Gausmann, R.

Seemann, W.

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Gausmann, R.; Seemann, W.A Note on a Nonlinear Model of a Piezoelectric RodIf piezoceramics are excited by weak electric fields a nonlinear behavior can be observed, if the excitation frequency isclose to a resonance frequency of the system. To derive a theoretical model nonlinear constitutive equations are used,to describe the longitudinal oscillations of a slender piezoceramic rod near the first resonance frequency. Hamilton’sprinciple is used to receive a variational principle for the piezoelectric rod. Introducing a Rayleigh Ritz ansatz withthe eigenfunctions of the linearized system to approximate the exact solution leads to nonlinear ordinary differentialequations. These equations are approximated with the method of harmonic balance. Finally it is possible to calculatethe amplitudes of the displacements numerically. As a result it is shown, that the Duffing type nonlinearities foundin measurements can be described with this model.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). The thickness to length ratio is assumed to be small,thus simple rod theory is used. Furthermore, the influence of the electrodesis neglected and the cross sectional area is assumed to be constant in thetheoretical model. To derive a variational principle for the piezoceramic, itis convenient to use HAMILTON’s principle for dielectric continua Figure 1: Model of the piezoceramicrodδ∫ t2t1L(u, ϕ) dt +∫ t2t1δW(u, ϕ) dt = 0, withL =∫V(T − H) dV. (1)In equation (1) H denotes the electric enthalpy density and T the kinetic energy density, which are given byT =12ρu̇2, H =12c0S23 +13c1S33 +14c2S43 − e0S3E3 −12e1S23E3 −13e2S33E3 −12εS33E23 . (2)The field quantities are the strain S3 and the electric field E3. In the second equation terms of higher order areintroduced to describe the nonlinear behavior of the piezoceramics. The stiffness constants are denoted by c0, c1,and c2, additionally the abbreviations e0 = c0d0, e1 = c0d1 + c1d0, e2 = c0d2 + c2d0 + c1d1 are introduced. Thepiezoelectric constants are named d0, d1 and d2. The nonlinear constitutive equations can be obtained by the partialderivatives of the electric enthalpy density with respect to the strain S3 and the electric field E3T3 =∂H∂S3= c0S3 + c1S23 + c2S33 − e0E3 − e1S3E3 − e2S23E3, (3)D3 = − ∂H∂E3= e0S3 +12e1S23 +13e2S33 + εS33E3. (4)In equation (3) and (4) T3 denotes the tension in longitudinal direction and D3 the dielectric displacement. Itis obvious that the stiffness and the piezoelectric coupling depend in a nonlinear way on the strain. Due to themechanical and electrical boundary conditions the virtual work δW in Hamilton’s principle (1) reduces to thematerial damping. From the second nonlinear constitutive equation (4) a relation for the electric field is derived.Applying the field equation E = −grad ϕ to the electric field, it is possible to receive the dielectric displacementdepending on the electric potential, the displacement and its derivatives. Introducing the variations of equation (2),the electric field, the electric displacement and their variations into Hamilton’s principle (1) an equation of theformA∫ t2t1∫ l0f (ϕ , ü , u̇,x , δu , δu,x) dx dt = 0 (5)PAMM · Proc. Appl. Math. Mech. 2, 64–65 (2003) / DOI 10.1002/pamm.200310018 First published in:EVA-STAR (Elektronisches Volltextarchiv – Scientific Articles Repository) http://digbib.ubka.uni-karlsruhe.de/volltexte/1000011091 is obtained. It is not possible to find an exact solution for this equation. Therefore, a Rayleigh-Ritz ansatz isused to get an approximate nonlinear ordinary differential equation. In this paper only the first eigenfunction of thelinearized problem is used as an approximation for the exact solution. For this simplified case a nonlinear ordinarydifferential equation is obtainedCmp̈ + Cdṗ + C5p5 + C4p4 + C3p3 + (C2aϕ + C2b)p2 + (C1aϕ + C1b)p = Cϕϕ, (6)with the coefficients Ci which depend on material properties and dimensions of the rod. It can be seen that theODE is inhomogeneous and of 5th order. Furthermore there is a parametric excitation by C2aϕ and C1aϕ due to thetime dependence of the excitation voltage ϕ(t). Solutions of this nonlinear differential equation may be calculatedby the method of harmonic balance. As in experiments only the fundamental and the second harmonic frequencyare observed, an ansatz of such a form is used to approximate the exact solution of (6). Additional a harmonicexcitation voltage ϕ(t) = ϕ̂ cos Ωt is assumed. Finally, a solution for the displacement amplitudes can be obtainednumerically and two transfer functions are defined. The intention is to receive a relation between the amplitudes ofthe first and second harmonic oscillation to the amplitude of the excitation voltage.2. ResultsTo calculate the influence of the nonlinear parameters in the constitutiveequations (3) and (4) the first transfer function is used. Certain materialparameters are chosen and the electric field strength is approximately 120V/mm, considered in the range of low level signals. The nonlinear materialparameters c1, c2, d1 and d2 are varied and their influence on the transferfunction is investigated. An excitation near the first resonance frequencyshows a nonlinear behavior in form of Duffing type nonlinearities. Frommeasurements it is known that the transfer curve inclines to the left nearthe first resonance frequency. To investigate the influence of the nonlinearparameters, each parameter is varied, while the others are set to zero. Thevalue of the inner damping is constant d= 0.45 Ns for all simulations. Itcan be shown, that c1, d1, positive c2 and positive d2 lead to curves V1(Ω),which incline to the right. For values of c2 and d2, which are negative, thecurves V1(Ω) incline to the left. This is the behavior which is observed inmeasurements.5 5.05 5.1 5.15 5.2 5.25 5.3x 10412345678x 10−8frequency [Hz]Transfer Function [m/V]Transfer Function for Amplitude 1φ= 0.1 kVφ= 1 kV  φ= 5 kV  φ= 10 kV φ= 20 kV Figure 2: Transfer functions V1(Ω) forvarying ϕ̂Figure 2 shows the influence on the transfer curve for varying excitation voltages. In this simulation the piezoelectricconstant is d2= −10−7 m/V and the other nonlinear parameters are set to zero. The typical Duffing type nonlin-earities can be observed. The curves incline to the left and the jump phenomena occurs. Moreover the difference atwhich the curves achieve their maximum can be seen. This behavior can be found in measurements, too.3. ConclusionsIn this paper a nonlinear model of a piezoceramic rod is presented. With the electric enthalpy density and kineticenergy density Hamilton’s principle for dielectric media is used to derive an integral form into which a Ritz ansatzis introduced. The resulting nonlinear ordinary differential equation is approximated by the method of harmonicbalance. The Duffing type nonlinearities, as they can be observed in measurements, can be modelled with thepresented nonlinear constitutive equations.4. References1 Seemann, W. and Gausmann, R. A note on the strong nonlinear behaviour of piezoceramics excited with a weak electricfield. SPIE Smart Structures and Materials, Vol. 4333 (2001).2 Seemann, W. and Gausmann, R. A refined model for a piezoelectric transformer. ASME 2001, DECT (2001).3 IEEE Standard on Piezoelectricity. ANSI/IEEE Standard 176 (1987).4 Tiersten, H.F. Linear Piezoelectric Plate Vibrations. Plenum Press, New York (1969).Dipl.-Ing. Rainer Gausmann, Prof. Dr.-Ing. Wolfgang Seemann, Department of Mechanical and ProcessEngineering, University of Kaiserslautern, P.O. Box 3049, 67653 Kaiserslautern, Germany Section 1: Linear and Nonlinear Oscillations 65            

A Note on a Nonlinear Model of a Piezoelectric Rod

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A Note on a Nonlinear Model of a Piezoelectric Rod

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