Nonlinear Effects of Piezoceramics Excited by Weak Electric Fields

There is a wide range of nonlinear effects which can be observed in piezoceramics.One example is the well-known butterfly hysteresis behavior for large stresses and strongelectric fields. For small stresses and weak electric fields, piezoceramics are usuallydescribed by linear constitutive equations around an operating point in the butterflyhysteresis curve. Nevertheless, typical nonlinear effects can be observed whenpiezoceramic actuators and structures with embedded piezoceramics are excited inresonance, even if the stresses and the electric field remain small.This was observed and described, e.g., by Beige and Schmidt [1, 2],who investigated longitudinal rod vibrations using thepiezoelectric 31-effect. They modeled these nonlinearities usinghigher-order quadratic and cubic elastic and electric terms.Typical nonlinear effects, e.g., dependence of the resonancefrequency on the amplitude, superharmonics in spectra and anonlinear relation between excitation voltage and vibrationamplitude were also observed by von Wagner et al. [3] inpiezo-beam systems. In this paper, the work is extended tolongitudinal vibrations of piezoceramic rods using thepiezoelectric 33-effect. The experiments with piezoelectriccylinders PIC 181 manufactured by PI-Ceramic, clearly exhibited not onlynonlinearities of the Duffing type, but also quadraticnonlinearities. These nonlinearities are modeled using an extendedelectric enthalpy density, including nonlinearquadratic and cubic elastic terms, piezoelectric terms anddielectric terms. The equations of motion for the system underconsideration are derived via the Ritz method usingHamilton's principle. Simple rod models for slendercylinders are used for the description of the piezoceramics. Theequations of motion are solved using perturbation techniques.Then, `nonlinear' parameters can be identified, and the numericalresults are compared to those obtained experimentally. Thenonlinear effects described in this paper may have stronginfluence on the relation between excitation voltage and responseamplitude whenever piezoceramic actuators and structures areexcited at resonance.