Theoretical and experimental investigation of coupled longitudinal-transverse nonlinear vibration for micro-positioning piezoelectric bending actuators

摘要

This paper investigates the nonlinear coupled longitudinal-transverse vibration of bimorph piezoelectric cantilever bending actuators. The previous dynamic models proposed for these actuators are chiefly associated with disregarding the longitudinal displacements coupling effect in the transverse vibration, which has led to inaccuracies in the dynamic behavior analysis of these systems. To overcome this deficiency, it was necessary to assume nonlinear material terms of degrees higher than two to match the experiments, which is a questionable assumption in micro-scale vibration. In the present study, considering nonlinear geometrical conditions and longitudinal coupling effect, it is shown that the theoretical model can match the experimental data with reasonable material nonlinearity. In this regard, this paper first develops a coherent nonlinear coupled approach by implementing Hamilton's principle considering quadratic geometrical, material and damping nonlinear terms. Additionally, an external soft harmonic excitation near the fundamental natural frequency of the system is applied. Then, the formulated non-dimensionalized PDE equations are discretized utilizing the Galerkin first mode approximation. A closed-form nonlinear solution is derived by employing the method of multiple scales. Then, the modelled system's nonlinear dynamic behavior is validated by experimental data. Numerical simulation is carried out as well to validate the theoretical model. The theory obtained from the method of multiple scales shows an error of 0.3 with experimental results.