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

multi-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 thestructures 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 actuatorfig. 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.the 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, 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.in 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.to 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 equationsaccording to elasticity theory, the geometric equations of the base structure, the elastic cover sheet and the multi-layer piezoelectric actuator can be respectively expressed as for the viscoelastic damping layer, if only the shear deformation is considered the geometric equation ism2.2 constitutive equationsthe constitutive equations of the base structure, the elastic cover sheet and the multi-layer piezoelectric actuator can be respectively presented as stress vector of thin-plate, elastic cover sheet and multi-layer piezoelectric actuator-elasticity matrix of thin-plate,elasticcover sheet and piezoelectric actuator2 .3 internal forces and momentsaccording to the bending theory of plate, the tension and the bending moment in the base structure are respectively in controllable constrained damping treatment2.4 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 elementwhere 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 respectivelyit needs to be noted that the third term on the left side of eq. (23a) and that on the left side of eq. (23b) characterize the stress concentration arising at the boundary of the multi-layer piezoelectric actuator. in this case, the stress concentration may influence the local stress distribution at the boundary of the piezoelectric actuator but has less effect on the global stress distribution and thereby can be neglected. thus eq. (23) can be rewritten in the matrix description asthe 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.denote the modal shape functions of the composite plate structure. thus the response of system can be approximately expressed in the form of modal superpositionit is obvious that eq. (25) is an integral-differential equation due to stiejtes integral term which characterizes the viscoelastic damping layer. in this section, ghm model11 is used to approximate stiejtes integral in the time domain, where characteristic of viscoelastic material is described by3 location placement of piezoelectric actuatorthe modal control force vector on the right side of eq. (25) can be expressed as eqs. (28h30), it is shown that the control forces depend on not only the geometric and material parameters of the composite plate structure, control voltage and the number n of the piezoelectric laminate used but also the location where the multi-layer piezoelectric actuator is placed. in addition, from eqs. (31) and (32) it can be observed that the in-plane vibrations of the base structure is uncontrollable for the multi-layer piezoelectric actuator. defining/; asit is obvious that the ith element inp represents the jth modal control force per unit voltage. the location placement of the multi-layer piezoelectric actuator can be optimized based on eq. (33) to maximize the modal control forces when the other physical parameters and modal shape functions are all determined.4 numerical simulationthe hybrid vibration control of a cantilever rectangular thin-plate through controllable constrained damping treatmentwith multi-layer piezoelectric actuator in numerically investigated in this section. the thin-plate is made from alloyed aluminum, the piezoelectric actuator from pmn piezocrystal ceramics, the elastic cover sheet from stainless steel and the viscoelastic layer from zn-1 viscoelastic material. the geometric and material parameters of the structure are shown in tables 1 and 2, respectively,the modal parameters it trie composite plate structure are firstly computed from eq. (27), where the shape functions of elastic cantilever plate are used to perform the modal transformation and the parameters of ghm model are selected as follows: ai=l, 6=4, coy= 000 rad/s, g=5 mpa. with the above method, the computed value of the first modal frequency is 18 hz, which nears reasonably the measured value 18.9 hz. here, only the first mode of the composite plate structure is considered as control object because the high frequency modes have been greatly attenuated by the constrained damping layer.the displacement feedback scheme is employed in active control design, i.e. based on eq. (35). defining a factor 8to characterize the vibration suppression effect per unit control voltage where ao and aq are respectively the response amplitudes for the uncontrolled and controlled cases, uf is the amplitude of control voltage. in order to illustrate the capability of the multi-layer piezoelectric actuator to produce greater actuating force under lower control voltage, the relation between 8 and n is numerically analyzed for the identical feedback gain and shown in fig. 5. it can be observed from fig. 5 that 8 is approximately the quadratic function of n, which agrees reasonably with the results showed in eq. (24). therefore, the more the number of the piezoelectric laminates used in the multi-layer piezoelectric actuator, the smaller the control voltage needed to achieve the same vibration suppression effect becomes.5 control experimentto validate the simulated results, the hybrid control experiment is performed in this section. the multi-layer piezoelectric actuator is fabricated through gluing four piezoelectric laminates together with epoxy. the geometric and material parameters of the experimental structure are the same as those used in simulation. fig. 6 shows the experimental setup. the multi-layer piezoelectric actuator is placed near the fixed side of structure where the large surface strain occurs because the first mode of the structure involves significant bending deformation. two sensors (b&k4370) are used in the experiment, one to provide the feedback output and the other to monitor the vibrational status of structure. the electromagnetic vibration exciter is used to exert the external load, where the excitation signal is generated by the signal generator (b&k1027). a voltage amplifier designed by authors is used to apply control voltage on the piezoelectric actuator. two low-pass filters are used to attenuate the high frequency noises. the active control is implemented in a digital processor (cpu 1 600 mhz)5.1 control experiment under harmonic excitationthe frequency of harmonic excitation is 18.9 hz, which is near the first modal frequency. the displacement proportional feedback scheme is employed in active control. fig. 7 shows the acceleration response time history of monitored point in the uncontrolled and controlled cases when n=3. it can be seen that structural vibration is suppressed significantly with a reduction of 56.65% in response amplitude. moreover, the relation between 8 and n is experimentally analyzed and also shown in fig. 5. it is seen from fig. 5 that the experimental and simulated results have a reasonable agreement for n=l4, which verifies the power of the multi-layer piezoelectric actuator.5.2 control experiment under sinusoidal swept excitationin this subsection, control experiment is performed under sinusoidal swept excitation (5-250 hz). the experimental conditions are the same as those in the case of harmonic excitation. fig. 8 gives the experimental frequency responses from the exci tation to monitoring signal for the uncontrolled and controlled cases. it can be observed from fig. 8 that the arnpiitudus of the first and the third modes ar