active control of nonlinear vibration of sandwich piezoelectric beams: a simplified approach

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Computers and Structures 86 (2008) 386–397

Active control of nonlinear vibration of sandwich piezoelectric beams:A simplified approach

S. Belouettar a,*, L. Azrar b, E.M. Daya c, V. Laptev a, M. Potier-Ferry c

a CRP Henri Tudor, LTI, Laboratoire de Technologies Industrielles, 70, Rue de Luxembourg, L-4221 Esch-sur-alzette, Luxembourgb Equipe de Modelisation Mathematique et Controle, Departement de Mathematiques, Faculte des Sciences et Techniques de Tanger,

BP 46, Universite Abdelmalek Essaadi, Tanger, Moroccoc LPMM, UMR CNRS 7554, Universite Paul Verlaine, Ile du Saulcy, F-57045 Metz, France

Available online 29 March 2007

Abstract

Nonlinear vibrations of piezoelectric/elastic/piezoelectric sandwich beams submitted to active control are studied in this paper. Theproportional and derivative potential feedback controls via sensor and actuator layers are used. Harmonic balance method and theGalerkin procedure are adopted. A complex amplitude equation governed by two complex parameters is derived accounting for the geo-metric nonlinearity and piezoelectric effects. The nonlinear frequency and loss-factor amplitude relationships with respect to the gainparameters are obtained. The feedback effects are analyzed for small and large vibration amplitudes of sandwich beams. The frequencyresponse curves are presented and discussed for various gain parameters.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Piezoelectric; Sandwich; Nonlinear; Sensor; Actuator; Active control; Vibrations; Loss-factor

1. Introduction

The lightweight and flexible structures are extensivelyused in aerospace engineering, civil and mechanical engi-neering. Because of the flexibility, the vibrations onceintroduced in the structure can grow up to large ampli-tudes. The effects of geometrically nonlinear deformationbecome prominent in the research of mechanical behaviorof structures and it is also hopeful to suppress those unde-sired vibrations. This led to extensive research in active andpassive vibration controls, more at small than at largeamplitudes. One approach to control or to suppress theundesired vibrations is to employ a control system with pie-zoelectric sensors and actuators. The early application ofpiezoelectric materials to control vibration was developedby Olson [1]. At present, this control meaner has been lar-gely applied in other engineering fields such as robots,rotor systems, antennas, high precision systems, medical

0045-7949/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruc.2007.02.009

* Corresponding author. Tel.: +352 545580 530; fax: +352 545580 501.E-mail address: [email protected] (S. Belouettar).

applications, to name only a few. Piezomaterials can bevery well integrated into lightweight structures and effi-ciently transform mechanical energy into electrical energyand vice versa. Owing to their good characteristics of light-weight and electromechanical coupling effects, piezoelectricsensors and actuators are often embedded in or attached onthe controlled structures. These structures are members ofa subset of controlled structures known as active or adap-tive structures.

In recent years, modeling and control have been per-formed on a variety of active structures including beamsand plates using analytical and numerical methods. Severalinvestigations have been devoted to get accurate modelingand analysis for vibration control and vibration suppres-sion of sandwich structures with piezoelectric materials.The vastness of the literature is obvious considering theinterdisciplinary nature of the subject [2–8]. Detailed sur-vey can be found in the paper of Benjeddou et al. [9,10].All these researchers, however, used linear models to inves-tigate the mechanical behaviors of piezocomposite orpiezolaminated structures. Thus, the possible nonlinear

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S. Belouettar et al. / Computers and Structures 86 (2008) 386–397 387

behaviors of these structures, when subjected to higherloads, are then missed. In fact, there are many nonlinearbehaviors that can appear in piezolaminated structures.In this paper, we are only interested in the geometricallynonlinear effects. Some researchers had afforded their bestto model and investigate the dynamic behaviors of beamsand plates with piezoelectric laminates. Moita et al. [11]used the updated Lagrangian formulation associated toNewton–Raphson technique. Gao and Shen [12] proposedan approach based on total Lagrange technique and virtualvelocity incremental principles. In these previous studies,finite element formulations have been proposed for thenonlinear transient vibration of composite structures withpiezoelectric materials.

It is well known that at large free and forced vibrations,the nonlinear frequency is highly amplitude dependent.Many researchers have analyzed the backbone and fre-quency amplitude dependences for beams, plates andshells. Based on harmonic balance method, free and forcednonlinear vibrations of beams and plates have been inves-tigated by Azrar et al. [13–15] using analytical and finiteelement methods. When a viscoelastic material damps thestructure, the loss-factor is also amplitude dependent.Sandwich beams and plates with a viscoelastic core havebeen investigated by Daya et al. [16,17]. Simplified ampli-tude frequency and amplitude loss-factor relationshipsare formulated for beams and plates. These papers pre-sented a mathematical modeling to passively damp beamsand plates and to easily analyze their behaviors at largeamplitudes. Based on a linear approximation of the electricpotential through the thickness, a modal approach estimat-ing the damping properties of sandwich beams has beenpresented by Duigou et al. [18,19].

The aim of this work is to develop a simplified and con-sistent theory to actively control sandwich beams at smalland large amplitudes. A simplified methodology is devel-oped for nonlinear vibration analysis of a piezoelectric–elastic–piezoelectric sandwich beams. The piezoelectric lay-ers playing roles of sensor and actuator respectively areconnected via direct proportional feedback control andvelocity feedback control law. Harmonic balance methodwith one mode Galerkin’s procedure is used. The frequencyresponse functions with respect to the gain and amplitudeparameters are obtained. A relationship between nonlinearamplitude, loss-factor and gain parameters is also formu-lated. The nonlinear vibration behaviors of piezoelectricsandwich beams can be investigated and optimal gainparameters may be obtained.

2. Mathematical modeling

The basis of our consideration here is to derive a simpleand consistent model combining the geometrical nonlineareffects and the feedback control of a piezoelectric/elastic/piezoelectric sandwich beam. The nonlinear effect pro-duced by the large transverse vibration amplitude ismodeled by a nonlinear strain–displacement relationship

of von karman type. The proportional and derivativepotential feedback controls via sensor and actuator layersare used. The resulting nonlinear dynamic equation is ana-lyzed by harmonic balance method. A complex amplitudeequation governed by two complex constants is derived.Due to its analytical nature, this kind of amplitude equa-tion is of particular value. The nonlinear frequency–ampli-tude relationships are obtained allowing the investigationof the nonlinear response of beams with various boundaryconditions. The feedback effects on the nonlinear frequencyand loss-factor can be analyzed at small and large vibrationamplitudes.

2.1. Kinematics of the model

Let us consider a slender sandwich beam with rectangu-lar cross section consisting of an elastic layer sandwichedbetween two piezoelectric layers as presented in Fig. 1.The upper and the lower layers play roles of sensor andactuator respectively and are connected via some feedbackcontrol laws. As a symmetric sandwich is of practical rele-vance a symmetric beam with respect to z = 0 is consid-ered. The thickness of the piezoelectric layers is hS ¼ hA

and the thickness of the elastic core layer is hc. The lengthof the beam is L while the width in the y direction is H. Thekinematics properties of the considered slender sandwichpiezoelectric–elastic–piezoelectric beam are described bythe classical laminate theory based on the Bernoullihypothesis

uðx; z; tÞ ¼ uðx; tÞ � zw;xðx; tÞvðx; z; tÞ � 0

wðx; z; tÞ ¼ wðx; tÞ

8><>: ð1Þ

According to the framework of small strain and finitedeflection, the geometrical nonlinearity effect is introducedby assuming moderate rotations [14,15,17]. The nonlinearstrain–displacement relationship is

e ¼ e0 � zw;xx; e0 ¼ u;x þ1

2w2;x ð2Þ

It is well known that in piezoelectric materials the elec-tric field and the deformation influence each other. Sucha property allows using the piezoelectric materials assensors and actuators for the vibration control. Moreprecisely, the latter relationship can be described by thefollowing constitutive relations, which characterize thecoupling effects between mechanical and electrical proper-ties as follows:

r ¼ ce� etE

D ¼ eeþ �E

�ð3Þ

where r; e; D and E are the stress tensor, strain tensor,electric displacement vector and electric field vector respec-tively. c, e and � are respectively the elasticity matrix,the piezoelectric matrix and the dielectric permittivity ten-sor. Orthotropic piezoelectric materials and an extension

hc

Piezoelectric

Elastic layer

Piezoelectric

Ah

Sh

z

x

H

0

Sz

Az

Sϕ

Aϕ

0=ϕ

HhS ss = HhS cc = HhS AA = As hh = SA zz −=

Fig. 1. Piezoelectric–elastic–piezoelectric sandwich beam.

388 S. Belouettar et al. / Computers and Structures 86 (2008) 386–397

mechanism are considered. Assuming that the stress tensoris uniaxial and the directions of the fields D and E are par-allel to oz, the reduced constitutive relations can be ex-pressed as [8]

r1

D3

� �¼

c�11 �e�31

e�31 ��33

� �e

E3

� �

��33 ¼ �33 þe2

33

c33; e�31 ¼ e31 � c13

c33e33; c�11 ¼ c11 �

c213

c33

8>><>>: ð4Þ

2.2. Feedback control law

Consider a piezoelectric layer (sensor or actuator)placed between z�, zþðz� < zþÞ, with the centerzC ¼ ðzþ þ z�Þ=2 and the thickness h. The piezoelectric sen-sor and actuator layers and the elastic beam are assumed tobe perfectly bounded. Therefore, the validity of our resultsdepends on this assumption. The electrostatic equilibriumequation, free of volume charge density oD3

oz ¼ 0, togetherwith the boundary condition (D3ðz�Þ ¼ 0 or D3ðzþÞ ¼ 0Þimply that D3ðzÞ � 0. Hence, the electric field in the sensor,depending on the displacement, is given by

E3ðzÞ ¼ �e�31

��33

e ¼ � e�31

��33

ðu;x þ1

2w2;x � zw;xxÞ ð5Þ

In order to deal with a potential E3 ¼ � ouoz , the difference of

the potentials on this piezoelectric layer can be written as

Du ¼ uðzþÞ � uðz�Þ ¼ �Z zþ

z�

E3ðzÞdz

¼ e�31hi

��33

ðu;x þ1

2w2;x � ziw;xxÞ; i ¼ S;A ð6Þ

To deal with an actuator, the quantity we are likely to con-trol is then the difference of the potentials. Therefore, basedon (5) E3(z) can be obtained from Du

E3ðzÞ ¼ �Duhiþ e�31

��33

ðz� ziÞw;xx; i ¼ S;A: ð7Þ

The core of the sandwich beam is assumed to be conductivewith a uniform potential fixed to zero. The potentials onthe upper and lower faces of the beam are respectivelydenoted uSðxÞ and uAðxÞ. Hence, the sensor’s potential isgiven by

uS ¼ DuS ¼e�31hS

��33

ðu;x þ1

2w2;x � zSw;xxÞ ð8Þ

The proportional and derivative feedback controls will beused. More precisely, the actuator’s potential uAðxÞ isassumed to depend on the sensor’s potential uSðxÞ by thefollowing control law:

uA ¼ GpuS þ Gd _uS ð9ÞThis leads to the direct and proportional feedback controlin the particular case when Gd = 0 and to the direct andproportional velocity feedback control when Gp = 0. It isthe scope of the present paper to analyze the effects of thesefeedback parameters on the dynamical behaviors of thesandwich beam.

Using Eqs. (7)–(9), the electric fields in the sensor andthe actuator are, respectively, given by [18,19]

ES3ðzÞ ¼ �

uS

hS

þ e�31

��33

ðz� zSÞw;xx;

EA3 ðzÞ ¼

uA

hA

þ e�31

��33

ðz� zAÞw;xx ð10Þ

in which uA and uS are z independent and zS ¼ ðhCþhSÞ=2 and zA ¼ �ðhC þ hAÞ=2. Let us note that the directand inverse piezoelectric coefficients have been taken intoaccount in these formulations and both of them intervenein the dynamic behavior of the sandwich beam.

2.3. Dynamic equation

Using the Hamilton’s principle and Eqs. (4) and (10),the equation of motion of the sandwich beam submittedto axial and lateral excitations FX and FZ is given by:Z

Vr1de ¼

ZV S

r1deþZ

V C

r1deþZ

V A

r1de

¼Z L

0

ðNde0 þMdw;xxÞdx

¼Z L

0

ðF X duþ F ZdwÞdx� ðqSÞ�Z L

0

ð€uduþ €wdwÞdx

ð11Þwhere ðqSÞ� ¼ qSSS þ qCSC þ qASA; de0 ¼ du;x þ w;xdw;x.

Integrating throughout the thickness and the width andassuming that the laminated layers are symmetric(hA = hS), the axial force N and the bending moment M

are given by

N ¼ ðESÞ�e0 � BNw;xx

�ðESÞpeGdð _u;x þ w;x _w;x � _w;xxzSÞ ð12aÞM ¼ �BMe0 þ ðEIÞ�w;xx

�ðESÞpezSGdð _u;x þ w;x _w;x � _w;xxzSÞ ð12bÞ

8>>><>>>:

ð12Þ


where

ðESÞ� ¼ ECSCþ 2c�11SSþðESÞpeð1�GpÞ; ðESÞpe ¼ SSðe�31Þ

2

��33

;

ð12cÞBN ¼ ðESÞpeð1�GpÞzS; BM ¼ ðESÞpeð1þGpÞzS; ð12dÞ

ðEIÞ� ¼ ECICþ 2c�11ðISþ SSz2SÞþðESÞpe

SS

ð2ISþð1þGpÞz2SSSÞ

ð12eÞ

The parameters (ES)* and (EI)* are the resulting exten-sional and bending stiffness of the piezo-sandwich beam.The rigidity terms BM and BN display coupling betweentransverse bending and axial stretching. These coupling ef-fects are introduced by the piezoelectric laminates and bythe proportional feedback. Decoupling behavior appearswhen (e31 = 0). Let us note that Eqs. (12a,b) are those ofa purely mechanical beam model of viscoelastic type. Nev-ertheless, this model is not classical because of non-conser-vative character of the control law. First, when Gd = 0 andGp 6¼ 0, the constitutive relation between (e0, w;xxÞ and(N,M) is not symmetric. Next, the damping due to thevelocity feedback parameter Gd is not of the same naturein traction and in bending: The latter property becomesobvious by considering an uncoupled and linearized ver-sion of the equivalent mechanical model

N ¼ ðESÞ�u;x � ðESÞpeGd _u;x ð13aÞM ¼ ðEIÞ�w;xx þ ESð Þpez

2SGd _w;xx ð13bÞ

(ð13Þ

For instance if Gd > 0, the control damps the bendingmodes while it destabilizes the traction modes. This spill-over phenomenon is common within control structures.This means that it is not possible to damp all the modeswith the described device. In this paper, only the controlof bending modes is studied and the destabilizing effectsin traction will be disregarded.

Applying the variational principle (11) to the displace-ments u and w, the following governing partial differentialequations are derived:

�N ;x þ ðqSÞ�€u ¼ F X ð14aÞM ;xx � ðNw;xÞ;x þ ðqSÞ�€w ¼ F z ð14bÞ

(ð14Þ

The numerical solution of the latter equations allows oneto investigate the dynamic behaviors of piezoelectric–elas-tic–piezoelectric beams under active control, lateral andaxial excitations. To deal only with lateral vibration char-acteristics of the piezo-sandwich beam, the axial forceand the axial displacement inertia may be neglected. Thesystem (14) is then reduced to

N ;x ¼ 0 ð15aÞM ;xx � Nw;xx þ ðqSÞ�€w ¼ F Z ð15bÞ

�ð15Þ

Eq. (15a) leads to a constant axial force ‘Nðx; tÞ ¼ NðtÞ’and the associated axial displacement partial differentialequation is given by

ðESÞ�u;x � ðESÞpeGd _u;x ¼ NðtÞ � ðESÞ�2

w2;x þ BNw;xx

þ ðESÞpeGdðw;x _w;x � _w;xxzSÞ ð16Þ

Without loss of generality the condition ‘uð0; tÞ ¼ 0’ is as-sumed. Integrating (16) from 0 to x, the time differentialequation associated to u(x, t) is then given by

ðESÞpeGd

ouðx; tÞot

� ðESÞ�uðx; tÞ

¼ �xNðtÞ þ 1

2ðESÞ�

Z x

0

w;sðs; tÞ2 ds� BN½w;xðx; tÞ

� w;xð0; tÞ� � ðESÞpeGd

Z x

0

w;sðs; tÞow;sðs; tÞ

otds

þ ðESÞpezsGd

ow;xðx; tÞot

� ow;xð0; tÞot

� �ð17Þ

This equation is a simple linear differential equation on u

and when its right hand side is known, it becomes easilysolved. The axial stretching force N(t) can be deduced withrespect to u and w by simply replacing x by L in (17).

When the axial displacement effects are neglected, theaxial force and bending moment can be expressed accord-ing to the transverse displacement only as

N ¼ 12ðESÞ�w2

;x � BNw;xx

�ðESÞpeGdðw;x _w;x � _w;xxzSÞ ð18aÞM ¼ � 1

2BMw2

;x þ ðEIÞ�w;xx

�ðESÞpeGdzSðw;x _w;x � _w;xxzSÞ ð18bÞ

8>>>><>>>>:

ð18Þ

Let us note that these formulations show additional termsfor the axial force and bending moment. N and M becomenot only nonlinear with respect to the transverse displace-ment w but also contain viscous terms. According to(15a) and integrating (18a) between the limits 0 and L,the nonlinear axial stretching force is given by

NðtÞ ¼ 1

2LðESÞ�

Z L

0

w2;x dx� BN

L

Z L

0

w;xx dx

�ðESÞpe

LGd

Z L

0

ðw;x _w;x � _w;xxzSÞdx ð19Þ

Remember that when the piezoelectric effects are neglected(e�31 ¼ 0Þ, the classical axial force corresponding to immov-able ends is obtained [13,14].

Based on the last simplifying assumption, the equationof motion is reduced to a nonlinear partial differentialequation on the transverse displacement only. The result-ing nonlinear equation of motion is then given by

ðqSÞ�€wþ ðEIÞ�w;xxxx �NðtÞw;xx � BMðw2;xx þw;xw;xxxÞ

� ðESÞpeGdzsð _w;xw;xxx þ 2 _w;xxw;xx þw;x _w;xxx � zs _w;xxxxÞ ¼ Fz

ð20ÞThis nonlinear partial differential equation governs thetransverse dynamic behavior of the piezoelectric sandwichbeams subjected to transverse excitation and piezoelectricpotential feedback when the axial displacement effects areneglected. The nonlinear free and forced vibration and


active vibration control of piezoelectric sandwich beamscan be analyzed by numerically solving (15) or Eq. (20).More generally, the axial effects can be investigated bysolving analytically or numerically Eq. (14).

3. Harmonic approximate solution

For the purpose of establishing simplified and usefulrelationships for active control vibration of sandwich pie-zoelectric beams, the structure is assumed to be excitedtransversally by a harmonic force F Zðx; tÞ ¼ f ðxÞ cosðxtÞ.f(x) may be concentrated, distributed or modal force. Inorder to obtain the simplest approximate formulation foractive control of nonlinear vibrations, the dependence intime and in space is specified in a very restrictive way.The harmonic balance method and one mode Galerkinapproximation in space are adopted. The deflection isassumed to be harmonic and proportional to a linear vibra-tion mode wn(x):

wðx; tÞ ¼ fAwnðxÞeixt þ CCg ¼ ðAeixt þ �Ae�ixtÞwnðxÞ; ð21Þ

where A is a complex unknown amplitude, x is the nonlin-ear frequency and CC indicates complex conjugate. wn(x) isassumed to be a real linear vibration mode of the sandwichbeam obtained by solving the linear eigenvalue problemresulting from (20) by neglecting the viscous and nonlinearterms, the excitation and the feedback terms. This modecan be analytically determined for classical boundary con-ditions or numerically by FE, for example, for more com-plex shapes and boundary conditions.

3.1. Amplitude equation

Assuming that the ends are immovable and inserting theharmonic decomposition (21) into (17), the harmonicdecomposition of the axial force is given by

NðtÞ ¼ ðESÞ�jAj2 F 2ðLÞ

L

þ �ðBN � ixðESÞpeGdzSÞAF 1ðLÞ

Leixt

�

þ 1

2ðESÞ� � ixðESÞpeGd

� �A2 F 2ðLÞ

Le2ixt þ CC

�ð22Þ

in which F 1ðxÞ ¼R x

0 wn;ssðsÞds; F 2ðxÞ ¼R x

0 ðwn;sðsÞÞ2 ds. The

axial displacement governing equation is formulated as

ðESÞpeGd

ouðx; tÞot

� ðESÞ�uðx; tÞ

¼ ðESÞ�jAj2F 2ðxÞ 1� x

L

� þ�ð�BN þ ixðESÞpeGdzSÞA 1� x

L

� F 1ðxÞeixt

þ 1


� �A2ð1� x

LÞF 2ðxÞe2ixt þ CC

�ð23Þ

Functions F1(x) and F2(x) are computed using the assumedvibration mode related the to the considered boundaryconditions. Based on the assumption (21), the general solu-tion of (20) is given by

uðx; tÞ ¼ CðxÞ expðESÞ�tðESÞpeGd

!� jAj2F 2ðxÞ 1� x

L

�

þ 1� xL

� AðBN � ixðESÞpeGdzSÞðESÞ� � ixðESÞpeGd

F 1ðxÞeixt

(

þA2�12ðESÞ� þ ixðESÞpe

ðESÞ� � 2ixðESÞpeGd

F 2ðxÞe2ixt þ CC

)ð24Þ

where C verifies the boundary conditions: Cð0Þ ¼CðLÞ ¼ 0. The first term in (24) illustrates the destabilisingeffect of the feedback parameters Gp and Gd. Because wefocus on modes with a predominant bending, we do notdiscuss this phenomenon. In practice, the feedback canbe balanced by the structural damping.

In order to derive a simplified nonlinear amplitude–fre-quency equation expressions (21), (22) and (24) are insertedinto Eq. (15b) and the harmonic balance method is used. Acomplex scalar amplitude equation, similar to the oneestablished in [16] for damped viscoelastic sandwich beam,is then obtained:

�x2MAþ KðxÞAþ KNLðxÞ�AA2 ¼ Q ð25Þ

in which

M ¼ aðqSÞ�KðxÞ ¼ bððEIÞ� þ ixz2

SðESÞpeGdÞð26aÞ

� cðBN � ixðESÞpeGdzSÞðBM þ ixðESÞpeGdzSÞ

ðESÞ� � ixðESÞpeGd

ð26bÞ

KNLðxÞ ¼ n3


� �ð26cÞ

Q ¼ F2

ð26dÞ

where

a ¼Z L

0

wnðxÞ2 dx; b ¼Z L

0

wn;xxxxðxÞwnðxÞdx;

F ¼Z L

0

f ðxÞwnðxÞdx

c ¼Z L

0

fð1� xLÞF 1ðxÞg;xxxwnðxÞdx;

n ¼ � F 2ðLÞL

Z L

0

wn;xxðxÞwnðxÞdx

ð27Þ

The constants a, b, c and n are simply determined using theassumed natural vibration mode and the constants M, K

and KNL are obtained in a straightforward manner. wnðxÞmay be any desired nth mode around which the analysisis needed. These real modes may be determined either ana-lytically or numerically for considered boundary condi-tions. Let us note that the excitation force FZ may be


concentrated, distributed uniform or modal harmonic forceby simply choosing accordingly f(x).

When the axial displacement effects are neglected, theresulting amplitude equation is similar to (25) with thesame coefficients except for the linear rigidity factor thatis reduced to

KðxÞ ¼ bfðEIÞ� þ ixz2SðESÞpeGdg ð28Þ

Based on the used harmonic balance method, the axialeffect appears only in the rhs of (26b). In order to take intoaccount the axial effects and the quadratic nonlinearitiesintroduced by the piezoelectric feedback control, harmonicbalance method with more harmonics may be used.

The amplitude Eq. (25) allows obtaining the complexvibration amplitude A as a response to the harmonic excita-tion force with the amplitude FZ and frequency x. By thisway, the control of free and forced nonlinear vibration anal-yses can be deduced from three modal constants M, K andKNL which represent mass, linear and nonlinear stiffnessrespectively. The later constant accounts for the couplingof piezoelectric and nonlinear geometrical effects. The pre-sented formulation is quite general and allows one to ana-lyze the proportional and derivative potential control ofpiezo-sandwich beams at small and large amplitudes withvarious boundary conditions. Only the transverse vibrationmode is needed and the vibration control in the vicinity ofthe associated resonance can be easily achieved.

3.2. Gain effects

The proportional and velocity gain effects can be ana-lyzed by considering the free amplitude equation

x2 ¼ KðxÞMþ KNLðxÞ

MjAj2 ð29Þ

Splitting the coefficients K(x) and KNL (x) into its real andimaginary parts, denoted by KRðxÞ þ iKIðxÞ andKR

NLðxÞ þ iKINLðxÞ, the complex nonlinear frequency x

can decomposed as

x2 ¼ X2NLð1þ igNLÞ ð30Þ

where the nonlinear real frequency XNL and loss-factor gNL

amplitude relationships are given by:

X2NL ¼ X2

L0 1þ KR1

MX2L0

þ a2

MX2L0

ðKRNL0 � GpKR

NL1Þ( )

ð31aÞ

gNL ¼ gL0

X2L0

X2NL

1þ KI1

KI0

� a2 KINL

KI0

� �ð31bÞ

in which a ¼ jAj, X2L0 ¼

KR0

M and gL0 ¼KI

0

KR0

.

The coefficients KR0 , KR

1 ðx;Gp;GdÞ, KI1ðx;Gp;GdÞ, KR

NL0,KR

NL1 and KINL are given in the Appendix A. In order to

demonstrate the feedback effects, the normalization withrespect to the sandwich beam without proportional effectis done. The used linear frequency XL0 and linear loss fac-tor gL0 correspond to ‘Gp = 0 and c = 0’.

When the axial effects are neglected, Eqs. (31) arereduced to

X2NL ¼ X2

L0 1þ a2CR þ G�p bz2S �

3

2na2

� �� ð32aÞ

gNL ¼ gL0

1� a2CI

1þ a2CR þ G�pðbz2S � 3

2na2Þ

ð32bÞ

in which

ðESÞ0 ¼ ECSC þ 2c�11SS þ ðESÞpe;

ðEIÞ0 ¼ ECIC þ 2c�11ðIS þ SSz2SÞ þ

ðESÞpe

SS

ð2IS þ z2SSSÞ

X2L0 ¼

baðEIÞ0ðqSÞeq

; CR ¼ 3

2

nbðESÞ0ðEIÞ0

CI ¼ nbz2

S

; ð33Þ

G�p ¼ Gp

ðESÞpe

MX2L0

; gL0 ¼ Gd

xz2SðESÞpe

ðEIÞ0Eqs. (32) and more generally Eq. (31) allow one to investi-gate the influence of the proportional and velocity feedbackparameters on the free nonlinear frequency and loss-factor.As expected, the loss factor is proportional to velocity feed-back parameter Gd and this later does not intervene in thenonlinear real frequency XNL.

Based on Eq. (31a) or (31b), the proportional feedbackparameter Gp can decrease the nonlinear frequency andthen increase the nonlinear loss-factor values with respectto the amplitude. This allows one to easily control theincrease or decrease of the damping in the structure atsmall and large amplitudes. Using Eq. (32), critical valuesof Gp can be reached at:

GCritp ¼ 1þ a2CR

32na2 � bz2

S

KR0

ðESÞpe

; GCrit Lp ¼ �1

bz2S

KR0

ðESÞpe

ð34Þ

where GCrit Lp is a particular critical value of Gp correspond-

ing to linear case. It can be also seen that at the amplitude

‘a ¼ zS

ffiffiffiffi2b3n

q’, the effects of the proportional feedback Gp on

the nonlinear frequency and the loss-factor disappear. Thisnew phenomenon is of particular interest because it intro-duces an effective change in the nonlinear behavior of thesandwich beam. A softening or a hardening effect can beobtained by simply choosing Gp.

4. Numerical results

Numerical results are obtained for simply supportedsandwich piezoelectric–elastic-piezoelectric beams as pre-sented in Fig. 1. The material and geometrical propertiesof the host beam and of the piezoelectric layers are givenin Table 1. The amplitude Eqs. (25) and (26) and the ana-lytical relationships formulated in (31) and (32) are used forthe control of the linear and nonlinear vibration behaviorsof sandwich beams. Only numerical computation of threecoefficients is needed and a large number of benchmarktests can be investigated. The proportional and derivativefeedback potential control is adopted and the active

Table 1Geometrical and material properties of the elastic beam and of thepiezoelectric layer

Elastic beam Piezoelectric layer

Length: L 1 m 1 mTotal thickness: h ¼ 0:01 m hC ¼ 5

6 h hS ¼ hA ¼ 112 h

Width H H ¼ 5 h H ¼ 5 hYoung’s modulus EC ¼ 6:9 � 1010 Pa –Mass density qC ¼ 2766 kg

m3 qS ¼ 7500 kgm3

c�11 – 6.98 · 1010 Pa

e�31 – �23:2 Cm2

��33 – 1:73� 10�8 Fm


control of linear and nonlinear vibration behaviors of thebeams can be easily achieved by the presented analyticalrelationships. The influence of the proportional and deriv-ative feedback effects upon the linear and nonlinear fre-quencies and loss-factors at small and large amplitudesare studied in detail for piezo sandwich beams.

The emphasis is first on the case when the axial effectsare neglected (c = 0). The normalization with respect tothe linear natural frequency X2

L0 ¼baðEIÞ0ðqSÞeq

, corresponding

to the uncontrolled sandwich beam is used in order to dem-onstrate the feedback parameter influences on the linearand nonlinear frequency amplitude responses. The effectof these parameters is first analyzed at small amplitudes(linear behavior). The derivative feedback parameter Gd

is proportional to the linear loss-factor and then governsthe damping effect actively introduced into the beam bythis kind of control. In order to show clearly the effect ofGp, the internal frequency is considered to be the natural

0 0.5 1 1.5 2 2.0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ω/ω

W(c

entr

e)/h

Linear case: f =200; G

Gp=0

Fig. 2. Forced linear frequency amplitude response around xL0 ¼ baðEI

ðqSÞ

frequency of the uncontrolled beam XL0 The proportionalparameter Gp intervenes in the linear frequency and also inthe linear loss-factor. Fig. 2 presents the deflection at thecentre of a simply supported sandwich beam with respectto the excitation frequency. These results are obtained forGp values varying from 0 to 100 and for an excitationamplitude f = 200 and Gd = 0.05. The effect of the para-meter Gp influences the bandwidth but leaves the amplitudepeak unchanged. As the frequency is normalized to the nat-ural frequency of the uncontrolled sandwich beam, theinfluence of Gp on the linear frequency of the controlledbeam is also shown in this figure. The effect of Gp on thebandwidth is clearly demonstrated in Fig. 3. The frequencyis normalized to the natural frequency of the controlledbeam. Some positive and negative values of Gp are selectedin order to clearly demonstrate its effect. As the more thefrequency range, the more the damping, the Gp can be usedto actively increase the damping in the sandwich beam asshown in Fig. 3. When the internal frequency is consideredto be the natural frequency of the controlled beamX2

L ¼ baðEIÞ�ðqSÞeq

and the axial effect is taken into account, the

influence of Gp is no longer as clear as in Figs. 2 and 3.In order t to consider the axial effect, Eq. (26c) is also used.In this case, (c 6¼ 0Þ, the influence of the control parametersGp and Gd can not be clearly identified. The linear responseof sandwich beam is presented in Fig. 4 for some values ofGp with (c 6¼ 0Þ, and without (c ¼ 0Þ, axial effect. The axialeffect is very small when Gp = 0 but can be strong related toGp and Gd as presented in this figure. Let us note that whenGd = 0, these is no damping effect induced by Gp. But,when a small Gd is introduced, Gp can be easily used to

5 3 3.5 4 4.5 5L0

d=0.05; Gp=0, ..., 100

Gp=100

Þ0eq

for various values of Gp. f = 200; Gd = 0.05 and Gp = 0, 10–100.

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ω/ωL

w(c

entr

e)/h

f=200; Gd=0.05; Gp=0, -2, -4, -6 ,-8, -10, 10, 100

Gp = 100

-2

0

10

Gp = 100

-6

-4

-8

-10

-10

-8

Fig. 3. Forced linear normalized frequency amplitude response around xL ¼ baðEIÞ�ðqSÞeq

for some positive and negative values of Gp. f = 200; Gd = 0.05 andGp ¼ 0, �2, �4, �6, �8, �10, 10, 100.

0 0 .5 1 1 .5 2 2 .50

0 .5

1

1 .5

2

2 .5

3

3 .5

4

4 .5

5

5 .5

ω /ωL

W(c

entr

e)/h

f = 200; Gd = 0.05; Gp = 0, 5, -10, 10

Gp= -10

1 0

Gp= 0

- 5

γ=0

γ # 0

Fig. 4. Forced linear frequency amplitude response around xL ¼ baðEIÞ�ðqSÞeq

at various values of Gp with axial effect (—– c 6¼ 0) and without axial effect.(––– c ¼ 0).


increase or decrease the active damping in the sandwichbeam.

The linear and nonlinear response of the controlledbeam at the centre is presented in Fig. 5 for Gp ¼ 0 andf = 0, 200 and 400. The damping Gd effect is clearly shownin linear and nonlinear behaviors. The difference betweenlinear and nonlinear behaviors is strong and the controlledbeam can be largely damped with a small parameter Gd.The vibration suppression of the beam at large amplitudes

can then be easily achieved by small derivative feedbackparameters Gd and the response is governed by a simpleamplitude Eq. (25). The nonlinear response is presentedin Fig. 6 for f = 700, Gp = 0 and Gd varying from 0.1 to0.5. The nonlinear backbone and forced vibration curvesof the uncontrolled beam are added for comparison. Theincrease of Gd tends to largely reduce the nonlinear effectand then to remove the snap through phenomenon. Withthe used harmonic balance method and disregarding the

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Linear

Nonlinear

f=0

f=400

f=200

Fig. 5. Linear and nonlinear forced frequency amplitude response around xL for various amplitudes of the excitation forces. (Gp ¼ 0, Gd ¼ 0:1Þ; f = 0,200, 400. Axial effect neglected (c = 0).

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

ω /ωL

W(c

entr

e)/h

f=700, Gp= 0; Gd=0, 0.05 0.1 0.2 0.4 0.5

Gd=0.2

Gd=0.1

Gd=0.05

Gd=0

Gd=0.4

Gd=0.5

f = 0

Fig. 6. Nonlinear frequency amplitude responses around xL for various velocity feedback parameters. f ¼ 700, Gp ¼ 0 and Gd ¼ 0; 0:05; 0:1; 0:2; 0:4; 0:5.Axial effect neglected (c ¼ 0).


axial effect, a loop effect is also observed in Fig. 6 for rela-tively large Gd.

The proportional feedback parameter has a strong effecton the nonlinear beam response. This effect is modeled bythe nonlinear free frequency–amplitude relationship. Thenonlinear free frequency XNL with respect to the amplitudeat the centre of the controlled beam is presented in Fig. 7when the axial effect is disregarded (c = 0). In this figure,the nonlinear frequency is normalized to the natural fre-quency of uncontrolled beam. The backbone curve corre-

sponding to Gp ¼ Gd ¼ f ¼ 0 is also presented forcomparison. The Gp effect on the linear and nonlinear fre-quency is clearly shown in this figure. The nonlinear behav-ior changes from the hard to the soft with respect to Gp.Invariant amplitude with respect to Gp is clearly shownand allows one to catch this new phenomenon. By intro-ducing a small Gd (Gd ¼ 0:1), the Gp effect on the nonlinearcontrolled beam is demonstrated in Fig. 8. In this figure,the nonlinear frequency is normalized to the controllednatural beam frequency. The hardening–softening changes

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

ω/ωL0

W(c

entr

e)/h

Gp=10Gp=20

Gp=0

Gp=0

Gp=60

Gp=30Gp=50

Gp=40

Gp=100

Fig. 7. Nonlinear free frequency XNL

XL0around xL0 with respect to the amplitude at various values of Gp. Gp ¼ 0–100 and without axial effect (c ¼ 0).

0 0.5 1 1 .5 2 2 .5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ω /ωL

W(c

entr

e)/h

f=400; G d=0.1; Gp=0, 10 20 30 40 50

Gp=0

50

30 20 Gp=1040

Fig. 8. Free and forced nonlinear frequency amplitude responses around xL at various values of Gp. f ¼ 400, Gd ¼ 0:1 and Gp ¼ 0; 10; . . . ; 50. Axial effectneglected (c = 0).


with respect to Gp are also obtained for damped structures.The snap through can be reduced or even suppressed butthe amplitude peak cannot be really reduced with Gp andparticularly when the axial effect is disregarded.

It has to be noted that terms coming from the axialcontribution are influencing the nonlinear frequency, loss-factor and then linear and nonlinear responses as demon-strated in Fig. 9. The responses at the centre of the beam

with respect to the normalized nonlinear frequency withand without axial effect are presented in this figure for Gd

0.1 and 0.2. It appears that for a more realistic modeling,the nonlinear effect can not be neglected and the nonlinearbehavior can be actively controlled by simply acting on Gp

and Gd. Following Eqs. (26), the contribution of the axialeffect appears in real and complex linear rigidities with cou-pling between Gp and Gd effects. Many numerical tests can

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

ω/ωL

W(c

entr

e)/h

f =400; Gp=0; Gd=0.1 , 0.2

LinearNonlinear

γ#0γ=0γ # 0

γ = 0

Fig. 9. Nonlinear frequency amplitude responses with and without axial effect. f ¼ 400, Gp ¼ 0 and Gd ¼ 0:1; 0.2 with axial effect (—- c 6¼ 0Þ and withoutaxial effect (––– c ¼ 0).


be investigated based on (25) and (26) for various beamboundary conditions and geometrical characteristics.

5. Conclusions

The active control of the linear and nonlinear vibrationsof sandwich piezoelectric–elastic-piezoelectric beams hasbeen investigated based on a proportional and derivativefeedback potential control and on a complex nonlinearamplitude equation. Similar to classical bifurcation equa-tion, this equation involves three modal constants: themass, the linear and nonlinear stiffness coefficients. Thefeedback and piezoelectric effects are taken into accountin the stiffness coefficients. These coefficients can be easilyobtained for various boundary conditions of sandwichbeams. Analytical relationships of the nonlinear fre-quency–amplitude and nonlinear loss-factor-amplitudeare given. The influence of the feedback parameters onthe linear and nonlinear vibrations of sandwich beams isdeeply analyzed. Linear and nonlinear dynamic behaviorsof sandwich beams with actuator and sensor piezoelectriclayers can be actively controlled by simply acting on twocontrol parameters.

Acknowledgements

This work has been supported by the European FP6STREP project CASSEM with the Contract No. 013517.Professor Azrar L. acknowledges the mobility supportfrom Luxembourgish National Research Fund (GrantNo. FNR/05/MA6/06).

Appendix A

ðESÞ0 ¼ ECSC þ 2c�11SS þ ðESÞpe;

ðEIÞ0 ¼ ECIC þ 2c�11ðIS þ SSz2SÞ þ

ðESÞpe

SS

ð2IS þ z2SSSÞ

The real and imaginary parts of the linear and nonlinearstiffness parameters are given by

KR0 ¼ bðEIÞ0; KI

0 ¼ xbz2SðESÞpeGd

KR1 ðx;Gp;GdÞ¼ ReðKðxÞÞ � KR

0

¼ bðESÞpez2SGp þ

�c

ðESÞ2� þ x2ðESÞ2peG2d

fBNBMðESÞ�

þ x2ðESÞ�ðESÞ2peG2dz2

S þ x2ðESÞ2peG2dzSðBM � BNÞg

KI1ðx;Gp;GdÞ¼ ImðKðxÞ � iKI

0Þ

¼ Gd

�cxðESÞpe

ðESÞ2� þ x2ðESÞ2peG2d

fBNBM þ x2ðESÞ2pez2SG2

d

þ ðESÞ�zSðBN � BMÞgKR

NL ¼ KRNL0 � GpKR

NL1

KRNL0 ¼

3

2nðESÞ0; KR

NL1 ¼3

2nðESÞpe

KINL ¼ xnðESÞpeGd

CR ¼ KRNL0

MX2L0

; CI ¼ KINL

KI0


Appendix B. Simply supported beam

The mode shape is wnðxÞ ¼ sinðnpxL Þ

For the first mode, one gets

a ¼ L2

; b ¼ 1

2

p4

L3; c ¼ 1

4

p4

L3; n ¼ 1

4

p4

L3

For distributed harmonic uniform force F = 2Lp f .

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active control of nonlinear vibration of sandwich piezoelectric beams: a simplified approach

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