FINITE ELEMENT MODELLING OF PIEZOELECTRIC ACTIVE STRUCTURES: SOME AP- PLICATIONS IN VIBROACOUSTICS. V. Piefort

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1 FINITE ELEMENT MODELLING OF PIEZOELECTRIC ACTIVE STRUCTURES: SOME AP- PLICATIONS IN VIBROACOUSTICS V Piefort Active Structures Laboratory, Université Libre de Bruxelles, Belgium ABSTRACT The use of piezoelectric materials as actuators and sensors for noise and vibration control and noise reduction has been demonstrated extensively over the past few years (1) The frequency response functions between the inputs and the outputs of a control system involving embedded distributed piezoelectric actuators and sensors in a shell structure are not easy to determine numerically The situation where they are nearly collocated is particularly critical, because the zeros of the frequency response functions are dominated by local effects which can only be accounted for by finite element analysis (7) The fundamental equations governing the equivalent piezoelectric loads and sensor output are derived for a plate starting from the linear piezoelectric constitutive equations The reciprocity between piezoactuation and piezosensing is pointed out A finite element formulation for an electromechanically coupled piezoelectric problem has been proposed (8) The frequency response functions between actuators and sensors are obtained using a state space model of the control system extracted from the dynamic finite element analysis The importance of the in-plane component is ilustrated by a cantilever plate with four nearly collocated piezoceramic patches One situation where this wor is particularly applicable is in the use of structure-borne sensors to measure sound power radiation These are preferable to microphones in vibroacoustic control because they do not introduce time delays in the control loop It can be shown that, at low frequency, there is a strong correlation between the sound power radiated by a baffled plate and its volume velocity Electrode shaping to achieve modal filtering or inherent integration (Quadratically Weighted Strain Integration Sensor) has been proposed (6, 12) This technique was implemented for active noise control: the ASAC (Active Structural AcousticControl) panel exhibits a colocated quadratically shaped piezoelectric actuator/sensor pair (3) This paper stresses the influence of the in-plane components on the zeros of the open-loop frequency response functions of such colocated control systems An alternative design is investigated A noise radiation sensor consisting of an array of independent piezoelectric patches connected to an adaptive linear combiner has been proposed (11); the piezoelectric patches are located at the nodes of a rectangular mesh This strategy can be used for reconstructing the volume displacement of a baffled plate with arbitrary boundary conditions 1 PIEZOLAMINATED PLATE The constitutive equations of a linear piezoelectric material read (5) {T } = c E ]{S} e] T {E} (1) {D} = e]{s} + ε S ]{E} (2) where {T } = {T 11 T 22 T 33 T 23 T 13 T 12 } T is the stress vector, {S}={S 11 S 22 S 33 2S 23 2S 13 2S 12 } T the deformation vector, {E} = {E 1 E 2 E 3 } the electric field, {D} = {D 1 D 2 D 3 } the electric displacement, c]the elasticity constants matrix, ε] the dielectric constants, e] the piezoelectric constants (superscripts E, S and T indicate values at E, S and T constant respectively) 11 Single Layer in Plane Stress We consider a shell structure with embedded piezoelectric patches covered with electrodes The piezoelectric patches are parallel to the mid-plane and orthotropic in their plane The electric field and electric displacement are assumed uniform across the thicness and aligned on the normal to the mid-plane (direction 3) With the plane stress hypothesis, the constitutive equations can be reduced to T 11 T 22 T 12 = c E] S 11 S 22 2S 12 e 31 e 32 E (3) D = {e 31 e 31 } {S} + ε S E (4) where c E] is the stiffness matrix of the piezoelectric material in its othotropy axes In writing Equ(3), it has been assumed that the piezoelectric principal axes are parallel to the structural othotropy axes

2 and that there is no piezoelectric contribution to the shear strain (e 36 =) This is the case for most commonly used piezoelectric materials in laminar designs (eg PZT, PVDF) The analytical form of c E] can be found in any textboo on composite materials 12 Laminate A laminate is formed from several layers bonded together to act as a single layer material (Fig1); the bond between two layers is assumed to be perfect, so that the displacements remain continuous across the bond Figure 1: Multilayered material According to the Kirchhoff hypothesis, a fiber normal to the mid-plane remains so after deformation It follows that: {S} = {S } + z {κ} (5) where {S } is the mid-plane deformation and {κ}, the mid-plane curvature The constitutive equations for layer in the global axes of coordinates read {T } = Q ] {S} R T ] 1 e 31 e 32 E (6) D = {e 31 e 32 } R S ] {S} + ε E (7) where R T ] 1 is the transformation matrix relating the stresses in the local coordinate system (LT) to the global one (xy) Similarly, R S ] is the transformation matrix relating the strains in the global coordinate system (xy) to the local one (LT) The stiffness matrix of layer in the global coordinate system, Q ], is related to c E] by Q ] = R T ] 1 c E ] R S] (8) As mentioned before, the electric field E is assumed uniform across the thicness h = z z 1 of layer Thus, we have E = φ /h, where φ is the difference of electric potential between the electrodes covering the surface on each side of the piezoelectric layer The global constitutive equations of the laminate, which relate the resultant in-plane forces {N} and bending moments {M}, to the mid-plane strain {S } and curvature {κ} and the potential applied to the various electrodes can now be derived by integrating Equ(6) over the thicness of the laminate { N M } A B = B D n =1 z z 1 or { N A B = M} B D where n =1 ] { S κ ] z I 3 I3 } + R T ] 1 ] { } S + κ ] I 3 z m I 3 z m = z 1 + z 2 R T ] 1 e 31 e 32 e 31 e 32 φ h dz (9) φ (1) (11) is the distance from the mid-plane of layer to the mid-plane of the laminate The first term in the right hand side of Equ(1) is the classical stiffness matrix of a composite laminate, where the extensional stiffness matrix A], the bending stiffness matrix D] and the extension/bending coupling matrix B] are related to the individual layers according to the classical relationships: A] = ] Q (z z 1 ) B] = 1 ] Q 2 (z2 z 1) 2 D] = 1 3 ] Q (z3 z 3 1) (12) The second term in the right hand side of Equ(1) expresses the piezoelectric loading Similarly, substituting Equ(5) into Equ(7), we get { } S φ D = {e 31 e 32 } R S ] I 3 z I 3 ] ε κ (13) h Since we have assumed that the electric displacement D is constant over the thicness of the piezoelectric layer, this equation can be averaged over the thicness, leading to { } S D ={e 31 e 32 } R S ] I 3 z m I 3 ] ε φ κ (14) h The classical Kirchhoff theory neglects the transverse shear strains Alternative theories which accomodate the transverse shear strains have been developed and

3 have been found more accurate for thic shells (4) In the Mindlin formulation, a fiber normal to the mid-plane remains straight, but no longer orthogonal to the mid-plane Assuming that there is no piezoelectric contribution to the transverse shear strain (e 34 = e 35 = ), which is the case for most commonly used piezoelectric materials in laminar designs (eg PZT, PVDF), the global constitutive equations of the piezoelectric Mindlin shell can be derived in a straightforward manner from equations (1) and (14) The Kirchhoff formulation is ept here for clarity reasons Figure 2: Piezoelectric load 13 Actuation: piezoelectric loads Equation (1) shows that a voltage φ applied between the electrodes of a piezoelectric patch produces in-plane loads and moments: { N M} = I 3 z m I 3 ] e 31 R T ] 1 e 32 φ (15) If the piezoelectric properties are isotropic in the plane (e 31 = e 32 ), we have 1 1 e 31 R T ] 1 1 = e 31 1 (16) It follows that {N} = {M} = N x N y N xy M x M y M xy 1 = e 31φ 1 1 = e 31z m φ 1 (17) (18) We note that the in-plane forces and the bending moments are both hydrostatic; they are independant of the orientation of the facet We therefore conclude that the piezoelectric loads result in a uniform in-plane load N p and bending moment M p acting normally to the contour of the electrode as indicated on Fig2: N p = e 31 φ, M p = e 31 z m φ (19) where z m is the distance from the mid-plane of the piezoelectric patch to the mid-plane of the plate 14 Sensing Consider a piezoelectric patch connected to a charge amplifier as on Fig3 The charge amplifier imposes a zero voltage between the electrodes and the output voltage is proportional to the electric charge: φ out = Q = 1 D d (2) C r C r Figure 3: Piezoelectric sensor where D is given by Equ(14) If the piezoelectric properties are isotropic in the plane (e 31 = e 32 ), we have e 31 {1 1 } R S ] = e 31 {1 1 } (21) and Equ(2) becomes φ out = e 31 ( S C x + S ) y d r ] +z m (κ x + κ y ) d (22) The first integral represents the contribution of the average membrane strains over the electrode and the second, the contribution of the average bending moment Using the Green integral a d = an dl (23) C the foregoing result can be transformed into φ out = e ] 31 u w n dl + z m C r n dl C C (24) where the integrals extend to the contour of the electrode The first term is the mid-plane displacement normal to the contour while the second is the slope of the mid-plane in the plane normal to the contour (Fig4) The comparison with Equ(19) shows a strong duality between actuation and sensing It is worth insisting that for both the actuator and the

4 sensor, it is not the shape of the piezoelectric patch that matters, but rather the shape of the electrodes Figure 4: Contribution to the output of the piezoelectric isotropic sensor (e 31 = e 32 ) 15 Finite element formulation The dynamic equations of a piezoelectric continuum can be derived from the Hamilton principle, in which the Lagrangian and the virtual wor are properly adapted to include the electrical contributions as well as the mechanical ones The potential energy density of a piezoelectric material includes contributions from the strain energy and from the electrostatic energy (13) H = 1 2 {S} T {T } {E} T {D} ] (25) Similarly, the virtual wor density reads δw = {δu} T {F } δφ σ (26) where {F } is the external force and σ is the electric charge From Equ(25) and (26), the analogy between electrical and mechanical variables can be deduced (Table 1) Mechanical Force {F } Displ {u} Stress {T } Strain {S} σ φ {D} {E} Electrical Charge Voltage Electric Displ Electric Field Table 1: Electromechanical analogy The variational principle governing the piezoelectric materials follows from the substitution of H and δw into the Hamilton principle (1) For the specific case of the piezoelectric plate, we can write the potential energy H = 1 {S T κ T } { } ] N E D d (27) 2 M Upon substituting Equ(1) and (14) into Equ(27), one gets the expression of the potential energy for a piezoelectric plate The electrical degrees of freedom are the voltages φ across the piezoelectric layers; it is assumed that the potential is constant over each element (this implies that the finite element mesh follows the shape of the electrodes) Introducing the matrix of the shape functions N ] (relating the displacement field to the nodal displacements {q}), and the matrix B] of their derivatives (relating the strain field to the nodal displacements), into the Hamilton principle and integrating by part with respect to time, we get = {δq} T mn ] T N ]d { q} ] + {δq} T B] T A B B]d {q} B D + {δq} T + { δφ } ] B] T E T E T z d m + { δφ } {δq} T E φ E z m B]d {q} ε /h d N ] T {P S }d {δq} T {P c } φ + { δφ } { σ } T (28) where we have introduced {E} = {e 31 e 32 } R S ] (29) and we have used the fact that R T ] 1 {e 31 e 32 } T = {E}T (3) P S ] and P c ] are respectively the external distributed forces and concentrated forces and { δφ } { σ } T = δφ σ is the electrical wor done by the external charges σ brought to the electrodes Equation (28) must be verified for any {δq} and {δφ} compatible with the boundary conditions; It follows that, for any element, we have M qq ]{ q} + K qq ]{q} + K qφ ]{φ} = {f} (31) K φq ] {q} + K φφ ]{φ} = {g} (32) where the element mass, stiffness, piezoelectric cou-

5 pling and capacitance matrices are defined as M qq ] = mn ] T N ]d (33) ] K qq ] = B] T A B B]d (34) B D ] K qφ ] = B] T E T E T z d (35) m K φφ ] = ε /h (36) K φq ] = K qφ ] T (37) and the external mechanical forces and electric charge: {f} = N ] T {P S }d + {P c } {g} = { σ } T The element coordinates {q} and {φ} are related to the global coordinates {Q} and {Φ} The assembly taes into account the equipotentiality condition of the electrodes; this reduces the number of electric variables to the number of electrodes Upon carrying out the assembly, we get the global system of equations M QQ ]{ Q} + K QQ ]{Q} + K QΦ ]{Φ} = {F } (38) K ΦQ ] {Q} + K ΦΦ ]{Φ} = {G} (39) where the global matrices can be derived in a straightforward manner from the element matrices (33) to (37) As for the element matrices, the global coupling matrices satisfy K ΦQ ] = K QΦ ] T The element used for the actual implementation is the Mindlin shell element from the commercial finite element pacage Samcef (Samtech sa) 16 State Space Model Equ(38) can be complemented with a damping term C]{ U} to obtain the full equation of dynamics and the sensor equation: {} = M]{ Q} + C]{ Q} +K QQ ]{Q} + K QΦ ]{Φ} (4) {G} = K ΦQ ] {Q} + K ΦΦ ]{Φ} (41) where {Q} represents the mechanical dof, {Φ} the electric potential dof, M] the inertial matrix, C] the damping matrix, K QQ ] the mechanical stiffness matrix, K QΦ ] = K ΦQ ] T the electromechanical coupling matrix and K ΦΦ ] the electric capacitance matrix Actuation is done by imposing a voltage {Φ} on the actuators and sensing by imposing {Φ} = {} and measuring the electric charges {G} appearing on the sensors Using a truncated modal decomposition (n decoupled modes) {Q} = Z]{x(t)}, where Z] represents the n modal shapes and {x(t)} the n modal amplitudes, Equ(4) and (41) become {} = M]Z]{ẍ} + C]Z]{ẋ} +K QQ ]Z]{x} + K (i) QΦ ]{Φ} (42) {G} = K (o) ΦQ ]Z]{x} + K ΦΦ]{Φ} (43) Left-multiplying Equ(42) by Z] T, using the orthogonality properties of the mode shapes Z] T M] Z] = diag(µ ) (44) Z] T K] Z] = diag(µ ω) 2 (45) and a classical damping Z] T C]Z] = diag(2ξ µ ω ) (46) the dynamic equations of the system in the state space representation finally read: { } ẋ ẍ = {G} = { } I ẋ 2ξ] 2 x ] µ 1 Z T K (i) {Φ} (47) QΦ ] { } K (o) T x QΦ Z + D ẋ HF ] {Φ} (48) where the modal shapes Z], the modal frequencies ] = diag(ω ), the modal masses µ] = diag(µ ), the modal electric charge on the sensor K (o) ΦQ ]Z], and the modal electric charge on the actuators Z] T K (i) QΦ ], representing the participation factor of the actuators to each mode, are obtained from a dynamic finite element analysis ξ] = diag(ξ ) are the modal classical damping ratios of the considered structure and D HF ] is the static contribution of the high frequency mode; its elements are given by D lm = d lm n =1 (K (o) ΦQ Z ) l (Z T K(i) QΦ ) m µ ω 2 (49) where d lm is the charge appearing on the l th sensor when a unit voltage is applied on the m th actuator and is obtained from a static finite element analysis Such a state space representation is easily implemented in a control oriented software allowing the designer to extract the various frequency response functions and use the available control design tools

6 Figure 5: Cantilever plate with piezoceramics Figure 7: Simulation results 2 APPLICATIONS: 21 Influence of the In-plane Component Consider the cantilever plate represented on Fig5; the steel plate is 5 mm thic and four piezoceramic strips of 25 µm thicness are bounded symmetrically as indicated in the figure, 15 mm from the clamp The size of the piezos is respectively 55 mm 25 mm for p 1 and p 3, and 55 mm 125 mm for p 2 and p 4 p 1 is used as actuator while the sensor is taen successively as p 2, p 3 and p 4 The experimental frequency response functions between a voltage applied to p 1 and the electric charge appearing successively on p 2, p 3 and p 4 when they are connected to a charge amplifier are shown on Fig6 22 ASAC Panel The ASAC (Active Structural Acoustical Control) (3) plate is a volume velocity control device based on the same principle as the QWSIS (Quadratically Weighted Strain Integrated Sensor) sensor (12) for both actuation and sensing It consists in a clamped 1 mm thic plate of aluminium (42 mm 32 mm) covered on both side with 5 mm thic piezoelectric PVDF film (4 mm 3 mm) That configuration exhibits a pair of colocated actuator/sensor The electrodes of both layers are milled in a quadratic shape Figure 8: Experimental setup and FE mesh Figure 6: Experimental results We note that the frequency response functions, particularly the location of the zeros, vary substancially from one configuration to the other This is because the frequency response functions of nearly collocated control systems are very much dependant on local effects, in particular the membrane strain in the thin steel plate between the piezo patches Figure 7 shows the numerical results, based on the finite element analysis, corresponding to the three sensor configurations; they agree reasonably well with the experiments For the actual laboratory model, the actuation and sensing layers electrodes present 24 strips The direction of smaller piezoelectric coupling coefficient (d 32 ) is perpendicular to the strips The experimental setup and the finite element mesh used are shown on Fig8 Since the performance of the control system is to a large extend related to the distance between the poles and the zeros of the open-loop frequency response function, these results were considered as disappointing, contrary to simplified analytical predictions which indicated far better performances (3) At first, this lac of performance was attributed to imperfect alignment of top and bottom layers (non colocated actuator/sensor pair) or to an electrical coupling due to the wiring

7 In fact, the finite element based simulations have shown that this lac of controllability is actually due to local membrane effects (9), neglected in the first analytical models together with the static contribution of the unmodelled high frequency modes (also called residual mode) In a first attempt to model the open-loop frequency response function of the ASAC panel using finite elements, the agreement of results with the experiment was rather unsatisfactory (Fig9, FE #1) It appeared soon that the boundary conditions were not those of a clamped plate: in the actual experiment, the plate was almost free to move in its plane The in-plane movement of the plate results in an even stronger influence of the membrane components and, therefore, in a stronger in-plane mechanical coupling between actuator and sensor This induces an important feedthrough term in the frequency response function: a substantial part of the strain induced by the actuator induces directly membrane strain in the sensor, without contributing to the transverse displacement which produces the volume velocity (useful control) The frequency response function of (Fig9, FE #2) was obtained by freeing the in-plane movement of the plate in the finite element model; it shows a very good agreement with the experimental result the worst possible configuration However, for the actuator and the sensor taen separately, the direction of the strips has no influence on their characteristics From this observation, the idea raised that the feedthrough component could be substantially reduced by using sensor and actuator strips perpendicular to each other Figure 1: FE mesh Figure 11: Open-loop frequency response functions Figure 9: Open loop frequency response 23 Alternative Design Note that, in the current design, the in-plane coupling is particularly strong because the direction of higher piezoelectric effect (e 31 e 32 ) for the sensor and the actuator are parallel; the most important strain is induced in a direction parallel to the direction of the strips and the sensor has the highest sensitivity to the strain in the direction of the strips The actuation and sensing strips being layed in the same direction for the ASAC plate, it is in By using sensor strips perpendicular to the actuator strips, the control device would then exhibit a cross-ply actuator/sensor architecture and the inplane feedthrough term would be greatly reduced The FE-based tools allow to modelize such architectures quite easily and to extract the corresponding frequency response functions to verify if this alternative is any better The mesh used is represented on Fig1; the sensor electrode forms a right angle with the actuator electrode The comparison of the frequency response functions between the voltage applied to the actuator layer and the charge measured on the sensor layer for the parallel and cross-ply architectures for two piezoelectric anisotropy ratios are represented on Fig11 Indeed, the distance between the poles and zeros of the frequency response function is much larger for the cross-ply configuration, as compared to the parallel configuration, and the distance increases when

8 the piezoelectric anisotropy ratio χ of the material decreases As a result, improved closed-loop performances may be expected from the cross-ply design This test case illustrates the situation of shell structures with embedded piezoelectric actuators and sensors where they are nearly collocated It stresses the importance of membrane components on the zeros of the frequency response function These local effects can easily be accounted for by the developped modelling tools based on finite elements 24 Array Sensor Figure 13: Experimental setup and FE mesh 124 cm, 4 mm thic) covered with an array of 4 by 8 piezoelectric patches (PZT mm 25 mm, 25 mm thic) A scanner laser interferometer was used to measure the velocity of an array of points over the window to deduce the volume velocity The excitation was provided by two shaers actuating the window directly Figure 12: Volume displacement sensor A noise radiation sensor consisting of an array of independent piezoelectric patches connected to an adaptive linear combiner was proposed in (2, 11) The coefficients of the linear combiner are adapted in such a way that the mean-square error between the reconstructed volume displacement (or velocity) and either numerical or experimental data is minimized The electric charges Q i induced on the various patches by the plate vibration are the independent inputs of a multiple input adaptive linear combiner The coefficients α i of the linear combiner are adapted in such a way that the mean-square error between the reconstructed volume displacement (or velocity) and either numerical or experimental data is minimized It must be noted that the same array sensor can also be used as modal filter by suitably adapting the coefficients α i of the linear combiner This strategy can be used for reconstructing the volume displacement of a baffled plate with arbitrary boundary conditions If the piezoelectric patches are connected to current amplifiers instead of charge amplifiers, the output signal becomes the volume velocity instead of the volume displacement The laboratory demonstration model (Fig13) consists of a simply supported glass plate (54 cm Figure 14: Freq Response Functions/Shaer #1 The finite element mesh used for the numerical anal-

9 ysis is represented on Fig13 The 3 first vibration modes were taen into account for the dynamic analysis Figure 14 shows the comparison between the frequency response functions between the excitation of Shaer #1 (in the center of the window) and, respectively, sensors 7, 14 and the volume velocity obtained by finite element analysis and experimentally 3 CONCLUSIONS The theory of piezolaminated plates has been developed; the fundamental equations governing the equivalent piezoelectric loads of a piezoelectric actuator and the output of a piezoelectric sensor have been derived A state space model has been obtained and the importance of the in-plane components in the open loop frequency response functions has been stressed Two applications of the developed tools in vibroacoustic control have been described and shown that good performances are achieved The importance of the in-plane components in the open-loop frequency response functions has been illustrated ACKNOWLEDGEMENTS This study has been supported by a research grant from the Région Wallonne, Direction Générale des Technologies, de la Recherche et de l Energie; The support of the IUAP-4/24 on Intelligent Mechatronic Systems is also anowledged; The technical assistance of Samtech sa is deeply appreciated 6 Lee CK and Moon FC, 199 Modal Sensors/Actuators Journal of Applied Mechanics, 57: Loix N, Piefort V and Preumont A, 1998 Modeling aspects of active structures with collocated piezoelectric actuators and sensors Benelux Quarterly Journal on Automatic Control (Journal a), 39 8 Piefort V, 21 Finite Element Modelling of Piezoelectric Active Structures PhD thesis, Université Libre de Bruxelles, Brussels, Belgium 9 Piefort V and Henrioulle K, 2 Modelling of Smart Structures with Colocated Piezoelectric Actuator/Sensor Pairs: Influence of the in-plane Components 5th International Conference on Computational Structures Technology, Leuven, Belgium 1 Preumont A, 1997 Vibration Control of Active Structures - An Introduction Kluwer Academic Publishers, Dordrecht, The Netherlands 11 Preumont A, François A and Dubru S, 1999 Piezoelectric Array Sensing for Real-Time, Broad- Band Sound Radiation Measurement Journal of Vibration and Acoustics, Rex J and Elliott SJ, 1992 The QWSIS - A New Sensor for Structural Radiation Control MOVIC-1, Yoohama 13 Tiersten HF, 1967 Hamilton s Principle For Linear Piezoelectric Media In Proceedings of the IEEE, REFERENCES 1 Alli H and Hughes TJR, 197 Finite Element Method for Piezoelectric Vibration International Journal for Numerical Methods in Engineering, 2: François A, De Man P and Preumont A, 21 Piezoelectric Array Sensing of Volume Displacement: A Hardware Demonstration Journal of Sound and Vibration, 244: Gardonio P, Lee Y, Elliott S and Debost S, 1999 Active Control of Sound Transmission Through a Panel with a Matched PVDF Sensor and Actuator Pair Active 99, Fort Lauderdale, Florida, USA 4 Hughes TJR, 1987 The Finite Element Method Prentice-Hall International Editions IEEE Standard on Piezoelectricity ANSI/IEEE Std