Nonlinear behavior in a piezoelectric resonator: a method of analysis

Abstract

Theories used for understanding nonlinear behavior of piezoelectric resonators are usually only valid for a given range of amplitudes. Thus, important discrepancies can sometimes be observed between theory and experiment. In this work, a simplified model of the resonator is assumed in order to extend the analysis of nonlinear behavior to any kind of nonlinear function, without a significant increase of mathematical complexity. Nevertheless, nonlinearities are considered to be weak enough to be taken as perturbations. An asymptotic method is used to obtain the first and second order perturbations of the response to an harmonic excitation applied to the system, and each one is separated into Fourier series. Nonlinearity is described by two functions-Phi, (S,D,S ,D ) and Psi (S,D,S ,D )-that must be added to the constitutive equations that give T and E as functions of S and D. These functions can be split into their symmetrical and antisymmetrical parts, which have different incidence over the perturbation terms. In order to simplify the problem, no mechanical excitation is considered, the electrical one is taken as strictly harmonic, and the current rather than the e.m.f. is taken as initial data. As an application example, this method is applied in order to find the second harmonic generation for a particular kind of nonlinearity.