Nonlinear principal resonance of a flexible piezoelectric thin plate under temperature effects

This study investigates the variation of nonlinear primary resonance in flexible piezoelectric plates under different thermal effects. Based on the von-Kármán large-deflection plate theory, the nonlinear vibration equations and the associated compatibility equations are derived using the fundamental principles of finite deformation elasticity in conjunction with the Bubnov-Galerkin method. The averaging method is subsequently applied to obtain the amplitude-frequency response equation for the plate under primary resonance conditions. The stability of the obtained solutions is analyed using the Routh-Hurwitz stability criterion. Numerical simulations are performed in MATLAB to explore the influence of various parameters, including plate thicknesses, aspect ratios, external excitations, and temperature difference, on the primary resonance response, with corresponding amplitude-frequency response curves presented accordingly. The results show that the instability region of the solution is more pronounced in the (1,1) mode, whereas it diminishes in the (2,1) and (1,2) modes. The response amplitude increases with greater external excitation, but decreases with rising temperature, plate thickness, or aspect ratio. Under primary resonance, two saddle-node bifurcation points exist. Outside the interval between these two jump points, the system exhibits a single-valued response; within the interval, a triple-valued response emerges, consisting of two stable solutions and one unstable solution. The trajectories near the two stable points converge, whereas the remaining point is a saddle point, around which the trajectories diverge.