A note on the nonlinear modeling of piezoelectric rods

Piezoceramic 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. Therefore nonlinear constitutive equations have to be used if the super harmonics in the measured velocity signals should be modeled. In this paper nonlinear constitutive equations are derived from the electric enthalpy density to describe the longitudinal oscillations of a piezoceramic rod. The kinetic energy considers the inertia effects in transversal direction so the model is not limited to the slender rod theory. With Hamilton's principle a variational principle is derived which is approximated using a Rayleigh Ritz ansatz with the first eigenfunction of the linearized system. The resulting nonlinear ODE is solved by application of the harmonic balance method. The amplitudes of displacement of the fundamental, the second and the third super harmonic oscillations are solved numerically. In a first step the results of the theoretical model are compared with measurements. The material parameters of the nonlinear constitutive equations are calculated using a parametric identification method. As a result it is shown that the theoretical model can describe the Duffing type nonlinearities found in measurements.