 119894 is denoted by the four nodes of the flat shell structuralelement



k119898119890

=

[[[[[[[

[

k11989811

k11989812

k11989813

k11989814

k11989821

k11989822

k11989823

k11989824

k11989831

k11989832

k11989833

k11989834

k11989841

k11989842

k11989843

k11989844

]]]]]]]

]

node 1

node 2

node 3

node 4

(2)




k119887119890=

[[[[[[[[

[

k11988711

k11988712

k11988713

k11988714

k11988721

k11988722

k11988723

k11988724

k11988731

k11988732

k11988733

k11988734

k11988741

k11988742

k11988743

k11988744

]]]]]]]]

]

node 1

node 2

node 3

node 4

(3)




k1015840119890

=

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

[

k11989811

0 0 k11989812

0 0 k11989813

0 0 k11989814

0 0

0 k11988711

0 0 k11988712

0 0 k11988713

0 0 k11988714

0

0 0 0 0 0 0 0 0 0 0 0 0

k11989821

0 0 k11989822

0 0 k11989823

0 0 k11989824

0 0

0 k11988721

0 0 k11988722

0 0 k11988723

0 0 k11988724

0

0 0 0 0 0 0 0 0 0 0 0 0

k11989831

0 0 k11989832

0 0 k11989833

0 0 k11989834

0 0

0 k11988731

0 0 k11988732

0 0 k11988733

0 0 k11988734

0

0 0 0 0 0 0 0 0 0 0 0 0

k11989841

0 0 k11989842

0 0 k11989843

0 0 k11989844

0 0

0 k11988741

0 0 k11988742

0 0 k11988743

0 0 k11988744

0

0 0 0 0 0 0 0 0 0 0 0 0

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

]

(4)





Τ =

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

[

120593 0 0 0 0 0 0 0

0 120593 0 0 0 0 0 0

0 0 120593 0 0 0 0 0

0 0 0 120593 0 0 0 0

0 0 0 0 120593 0 0 0

0 0 0 0 0 120593 0 0

0 0 0 0 0 0 120593 0

0 0 0 0 0 0 0 120593

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

]

120593 =

[[[[

[

cos (1199091015840 119909) cos (1199091015840 119910) cos (1199091015840 119911)

cos (1199101015840 119909) cos (1199101015840 119910) cos (1199101015840 119911)

cos (1199111015840 119909) cos (1199111015840 119910) cos (1199111015840 119911)

]]]]

]

(5)



k119890= Τ

Tk1015840119890ΤT (6)



120575int

1199052

1199051

(119879119890minus 119880119890+ 119882119890) 119889119905 = 0 (7)




M119890 q119890 + (Κ119890

119901+ Κ119890

119898) q119890 minus Κ119890

119898119886119881119890

119886119911+ Κ119890

119898119904119881119890

119904119911

= 119865119890

119891

(8)

Κ119890

119886119898q119890 + Κ119890

119886119886119881119890

119886119911= minus∬

119860119890

120590119890

119886119889119909 119889119910 (9)

Κ119890

119904119898q119890 + Κ119890

119904119904119881119890

119904119911= 0 (10)

M119890 = sum

ℎ=119898119904119886

int

119881119890

ℎ

120588ℎ [N]

T[N] 119889119881 (11)

Κ119890

119898= int

119881119890

119898

1199112[B119906]T[D119898] [B119906] 119889119881 (12)


Κ119890

119901= int

119881119890

119904+119881119890

119886

1199112[B119906]T[S119864] [B

119906] 119889119881 (13)

Κ119890

119898119886= (Κ119890

119886119898)T= int

119881119890

119886

119911 [B119906]T[e]T

1199052

119889119881 (14)

Κ119890

119898119904= (Κ119890

119904119898)T= int

119881119890

119904

119911 [B119906]T[e]T

1199052

119889119881 (15)

Κ119890

119886119886= Κ119890

119904119904= int

119881119890

119886

120576119864

33

1199052

2

119889119881 (16)

F119890119891 = ∬

119860119890

NTf 119889119909 119889119910 (17)



119904119911and 119881

119890




where

Κ119890= Κ119890

119898+ Κ119890

119901+ (Κ119890

119898119886)T(Κ119890

119886119886)minus1

Κ119886119898

+ (Κ119890

119898119904)T(Κ119890

119904119904)minus1

Κ119904119898

(19)


F119890control = Κ119890

119898119886(Κ119890

119886119886)minus1

∬

119860119890

1205890

1199052

119881119890

119886119911119889119909 119889119910 (20)






find 119889119904

119894 119889119888

119896(119894 = 1 2 119873 119896 = 1 2 119872)

min 119878 =

119873

sum

119894=1

4120588119894119886119894119887119894119905119894

st 119892 (119889119904

119894 119889119888

119896)119895le 0 (119895 = 1 2 119901)

(22)

where 119889119904

119894and 119889

119888



119894

119886119894 119887119894 and 119905



119894 119889119888




119894indi-


119889119904

119894isin 0 1 (23)






119889119888

119896=

119899119886 119899119886isin [1 119878]

119901119899119886

119901119899119886

isin [1119873]

119877119899119886

119877119899119886

isin [119877119897 119877119906]

(24)

where 119899119886 119901119899119886

and 119877119899119886







119891119868

1minus 119891119877

1le 0 (25)

where 119891119868

1and 119891

119877




119881119886

119896minus 119881119886

119896le 0 (26)

where 119881119886

119896and 119881

119886




119877119897

119896le 119877119896le 119877119906

119896 (27)

where 119877119897

119896and 119877

119906






y119888(119905)

y119903(119905)

+ [

Λ119888

0

0 Λ119903

]

y119888(119905)

y119903(119905)

=

120601T119888

120601T119903

Bu (119905) (28)

120601119888= [1206011 120601

2sdot sdot sdot 120601119899119898]

120593T119894119872120593119895= 120575119894119895

Λ119888= diag (1205962

1 1205962

2 120596

2

119899119898

)

(29)

where y119888(119905) and y





matrix form

Z (119905) = AZ (119905) + B1015840u (119905) (30)

Z (119905) = [

y119888(119905)

y119888(119905)

]

A = [

0119899119898times119899119898

I119899119898times119899119898

minusΛc 0119899119898times119899119898

]

B1015840 = [

0119899119898times119899119886

120593T119888B

]

(31)


119898in





119869 =1

2int

infin

0

(ZT(119905)QZ (119905) + uT (119905)Ru (119905)) 119889119905

997888rarr min(33)

Q = [

Λ119888

0119899119898times119899119898

0119899119898times119899119898

Ι119899119898times119899119898

]

R =

[[[

[

1199031

0

d

0 119903119899119886

]]]

]

(34)



where P is a 2119899119898





Z (119905) = AZ (119905)


(37)







eval (119889119904119894 119889119888

119896) =

1


sum119901

119895=1119863119895

119901) (38)


= (119891119877

1minus 119891119868

1)119891119868

1 the other 119863


method as1198631







Begin k = 1


Binary decode







k = k + 1


Yes End

No





03

02

01

0

02 0 minus02 0

02

04

06






119901= 63times10



11988931

= 11988932


120576 = 15 nfm



03

02

01

0

02 0 minus02 0

02

04

06



1198911Hz 119881

1V 119881

2V 119881

3V 119881

4V 119878 (Kg)

1900 39208 39208 62244 62244 73553



=










03

02

01

0

020 minus02

0

02

04

06


14

12

10

8

60 20 40 60 80 100 120 140 160 180 200

S(k

g)




= 09789 times 10minus12 and 119877

11988634

=











15

1

05

0

minus05

minus1

minus150 005 01 015 02502 03 035 04 045 05

t (s)

The t

ime r

espo

nse (

m)

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol

times10minus4


0

0

02

04

06

08







0

02

04

06

08


0

02

04

06

08



1198911Hz 119881

1V 119881

2V 119881

3V 119881

4V 119878 (Kg)

2886 8842 8842 24806 24806 78205


=


11988634





(kg)

9

85

8

750 20 40 60 80 100



25

2

15

1

05

0

minus05

minus1

minus15

minus2

minus250 01 02 03 04 05

t (s)

The t

ime r

espo

nse (

m)

times10minus7

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol


8 Conclusions





Acknowledgments


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











k119898119890

=

[[[[[[[

[

k11989811

k11989812

k11989813

k11989814

k11989821

k11989822

k11989823

k11989824

k11989831

k11989832

k11989833

k11989834

k11989841

k11989842

k11989843

k11989844

]]]]]]]

]

node 1

node 2

node 3

node 4

(2)




k119887119890=

[[[[[[[[

[

k11988711

k11988712

k11988713

k11988714

k11988721

k11988722

k11988723

k11988724

k11988731

k11988732

k11988733

k11988734

k11988741

k11988742

k11988743

k11988744

]]]]]]]]

]

node 1

node 2

node 3

node 4

(3)




k1015840119890

=

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

[

k11989811

0 0 k11989812

0 0 k11989813

0 0 k11989814

0 0

0 k11988711

0 0 k11988712

0 0 k11988713

0 0 k11988714

0

0 0 0 0 0 0 0 0 0 0 0 0

k11989821

0 0 k11989822

0 0 k11989823

0 0 k11989824

0 0

0 k11988721

0 0 k11988722

0 0 k11988723

0 0 k11988724

0

0 0 0 0 0 0 0 0 0 0 0 0

k11989831

0 0 k11989832

0 0 k11989833

0 0 k11989834

0 0

0 k11988731

0 0 k11988732

0 0 k11988733

0 0 k11988734

0

0 0 0 0 0 0 0 0 0 0 0 0

k11989841

0 0 k11989842

0 0 k11989843

0 0 k11989844

0 0

0 k11988741

0 0 k11988742

0 0 k11988743

0 0 k11988744

0

0 0 0 0 0 0 0 0 0 0 0 0

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

]

(4)





Τ =

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

[

120593 0 0 0 0 0 0 0

0 120593 0 0 0 0 0 0

0 0 120593 0 0 0 0 0

0 0 0 120593 0 0 0 0

0 0 0 0 120593 0 0 0

0 0 0 0 0 120593 0 0

0 0 0 0 0 0 120593 0

0 0 0 0 0 0 0 120593

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

]

120593 =

[[[[

[

cos (1199091015840 119909) cos (1199091015840 119910) cos (1199091015840 119911)

cos (1199101015840 119909) cos (1199101015840 119910) cos (1199101015840 119911)

cos (1199111015840 119909) cos (1199111015840 119910) cos (1199111015840 119911)

]]]]

]

(5)



k119890= Τ

Tk1015840119890ΤT (6)



120575int

1199052

1199051

(119879119890minus 119880119890+ 119882119890) 119889119905 = 0 (7)




M119890 q119890 + (Κ119890

119901+ Κ119890

119898) q119890 minus Κ119890

119898119886119881119890

119886119911+ Κ119890

119898119904119881119890

119904119911

= 119865119890

119891

(8)

Κ119890

119886119898q119890 + Κ119890

119886119886119881119890

119886119911= minus∬

119860119890

120590119890

119886119889119909 119889119910 (9)

Κ119890

119904119898q119890 + Κ119890

119904119904119881119890

119904119911= 0 (10)

M119890 = sum

ℎ=119898119904119886

int

119881119890

ℎ

120588ℎ [N]

T[N] 119889119881 (11)

Κ119890

119898= int

119881119890

119898

1199112[B119906]T[D119898] [B119906] 119889119881 (12)


Κ119890

119901= int

119881119890

119904+119881119890

119886

1199112[B119906]T[S119864] [B

119906] 119889119881 (13)

Κ119890

119898119886= (Κ119890

119886119898)T= int

119881119890

119886

119911 [B119906]T[e]T

1199052

119889119881 (14)

Κ119890

119898119904= (Κ119890

119904119898)T= int

119881119890

119904

119911 [B119906]T[e]T

1199052

119889119881 (15)

Κ119890

119886119886= Κ119890

119904119904= int

119881119890

119886

120576119864

33

1199052

2

119889119881 (16)

F119890119891 = ∬

119860119890

NTf 119889119909 119889119910 (17)



119904119911and 119881

119890




where

Κ119890= Κ119890

119898+ Κ119890

119901+ (Κ119890

119898119886)T(Κ119890

119886119886)minus1

Κ119886119898

+ (Κ119890

119898119904)T(Κ119890

119904119904)minus1

Κ119904119898

(19)


F119890control = Κ119890

119898119886(Κ119890

119886119886)minus1

∬

119860119890

1205890

1199052

119881119890

119886119911119889119909 119889119910 (20)






find 119889119904

119894 119889119888

119896(119894 = 1 2 119873 119896 = 1 2 119872)

min 119878 =

119873

sum

119894=1

4120588119894119886119894119887119894119905119894

st 119892 (119889119904

119894 119889119888

119896)119895le 0 (119895 = 1 2 119901)

(22)

where 119889119904

119894and 119889

119888



119894

119886119894 119887119894 and 119905



119894 119889119888




119894indi-


119889119904

119894isin 0 1 (23)






119889119888

119896=

119899119886 119899119886isin [1 119878]

119901119899119886

119901119899119886

isin [1119873]

119877119899119886

119877119899119886

isin [119877119897 119877119906]

(24)

where 119899119886 119901119899119886

and 119877119899119886







119891119868

1minus 119891119877

1le 0 (25)

where 119891119868

1and 119891

119877




119881119886

119896minus 119881119886

119896le 0 (26)

where 119881119886

119896and 119881

119886




119877119897

119896le 119877119896le 119877119906

119896 (27)

where 119877119897

119896and 119877

119906






y119888(119905)

y119903(119905)

+ [

Λ119888

0

0 Λ119903

]

y119888(119905)

y119903(119905)

=

120601T119888

120601T119903

Bu (119905) (28)

120601119888= [1206011 120601

2sdot sdot sdot 120601119899119898]

120593T119894119872120593119895= 120575119894119895

Λ119888= diag (1205962

1 1205962

2 120596

2

119899119898

)

(29)

where y119888(119905) and y





matrix form

Z (119905) = AZ (119905) + B1015840u (119905) (30)

Z (119905) = [

y119888(119905)

y119888(119905)

]

A = [

0119899119898times119899119898

I119899119898times119899119898

minusΛc 0119899119898times119899119898

]

B1015840 = [

0119899119898times119899119886

120593T119888B

]

(31)


119898in





119869 =1

2int

infin

0

(ZT(119905)QZ (119905) + uT (119905)Ru (119905)) 119889119905

997888rarr min(33)

Q = [

Λ119888

0119899119898times119899119898

0119899119898times119899119898

Ι119899119898times119899119898

]

R =

[[[

[

1199031

0

d

0 119903119899119886

]]]

]

(34)



where P is a 2119899119898





Z (119905) = AZ (119905)


(37)







eval (119889119904119894 119889119888

119896) =

1


sum119901

119895=1119863119895

119901) (38)


= (119891119877

1minus 119891119868

1)119891119868

1 the other 119863


method as1198631







Begin k = 1


Binary decode







k = k + 1


Yes End

No





03

02

01

0

02 0 minus02 0

02

04

06






119901= 63times10



11988931

= 11988932


120576 = 15 nfm



03

02

01

0

02 0 minus02 0

02

04

06



1198911Hz 119881

1V 119881

2V 119881

3V 119881

4V 119878 (Kg)

1900 39208 39208 62244 62244 73553



=










03

02

01

0

020 minus02

0

02

04

06


14

12

10

8

60 20 40 60 80 100 120 140 160 180 200

S(k

g)




= 09789 times 10minus12 and 119877

11988634

=











15

1

05

0

minus05

minus1

minus150 005 01 015 02502 03 035 04 045 05

t (s)

The t

ime r

espo

nse (

m)

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol

times10minus4


0

0

02

04

06

08







0

02

04

06

08


0

02

04

06

08



1198911Hz 119881

1V 119881

2V 119881

3V 119881

4V 119878 (Kg)

2886 8842 8842 24806 24806 78205


=


11988634





(kg)

9

85

8

750 20 40 60 80 100



25

2

15

1

05

0

minus05

minus1

minus15

minus2

minus250 01 02 03 04 05

t (s)

The t

ime r

espo

nse (

m)

times10minus7

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol


8 Conclusions





Acknowledgments


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Κ119890

119901= int

119881119890

119904+119881119890

119886

1199112[B119906]T[S119864] [B

119906] 119889119881 (13)

Κ119890

119898119886= (Κ119890

119886119898)T= int

119881119890

119886

119911 [B119906]T[e]T

1199052

119889119881 (14)

Κ119890

119898119904= (Κ119890

119904119898)T= int

119881119890

119904

119911 [B119906]T[e]T

1199052

119889119881 (15)

Κ119890

119886119886= Κ119890

119904119904= int

119881119890

119886

120576119864

33

1199052

2

119889119881 (16)

F119890119891 = ∬

119860119890

NTf 119889119909 119889119910 (17)



119904119911and 119881

119890




where

Κ119890= Κ119890

119898+ Κ119890

119901+ (Κ119890

119898119886)T(Κ119890

119886119886)minus1

Κ119886119898

+ (Κ119890

119898119904)T(Κ119890

119904119904)minus1

Κ119904119898

(19)


F119890control = Κ119890

119898119886(Κ119890

119886119886)minus1

∬

119860119890

1205890

1199052

119881119890

119886119911119889119909 119889119910 (20)






find 119889119904

119894 119889119888

119896(119894 = 1 2 119873 119896 = 1 2 119872)

min 119878 =

119873

sum

119894=1

4120588119894119886119894119887119894119905119894

st 119892 (119889119904

119894 119889119888

119896)119895le 0 (119895 = 1 2 119901)

(22)

where 119889119904

119894and 119889

119888



119894

119886119894 119887119894 and 119905



119894 119889119888




119894indi-


119889119904

119894isin 0 1 (23)






119889119888

119896=

119899119886 119899119886isin [1 119878]

119901119899119886

119901119899119886

isin [1119873]

119877119899119886

119877119899119886

isin [119877119897 119877119906]

(24)

where 119899119886 119901119899119886

and 119877119899119886







119891119868

1minus 119891119877

1le 0 (25)

where 119891119868

1and 119891

119877




119881119886

119896minus 119881119886

119896le 0 (26)

where 119881119886

119896and 119881

119886




119877119897

119896le 119877119896le 119877119906

119896 (27)

where 119877119897

119896and 119877

119906






y119888(119905)

y119903(119905)

+ [

Λ119888

0

0 Λ119903

]

y119888(119905)

y119903(119905)

=

120601T119888

120601T119903

Bu (119905) (28)

120601119888= [1206011 120601

2sdot sdot sdot 120601119899119898]

120593T119894119872120593119895= 120575119894119895

Λ119888= diag (1205962

1 1205962

2 120596

2

119899119898

)

(29)

where y119888(119905) and y





matrix form

Z (119905) = AZ (119905) + B1015840u (119905) (30)

Z (119905) = [

y119888(119905)

y119888(119905)

]

A = [

0119899119898times119899119898

I119899119898times119899119898

minusΛc 0119899119898times119899119898

]

B1015840 = [

0119899119898times119899119886

120593T119888B

]

(31)


119898in





119869 =1

2int

infin

0

(ZT(119905)QZ (119905) + uT (119905)Ru (119905)) 119889119905

997888rarr min(33)

Q = [

Λ119888

0119899119898times119899119898

0119899119898times119899119898

Ι119899119898times119899119898

]

R =

[[[

[

1199031

0

d

0 119903119899119886

]]]

]

(34)



where P is a 2119899119898





Z (119905) = AZ (119905)


(37)







eval (119889119904119894 119889119888

119896) =

1


sum119901

119895=1119863119895

119901) (38)


= (119891119877

1minus 119891119868

1)119891119868

1 the other 119863


method as1198631







Begin k = 1


Binary decode







k = k + 1


Yes End

No





03

02

01

0

02 0 minus02 0

02

04

06






119901= 63times10



11988931

= 11988932


120576 = 15 nfm



03

02

01

0

02 0 minus02 0

02

04

06



1198911Hz 119881

1V 119881

2V 119881

3V 119881

4V 119878 (Kg)

1900 39208 39208 62244 62244 73553



=










03

02

01

0

020 minus02

0

02

04

06


14

12

10

8

60 20 40 60 80 100 120 140 160 180 200

S(k

g)




= 09789 times 10minus12 and 119877

11988634

=











15

1

05

0

minus05

minus1

minus150 005 01 015 02502 03 035 04 045 05

t (s)

The t

ime r

espo

nse (

m)

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol

times10minus4


0

0

02

04

06

08







0

02

04

06

08


0

02

04

06

08



1198911Hz 119881

1V 119881

2V 119881

3V 119881

4V 119878 (Kg)

2886 8842 8842 24806 24806 78205


=


11988634





(kg)

9

85

8

750 20 40 60 80 100



25

2

15

1

05

0

minus05

minus1

minus15

minus2

minus250 01 02 03 04 05

t (s)

The t

ime r

espo

nse (

m)

times10minus7

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol


8 Conclusions





Acknowledgments


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119891119868

1minus 119891119877

1le 0 (25)

where 119891119868

1and 119891

119877




119881119886

119896minus 119881119886

119896le 0 (26)

where 119881119886

119896and 119881

119886




119877119897

119896le 119877119896le 119877119906

119896 (27)

where 119877119897

119896and 119877

119906






y119888(119905)

y119903(119905)

+ [

Λ119888

0

0 Λ119903

]

y119888(119905)

y119903(119905)

=

120601T119888

120601T119903

Bu (119905) (28)

120601119888= [1206011 120601

2sdot sdot sdot 120601119899119898]

120593T119894119872120593119895= 120575119894119895

Λ119888= diag (1205962

1 1205962

2 120596

2

119899119898

)

(29)

where y119888(119905) and y





matrix form

Z (119905) = AZ (119905) + B1015840u (119905) (30)

Z (119905) = [

y119888(119905)

y119888(119905)

]

A = [

0119899119898times119899119898

I119899119898times119899119898

minusΛc 0119899119898times119899119898

]

B1015840 = [

0119899119898times119899119886

120593T119888B

]

(31)


119898in





119869 =1

2int

infin

0

(ZT(119905)QZ (119905) + uT (119905)Ru (119905)) 119889119905

997888rarr min(33)

Q = [

Λ119888

0119899119898times119899119898

0119899119898times119899119898

Ι119899119898times119899119898

]

R =

[[[

[

1199031

0

d

0 119903119899119886

]]]

]

(34)



where P is a 2119899119898





Z (119905) = AZ (119905)


(37)







eval (119889119904119894 119889119888

119896) =

1


sum119901

119895=1119863119895

119901) (38)


= (119891119877

1minus 119891119868

1)119891119868

1 the other 119863


method as1198631







Begin k = 1


Binary decode







k = k + 1


Yes End

No





03

02

01

0

02 0 minus02 0

02

04

06






119901= 63times10



11988931

= 11988932


120576 = 15 nfm



03

02

01

0

02 0 minus02 0

02

04

06



1198911Hz 119881

1V 119881

2V 119881

3V 119881

4V 119878 (Kg)

1900 39208 39208 62244 62244 73553



=










03

02

01

0

020 minus02

0

02

04

06


14

12

10

8

60 20 40 60 80 100 120 140 160 180 200

S(k

g)




= 09789 times 10minus12 and 119877

11988634

=











15

1

05

0

minus05

minus1

minus150 005 01 015 02502 03 035 04 045 05

t (s)

The t

ime r

espo

nse (

m)

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol

times10minus4


0

0

02

04

06

08







0

02

04

06

08


0

02

04

06

08



1198911Hz 119881

1V 119881

2V 119881

3V 119881

4V 119878 (Kg)

2886 8842 8842 24806 24806 78205


=


11988634





(kg)

9

85

8

750 20 40 60 80 100



25

2

15

1

05

0

minus05

minus1

minus15

minus2

minus250 01 02 03 04 05

t (s)

The t

ime r

espo

nse (

m)

times10minus7

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol


8 Conclusions





Acknowledgments


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














eval (119889119904119894 119889119888

119896) =

1


sum119901

119895=1119863119895

119901) (38)


= (119891119877

1minus 119891119868

1)119891119868

1 the other 119863


method as1198631







Begin k = 1


Binary decode







k = k + 1


Yes End

No





03

02

01

0

02 0 minus02 0

02

04

06






119901= 63times10



11988931

= 11988932


120576 = 15 nfm



03

02

01

0

02 0 minus02 0

02

04

06



1198911Hz 119881

1V 119881

2V 119881

3V 119881

4V 119878 (Kg)

1900 39208 39208 62244 62244 73553



=










03

02

01

0

020 minus02

0

02

04

06


14

12

10

8

60 20 40 60 80 100 120 140 160 180 200

S(k

g)




= 09789 times 10minus12 and 119877

11988634

=











15

1

05

0

minus05

minus1

minus150 005 01 015 02502 03 035 04 045 05

t (s)

The t

ime r

espo

nse (

m)

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol

times10minus4


0

0

02

04

06

08







0

02

04

06

08


0

02

04

06

08



1198911Hz 119881

1V 119881

2V 119881

3V 119881

4V 119878 (Kg)

2886 8842 8842 24806 24806 78205


=


11988634





(kg)

9

85

8

750 20 40 60 80 100



25

2

15

1

05

0

minus05

minus1

minus15

minus2

minus250 01 02 03 04 05

t (s)

The t

ime r

espo

nse (

m)

times10minus7

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol


8 Conclusions





Acknowledgments


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Begin k = 1


Binary decode







k = k + 1


Yes End

No





03

02

01

0

02 0 minus02 0

02

04

06






119901= 63times10



11988931

= 11988932


120576 = 15 nfm



03

02

01

0

02 0 minus02 0

02

04

06



1198911Hz 119881

1V 119881

2V 119881

3V 119881

4V 119878 (Kg)

1900 39208 39208 62244 62244 73553



=










03

02

01

0

020 minus02

0

02

04

06


14

12

10

8

60 20 40 60 80 100 120 140 160 180 200

S(k

g)




= 09789 times 10minus12 and 119877

11988634

=











15

1

05

0

minus05

minus1

minus150 005 01 015 02502 03 035 04 045 05

t (s)

The t

ime r

espo

nse (

m)

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol

times10minus4


0

0

02

04

06

08







0

02

04

06

08


0

02

04

06

08



1198911Hz 119881

1V 119881

2V 119881

3V 119881

4V 119878 (Kg)

2886 8842 8842 24806 24806 78205


=


11988634





(kg)

9

85

8

750 20 40 60 80 100



25

2

15

1

05

0

minus05

minus1

minus15

minus2

minus250 01 02 03 04 05

t (s)

The t

ime r

espo

nse (

m)

times10minus7

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol


8 Conclusions





Acknowledgments


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










03

02

01

0

02 0 minus02 0

02

04

06



1198911Hz 119881

1V 119881

2V 119881

3V 119881

4V 119878 (Kg)

1900 39208 39208 62244 62244 73553



=










03

02

01

0

020 minus02

0

02

04

06


14

12

10

8

60 20 40 60 80 100 120 140 160 180 200

S(k

g)




= 09789 times 10minus12 and 119877

11988634

=











15

1

05

0

minus05

minus1

minus150 005 01 015 02502 03 035 04 045 05

t (s)

The t

ime r

espo

nse (

m)

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol

times10minus4


0

0

02

04

06

08







0

02

04

06

08


0

02

04

06

08



1198911Hz 119881

1V 119881

2V 119881

3V 119881

4V 119878 (Kg)

2886 8842 8842 24806 24806 78205


=


11988634





(kg)

9

85

8

750 20 40 60 80 100



25

2

15

1

05

0

minus05

minus1

minus15

minus2

minus250 01 02 03 04 05

t (s)

The t

ime r

espo

nse (

m)

times10minus7

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol


8 Conclusions





Acknowledgments


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










15

1

05

0

minus05

minus1

minus150 005 01 015 02502 03 035 04 045 05

t (s)

The t

ime r

espo

nse (

m)

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol

times10minus4


0

0

02

04

06

08







0

02

04

06

08


0

02

04

06

08



1198911Hz 119881

1V 119881

2V 119881

3V 119881

4V 119878 (Kg)

2886 8842 8842 24806 24806 78205


=


11988634





(kg)

9

85

8

750 20 40 60 80 100



25

2

15

1

05

0

minus05

minus1

minus15

minus2

minus250 01 02 03 04 05

t (s)

The t

ime r

espo

nse (

m)

times10minus7

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol


8 Conclusions





Acknowledgments


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(kg)

9

85

8

750 20 40 60 80 100



25

2

15

1

05

0

minus05

minus1

minus15

minus2

minus250 01 02 03 04 05

t (s)

The t

ime r

espo

nse (

m)

times10minus7

Unc

ontro

lIn

tegr

ated

optim

izat

ion

cont

rol


8 Conclusions





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 Integrated Topology Optimization of...

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