MULTI-LAYER PIEZOELECTRIC ACTUATOR AND ITS APPLICATION IN CONTROLLABLE CONSTRAINED DAMPING TREATMENT (2).doc

94chinese journal of mechanical engineeringvol.20, no.4, 2007zhang xinongxieshilinzhang yahongschool of aerospace,xt an jiaotong university,xian 710049, chinamulti-layer piezoelectric actuator and its application in controllable constrained damping treatment*abstract: a kind of novel multi-layer piezoelectric actuator is proposed and integrated with controllable constrained damping treatment to perform hybrid vibration control. the governing equation of the system is derived based on the constitutive equations of elastic, viscoelastic and piezoelectric materials, which shows that the magnitude of control force exerted by multi-layer piezoelectric actuator is the quadratic function of the number of piezoelectric laminates used but in direct proportion to control voltage. this means that the multi-layer actuator can produce ijrsater actuating force than that by piezoelectric laminate actuator with the same area under the idsctiod control voltage. the optimal location placement of the multi-layer piezmlectric actuator ir. aiso cisaissed. as an example, the hybrid vibration control of a cantilever rectangular thin-plate is nunarically simulated and carried out experimentally. the simulated and experimental results validate the power of multi-layer piezoelectric actuator and indicate that the ir/escnt hybrid dai.ping technique can effectively suppress the low frequency modal vibration of the expoiinraxta! tsin-plate structure. key words: vibration control mi !*j-iayer piezoelectric actuatorcontrollable constrained damping treatment hybrid damping0 introductionit is well-known that control performance can be improved by increasing the control voltage applied on piezoelectric actuator. however, higher control voltage will generally lead to higher control cost and lower economic efficiency. in addition, it may lower the stability and reliability of control system in practice because the piezoelectric actuator has the risk of being punctured by exorbitant control voltage. therefore control voltage and control performance should be considered simultaneously in vibration control design. this is especially true for the vibration control of astronautic and aeronautic structures, where the control voltage must be at the reasonably low level in order to ensure the safety and reliability of system.the hybrid damping techniques integrating active action of piezoelectric material with damping characteristic of viscoelastic material have been fully investigated in recent years11. baz, et al25, proposed a hybrid damping scheme called active constrained damping layer, where the piezoelectric layer serves as both the cover sheet of viscoelastic damping layer and the actuator. active constrained damping layer can actively enhance the shear damping of viscoelastic layer by means of inverse piezoelectric effect of the piezoelectric layer and may consequently control the low frequency vibrations of thin-walled structure well. stanway, et al16, has made a good state-of-the-art review on active constrained layer damping. however, it suffered from such disadvantages as requirement of the piezoelectric layer with bigger area, be difficult to apply distributed electric field and low economic efficiency in vibration control of larger structures. zhang, et al791, have developed a self-contained hybrid damping design called controllable constrained damping layer, where the piezoelectric laminate actuator was bonded with surface of the elastic cover sheet to reinforce the shear damping of the viscoelastic layer by deforming the elastic cover sheet. in this way, the piezoelectric laminates with smaller area instead of the piezoelectric layer with bigger area were enough for the vibration control of larger thin-walled structures and thereby the application of distributed electric field could be circumvented. especially, it provided a preferable hybrid manner for vibration control of the* this project is supported by national natural science foundation of china (no. 50275114, no. 10476020). received october 12, 2006; received in revised form april 16, 2007; accepted april 26,2007structures that have been previously covered with the constrained damping layer at the viewpoint of economy. zhang, et al10, has investigated the hybrid vibration control of structure through controllable constrained damping treatment, where its feasibility and effectiveness were validated experimentally.in this paper, a kind of novel multi-layer piezoelectric actuator is designed to produce great actuating force under low control voltage. it is integrated with controllable constrained damping treatment to perform hybrid vibration control of thin-plate structure. the governing equation of the system is deduced based on the constitutive equations of elastic, viscoelastic and piezoelectric materials and thereafter reduced by employing galerkin method and ghm model1 describing characteristic of viscoelastic material. the optimal location placement of the multi-layer piezoelectric actuator is also discussed. the hybrid vibration control of a cantilever rectangular thin-plate structure is numerically simulated and carried out experimentally, where how the vibration control performance depend on the control voltage and the number of piezoelectric laminates used is investigated and the power of the multi-layer piezoelectric actuator is validated. conclusions are presented at the end of the paper.1 multi-layer piezoelectric actuatorpositive pole facefig. 1 shows the configuration of a common piezoelectric laminate actuator that is polarized in the z direction. it can apply control force or moment on the controlled structure under electric field in terms of the inverse piezoelectric effect.negative pole facefig. 1 configuration of piezoelectric laminate actuatorthe multi-layer piezoelectric actuator is made up of some piezoelectric laminates with the same geometric and material parameters, which are adhered to one another as shown in fig. 2. all piezoelectric patches are polarized along the normal direction and glued each other through the surfaces with the same polarity,19 4 20 7 ch na acad mic journal electronic publish hvtp /echinese journal of mechanical engineering.95.which is arranged so that each piezoelectric laminate is driven with the identical electric field. besides that, each piezoelectric laminate should be placed so as to deform in the same direction under the identical electric field. with this configuration, one can expect that the multi-layer piezoelectric actuator generates greater actuation force or moment on controlled structure through in-plane deformations of all piezoelectric patches in terms of inverse piezoelectric effect than a single piezoelectric patch actuator under the same control voltage. this will be illustrated in the following sections. it needs to be pointed out that working principle of the multi-layer piezoelectric actuator is different from that of traditional piezoelectric stack actuator which exerts control force through out-of-plane deformation.negative pole of control voltagepositive pole of control voltagefig. 2 configuration of multi-layer piezoelectric actuator2 modelling of systemmulti-layer piezoelectric actuatorin this section, the hybrid vibration control of thin-plate through controllable constrained damping treatment with the multi-layer piezoelectric actuator is investigated, which is different from the cases discussed by zhang, et al1791, where only piezoelectric single laminate actuator is considered. the present hybrid control manner is illustrated in fig. 3, where controllable constrained damping layer is made up of base structure (thin-plate), viscoelastic damping layer, elastic cover sheet and multi-layer piezoelectric actuator.elastic cover sheet viscoelastic damping layerbase structure( thin-plate)fig. 3 configuration of controllable constrained damping layer with multi-layer piezoelectric actuatorto derive the governing equation of the system, the following assumptions are made: only the elastic deformation occurs for every layer. the rotational inertia of every layer can be neglected. only the damp resulting from the shear deformation of viscoelastic layer is considered. the displacement of every layer is same in the z direction. the structural displacements in all interfaces are continuous. each piezoelectric laminate in the multi-layer piezoelectric actuator is driven by the identical control voltage. the control voltage is uniformly applied on the piezoelectric actuator in both the x and y directions. the influences of all adhesive layers can be neglected. in addition, it is supposed that n piezoelectric laminates with the same geometric and material parameters are used to fabricate the multi-layer piezoelectric actuator. therefore we have pa=pe, ecj=ee, strst (i=l, ,n), where pe, ec, se are mass den-sity.modulus of elasticity and thickness of the piezoelectric laminate respectively.2.1 geometric equations(1) (2) (3)according to elasticity theory, the geometric equations of the base structure, the elastic cover sheet and the multi-layer piezoelectric actuator can be respectively expressed aswhere b, %, 4strain vector of thin-plate, elastic cover sheet and multi-layer piezoelectric actuator mid-plane displacement vector of thin-plate, elastic cover sheet and multi-layer piezoelectric actuatorl1jl2differential operators- 0 -a-ox . dy0 a l dy fa)r fa2 a2 . d2 k =k dx2 dy2 dxdy,k =-l2wfor the viscoelastic damping layer, if only the shear deformation is considered the geometric equation ismrw =(4)rj= uhssl, i -i) 0.vd;where 3differential operator3 =dx dya = l +18.4. 4. 4thickness of thin-plate, elastic cover sheet,sviscoelastic layerwdisplacement of thin-plate in the 1directiony/strain vector of viscoelastic damping layer2.2 constitutive equationsthe constitutive equations of the base structure, the elastic cover sheet and the multi-layer piezoelectric actuator can be respectively presented as(5) (6) (7)h = a,b .=j,where oi, cmodulus of elasticity of tktn-piate, elastic coversheet(13)vb,vcpoisson ratio of thin-plate, elastic cover sheetr(x,y) is used to describe the location where the multi-layer pie-c3i, enpiezoelectric coefficient in the x and y direction zoelectric actuator is placed, andj-r(x, y) = 8(x -x,)-s(x- x2 )h(y -y)-h(y-y2)slr(x,y) = hx-xl)-hx-x2)s(y-yl)-sy-y2)dywhere 5(x)dirac delta functionh(x)unit step function2.4 governing equationsfig. 4 shows the differential elements of the composite plate structure. from fig. 4a, according to newtons second law, the force and moment equilibrium equations for differential element2 j internal forces and moments* u 1 j 1 . jr *l 1 1. _, jaccording to the bending theory of plate, the tension and the of ,me,lammated plate *t u of *e elastic cover sheet and*b=ctbxyotc = (*,*y)tff. = (ff =vb1 01-k20vo -*2 jf1m, odrespectivelycn. ci2. c22elastic coefficient of piezoelectric materialfor the viscoelastic layer, the shear stress can be formulated as,1 . 1 .r =goy, = kligw+gdc-godbwhere g(f)relaxation function of viscoelastic material-stiejtes integralbending moment in the base structure are respectivelyrstress vector of viscoelastic damping layer?cz)cvt +(.v, -|a- i*(*,jo (idvjfc=(a/; a/c =(12)where(8) = (z,-z;-,)-(14)multi-layer piezoelectric actuator can be presented as follows z,tfic -r +fc =pcsc + npcscr(x,y)dc(9)(10)a/b=(a/b m m?)t = jlatbzdzfrom eqs. (1) and (5), we haveflb=sbdbl,dbbj2 b-2where the elastic cover sheet and the multi-layer piezoelectric actuator are treated as a laminated plate, thus we have dc=dt. basing on the laminated plate theory, the corresponding tension and bending moment are respectively*yf _f =(fx fy fy =atciz + r(x,y)ftjar,dz =*,t_%mc +-%t +pc +qc = pcsc +npcser(x,y)* (15)where fc=(fc f/)tfrom fig. 4b, the mechanics equilibrium equations for differential element of the viscoelastic layer can be given as(16) (17)(18)sbs+/b-.pc+?s=m*where fsta=(f f)tsubstituting eq. (17) into eq. (16) we haves&*+p*-pc+ y)w (23a) (scljdcll+raldel,)dc-rldtl2w+ / u ? f(a) differential element of the elastic cover sheet and the actuator as laminated plates*yrstea?-x fshs/ /9, /1 1c/r, /fojt/bb(b) differential elemen. of viscoelastic damping layer+d-+d,9yv/3y. v+mdmbm*?7?uhligw-godc +gdb +(23b)k=pa+npar(x,ym6h$di,ldhl1gw+godc -i(23c)gdb + fb=pbsbdb o.where #,. a. amass density of thin-plate, cover sheet,viscoelastic layerfh, fc, ftin-plane disturbance force vector of thin-plate, elastic cover sheet and piezoelectricactuator?b. 9c. 1c y)wsubstituting eq. (8) and eqs. (9)(12) into eq. (14), eq. (19) and eq. (22) respectively to give the governing equations of the system as followskd*jml,1 -1s.5.1-w,_1_s.xgo(l2r)( c3c3si12 c 12 3 ew+/?!tdevc +rqvl,eu(t)tn(ns, + 5c)2/7lrfneuit) 0(24)4-20 china a e ic our 1 el t oni ublishi h arghs re t:/ ww. n98zhang xinong, et al: multi-layer piezoelectric actuator and its application in controllable constrained damping treatmentwhere sh = + &+ h 2 s 2p = nptstr + pcsc + psss + pbsbthe second term on the right side of eq. (24) represents the control force produced by the multi-layer piezoelectric actuator, which is the second-order polynomial function of the number of piezoelectric laminates used in the multi-layer piezoelectric actuator but in direct proportion to control voltage. therefore it is obvious from eq. (24) that the multi-layer piezoelectric actuator (the case of n) may produce greater control force than that by single piezoelectric laminate actuator (the case of =1) under the same control voltage.2.5 model reductionit is shown from eq. (24) that the governing equation of the system is made up of a group of partial differential equations, which can not be directly used in control design. therefore model reductions should be performed to transform these partial differential equations into ordinary differential ones.let wjx,y), ucj(x,y), vc.y), utfcy), vfay) 0=1,-, ao denote the modal shape functions of the composite plate structure. thus the response of system can be approximately expressed in the form of modal superpositionx,y,t) = flhr(x,y)aj(t)(*.-.o=5x(*,rt4(ou,(x,y,t) = iux,y)rji(t)y-inemploying galerkin method12 to perform the model reduction, we havedamping matrix, respectively, is the augment modal vectoro),* = (rbzt)trz(s) =.2 8s + 2lals + (a;00),*g+g(l + )c -agjfl =0002f,rc8wgck =8 /u =8v =3 location placement of piezoelectric actuatorthe modal control force vector on the right side of eq. (25) can be expressed asg=(1.-.,.fl.-./.r1.-.-.,.-.)twhered