Anomalous behavior of nonlinearity in a piezoelectric resonator

Theories used to understand the nonlinear behavior of piezoelectric resonators are usually valid only for a given range of amplitudes. So, relevant discrepancies can usually be observed between theory and experiments. As an example, the shift of the resonant frequency or the rise of the inverse of the quality factor are not always proportional to the square of the amplitude, as theories predict. An oversimplified model of the resonator is assumed in this work, in order to show the meaning of such discrepancies. A single degree of freedom is taken, so electromechanical coupling is not considered. A nonlinear term g(u, u˙) of any type is included as a perturbation. An asymptotic method is used in order to get the first and second order perturbations of the response to an harmonic force applied to the system, and each one is separated into Fourier series. Splitting the function g into its symmetrical and antisymmetrical parts (g_s and g_A), the term g_A gives first order perturbation of odd frequencies (direct effect) while g_s gives even frequencies in first order, but odd frequencies in its second order (indirect effect). The increase of the impedance of the resonator (perturbation at the main frequency) can be obtained as some integral of function g, suggesting that sometimes this may take the form A^5/2 . Experimental fact that resistance increment ΔR and reactance increment ΔX depend on the amplitude in a similar way, so that the rate m=ΔX/ΔR is a constant, implies also a restriction in the form of the function g(u, u˙). The empirical relation between m and the quality factor could be explained if the contribution of g_s was greater than the g_A one