PIEZOELECTRIC materials are widely used for structural

Transcription

1 Parametri Analysis o the Vibration Control o Sandwih Beams Through Shear-Based Piezoeletri Atuation M. A. TRINDADE,* A. BENJEDDOU AND R. OHAYON Strutural Mehanis and Coupled Systems Laboratory, Conservatoire National des Arts et Métiers,, rue Conté, 753, Paris, Frane ABSTRACT: This paper presents a omparative numerial analysis o shear and extension atuation mehanisms or the bending vibrations ontrol o sandwih beams. The extension atuation mehanism denotes the use o through-thikness poled piezoeletri atuators bonded on the suraes o the struture suh that, when submitted to a through-thikness applied eletri potential, these atuators produe axial stresses or strains. The shear atuation mehanism, in the ontrary, is obtained through an embedded longitudinally poled piezoeletri atuator that, subjeted to the same eletri potential, produes shear stresses or strains. Theoretial and inite element models o a sandwih beam, apable o dealing with both mehanisms, are presented. The models are based on Bernoulli-Euler assumptions or the surae layers and Timoshenko ones or the ore. An optimal state eedbak ontrol law is used to maximize the damping o the irst our natural modes o the sandwih beam. The inluene o important parameters variation, suh as atuator thikness and struture/atuator modulus ratio, on the perormane o the ontrol system is analyzed under limited input voltage and indued beam tip transverse deletion. Results suggest that shear atuators an be more eetive than extension ones or the ontrol o bending vibrations. INTRODUCTION PIEZOELECTRIC materials are widely used or strutural vibration ontrol. Commonly, they are bonded on the surae o the struture and, when ativated by an applied eletri ield, their indued membrane deormation ontrols the vibrations o the struture. In this ase, onstant through-thikness eletri ields are imposed to a transversely poled piezoeletri atuator, using the so-alled e 31 piezoeletri onstant. This deines the extension atuation mehanism whih has been widely used on either ative ontrol appliations (Chandra and Chopra, 1993; Crawley and Anderson, 199) or hybrid ative-passive damping treatments (Baz, 1997; Huang, Inman and Austin, 1996; Tomlinson, 1996; Varadan, Lim and Varadan, 1996). Reently, developments in omposites design have brought attention to the use o embedded atuators. Although extension atuators an be embedded to produe torsional deormation (Bent, Hagood and Rodgers, 1995) using Piezoeletri Fiber Composites with or without Interdigitated Eletrodes (Hagood et al. 1993), or the ontrol o bending vibrations, they are not optimal on embedded onigurations. Some reent works presented shear atuators, that are longitudinally poled and, when subjeted to transverse eletri ield, present shear deormations through the so-alled e 15 piezoeletri onstant. This leads to the less known shear atuation mehanism. In at, the shear mode may also be obtained by applying axial eletri ields on standard transversely poled piezoeletri atuators. However, putting *Author to whom orrespondene should be addressed. ondutors on side suraes is a diiult task and leads to small axial eletri ields or plate-type atuators. So that one should preer to apply transverse eletri ields on axially poled atuators. Figure 1 illustrates both atuation mehanisms. Hene, a omparative numerial stati analysis using a ommerial inite element ode has been perormed by Sun and Zhang (1995), who proposed also a theoretial model or shear-based atuators (Zhang and Sun, 1996). It was shown that embedded shear atuators are subjeted to lower stresses than surae-mounted extension atuators, under atuation. Shear atuation mehanisms were also studied by the present authors. A sandwih beam inite element, using the mean and relative axial displaements o the ore skins as main parameters, was developed and validated (Benjeddou, Trindade and Ohayon, 1997). A omparison o extension and shear atuation mehanisms in stati and ree-vibration analysis was then arried out using this element disretization (Benjeddou, Trindade and Ohayon, ). It showed that or bending, shear atuators indue distributed atuation moments in the struture [Figure 1(b)] unlike extension atuators whih indue boundary point ores [Figure 1(a)]. Thereore, it is proposed that the shear atuation mehanism may lead to less problems o debonding in atuators boundaries and to minor dependene o the ontrol perormane on atuators position and length. To provide a better understanding o the energy dissipation harateristis o both mehanisms, another sandwih beam inite element was developed, using the surae layers mean and relative axial displaements as independent variables (Benjeddou, Trindade and Ohayon, 1999b). Comparisons between the two inite elements showed that the seond one presents better and aster JOURNAL OF INTELLIGENT MATERIAL SYSTEMS AND STRUCTURES, Vol. 1 May /99/ $1./ DOI: 1.116/UB6-GCGT-1PP6-HKY Tehnomi Publishing Co., In. 377

2 378 M. A. TRINDADE, A.BENJEDDOU AND R. OHAYON Figure 1. Piezoeletri extension and shear atuation mehanisms. onvergene (Benjeddou, Trindade and Ohayon, 1999a). That is why it was retained or urrent and uture researhes. Using the strain-indued piezoeletri oupling onstant d 15, Kim et al. (1997) have reently proposed omposite piezoeletri assemblies or shear-based torsional atuators or the prodution o large angular displaement and torque. They disussed atuator designs and assembly methods, material preparation, poling proedures, test results or joint strengths, and atuator output apabilities. It was pointed out that ommerially available PZT piezoeletris are optimized or their extension response but not or their shear behavior. This paper aims to present a omparative study o shear and extension atuation mehanisms or strutural bending vibration ontrol. Theoretial and inite element models together with an LQR optimal ontrol strategy are presented. Then, under limited input voltage and indued beam tip transverse deletion, these are used to study ontrol perormanes o both mehanisms through parameters variations, suh as atuator thikness, struture/atuator modulus ratio and ore illing material properties. THEORETICAL FORMULATION Two onigurations o a symmetri three-layer sandwih beam are onsidered. In the irst one, an elasti entral ore is sandwihed between two transversely poled piezoeletri layers [Figure 1(a)], whereas, in the seond one, two elasti layers sandwih a longitudinally poled piezoeletri ore [Figure 1(b)]. For both ases, a transverse eletri ield is applied to piezoeletri layers, whih have eletrodes on top and bottom skins. However, elasti layers are assumed insulated. All layers are assumed peretly bonded and in plane stress state. Top and bottom layers are assumed to behave as Bernoulli-Euler beams, whereas Timoshenko theory is retained or the entral ore to allow shear deormation. This is neessary or the shear atuation mehanism. Loal axes are attahed to surae layers at their let end enters, and a global one is attahed to the let end enter o the beam, so that beam entroidal and elasti axes oinide with the x-axis. The length, width and thikness o the beam are denoted by L, b and h, respetively. a, b, indies indiate top, bottom and ore layers quantities and index is used or surae layer parameters. The geometrial and kinematis desriptions o the sandwih beam are given in Figure. Mehanial Displaements and Strains Starting with linear axial displaements or eah layer and enoring the interae displaement ontinuities, the ollowing expressions or the surae layers and ore axial displaements are obtained Ê u ˆ uˆ k = Áu ± -( z- zk) w ; Ë k = a, b ( + or k = a and - or k = b) Ê u ˆ h uˆ = u + z Á + λw ; λ = Ëh h Where z a = (h a + h )/, z b =-(h b + h )/ and w is the irst derivative o the transverse deletion w, supposed onstant through-thikness. u and u are the mean and relative axial displaements o the surae layers, deined by, ua + ub u = ; u = ua -u here, u a and u b are mid-plane displaements o the top and bottom layers (Figure ). Figure. Geometrial and kinematis desriptions o the sandwih beam. b (1) ()

3 Parametri Analysis o the Vibration Control o Sandwih Beams Through Shear-Based Piezoeletri Atuation 379 From the above displaements and usual strain-displaement relations, layers strains an be written as m b m b k1 = k + z- zk k 1 = + z ε ε ( ) ε ; ε ε ε ; * Ïσ1 È 33 Ïε 1 Í Ìσ5 = Í 55 -e15ìε5 D Í 3 e E Ó Î Ó 3 (6) where s È u ε5 = ε = Í + ( λ+ 1) w Îh (3) where * = The mehanial parameter l ouples the bending behavior o the surae layers to that o the ore. It is an important variable or parameter studies. Piezoeletri Plane-Stress Redued Constitutive Equations and Eletri Potentials The piezoeletri layers are onsidered to be linear orthotropi piezoeletri materials with material symmetry axes parallel to the beam axes. jl, e ml and mm (j,l = 1,...,6;m = 1,, 3) denote their elasti, piezoeletri and dieletri onstants. For extension atuation mehanism, supposing a plane-stress state (s 3 = ), the three-dimensional linear onstitutive equations o an orthotropi piezoeletri layer an be redued to (or redution details, see Benjeddou, Trindade and Ohayon, 1997) where ε m k u b = u ± ; εk = - w ; k = a( + ), b( -); m b Êu ˆ ε = u ; ε = Á + λw Ëh * * Ïσ1 È 11 -e 31 Ïε1 Ì = Í Ì D * * Ó 3 ÍÎe E Ó 3 * 13 * 13 * e33 11 = 11 - ; 31 = 31-33; 33 = e e e s 1, e 1, D 3 and E 3 are axial stress and strain, and transverse eletri displaement and ield. Notations o the IEEE Standard on Piezoeletriity (1987) are retained here. Notie that the piezoeletri eet ouples only axial strain and transverse eletri ield, haraterizing an extension atuation mehanism. For the shear atuation mehanism, it an be shown (or details see Benjeddou, Trindade and Ohayon, 1997) that, ater oordinate transormations (Bent, Hagood and Rodgers, 1995; Hagood et al., 1993) so that axial and transverse indies interhange, the three-dimensional linear onstitutive equations o the orthotropi piezoeletri ore redue to, (4) (5) s 5 and e 5 are transverse shear stress and strain. Here, the piezoeletri eet ouples shear strain and transverse eletri ield, haraterizing a shear atuation mehanism. The ombination o the strain-displaement relations obtained rom Equations (1) (4), and the redued onstitutive Equations (5) or (6), then integration o the eletrostati equilibrium equation, ree o volumi harge density, allow us to write the ollowing eletri potential orms or piezoeletri surae layers j k (extension atuation mehanism), and or a piezoeletri ore j (shear atuation mehanism), respetively where ϕ = + z- z È + - w Î * ϕ k 4( z- z ) h k e31 k ϕk ( k) Í1 h 8 * Í h ϕ are the mean and the dierene o the presribed eletri potentials on top ( ϕ + i ) and bottom ( ϕ - i ) skins o the i-th layer. The last term in Equation (7) represents the quadrati indued potential, oten negleted in the literature (Rahmoune et al., 1998). Variational Formulation ϕ = ϕ + z h ϕi + ϕi + - ϕi = ; ϕ i = ϕi - ϕi ; i = a, b, In order to study the eets o the eletromehanial oupling on the dynamis o the sandwih beam, let us start rom the ollowing variational ormulation o the problem in terms o the unknown ields u, u, w and ϕ δt - δh + δw = ; " δu, δu, δw, δϕ where dt, dh and dw are the virtual variations o kineti energy Tuuw (,, ), eletromehanial energy Huuwϕ (,,, ) and work done by applied mehanial loads Wuuw (,, ), respetively. Here only the atuation problem is onsidered, that is, ϕ i are given. Thus, only u, u and w must be retained as independent and unknown ields. Note that these ields are time and spae-dependent, whereas their variations (7) (8) (9)

4 38 M. A. TRINDADE, A.BENJEDDOU AND R. OHAYON ( δu, δu, δw) are only spae-dependent. Thereore, the virtual variations in Equation (9) are now detailed in terms o these three main variables, only. It is important to notie that the Equation (9) must be omplemented by initial onditions. To provide a better understanding o the extension and shear atuation mehanisms, surae layers and ore ontributions ( δh and δh) to the eletromehanial energy variation are studied separately (1) Deomposing these two variations into mehanial and eletromehanial ontributions, the surae layers ontribution dh is written as where, and the ore ontribution dh as where, δh = δh + δh δh = δh -δh (11) (1) In Equations (11) and (1), I i, A i are moment area and area o the i-th layer. Notie that the indued potential leads to an augmentation o the surae layers bending stiness through From Equation (1), dh me an be interpreted as a distributed moment e15 Aϕ / h indued by the applied eletri di- erene o potential ϕ to the ore layer [Figure 1(b)]. A transverse shear strain [ u / h + ( λ + 1) w ] is then produed. Sine, only the bending atuation will be onsidered here, surae imposed potentials are o opposite signs ( ϕ a =- ϕ b = ϕ ). Hene, dh me, given in Equation (11), redues to, m me L Ï * 1 * δhm = Ú Ì 11 Au δu + 11 Au δu + 11I w δw dx Ó 4 A È δu δh e ( ϕ ϕ ) δu ( ϕ ϕ ) dx δh m L * me =- Ú 31 a + b + a - b h Í Î δh = δh -δh m me Ï * * Êu ˆÊδu ˆ 33 Au δu + 33I + λw + λδw L Á Ëh Á Ë h = Ú Ì dx È u Èδu + 55A + ( λ+ 1) w + ( λ+ 1) δw Í Í Îh Îh Ó L ϕ Èδu δhme =- Ú e 15A Í + ( λ+ 1) δw dx h Îh * * ( e31 ) 11 = 11 + * 33 (13) Thereore, δh me is interpreted as the virtual work o * boundary point atuation trations e31 Aϕ / h indued by the applied opposite dierene o potentials ϕ on the surae layers [Figure 1(a)]. Only relative axial displaement or strain o the surae layers is produed. Comparing dh me in Equation (1) to δh me in Equation (13), one an notie that the extension atuation mehanism produes boundary point ores (trations/ompressions), whereas the shear atuation mehanism indues distributed moments (Figure 1). Hene, one an expet that the latter avoids the ommon singularity problems at the boundaries o onventional extension atuators. Variations o the kineti energy and work due to applied mehanial loads o the sandwih beam or both mehanisms, written in terms o the main variables, are and H e A ϕ u dx e A ϕ δ δ δ u L * * me =- Ú 31 =- 31 h h Ï( ρa + ρa) u δu ÈÊ I ˆ u I δu + ÍÁρA + 4ρ + ρ λw L ÍË h h δt = Î Ú Ì dx + ( ρa + ρa) w δw È u + ρiλ + ( ρi+ ρiλ ) w δw Í h Ó Î (14) È Êna - nb mˆ L Í( na + nb + n) δu + Á + u h δ δw = Ë Ú Í dx Í ÍÎ- ( ma + mb - λm) δw + ( qa + qb + q) δw L (15) È ÊNa - Nb Mˆ Í( Na + Nb + N) δu + Á + δu h + Í Ë Í ÍÎ- ( Ma + Mb - λm) δw + ( Qa + Qb + Q) δw where r i is the mass density o the i-th layer. n i, m i, q i and N i, M i, Q i are distributed and point normal, moment and shear stress resultants. For the extension atuation mehanism, dh me vanishes sine the ore is not piezoeletri. Thus, the variational Equation (9) redues to δhm + δhm - δt = δw + δhme ; " δu, δu, δw (16) Similarly, or the shear atuation mehanism, dh me vanishes sine the surae layers are elasti. The variational Equation (9) is then δhm + δhm - δt = δw + δhme ; " δu, δu, δw L (17)

5 Parametri Analysis o the Vibration Control o Sandwih Beams Through Shear-Based Piezoeletri Atuation 381 where dh m is similar to δ but with 11 = 11*. FINITE ELEMENT DISCRETIZATION The standard inite element method is ollowed to disretize the variational problems Equations (16) and (17). The variables u and u are interpolated by Lagrange linear shape untions and w by Hermite ubi ones. For the shear atuation mehanism, the disretized equations o motion an be written as, (18) where q = [ u1, u 1, w1, w1, u, u, w, w ] is the vetor o degrees o reedom and qq, the orresponding veloity and aeleration. M is the mass matrix obtained rom the disretization o dt. C v is a global visous damping matrix aounting or materials damping. K and K are the surae layers and ore stiness matries obtained rom the disretization o δh m (with 11 = 11* ) and dh m given in Equations (11) and (1), respetively. Feϕ and F m are the indued eletri and mehanial load vetors dedued rom disretization o dh me in Equation (1) and dw, respetively. For the extension atuation mehanism, the disretization o Equation (17) leads to the ollowing disretized equations o motion (19) where K is the stiness matrix o the piezoeletri surae layers and Feϕ is the indued eletri ore vetor obtained rom the disretization o Equation (13). All matries and vetors o Equations (18) and (19) were integrated analytially and implemented in MATLAB sotware. A omprehensive review synthesis on the piezoeletri inite element literature was given by Benjeddou (). CONTROL STRATEGY H m Mq + C q + ( K + K ) q = F + F ϕ v m e Mq + C q + ( K + K ) q = F + F ϕ v m e Prior to the presentation o the vibration ontrol strategy, linear seond order matriial Equations (18) and (19), are written in state-spae orm, with state vetor x, input vetor u and output vetor y, Ïx = Ajx+ Bju+ Bp; j =, Ïq Ì ; x = Ì () Óy = Cx Óq A j and B j (j =,) represent system and ontrol matries o the extension () and shear () atuation mehanisms, respetively. B p represents the perturbations vetor and C, the state output matrix. These have the ollowing expressions, A È I È I = Í -1 ; A = Í -1 ÍÎ - M ( K + K) ÍÎ - M ( K + K) (1) È È È B = Í -1 ; B = ; -1 Bp = -1 ÍÎ M Fe ÍÎM Fe ÍÎM Fm () For both onigurations (surae-mounted or sandwih), it is supposed that the ontrol atuation is done by the piezoeletri atuators only. Thereore, B j and B p are olumn vetors and u is a salar, representing the imposed voltage u = ϕ or the extension atuation mehanism and u = ϕ or the shear one. The design o the ontroller is based on LQR ull state eedbak, i.e., the ontrol voltage u is proportional to the state vetor x, u=- K x (3) where K g is a row vetor representing the ontrol gain. Substituting Equation (3) in the unontrolled state Equations (), the resulting ontrolled ones an be written as x = ( A - B K ) x+ B y = Cx (4) The system is then ontrolled by a modiiation o the matrix A j, whih beomes A j - B j K g. Thereore, the ontrol ation may stabilize the system by hanging its vibration harateristis, suh as the damping o some hosen poles, as explained in the ollowing setion. In order to interpret these results in a strutural mehanis approah, Equations (18) and (19) an be rewritten, taking into aount Equations (1), () and (4), as or the shear atuation mehanism, and (5) (6) or the extension one. The row vetors K d and K p are obtained rom deomposition o the gain row vetor into proportional and derivative omponents, i.e., K g = [K p K d ]. One an notie that the ontrol law supplies a stiness matrix Kˆ = FjeKp and a damping matrix Cˆ = FjeKd in addition to the atual strutural stiness matries and eventual initial visous damping matrix C v. The uniied resulting system may be represented by the general orm (7) where K= K + K or the extension atuation mehanism and K= K + K or the shear one. It is worthwhile to ompare the original unontrolled systems (18) and (19) with the orresponding ontrolled systems (5) and (6). The LQR ontrol strategy is implemented using MATLAB Control Toolbox. g j j g p Mq + ( C + F K ) q + ( K + K + F K ) q = F v e d e p m Mq + ( C + F K ) q + ( K + K + F K ) q = F v e d e p m Mq + ( C + Cˆ ) q + ( K + Kˆ ) q = F v m

6 38 M. A. TRINDADE, A.BENJEDDOU AND R. OHAYON NUMERICAL RESULTS This setion aims to present a omparative numerial analysis o shear and extension atuation mehanisms. To this end, the present inite element model is used to evaluate bending vibration harateristis o both mehanisms (Figure 3), under variations o several parameters suh as, atuator thikness, struture/atuator modulus ratio, oam stiness and number o atuators. The geometrial data o the beam, aording to Figure 3, are L = mm, h = 4 mm, t =.5 mm, d =3mm,a = 3 mm. The shear atuator thikness is t leading to equivalent surae-mounted and sandwih onigurations. Aluminum properties are: Young s modulus E b = 7.3 GPa, Poisson s ratio n=.35, density r b = 71 kg m -3. Those o the oam are: Young s modulus E = 35.3 MPa, shear modulus G = 1.7 MPa, density r = 3 kg m -3 ; and, or the PZT-5H: * * 11 = 55 = 3 GPa, density r p = 75 kg m = E p = 69.8 GPa, * -, piezoeletri oupling onstants e 31 =-3. C m, e 15 = 17 C m -, and dieletri onstant * 33 = F m. An initial visous damping o.1% was assumed. The ontrol gain vetor K g is evaluated using LQR optimal ontrol algorithm. The ponderation matries Q and R are onsidered to be Q =giand R = I, giving the same ontrol weight g or all states. One to three atuators are onsidered, eah o them having same length a and being at position d k = 15(3k-1) mm (k = 1,,3). By deault, three atuators are onsidered. The numerial analysis, presented here, onsists in evaluation o the ative damping, or eah parameter variation, supplied by both mehanisms or the irst our natural bending max max Figure 4. Algorithm or ontrol gain evaluation (ϕ = ϕ = V). modes o the sandwih beam. Thereore, to ompare dierent onigurations with dierent parameters, two basi parameters are ixed: the exitation F m and the maximum ontrol voltage ϕ max. The ree end o the beam is exited by a transverse impulse exitation, whih produes a maximum open-loop deletion o w(l) = 5 mm. An iterative algorithm, shown in Figure 4, was developed to evaluate the ontrol gain, suh that the maximum supplied voltage is max ϕ = 1 V. Sine the same eletri ield must be imposed to both atuation mehanisms, the maximum voltage or the shear atuators is the double o that o extension ones, i.e., max ϕ = V. Limited to these voltages, the damping o the two onigurations is evaluated. In the irst ase, the atuator thikness t is varied in the range [.1,.5] mm (note that the shear atuator thikness is t). As it an be seen in Figure 5, the shear atuation mehanism (SAM) provides muh larger damping ators or very thin atuators. Moreover, one an see that the eetiveness o the extension atuation mehanism (EAM) is almost independent o the thikness o the atuator, whereas, the shear one is better or a thikness t <. mm. Figure 3. Cantilever beam, shear and extension atuation onigurations. Figure 5. Variation o irst our natural bending modes damping with atuator thikness.

7 Parametri Analysis o the Vibration Control o Sandwih Beams Through Shear-Based Piezoeletri Atuation 383 Figure 6. Variation o irst our natural bending modes damping with struture/atuator modulus ratio. Next, the eet o the struture stiness on the ative damping is analyzed. To this end, the damping ator is evaluated or several struture/atuator Young s modulus ratios (E b /E p ). The results, presented in Figure 6, suggest a superiority o the shear atuation mehanism over the extension one or soter strutures (E b /E p < 1). Moreover, shear atuators perormane is highly dependent on struture stiness, whereas, that o extension ones is not, even i the results suggest an optimal medium stiness ratio (E b /E p =.). It is important to note that these results are subjeted to a variation in the maximum open-loop deletion amplitude sine, as the stiness o the beam dereases, this amplitude inreases or a ixed impulse magnitude. Sine the shear atuation mehanism requires the use o an extra-material (here, a oam) to over the rest o the ore Figure 8. Variation o the irst our natural bending modes damping with number o atuators. layer, it is important to investigate the inluene o its material properties. Figure 7 presents the variation o the damping ator with the oam modulus multiplying ator (E = 35.3 MPa, G = 1.7 MPa). It indiates that, generally, the inrease o the oam stiness dereases damping. However, although the output weight g is the same or all modes, eah mode damping presents a dierent optimal oam stiness, e.g., the third mode damping is optimal or a relatively rigid oam ( = 3). This means that sot oams may improve the ontrol o some modes, but not o all o them. The ontrol perormane is also dependent on the number o atuators. Generally, several atuators will outperorm a single one. Moreover, as the position o atuator deines whih modes an be well ontrolled, several atuators may provide damping over larger requeny range. Figure 8 presents the damping o the irst our natural bending modes using one, two and three atuators. It shows that, as expeted, the inrease in the number o atuators inreases the damping ator o all modes. For the extension atuation mehanism, the variation o modal damping with the number o atuators is almost linear. However, or the shear atuation mehanism, although the variation o irst and ourth modal damping are also almost linear (approximately +.% per atuator), or the seond and third modes, the inlusion o a seond atuator does not inrease muh the damping (+.1%) ompared to that o a third one (+.8%). CONCLUSIONS Figure 7. Variation o irst our natural bending modes damping with oam modulus ator. Theoretial and inite element models o an adaptive sandwih beam, apable o dealing with both shear and extension atuation mehanisms, were presented and used to ompare ative damping perormanes o suh mehanisms or the ontrol o strutural bending vibrations o smart beams. It was shown that, or bending atuation, shear atuators indue distributed moments, unlike extension ones whih pro-

8 384 M. A. TRINDADE, A.BENJEDDOU AND R. OHAYON due boundary point ores, prediting less problems o debonding and singularities, and better ontrollability. Using a LQR optimal ontrol law, the inluene o important geometrial and material properties on the ative damping o the beam was analyzed under maximum applied voltage o ϕ = ϕ = V and indued tip transverse deletion o 5 mm. Finite element results show that shear atuators may be better in produing ative damping than the generally used extension ones. The shear atuation mehanism was shown to be optimal or a range o thin atuators (t <. mm) and relatively sot strutures. It was observed that the hoie o the ore illing material is important or the shear atuation mehanism perormane, with advantages or soter materials. The inrease o the number o atuators enhanes the average damping over a broader requeny range. Finally, due to lower stresses in the atuator and better ontrollability properties, shear atuators are suitable or the vibration ontrol o strutures with embedded atuators. The present study has been extended to the inlusion o embedded shear and/or extension sensors and hybrid piezoeletri-visoelasti damping treatments (Trindade, Benjeddou and Ohayon, ). NOMENCLATURE A i = ross-setion area o the layer i A = state-spae system matrix a = piezoeletri atuators length B = state-spae ontrol input matrix B p = state-spae perturbation vetor b,l = beam width and length, respetively C = state-spae output matrix C v = visous damping matrix Ĉ = ontrol supplied damping matrix jl,e kl, kk = elasti, piezoeletri and dieletri onstants, respetively dh = virtual variation o eletromehanial energy dt = virtual variation o kineti energy dw = virtual variation o external loads work d = piezoeletri atuators enter position E 3,D 3 = transverse eletrial ield and displaement, b ε i m ε i respetively = bending strain o layer i = axial strain at enterline o layer i (membrane strain) e 5 = shear strain o layer e i1 = axial strain o layer i F m = mehanial loads vetor F e = indued eletrial loads vetor ϕ i = mean o applied eletri potentials on the layer i ϕ i = dierene o applied eletri potentials on the layer i j i = eletri potential in the layer i + - i K p = proportional ontrol gain matrix ˆK = ontrol supplied stiness matrix M = mass matrix N i,m i,q i = point normal, moment and shear resultants on layer i, respetively n i,m i,q i = distributed normal, moment and shear resultants on layer i, respetively Q = LQR state ponderation matrix q = degrees o reedom vetor R = LQR input ponderation matrix r i = mass density o the layer i s 1,s 5 = axial and shear stresses, respetively Subsripts ϕ, ϕi = eletri potential at the top and bottom skins o the layer i, respetively g=lqr state ponderation ator h i = thikness o layer i I i = ross-setion moment area o the layer i K = stiness matrix K d = derivative ontrol gain matrix K g = ontrol gain matrix u u1, u 1, w1, w 1 u, u, w, w t = piezoeletri atuators thikness = mean o the axial displaements o surae layers enterlines = displaements and rotation o element node 1 = displaements and rotation o element node u i = axial displaement o the enterline o the u layer i = dierene between the axial displaements o surae layers enterlines = axial displaement o the layer i uˆi w = transverse displaement o beam enterline x = state vetor x,z = axial and transverse oordinates y = state-spae output vetor z k = distane to enterline o surae layer k (k = a,b) e,me = state or eletrial or mehanial-eletrial oupling ontributions (piezoeletri) = states or quantities related to sandwih surae layers i = states or beam layers a, b or j = states or quantities related to extension () and shear () atuation mehanisms k = states or surae layers a or b m = states or mehanial ontributions Supersripts * = states or modiied material onstants b = states or bending ontributions = states or ore material onstants

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