Modeling of smart piezoelectric shell structures with finite elements ISMA 25 - Leuven - Belgium -September 2000 V Piefort & A Preumont Active Structures Laboratory, Université Libre de Bruxelles, Belgium e-mail: scmero@ulbacbe Abstract The design of control systems involving piezoelectric actuators and sensors requires an accurate nowledge of the transfer functions between the inputs and the outputs of the system These are not easy to determine numerically, particularly for shell structures with embedded distributed actuators and sensors The situation where they are nearly collocated is particularly critical, because the zeros of the transfer functions are dominated by local effects which can only be accounted for by finite elements 5] In the first part, this paper develops the theory of piezolaminated plates The fundamental equations governing the equivalent piezoelectric loads and sensor output are derived The reciprocity between piezoactuation and piezosensing is pointed out Piezoelectric finite elements are developed based on Samcef shell elements The interfacing with a control oriented software environment is discussed in the second part, where some applications are presented 1 Piezolaminated plate The constitutive equations of a linear piezoelectric material read 4] {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 S 11 e 31 T 22 T = c E] S 22 12 2S e 32 12 0 E (3) D = {e 31 e 31 0} {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 and that there is no piezoelectric contribution to the shear strain (e 36 =0) 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 According to the Kirchhoff hypothesis, a fiber normal to the mid-plane remains so after deformation It
follows that: Figure 1: Multilayered material {S} = {S 0 } + z {κ} (5) where {S 0 } is the mid-plane deformation and {κ}, the mid-plane curvature The constitutive equations for layer in the global axes of coordinates read {T } = e 31 Q {S} R T ] 1 e 32 E (6) D = {e 31 e 32 0} 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 0 } 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 ] { } S0 + κ n =1 z z 1 I3 z I 3 ] R T ] 1 e 31 e 32 φ h dz (9) or { } { } N A B S0 = + M B D κ n I e 31 3 R =1 z m I T ] 1 e 32 φ (10) 3 where z m = z 1 + z (11) 2 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(10) 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] = B] = 1 2 D] = 1 3 Q (z z 1 ) Q (z2 z 1) 2 Q (z3 z 1) 3 (12) The second term in the right hand side of Equ(10) expresses the piezoelectric loading Similarly, substituting Equ(5) into Equ(7), we get { } S0 φ D = {e 31 e 32 0} R S ] I 3 z I 3 ] ε κ h (13) 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 { } S0 D ={e 31 e 32 0} R S ] I 3 z m I 3 ] ε φ κ h (14) The classical Kirchhoff theory neglects the transverse shear strains Alternative theories which accomodate the transverse shear strains have been developed and have been found more accurate for thic shells 3] 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 = 0), 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 (10) and (14) The Kirchhoff formulation is ept here for clarity reasons 13 Actuation: piezoelectric loads Equation (10) shows that a voltage φ applied between the electrodes of a piezoelectric patch produces inplane 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 (20) C r C r Figure 2: Piezoelectric load 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 0} R S ] = e 31 {1 1 0} (21) and Equ(20) becomes φ out = e 31 C r +z m ( ) Sx 0 + Sy 0 d ] (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) the foregoing result can be transformed into φ out = e ] 31 u 0 w n dl + z m C r C C n dl 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 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 8] H = 1 2 ] {S} T {T } {E} T {D} Similarly, the virtual wor density reads (25) 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 { } { } ] S0 T κ T N E D d (27) 2 M Upon substituting Equ(10) 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 0 = {δq} T + {δq} T + {δq} T mn ] T N ]d { q} B] T A B B]d {q} B D B] T E T E T z d m φ + { δφ } E E z m B]d {q} 0 + { δφ } ε /h d φ 0 {δq} T N ] T {P S }d {δq} T {P c } + { δφ } { σ } T (28) where we have introduced {E} = {e 31 e 32 0} R S ] (29) and we have used the fact that R T ] 1 {e 31 e 32 0} T = {E}T (30) 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 coupling 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 0 K φφ ] = ε /h (36) 0 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) 2 Applications 21 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: {0} = M]{ Q} + C]{ Q} +K QQ ]{Q} + K QΦ ]{Φ} (40) {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} = {0} 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(40) and (41) become {0} = 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 } { } 0 I x = ẍ 2 2ξ ẋ ] 0 µ 1 Z T K (i) {Φ}(47) QΦ ] { } {G} = K (o) T x QΦ Z 0 + 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 transfer functions and use the control design tools membrane strain in the thin steel plate between the piezo patches Figure 7 shows the numerical results, 22 Influence of in-plane contribution Figure 7: Simulation results based on the finite element analysis, corresponding to the three sensor configurations; they agree reasonably well with the experiments Figure 5: Cantilever plate with piezoceramics Consider the cantilever plate represented on Fig5; the steel plate is 05 mm thic and four piezoceramic strips of 250 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 transfer 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 23 ASAC panel The ASAC (Active Structural Acoustical Control) 2] plate is a volume velocity control device based on the same principle as the QWSIS (Quadratically Weighted Strain Integrated Sensor) sensor 7] for both actuation and sensing It consists in a clamped 1 mm thic plate of aluminium (420 mm 320 mm) covered on both side with 05 mm thic piezoelectric PVDF film (400 mm 300 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 transfer functions, particularly the location of the zeros, vary substancially from one configuration to the other This is because the transfer functions of nearly collocated control systems are very much dependant on local effects, in particular the 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 In a first attempt to modelize this control device analytically 2], the agreement with experiments was rather poor The experimental setup does not show the
good controllability characteristics it was predicted The foreseen explanations for the poor controllability of the actual experimental setup were either that the top and bottom layers were not perfectly aligned and therefore did not form a colocalized actuator/sensor pair or that there was a electrical coupling between actuator and sensor layers 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 (Fig10) Figure 10: Volume displacement sensor Figure 9: Open loop transfer functions The finite element simulations have shown that this lac of controllability is actually due to local membrane effects, neglected in the first analytical models The in-plane movement of the plate (not fixed in the actual experiment) results in a stronger influence of the membrane component and, therefore, in a stronger in-plane mechanical coupling between actuator and sensor This induces an important feedthrough term in the transfer function The results of the FE simulation are shown on Fig9; they show a very good agreement with the experiments 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 transfer function These local effects can easily be accounted for by the developped modelling tools based on finite elements 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 24 ALC sensor A noise radiation sensor consisting of an array of independent piezoelectric patches connected to an adaptive linear combiner was proposed in 6] 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 Figure 11: Experimental setup and FE mesh The laboratory demonstration model (Fig11) consists of a simply supported glass plate (54 cm 124 cm, 4 mm thic) covered with an array of 4 by 8 piezoelectric patches (PZT - 1375 mm 25 mm, 025 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 The finite element mesh used for the numerical analysis is represented on Fig11 The 30 first vibration modes were taen into account for the dynamic analysis Figure 12 shows the comparison between the transfer 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 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 transfer functions has been stressed Two applications in vibroacoustics have been presented The finite element results shown very good agreement with experimental datas Acnowledgements 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 and the collaboration of K Henrioulle (KUL- PMA) are also acnowledged The technical assistance of Samtech sa is deeply appreciated The ASAC panel was developed and built by P Gardonio from ISVR (UK) and S Debost from Thomson (F) under the EC Brite Euram research project DAFNOR supported by the Directorate General for Science, Research and Development of the CEC References 1 H Alli, T J R Hughes Finite element method for piezoelectric vibration International Journal for Numerical Methods in Engineering, Vol 2, pp 151 157, (1970) 2 P Gardonio, Y Lee, S Elliot, S Debost Active control of sound transmission through a panel with a matched PVDF sensor and actuator pair Active 99, Fort Lauderdale, Florida, USA, (December 1999) 3 T J R Hughes The Finite Element Method Prentice-Hall International Editions, (1987) 4 IEEE standard on piezoelectricity ANSI/IEEE Std 176-1987, (January 1988) Figure 12: Transfer Functions /Shaer #1 3 Conclusions 5 A Preumont Vibration Control of Active Structures - An Introduction Kluwer Academic Publishers, (1997) 6 A Preumont, A François, S Dubru Piezoelectric array sensing for real-time, broad-band sound radiation measurement Journal of Vibration and Acoustics, Vol 121, (Oct 1999) 7 J Rex, S J Elliott The QWSIS - a new sensor for structural radiation control MOVIC-1, Yoohama, (September 1992) 8 H F Tiersten Hamilton s principle for linear piezoelectric media In Proceedings of the IEEE, pp 1523 1524 (August 1967)