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 plate large deflection theory, the nonlinear vibration equations and compatibility equations of the system are derived using the fundamental theory of finite deformation in elasticity together with the Bubnov-Galerkin method. The averaging method is then employed to obtain the amplitude-frequency response equation of the flexible piezoelectric plate under primary resonance conditions. The stability of the obtained solutions is analyzed according to the Routh-Hurwitz stability criterion. Numerical simulations are performed in MATLAB to examine the influence of different parameters on the primary resonance response of the plate, and the corresponding amplitude-frequency response curves are obtained under different plate thicknesses, aspect ratios, external excitations, and temperature differences. The results show that the instability region of the solution is more pronounced in the (1,1) mode, whereas it decreases in the (2,1) and (1,2) modes. As the external excitation increases, the system’s response amplitude increases, whereas it decreases with increasing temperature, plate thickness, or aspect ratio. Under primary resonance, two saddle–node bifurcation points exist in the system. The intervals outside the two jump points correspond to a single-valued response, whereas the interval between the two jump points corresponds to a triple-valued response consisting of two stable solutions and one unstable solution. The trajectories near the two stable points converge, whereas the other point is a saddle point, around which the trajectories diverge.