Gausmann, R.

Seemann, W.

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Gausmann, Rainer; Seemann, WolfgangA note on the nonlinear modeling of piezoelectric rodsPiezoceramic materials are usually calculated using linear constitutive equations if the exciting electric field is small.If the structure is excited near a resonance frequency the assumption of a linear material becomes invalid. Thereforenonlinear constitutive equations have to be used if the super harmonics in the measured velocity signals should bemodeled. In this paper nonlinear constitutive equations are derived from the electric enthalpy density to describethe longitudinal oscillations of a piezoceramic rod. The kinetic energy considers the inertia effects in transversaldirection so the model is not limited to the slender rod theory. With Hamilton’s principle a variational principle isderived which is approximated using a Rayleigh Ritz ansatz with the first eigenfunction of the linearized system. Theresulting nonlinear ODE is solved by application of the harmonic balance method. The amplitudes of displacementof the fundamental, the second and the third super harmonic oscillations are solved numerically. In a first stepthe results of the theoretical model are compared with measurements. The material parameters of the nonlinearconstitutive equations are calculated using a parametric identification method. As a result it is shown that thetheoretical model can describe the Duffing type nonlinearities found in measurements.1. Theoretical ModelThe model of the piezoceramic rod is shown in figure 1. A coordinate sys-tem is introduced such that the 3-axis is aligned with the poling direction.The left and right end of the rod are stress-free (mechanical boundary con-ditions) and the rod is excited by a harmonic electric voltage ϕ(t) (electricalboundary condition). To derive a variational principle for the piezoceramic,it is convenient to use Hamilton’s principle for dielectric mediaδt1∫t0L dt +t1∫t0δW dt = 0, L = T − H (1)Figure 1: Model of the piezoceramicrodin which L is a modified Lagrangefunction and δW the virtual work [3]. From measurements it is known, that thelongitudinal oscillations of a piezoelectric rod show nonlinear behavior if the rod is excited near a resonance frequencyeven at low electric fields. To model the observed jump phenomenon and the superharmonics an uniaxial state ofstresses is assumed. Furthermore the electric displacement in transversal direction will be neglected. Therefore theelectric enthalphy can be written asH =∫V14c2S43 + (13c1 − 13e2E3)S33 + (12c33 − 12e1E3)S23 − e33E3S3 −12ε33E23dV. (2)In this equation the independent variables are the electric field E3 and the strain S3, where the index ()3 denotesthe polarization axis of the rod. Furthermore the following abbreviations are usedc33 = cE33 −2(cE13)2cE11 + cE12, e33 = e33 − 2cE13e31cE11 + cE12, ε33 = εS33 +2e231cE11 + cE12, (3)in which cij denotes the stiffness tensor components, eij the piezoelectric tensor components and εij the permittivitytensor components, in case the Voigt notation is used. By assuming a linear velocity in transversal direction, thekinetic energy can be written asT =∫V12ρu̇2 +12ρ[(yṠ1)2 + (zṠ2)2]dV. (4)A detailed derivation of the above potential functions can be found in [1] and [2]. Nevertheless the nonlinearPAMM · Proc. Appl. Math. Mech. 3, 100–101 (2003) / DOI 10.1002/pamm.200310327 First published in:EVA-STAR (Elektronisches Volltextarchiv – Scientific Articles Repository) http://digbib.ubka.uni-karlsruhe.de/volltexte/1000011092constitutive equations, which are based on equation (2) are given byT3 =∂H∂S3= c2S33 + (c1 − e2E3)S23 + (c33 − e1E3)S3 − e33E3 (5)D3 = − ∂H∂E3=13e2S33 +12e1S23 + e33S3 + ε33E3, (6)with the stress T3 and the electric displacement D3 in poling direction.Introducing the potential functions into the variational principle (1) and using Gauss law for dielectric mediadivD = 0 and the field equation E = −gradϕ, an equation is obtained, which depends only on the displacementu(x, t) and the electric potential ϕ(t) and various derivatives of u(x, t) and ϕ(t) with respect to the time t andthe coordinate x. The variational principle can not be solved analytically. Therefore a first order Rayleigh-Ritzansatz is introduced by using the first eigenfunction of the linearized systemu(x, t) = U(x)p(t). (7)Obtained is a nonlinear ordinary differential equation of the formC1 p̈(t) + C2 p̈(t)p(t) + C3 p̈(t)p(t)2 + C4 p̈(t)p(t)3 + C5 p̈(t)p(t)4 + C6 ṗ(t) + C7 ṗ(t)2p(t) +C8 ṗ(t)2p(t)2 + C9 ṗ(t)2p(t)3 + C10 ṗ(t)2 + C11 p(t)5 + C12 p(t)4 + C13 p(t)3 + C14 p(t)2 +C15 p(t)2ϕ(t) + C16︸︷︷︸=0p(t)2ϕ̈(t) + C17 p(t) + C18 p(t)ϕ(t) + C19︸︷︷︸=0p(t)ϕ̈(t) + C20 ϕ̈(t) = ϕ(t). (8)It can be seen, that a parameter excitation by the voltage ϕ(t) occurs. The harmonic balance method is used to solvethe nonlinear ODE. The fundamental, the first and second higher harmonics are taken into account. The resultingequations are solved numerically for the amplitudes of the displacement functions.2. ResultsThe comparison of the fundamental harmonic oscillation with a measure-ment for a low voltage excitation is shown in figure 2. The measurements arecarried out at a piezoceramic rod (diameter 25mm, length 20mm, materialPIC141 PI-Ceramic, Germany). The material parameters used for the simu-lation are determined by using the Nelder-Mead simplex algorithm wherethe objective function is the L2-norm of the difference between measurementand simulation data. As one can see it is possible to describe the Duffing-type nonlinearities with the presented model. The figure shows the sweep upmode, which means that the frequency is increased, while the amplitudes ofvelocity are measured with a gain-phase analyzer. It is well known, that thefrequency at which the jump occurs depends on the sweep mode. Further-more it can be seen, that the amplitude function inclines to the left, whichis the so called softening effect. The material parameters used for the abovesimulation are in SI-units: c1 = 1.2e−4, c2 = −9e14, e1 = 1e−4, e2 = −1e8.The main parameters, which are responsible for the nonlinear behavior arethe piezoelectric parameter e2 and the stiffness parameter c2. Both valueslead the curve incline to left in case of negative values. A detailed discussionof the parameter identification will be published soon. Future work will fo-cus on the improvement of the parameter identification and the applicationof the model to a piezoelectric transformer [1].6.5 6.55 6.6 6.65 6.7 6.75 6.8x 10400.511.522.533.54x 10−7frequency [1/s]amplitude [m]fundamental oscillationmeasurementsimulationFigure 2: Comparison of the funda-mental amplitudes (simulation / mea-surement)3. References1 Seemann, W., Gausmann R. A refined model for a piezoelectric transformer. Proceedings of DETC’01, ASME 2001,VIB-21485, Pittsburgh, (2001).2 Seemann, W. and Gausmann, R. A note on the strong nonlinear behavior of piezoceramics excited with a weak electricfield. SPIE Smart Structures and Materials, Vol. 4333 (2001).3 Tiersten, H.F. Linear Piezoelectric Plate Vibrations. Plenum Press, New York (1969).Dipl.-Ing. Rainer Gausmann, Prof. Dr.-Ing. Wolfgang Seemann, University of Kaiserslautern,Department of Mechanical and Process Engineering, PO Box 3049, 67653 Kaiserslautern, GermanySection 1: Linear and nonlinear oscillations 101             

A Note on the Nonlinear Modelling of Piezoelectric Rods

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A Note on the Nonlinear Modelling of Piezoelectric Rods

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