Computers and Structures 128 (2013) 101–115

Contents lists available at ScienceDirect

Computers and Structures

journal homepage: www.elsevier .com/locate /compstruc

Optimal discrete piezoelectric patch allocation on composite structuresfor vibration control based on GA and modal LQR

0045-7949/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compstruc.2013.07.003

⇑ Corresponding author. Tel.: +55 (51) 3308 3681; fax: +55 (51) 3308 3222.E-mail addresses: sl.schulz@uepg.br (S.L. Schulz), herbert@mecanica.ufrgs.br

(H.M. Gomes), amawruch@ufrgs.br (A.M. Awruch).1 Tel.: +55 (51) 3308 3486; fax: +55 (51) 3308 3999.

Sérgio L. Schulz a,1, Herbert M. Gomes b,⇑, Armando M. Awruch a,1

a Graduate Program in Civil Engineering, Federal University of Rio Grande do Sul, Rua Osvaldo Aranha, 99, 3� Andar, 90035-190 Porto Alegre, RS, Brazilb Graduate Program in Mechanical Engineering, Federal University of Rio Grande do Sul, Rua Sarmento Leite, 425, Room 202, 2� Andar, 90050-170 Porto Alegre, RS, Brazil

a r t i c l e i n f o a b s t r a c t

Article history:Received 18 April 2013Accepted 11 July 2013

Keywords:Structural vibration controlLinear quadratic regulator (LQR)Composite materialsGenetic algorithm (GA)Optimal piezoelectric placement

The optimization of piezoelectric patches allocation in composite structures is analysed in this paper. Thefinite element method and a linear quadratic regulator are used to study the electro-mechanical behav-iour and the gain calculation. Due to the discrete nature of the problem, a simple binary Genetic algo-rithm is used as an optimization tool. Three examples are presented related to the optimal allocationbased on Lyapunov functional. The PSD (Power Spectral Density) of the state space variables as well asinput voltages are presented in order to identify the controlled modes and to show the effective attenu-ation obtained due to control of specific modes.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction The total weighted energy method was proposed by Ang et al.

In recent years, piezoelectric materials has been used in severalareas and in a wide variety of problems related to vibration control,noise reduction or elimination, shape control and positioning con-trol [1–5].

In vibration control, active control of lightweight structuresmay be the best solution due to the low damping (passive control).In any case, it is always desirable to perform the active controlminimizing the control forces (which are constrained by theactuator force limits) and the attenuation of the structural vibra-tion, although these two aspects lead to a conflicting task. Theproper choice of the free parameters of the control algorithm aswell as the positioning and number of piezoelectric actuators playthe main role in achieving this goal. Modal control allows therational use of the energy spent in control, since it is possible tocontrol only the most important vibration modes (having thehigher energy content) and neglecting the other ones.

Ning [6] presented an optimal design method with respect tothe number and placement of piezoelectric patch actuators inactive modal vibration control on a plate using a genetic algorithm(GA). This author used the eigenvalue distribution of the energymatrix of the control input force as the function to determine theoptimal number and positions of the patches, concluding that theinitial disturbance conditions is the key factor.

[7] to obtain the weighting matrices for the modal LQR (linear qua-dratic regulator) control. The correct choice of the weighting matri-ces can generate vibration attenuations, which are proportional tothe input voltages. Then, a compromise between vibration attenu-ation and input voltage should be obtained. A balanced vibrationcontrol and low input cost may be attained considering three de-sign variables: the total kinetic energy, the strain energy and theinput energy. The paper highlighted the advantages in using modalcontrol analysis due to reduction of computational cost.

Roy and Chakraborty [8,9], Satpathy [10] and Chakraborty andRoy [11] presented the active vibration control of smart FRP (fibrereinforced polymer) composite plate and shell structures. Theyused a layered plate or shell finite element and an improved GAto optimize the positioning of piezoelectric patches and theweighting matrices Q and R for the control. The reasoning forchoosing best allocation among patch positions was based on thedamping ratio of the actual responses.

Based on a controllability index X, Wang and Wang [12,13]used a binary coded GA to find the optimal placement of a previ-ously defined number of patches. Displacements and input volt-ages time histories are presented. As it is shown, in some casesthe optimal solution differs from the intuitive positioning basedon the mode shapes.

Several authors addressed the problem of allocation of thepiezoelectric patches. Araújo et al. [2] used a Direct MultisearchMethod with topology optimization of composite plates in orderto reduce the modal loss factor by the co-located negative velocityfeedback control. They apply the proposed methodology to a

102 S.L. Schulz et al. / Computers and Structures 128 (2013) 101–115

composite plate with coarse mesh refinement. Bachmann et al. [3]used a strain based optimization approach to allocate the patches.A MATLAB routine was also used allowing an exhaustive search ofthe best allocations for the patches (like a Monte Carlo sampling).

Potami [1] investigated three different approaches for the opti-mal positioning of sensors and actuators. All the proposed ap-proaches present some level of heuristic procedures to handlethe problem. For the first approach, multiple sensors/actuatorsare simultaneously allocated. In the second approach, the sensor/actuators are placed in pairs, taking into account the influence ofthe spatial distribution of disturbances. Finally, the third approachprovides a solution to the actuator location problem by incorporat-ing considerations with respect to the preferred spatial regionswithin the flexible structure. All investigations are performed rank-ing the candidate positions based on a controllability index.

The proposed paper brings a discrete patch allocation whichdoes not use a controllability index as usual in the revised papers.The metric used to highlight the suitability of the allocation isbased directly in the Lyapunov functional, since it represents a bal-ance between vibration suppression (remaining kinetic energy)and application of forces originated by the actuators (applied en-ergy for vibration or displacement control). As it is assumed a dis-crete allocation and a predefined number of patches, the obtainedresults are not expected to be a global optimum, but it may be con-sidered the optimal solution within those constraints.

2. Finite element formulation

2.1. The element GPL-T9 (generalised point and line compatibilitytriangular finite element with 9 DoF) for slender plates and shells

The incremental equilibrium equations using the finite elementmethod (FEM) is given by [14,15]

½tK�fDug ¼ ftþDtRg � ftFg ð1Þ

where {Du} is the vector containing the incremental nodal displace-ments and rotations, which is given by

fDug ¼fDu1gfDu2gfDu3g

8><>:

9>=>;; fDuig ¼ fDuxi Duyi Dwi Dhxi Dhyi DhzigT

ð2Þ

with (i = 1,2,3), tþDtR is the vector of external nodal forces, tF is theinternal force vector and ½tK� is the stiffness matrix.

Taking into account coupling of membrane and bending effectsfor slender shells and plates, the following system of equations isobtained:

½Km� ½Kmb�½Kbm� ½Kb�

� � fDumgfDubg

� �¼ ftþDtRmg � ftFmg

ftþDtRbg � ftFbg

� �ð3Þ

where the stiffness matrices due to effects of membrane [Km], mem-brane-bending coupling [Kmb], [Kbm] and bending [Kb], are defined by

½Km� ¼Z

t A½Bm�T ½Dm�½Bm�tdA;

½Kmb� ¼Z

t A½Bm�T ½Dmb�½Bb�tdA;

½Kbm� ¼Z

t A½Bb�T ½Dbm�½Bm�tdA ¼ ½Kmb�T ;

½Kb� ¼Z

t A½Bb�T ½Db�½Bb�tdA:

ð4Þ

where [B] and [D] are the strain–displacement matrix and constitu-tive matrix, respectively, and m stands for membrane effects, brepresents bending effects and mb indicates membrane-bendingcoupling effects.

These matrices are explicitly defined in the developed finiteelement code. The external nodal force vector referred to themembrane {tþDtRm} and bending effects {tþDtRb} are

tþDtRm� �

¼Z

t A½Hm�T

tþDtRx� �

tþDtRy� �

( )tdA;

tþDtRb

� �¼Z

t A½Hb�T tþDtRz

� �tdA;

ð5Þ

where [H] is the shape function matrix, ftþDtRxg, ftþDtRyg and ftþDtRzgare the external nodal force vectors in the x, y and z direction, beingthe internal membrane and bending force vectors in the time t,ftFmg and ftFbg, respectively, given by

ftFmg ¼Z

t A½Bm�TftNgtdAþ

Zt A½Bm�TftNMgtdA;

ftFbg ¼Z

t A½Bb�TftMgtdAþ

Zt A½Bb�TftMNgtdA:

ð6Þ

where {N}, {M} and {NM} or {MN} are vectors of membrane force,bending per unit length and bending-membrane coupling,respectively.

For dynamic analysis, the equilibrium equation may be writtenas

½M�ftþDt €ug þ ½C�ftþDt _ug þ ½tK�fDug ¼ fftþDtRg � ftFgg ð7Þ

The consistent mass matrix [M] is given by

½M� ¼Xn

k¼1

hkqk

ZA½H�T ½H�dA ð8Þ

where n is the total number of composite layers, hk is the thicknessof the kth layer, qk is the specific mass of the kth layer and the com-plete interpolation matrix [H] is given by

½H� ¼Li 0 Huhi

0 0 00 Li Hvhi

0 0 00 0 0 Hi Hxi Hyi

264

375ði ¼ 1;2;3Þ ð9Þ

The damping matrix [C] can be evaluated using the Rayleighmodel½C� ¼ aR½M� þ bR½K� ð10Þ

where constants are determined with eigenvalues and damping ra-tios corresponding to two modes. More details may be found in Iso-ldi et al. [14].

2.2. Embedded piezoelectric material

If plies of piezoelectric material are added to the laminatedcomposite material (as actuators and/or sensors), the electrical po-tential field must be included as an additional degree of freedomper node and per piezoelectric layer. The electric potential fieldincrement is given by [16]tþDt

0 / ¼ t0/ þ D/ ð11Þ

Increment of the electric displacement vector is evaluated asfollows:

tþDt0 Ekt þ Dt ¼ � @ð

t0/ þ D/Þ@0xk

¼ � @t0/

@0xk� @D/@0xk

¼ t0E1 � ck ð12Þ

where t0E1 is the gradient of the electric potential field. Then, the

equations for the incremental electric displacement field is given by

fDEg ¼DEx

DEy

DEz

8><>:

9>=>; ¼ �

D/;x

D/;y

D/;z

8><>:

9>=>; ¼ �

@D/@x@D/@y

@D/@z

8>><>>:

9>>=>>; ¼ �

00@D/@z

8><>:

9>=>; ¼ �

00D/hp

8><>:

9>=>;ð13Þ

When the finite element method is used, Eq. (13) is given by

S.L. Schulz et al. / Computers and Structures 128 (2013) 101–115 103

f�DEg ¼ ½B/�fD/g ¼

0 00 0

1=hpi0

0 00 00 1=hps

2666666664

3777777775

D/pi

D/ps

( )ð14Þ

where matrix [B/] contains derivatives of the interpolationfunctions, hpi

and hpsare the thickness of the top and bottom

piezoelectric layers, respectively, while D/piand D/ps

are thecorresponding incremental values of the electric potentials fields.

In this way, the incremental equation becomes:

½Muu� ½0�

½0� ½0�

" #ftþDt €ug

ftþDt €/g

( )þ½cuu� ½0�

½0� ½0�

" #ftþDt _ug

ftþDt _/g

( )

þ½tKuu� ½Ku/�

½K/u� ½K//�

" #fDug

fD/g

( )¼

ftþDtRug � ftFug

ftþDtR/g � ftF/g

( ); ð15Þ

where mechanical mass, damping and stiffness matrices are ob-tained taking into account the piezoelectric layers. The electric stiff-ness is defined by

½K//� ¼ �Z

t V½tB/�

T ½ni� ½0�½0� ½ns�

� �½tB/�tdV ¼ �

Ain33ihi

0

0 Asn33shs

24

35 ð16Þ

where A is the element area, h is the element thickness, n33 is thedielectric constant and the subscripts i and s are referred tothe bottom and top piezoelectric layers, respectively. Theelectro-mechanical coupling stiffness matrix [Ku/] = [Ku/]T is givenby

½Ku/� ¼Z

t V

½tBm� ½tBm�½0� ½0�

" #T ½e1�i ½0�½0� ½e1�s

" #T

½tB/�tdV

þZ

t Vz½0� ½0�½tBb� ½tBb�

" #T ½e1�i ½0�½0� ½e1�s

" #T

½tB/�tdV ; ð17Þ

where z is the coordinate normal to the element surface and [e1],the piezoelectric coupling matrix in the local coordinate system,which is given by

½e1� ¼f0gf0gfe1g

264

375 ¼

0 0 00 0 0

e31 e32 e36

264

375 ð18Þ

Evaluation of the vector of internal mechanical forces tFu iscomputed considering the piezoelectric layers, while the vectorof external nodal electric loads tþDtRu is evaluated as

ftþDtR/g ¼Z

tþDt A

qi

qs

� �tþDtdA ð19Þ

where qi and qs are the electric charges at the bottom and top pie-zoelectric layers of the piezoelectric elements, respectively.

The internal electric force vector ftF/g is given by

ftF/g ¼Z

t V½tB/�

T ftDgi

ftDgs

� �tdV ð20Þ

where the electrical displacements ftDgi and ftDgs of the bottomand top piezoelectric layers are given, in an updated Lagrangian for-mulation by

ftDg ¼ det½tþDtt F �½tþDt

t F ��1fttDg ð21Þ

where fttDg is evaluated as follows

fttDg ¼

00

fe31 e32 e36gk½T��Tk ½I3� zmk½I3�½ �

te0

tj

( )� nk

t/khk

8>>><>>>:

9>>>=>>>;

ð22Þ

where e31, e32 and e36 are the piezoelectric constants in the localcoordinate system, n is the dielectric constant, [I3] is a 3 � 3 identitymatrix, e and j are the strains and curvatures, respectively, assumedconstants at the layer thickness, k stands for a piezoelectric layer atposition zk from the element mean surface.

If damping is not considered, the dynamic equilibrium equa-tions are separated as

½Muu�ftþDt €ug þ ½tKuu�fDug þ ½Ku/�fD/g ¼ ftþDtRug � ftFug ð23Þ

½K/u�fDug þ ½K//�fD/g ¼ ftþDtR/g � ftF/g: ð24Þ

Isolating {D/}, the following expression is obtained:

½Muu�ftþDt €ug þ ½½tKuu� � ½Ku/�½K//��1½K/u��fDug

¼ ftþDtRug � ftFug � ½Ku/�½K//��1fftþDtR/g � ftF/gg: ð25Þ

If there is not any actuator, and thus there is not an external ap-plied electric field, the sensor voltage can be calculated by the fol-lowing expression:

fD/gs ¼ �½K//��1s ff

tF/g þ ½K/u�sfDugsg ð26Þ

where the subscript ‘‘s’’ refers to the use of only piezoelectricsensors.

In a general case, the following equations are valid instead:

½Muu�f€ug þ ½Kuu�fug þ ½Ku/�f/g ¼ fRug ð27Þ

½K/u�fug þ ½K//�f/g ¼ fR/g ð28Þ

½Muu�f€ug þ ½Kuu� � ½Ku/�½K//��1½K/u�h i

fug

¼ fRug � ½Ku/�½K//��1fR/g ð29Þ

f/gs ¼ �½K//��1s ½K/u�sfugs ð30Þ

3. Full state feedback modal control

For the full state feedback modal control, relationship betweenelectric charges and modal state variables is given by the followingexpression [17]:

fqg ¼ ½U�fgg ð31Þ

Then,

fqgf _qg

� �¼½U� ½0�½0� ½U�

� � fggf _gg

� �fxg ¼ ½N�fxmg;

ð32Þ

where {xm} is the vector of the state space variables composed bythe modal variable vector fgg ¼ fg1; g2; . . . ; gng

T and the respec-tive time derivatives f _gg ¼ f _g1; _g2; . . . ; _gngT . [U] is a matrix, whichis organised in such a way that each column contains an eigenvec-tor, obtained from the eigenvalue–eigenvector problem

½½K� � ½X�½M��½U� ¼ f0g; ð33Þ

where [X] is a diagonal matrix.

½X� ¼ diagðx21;x

22; . . . ;x2

nÞ; ð34Þ

where fx21; x2

2;x23; . . . ; x2

ng are the square values of circularfrequencies, and the matrix ½U� ¼ ½fU1g; fUg2; . . . ; fUng� containsin each column fUig ¼ f/1i

; /2i; . . . ; /ni

gT , the eigenvector

104 S.L. Schulz et al. / Computers and Structures 128 (2013) 101–115

corresponding to the eigenvalue x2i . Taking into account Eq. (31),

the following equation of motion is obtained:

½I�f€gg þ ½K�f _gg þ ½X�fgg ¼ ½½U�T ½M�½U���1½U�T ½F�fug; ð35Þ

where [I] is the identity matrix and [K] a diagonal matrix with thediagonal terms defined as f2n1x1; 2n2x2;2n3x3; . . . ; 2nnxng,where ni is the damping ratio of the ith mode shape. The equations,in the state space form, become [17,18]:

f _xmg ¼ ½Am�fxmg þ ½Bum �fugfyg ¼ ½Cm�fxmg þ ½Dm�fug;

ð36Þ

where matrices [Am], ½Bum �, [Cm] and [Dm] are defined as follows:

½Am� ¼½0� ½I��½X� �½K�

� �; ½Bum � ¼

½0�

½½U�T ½M�½U���1½U�T ½F�

" #;

½Cm� ¼ �½X� �½K�½ �; ½Dm� ¼ ½½½U�T ½M�½U���1½U�T ½F��:

ð37Þ

[Cm] and [Dm] can be defined according to the desired output vari-ables (modal state space variables, control forces, acceleration, etc.).

In the case of optimal control, the following Lyapunov quadraticfunctional, to be minimized, is defined:

Jm ¼ ð1=2ÞZ 1

0ðfxmgT ½Q m�fxmg þ fugT ½Rm�fugÞdt; ð38Þ

Then, the following equations hold:

fxmg ¼ ½N��1fxgfggf _gg

� �¼ ½U��1 ½0�

½0� ½U��1

" #fqgf _qg

� �;

ð39Þ

Modal weighting matrices [Qm] and [Rm] are related to the well-known traditional weighting matrices [Q] and [R], respectively, by

½Qm� ¼ ½½U��T ½Q �½U��1� and ½Rm� ¼ ½½U��T ½R�½U��1� ð40Þ

The following assumption is used to make easier the tuning ofthe modal LQR control behaviour (loop shaping), as stated by Royand Chakraborty [8].

½Q m� ¼a1½x2

i � ½0�½0� a2½I�

" #and ½Rm� ¼ c½I� ð41Þ

where a1, a2 and c are parameters to be set by the user or obtainedby optimization. The use of full non-negative and positive definite Qand R matrices is still focus of research and is beyond the scope ofthis paper.

The input forces are defined by the relation:

fug ¼ �½Gm�fxmg ¼ �½Gm�fggf _gg

� �; ð42Þ

where [Gm], the modal gain matrix, is given by½Gm� ¼ ½½Rm��1½Bum �

T ½Sm��; and obtained solving the following Ricattiequation in the modal state space:

½Sm�½Am� þ ½Am�T ½Sm� � ½Sm�½Bum �½Rm��1½Bum �T ½Sm� þ ½Q m� ¼ ½0�: ð43Þ

Moreover, when considering the electric field as control forces,the equations in the state space form are transformed to:

f _xmg¼½0� ½I��½X� �½K�

� � g_g

� �

þ½0�

½½U�T ½M�½U���1½U�T ½F�½Ku/�½K//��1½Fp�

" #fvg

þ½0�

½½U�T ½M�½U���1½U�T ½Fl�

" #fwgf _xmg¼ ½Am�fxgþ½B�um

�fvg; ð44Þ

with the control forces {u} given indirectly by the transformation tothe applied electric field {v} by the equation

fvg ¼ �½G�m�fxmg ð45Þ

and using the modal gain matrix evaluated by

½G�m� ¼ ½½R��1½B�um

�T ½S�m�� ; ð46Þ

where ½S�m� is the solution of the Ricatti equation in the modal space.Assuming that there are most important modes in the system to

be controlled, Eq. (31) can be truncated using just those mostimportant modes

fqig ¼Xnmod

j¼1

½Uij� fgjg or fqg ¼ ½U�fgg; ð47Þ

where nmod is the number of modes that are retained.

4. Genetic algorithm

There are several mathematical programming methods that canbe used to solve the optimization problem for discrete patch allo-cation in plate and shell composite structures. Most of the optimi-zation methods often search solutions in the neighbourhood of aninitial point, through gradient calculus. But if the problem showsmore than one local optimum, the result will depend on the choiceof the initial point, and the global optimum might not be found.Furthermore, when an objective function and its constraints showseveral local minima (or maxima), using of gradient calculus mightbecome hard and unstable.

Many engineering problems are complex, nonlinear, and de-scribed by functions not always differentiable. It is often conve-nient to make use of stochastic optimization methods, such as agenetic algorithm (GA), which undertakes a set of actions thatsearch for an optimum solution almost randomly, and it is not nec-essary to know the gradient of the objective function with respectto design variables.

4.1. Binary coded genetic algorithm

Genetic algorithms (GAs) were created by Holland [19], and la-ter improved by several researchers, such as Goldberg [20]. Withthe GAs, the search of the optimum solution begins with a popu-lation of different starting points inside the function domainbeing studied; these points are called individuals. The main dif-ference between classical gradient based methods and the GAsis the fact that the GAs do not get stuck easily around local opti-ma, since they work with a population of searching points, cover-ing a broad search space using function values instead of functionderivatives. Initially, a set of individuals defines a population, inwhich each individual has an associated fitness value. A new pop-ulation is developed through genetic operations such as crossoverand mutation, getting to a new generation by using the Darwinprinciples of reproduction and survival of the fittest individuals.The fitness value is calculated by starting from the objective func-tion value that will be optimized. The process will arrive to itsend when a maximum number of generations is reached, or suc-cessive optimum values do not change significantly accordingly toa predefined tolerance. The main steps of a GA are described asfollows.

(1) Creation of the Initial Population, which is generated ran-domly. It must be taken into account that if the initial pop-ulation is small, probably it will not represent some regionsof the search space.

Fig. 1. (a) Bimorph PVDF beam dimensions and (b) the obtained optimal allocation for two piezoelectric patches using GA.

0.50.40.30.20.10t [s]

-0.008

-0.006

-0.004

-0.002

0

w(t)

[m]

UncontrolledControlled with all patchesControlled with Patches 1 and 2

Fig. 2. Vertical displacements at the tip of the uncontrolled beam, with the activation of all patches and with only patches 1 and 2 activated.

S.L. Schulz et al. / Computers and Structures 128 (2013) 101–115 105

(2) Objective Function Evaluation. Each individual in a popu-lation is evaluated. A typical optimization problem showsdesign variable constraints which are used as an initialdomain for variable generations of individuals. Thefitness of each individual will be defined by the objectivefunction and constraints. There are two approaches todeal with constraints: (a) to penalize the fitness functionby defining how much the solution violates the con-straint, or (b) to convert each constraint into differentobjective functions and carry out multi-objectiveoptimization.

(3) Scaling of the Objective Function. This step can be defined asthe final fitness value evaluation. The objective function isnot always suitable for being used as a fitness function espe-cially when the individual’s objective functions are very sim-ilar near optimal solution. To sum up, the objective functionshould allow all the individuals to have appropriate chancesto be selected by the genetic operations by a suitable scalingof their values.

(4) Selection of Individuals. Individuals that will be parents inorder to generate the offspring, which in your turn, will com-pose the next generation are selected. The Roulette WheelMethod, described by Goldberg [20] was used in this paperto perform the corresponding selection.

(5) Crossover, which is the combination of genetic informationbetween two individuals (parents), resulting in a new indi-vidual (child or offspring) for the next generation, accordingto a crossover probability (80% was adopted in this paper).

(6) Mutation, which is a genetic operator that modifies thegenes of the individual that will be part of the next genera-tion, according to a mutation probability, and improving thechromosome diversity of the population, but destroying partof the genetic information kept in the offspring. Therefore, asin nature, a low mutation probability shall be used (1% wasadopted in this paper).

(7) Elitism, a procedure that copies the best individual in thepopulation into the next generation without going throughthe selection step, assuring the convergence.

0 100 200 300 400 500f [Hz]

-200

-160

-120

-80

-40

PSD

(w(t)

)[dB

](ref

=1m

)UncontrolledControlled with all patchesControlled with Patches 1 and 2

Fig. 3. Displacements Power Spectral Density (PSD) attenuation (referred to 1 m) for the uncontrolled beam, controlled activating all patches or only when patches 1 and 2are activated.

0 0.1 0.2 0.3 0.4 0.5t [s]

-1000

-500

0

500

1000

1500

v(t)

[ V]

Controlled with all patchesControlled with Patches 1 and 2

Fig. 4. Applied voltage to the beam using all patches or when only patches 1 and 2 are activated.

106 S.L. Schulz et al. / Computers and Structures 128 (2013) 101–115

When working with binary coded genetic algorithms, each ofthe real parameters bi to be optimized is translated into binarycodes by the following equation:

s ¼ binn round ð2n � 1Þ ½biðkÞ � PminðkÞ�½PmaxðkÞ � PminðkÞ�

� �ð48Þ

where binn indicates a binary translation to a string s of n bits, P(k)means the range of maximum and minimum values allowed foreach design variable.

To transform the binary codes to real values, the followingequation is used in the sequence:

biðkÞ ¼ PðkÞmin þ bin�1ðsÞ PðkÞmax � PðkÞmin

2n � 1ð49Þ

where bin�1(s) means the translation of the binary coded values torespective real ones.

With this formulation, it is implicit that the design variablemapping has a resolution of ½PðkÞmax � PðkÞmin�=ð2

n � 1Þ . This

0 100 200 300 400 500f [Hz]

-40

0

40

80

120

160

PSD

(v(t)

)[dB

](ref

=1V)

Controlled with all patchesControlled with patches 1 and 2

Fig. 5. Voltage Power Spectral Density (PSD) attenuation (referred to 1 V) for the controlled beam with activation of all patches or when only patches 1 and 2 are activated.

Fig. 6. Simply supported piezoelectric composite square plate. (a) Roy and Chakraborty [9] piezoelectric patch configuration (configuration 1). (b) Optimal piezoelectricconfiguration obtained using GA (configuration 2).

Table 1Material properties for the simply supported piezoelectric composite square plate.

Properties Graphite epoxy laminate PZT

E1 172.5 GPa 63.0 GPaE2 = E3 6.9 GPa 63.0 GPaG12 = G13 3.45 GPa 24.6 GPaG23 1.38 GPa 24.6 GPam12 = m13 = m23 0.25 0.28P 1600 kg/m3 7600 kg/m3

e31 = e32 0.0 10.62 C/m2

n11 = n22 = n33 0.0 0.1555 � 10�7 F/m

S.L. Schulz et al. / Computers and Structures 128 (2013) 101–115 107

restricts the search space of the real parameters to a discrete range.In this paper, the design parameters are the position of the patchesof piezoelectric material (integer values). The number of patches islimited by a number defined by the user and the patch position isattached to the discretization of the finite element mesh. The num-ber of the bits for each design variable (patch position) is definedsuch that the number of bits combination is greater than the num-ber of finite element positions. Problems related to the resolutionof the binary codification are not faced since the patch locationsare integer values.

In order to compare the suitability of two arrangements ofpatch position, the value of Lyapunov quadratic functional wasused as a metric, since this accounts for the vibration attenuationof the state space variables as well as for the applied input voltage(control forces) as indicated in Eq. (50)

Jm ¼ ð1=2ÞZ 1

0ðfxmgT ½Q m�fxmg þ fugT ½Rm�fugÞdt ð50Þ

108 S.L. Schulz et al. / Computers and Structures 128 (2013) 101–115

5. Numerical examples

5.1. Piezoelectric patch optimal placement in a cantilever beam(Bimorph beam)

The developed methodology is validated in a simple problem.This example compares the bimorph cantilever beam with thebenchmark problem proposed by Sze and Yao [21]. The beam iscomposed by two PVDF (PolyVinylidene DiFluoride) layers lami-nated together and subjected to an external unit voltage that in-duces bending. In all numerical examples, the equipotentialelectrode condition for patches in the same surface was alwaysconsidered. This means only one control voltage channel is usedto control the structures. It is obvious that several control voltagechannels can be used, and the presented formulation accountsfor this possibility as can be seen in Eqs. (45) and (46). However,in order to be compared with literature results (that use othermethodologies to allocation of the patches), the examples were de-signed in the same way they are presented in the literature.

The finite element mesh has 10 triangular GPL-T9 finiteelements as indicated in Fig. 1(a). The dimensions of the beamare defined in the same figure. The stacking sequence of thelaminate is p/[0/0]s/p, where there is no composite material butonly PVDF material. Symmetrical voltages can be applied the topand the bottom piezoelectric materials, in order to bend the beam.The mechanical, electrical and coupled material properties are:E1 ¼ E2 ¼ 2 GPa; (Young Modulus), m12 = 0.29, (Poisson coefficient),G12 ¼ 0:7752 GPa (Shear Modulus), q ¼ 1800 kg=m3 (mass den-sity), e31 ¼ e32 ¼ 0:046 C=m2 and e36 = 0 (piezoelectric constants),n33 ¼ 106:2� 10�12 F=m (dielectric constant). The damping ratiosare n1 ¼ 1� 10�3; n2 ¼ 2� 10�3; n3 ¼ 2� 10�3 for the first threevibration modes.

An initial static vertical force F = 10�2N is applied at the tip ofthe beam, remaining until the end of the analysis. This load

200.0100.00

-4E-007

-2E-007

0

2E-007

4E-007

w(t)

[ m]

Fig. 7. Vertical displacements at the centre of the uncontrolled plate presented by Roy aconfiguration 1, controlled plate using the optimum solution obtained by GA (configura

generates a final vertical displacement of w ¼ �3:138� 10�7 m atthe tip of the deflected beam in the steady state condition. Thisforce is applied as an initial perturbation in order to generatevibrations to be controlled. A complete accuracy validation of thismodel can be found in Isoldi et al. [14]. The linear quadratic regu-lator parameters a1, a2 and c were tuned and fixed by an optimiza-tion procedure employing GA and using the Lyapunov functional asthe cost function. It resulted in the values 1.0, 1.0 and 10�6, respec-tively, so all vibration modes are supposed to be controlled usingthe same balance between the remaining kinetic energy in the sys-tem and the energy applied by the control forces.

It is imposed that only two square patches of 5 mm (a patch iscomposed of two contiguous triangular finite elements) can beactivated by the applied voltage and the other ones are turnedoff. In this example, it is sought the optimum position of thesetwo patches in order to minimize the Lyapunov Functional. Theexpected solution to this case is the activation of triangular finiteelements 1, 2, 6 and 7 near the clamped end corresponding tothe activated patches positions 1 and 2.

The genetic algorithm was applied using 20 individuals and 10generations, adopting for crossover and mutation probabilities 80%and 1%, respectively. Each individual has two genes indicating theposition for each of the patches. Since the patch is formed by twocontiguous triangular finite elements, there are five possible posi-tions to allocate the patches. The number of bits for each gene wasassumed as being 3, since this enables decoded values rangingfrom 1 to 8. Fig. 1(b) shows the obtained optimum position.

As depicted in Fig. 2, the vertical displacement at the tip of thebeam is considerably more attenuated when all patches areactivated with respect to the case when only patches 1 and 2 areactivated in the obtained optimum position. The same figureshows that the final displacement converges towards the staticoffset w ¼ �3:138� 10�7 m as expected. It can be noticed also thatin the last case there is a delay in the vibration attenuation, which

00.0400.0300.0 5t [s]

Uncontrolled with Roy and Chakraborty (2009b)configuration1

Uncontrolledconfiguration1Controlledconfiguration1Controlled with optimum GA placementconfiguration2

nd Chakraborty [9], uncontrolled plate using configuration 1, controlled plate usingtion 2).

(a) 1st. mode shape (f=3102 Hz). (b) 2nd. mode shape (f=10852 Hz).

(c) 3rd. mode shape (f=24413Hz). (d) 4th. Mode shape (f=27315 Hz). Fig. 8. The first 4 vibration mode shapes for the plate example.

S.L. Schulz et al. / Computers and Structures 128 (2013) 101–115 109

is expected since only two patches were activated, which impairsthe control performance.

Fig. 3 shows the corresponding PSD (Power Spectral Density)attenuation for the vertical displacements at the tip of the beam(in dB and referred to 1 m). It can be noticed that for the first mode,the attenuation obtained with only patches 1 and 2 is not so effec-tive than that obtained using all the patches. The attenuation forthe 1st vibration mode is approximately �60 dB with patches 1and 2 activated and �80 dB for the case where all patches areactivated.

Fig. 4 shows the applied voltage for the case where all patchesare activated and for the case where only patches 1 and 2 are acti-vated. It can be noticed that for the last case the final mean value ofthe voltage is larger. This fact is explained because the applied loadF will impose larger applied voltages to patches 1 and 2. A delay inthe attenuation of applied voltage when using only two patches isalso observed in Fig. 4.

Fig. 5 shows the corresponding PSD attenuation for the appliedvoltage (in dB and referred to 1 V). Related to the case where allpatches are activated, Fig 5 shows that there is an applied voltageamplification using patches 1 and 2. This amplification is about+3 dB for first the mode and about +20 dB for the second mode.

This last case was expected as indicated in the Fig. 4, where it isnoticeable higher frequency content in the applied voltage.

Although the previous plots are based on tip deflection of thebeam, the overall behaviour is confirmed. In this case, the J (Lyapu-nov) value of the controlled beam with 2 patches is lower than theJ value of the controlled one with all patches activated, but greaterthan any other J value of controlled beam with 2 patches in othersites.

5.2. Piezoelectric patch optimal placement on a simply supportedpiezoelectric composite square plate

Fig. 6 shows a simply supported piezoelectric composite squareplate. This plate was studied by Roy and Chakraborty [9]. Thethickness of the plate is 7 mm and it is composed by Graphite/Epoxy layers of 1.5 mm, including some attached piezoelectricpatches of 0.5 mm at the top and bottom of the plate. The stackingsequence of the laminated structure is given by p/[0/90]s/p. Eachply has 0.75 mm and the piezoelectric patches have 0.5 mm,resulting in a total thickness equal to 2.5 mm. The length of theplate is a = 0.1 m. A finite element regular mesh of 200 triangularelements was generated and the positions of the patches are

0 10000 20000 30000f [Hz]

-440

-400

-360

-320

-280

-240

-200PS

D( w

( t))[

dB](

ref=

1m)

Unontrolledconfiguration 1Controlledconfiguration 1Controlled with optimum GA placementconfiguration 2

Fig. 9. Displacements Power Spectral Density attenuation (referred to 1 m) for the uncontrolled plate, controlled with configuration 1 and controlled with configurationobtained by this work (configuration 2).

0 0.001 0.002 0.003 0.004 0.005t [s]

-8

-4

0

4

8

v(t)

[ V]

Controlledconfiguration 1Controlled with optimum GA placementconfiguration 2

Fig. 10. Applied voltage to the plate using configuration 1 and configuration 2 (which was obtained with this work).

110 S.L. Schulz et al. / Computers and Structures 128 (2013) 101–115

defined by a mesh of 100 positions as indicated in Fig. 6. A verticalimpulsive load of 10 N is applied in the centre of the plate during14.04 ls in order to generate an initial vibration perturbation to

be later controlled. In this case, the first vibration mode has moreimportance since the excitation is based on a suddenly appliedimpact force that is removed quickly (Dirac delta function) in the

Fig. 11. (a) Dimensions of a simply supported piezoelectric composite spherical shell. (b)Allocation of 8 pairs of collocated patches based on mode shapes (configuration 1).(c) Allocation of 6 pairs of collocated patches based on maximum controllability index (configuration 2). (d) Allocation of 8 pairs of collocated patches obtained in the presentstudy with GA (configuration 3).

S.L. Schulz et al. / Computers and Structures 128 (2013) 101–115 111

centre of the plate. It is known that this excites all vibration modesand more pronouncedly the 1st mode. It was considered 4vibration modes in the modal control with the following dampingratios: f1 = 0.009, f2 = 0.0049, fi ¼ 0:002 i ¼ 3;4:. Material proper-ties for the laminate and PZT are defined in Table 1.

Eight pairs of collocated patches (top/bottom actuators/sensors)have been considered in order to control the vibrations. It is soughtthe optimum position of these eight pairs of collocated patches inorder to attenuate vibrations. The same heuristic parameters of theprevious example were used in this case. In this example all thepatches are allocated independently. Since a patch is formed bytwo contiguous triangular finite elements, then there are 100 pos-sible positions to allocate the patches. The number of bits for eachgene was assumed 7 in order to enable position decoded valuesthat ranges from 1 to 128.

Fig. 6(a) indicates by dashed rectangles the collocated piezo-electric patches configuration adopted by Roy and Chakraborty[9], while Fig. 6(b) shows the configuration obtained by the presentstudy. Fig. 7 presents the displacement at the centre of the platewithout control (indicated by Roy and Chakraborty [9]) and thatobtained by the present FEM formulation. There is a good agree-ment between these two curves. In the same figure, it is shownthe displacement in the controlled plate for the patch position indi-cated by Roy and Chakraborty [9] (configuration 1) and that ob-tained by the present work (configuration 2). There is a fastervibration attenuation using patches in the centre of the plate(configuration 2) compared to the attenuation obtained withconfiguration 1 [9].

Fig. 8 shows the normalised first 4 plate vibration modes withthe patches in the obtained optimal allocation. It can be noticed

that 1st and 4th modes have symmetry and the 2nd and 3rd modespresented anti-symmetry. This confirms that the patch stiffnessand mass are negligible regarding the overall modal behaviour.

Fig. 9 shows the corresponding PSD attenuation of the verticaldisplacements (in dB and referred to 1 m) in the case of configura-tion 1 and that obtained by this paper (configuration 2). Fig. 9 alsoshows the other excited modes (peaks) due to impact force that arepresented in the signal. Unfortunately, Power Spectral Densities forthe literature results are not possible to be plotted and comparedsince only time histories of the central deflections are availablein the literature.

It can be noticed that the attenuation for the first vibrationmode in configuration 1 is about �60 dB, while for configuration2 is about �40 dB. The second vibration mode is completely atten-uated with configuration 1 and 2. The third vibration mode isattenuated and shifted. The attenuation of this mode is about�20 dB and�60 dB for configuration 1 and configuration 2, respec-tively. The fourth mode is completely attenuated with configura-tion 1 but amplified +60 dB for configuration 2, indicating thatspill over occurs for this mode.

Fig. 10 shows the applied voltage in the patches for the con-figuration 1 and configuration 2. In the case of configuration 1,the voltages are larger than in the case of configuration 2;however the attenuation of the applied voltage is faster in con-figuration 1 than in configuration 2. Since the objective functionis based on the value of Jm, this explains the apparent inconsis-tency in the suitability of configuration 2. As the value of Jm

accounts for displacements and control forces, the minimum ofthe sum of these values defines the suitability of the patchesarrangement.

0 0.0004 0.0008 0.0012 0.0016 0.002t[s]

-8E-007

-4E-007

0

4E-007

8E-007w

(t)[

m]

UncontrolledwithRoyandChakraborty(2009a)configuration1

Uncontrolled configuration 1Controlled configuration 1Controlled with max. controllabilityconfiguration 2Controlled with optimum GA placementconfiguration 3

Fig. 12. Vertical displacements at the centre of the uncontrolled shell presented by Roy and Chakraborty [8], uncontrolled shell using configuration 1, controlled plate usingconfiguration 1, controlled plate using configuration 2, controlled plate using the optimum solution obtained with GA (configuration 3).

0 10000 20000 30000 40000 50000f [Hz]

-360

-320

-280

-240

-200

PSD

(w(t

))[ d

B](

ref=

1m)

Unontrolled configuration 1Controlled configuration 1Controlled with max. controllabilityconfiguration 2Controlled with optimum GA placementconfiguration 3

Fig. 13. Power Spectral Density attenuation (referred to 1 m) for the uncontrolled shell, controlled with configuration 1, configuration 2 and configuration 3.

112 S.L. Schulz et al. / Computers and Structures 128 (2013) 101–115

0.0004 0.0008 0.0012 0.0016 0.002t [s]

-30

-20

-10

0

10

20

30v(

t)[V

]

Royand Chakraborty (2009a) max controllabilityconfiguration 2Controlledconfiguration1Controlled with max.controllabilityconfiguration 2Controlled with optimum GA placementconfiguration 3

Fig. 14. Applied voltage to the shell using configuration 1, configuration 2 (given by Roy and Chakraborty, [8] and the present formulation) and configuration 3.

0 1E-005 2E-005 3E-005 4E-005t [s]

-400

-200

0

200

400

v(t)

[ V]

RoyandChakraborty(2009a)maxcontrollabilityconfiguration2Controlledconfiguration 1Controlled with max.controllabilityconfiguration 2Controlled with optimum GA placementconfiguration 3

Fig. 15. Zoom of the applied voltage to the shell using configuration 1, configuration 2 (given by Roy and Chakraborty, [8] and the present formulation) and configuration 3.

S.L. Schulz et al. / Computers and Structures 128 (2013) 101–115 113

114 S.L. Schulz et al. / Computers and Structures 128 (2013) 101–115

5.3. Piezoelectric patch optimal placement on a simply supportedpiezoelectric composite shell

This example is related to the optimum placement of the sameeight pairs of piezoelectric patches (similar to the previous exam-ple) in a simply supported laminated composite spherical shell.The laminated shell has a length of a = 0.04 m and the radius isR = 0.12 m. The stacking sequence is p/[0/90]s/p. Each ply has0.75 mm and the piezoelectric patches have 0.5 mm, resulting ina total thickness equal to 2.5 mm. A regular mesh of 200 triangularfinite elements was generated and the positions of the patches aredefined by a mesh of 100 positions as indicated by Fig. 11(a).

A vertical impulsive load of 10 N is applied in the centre of theplate during 4.6572 ls in order to generate an initial vibration per-turbation. It will be used eight vibration modes to control vibra-tion, which have the following damping ratios: f1 = 0.009,f2 = 0.0049, fi ¼ 0:002; i ¼ 3;8. The material properties for thelaminate and PZT are defined in Table 1. Eight pairs of collocatedpatches have been considered and their optimum position issought in order to control the vibrations. The same heuristicparameters of the previous example were used in this case.

Roy and Chakraborty [8] also studied this example. In their pa-per, the optimal allocation is first solved, based on the maximumcontrollability index (that is function of B matrix relating controlforces and input voltages), and then, starting from this optimumposition, a GA was used to tune the Q and R matrices in order tominimize the functional J .They also investigated the patch alloca-tion indicated in Fig. 11(b) (Configuration 1), based on vibrationmode shapes and Fig. 11(c) (Configuration 2), based on maximumcontrollability index (with only six pairs of collocated patches).

In the present paper, the configuration indicated by Fig. 11(d)(configuration 3), with patches allocated at the centre of the shell,was found by the optimization with GA. Configuration 3 will beconfronted with the two other configurations.

Fig. 12 shows time histories for the vertical displacement at thecentre of the shell without control, indicated by Roy and Chakr-aborty [8] and results applying the formulation of the presentwork. It can be noticed a good agreement between these twocurves. In the same figure, displacements time histories at the cen-tre of the shell in the case of patch allocation of configuration 1,configuration 2 and configuration 3 (which was obtained by thispaper using a GA optimization) are shown. The magnitude of thedisplacements is slightly lower for the configuration 3 than thoseobtained in configuration 1 and configuration 2. It is confirmedin Fig. 13 (displacement PSD attenuation) showing, for the firstvibration mode, an attenuation of�35 dB for configuration 3, whilefor the configuration 1 and configuration 2, attenuations are�30 dB and �25 dB, respectively.

Fig. 14 shows the applied voltages obtained by Roy and Chakr-aborty [8] using maximum controllability index (configuration 2given by the paper and with the present formulation) and configu-ration 1 and 3. The results for configuration 1 and configuration 3are very similar, but configuration 3 attenuates faster when com-pared with configuration 1. A zoom of the applied voltage is shownin Fig. 15. It can be observed a peak of 350 V for configuration 1 andconfiguration 2 during the time where the load is acting, while thepeak value for configuration 3 is about 100 V. This behaviour isconfirmed for the values of the objective function Jm are evaluated.

Although the previous plots are based in the central deflectionof the shell, the overall behaviour is confirmed. The J (Lyapunov)value of the controlled shell with 8 patches in the central position(configuration 3) is lower than the J value of the controlled shellwith patches in configuration 2 and configuration 1. Nevertheless,it should be emphasized that this optimal allocation resembles theboundary conditions, applied loads (the way and sites where theyare applied), damping ratio for each mode and limited number of

piezoelectric patches which reflects the example presented byRoy and Chakraborty [8]. For other cases, new analysis must bedone.

6. Conclusions and final remarks

This paper addressed the problem of optimum piezoelectricpatches allocation in composite beams, plates and shells. A trian-gular GPL-T9 finite element was used and the main steps relatedto the addition of piezoelectric patches using this type of finite ele-ment were described. A brief review of modal control theory forlinear quadratic regulator was presented focusing on the necessarymodifications in order to incorporate the piezoelectric patches.

The used GA as optimization tool in the patch allocation taskwas briefly explained as well as the cost function used in orderto differentiate the suitability between different patch positions.

As a first example, a benchmark problem, the bimorph beam,was used to validate the formulation and implementation. It wasoptimized the positioning of two PVDF piezoelectric patches in or-der to attenuate vibrations and applied voltages in the beam. Asexpected, positions near the clamped end were found as the bestones for the control based on Lyapunov values.

Then, a more elaborated example regarding the optimum allo-cation of 8 pairs of collocated piezoelectric patches on a simplysupported composite plate was investigated. It was used only thefirst four vibration modes. The GA optimization resulted in piezo-electric patches positioned around the centre of the plate, showinga configuration other than that reported by other authors (Roy andChakraborty [9]) using the controllability index to indicate the bestpositions and only six patches as actuators, acting in only one face.The configuration indicated by these authors was investigated andcompared with that obtained in this work. The comparison indi-cates that the solution obtained by this paper presents a betterbehaviour.

Finally, a similar problem, related to the optimum allocation of8 pairs of collocated piezoelectric patches on a simply supportedshell was addressed. Similarly, the GA optimization resulted inthe best position being around the centre of the shell. This config-uration also differs from that stated by other authors (Roy and Cha-kraborty [8]) based on the controllability index. Again, the solutionobtained by this paper was confronted with that reported by otherauthors, showing that the first one presented a better behaviour.

Acknowledgements

The authors wish to thank CNPq and CAPES for their financialsupport.

References

[1] Potami R. Optimal sensor/actuator placement and switching schemes forcontrol of flexible structures, Ph.D. thesis, Worcester Polytechnic Institute;2008.

[2] Araujo AL, Madeira JFA, Mota Soares CM, Mota Soares CA. Optimal design foractive damping in sandwich structures using the direct multisearch method.Compos Struct 2013. doi: 10.1016/j.compstruct.2013.04.044..

[3] Bachmann F, Bergamini AE, Ermanni P. Optimum piezoelectric patchpositioning: a strain energy-based finite element approach. J Intell MaterSyst Struct 2012;23(14):1575–91. http://dx.doi.org/10.1177/1045389X12447985.

[4] Belloli A, Ermanni P. Optimum placement of piezoelectric ceramic modules forvibration suppression of highly constrained structures. Smart Materials Struct2007;16:1662–71. http://dx.doi.org/10.1088/0964-1726/16/5/019.

[5] Godoy T, Trindade MA. Modeling and analysis of laminate composite plateswith embedded active-passive piezoelectric networks. In Proceedings of ISMAinternational conference on noise and vibration engineering including USD2010;5:335-49.

[6] Ning HH. Optimal number and placements of piezoelectric patch actuators instructural active vibration control. Eng Comput 2004;21(6):651–65. http://dx.doi.org/10.1108/02644400410545218.

S.L. Schulz et al. / Computers and Structures 128 (2013) 101–115 115

[7] Ang KK, Wang SY, Quek ST. Weighted energy linear quadratic regulator vibrationcontrol of piezoelectric composite plates. Smart Materials Struct2002;11:98–106. stacks.iop.org/SMS/11/98. doi: 10.1088/0964-1726/11/1/311.

[8] Roy R, Chakraborty D. Optimal vibration control of smart fiber reinforcedcomposite shell structures using improved genetic algorithm. J Sound Vib2009;319:15–40. http://dx.doi.org/10.1016/j.jsv.2008.05.037.

[9] Roy R, Chakraborty D. GA-LQR based optimal vibration control of smart FRPcomposite structures with bonded PZT patches. J Reinf Plast Compos2009;28(11):1383–404. http://dx.doi.org/10.1177/0731684408089506.

[10] Satpathy A. Optimal placement of actuators in fiber reinforced polymercomposite shell structures using genetic algorithm. Final work for the Bachelorof Technology Degree in Mechanical Engineering at National Institute ofTechnology, Rourkela, India: Deemed University; 2009.

[11] Chakraborty D, Roy T. Control of smart fiber reinforced polymer sphericalshells. World J Eng 2010;7(3):251–2.

[12] Wang Q, Wang CM. Optimal placement and size of piezoelectric patches onbeams from the controllability perspective. Smart Materials Struct2000;9:558–67. doi:10.1088/0964-1726/9/4/320..

[13] Wang Q, Wang CM. A controllability index for optimal design of piezoelectricactuators in vibration control of beam structures. J Sound Vib2001;242(3):507–18. http://dx.doi.org/10.1006/jsvi.2000.3357.

[14] Isoldi LA, Awruch AM, Morsch IB, Teixeira PRF. Geometrically nonlinear staticand dynamic analysis of composite laminates shells with triangular finiteelement. J Braz Soc Mech Sci Eng 2008;30(1):83–92. http://dx.doi.org/10.1590/S1678-58782008000100012.

[15] Jones RM. Mechanics of composite materials. 2nd ed. New York: Taylor &Francis; 1999.

[16] Leo DJ. Engineering analysis of smart material systems. New York: John Wiley& Sons; 2007.

[17] Preumont A, Seto K. Active control of structures. Chichester: John Wiley &Sons; 2008.

[18] Ogata K. Modern control engineering. 5th ed. Prentice Hall; 2009. p. 905.[19] Holland JH. Adaptation in natural and artificial systems. Ann Arbor,

Michigan: University of Michigan Press; 1975.[20] Goldberg DE. Genetic algorithms in search optimization and machine

learning. Addison-Wesley Publishing Company Inc.; 1989.[21] Sze KY, Yao LQ. Modelling smart structures with segmented piezoelectric

sensors and actuators. J Sound Vib 2000;235(3):495–520. http://dx.doi.org/10.1006/jsvi.2000.2944.