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柔性压电薄板在温度效应下的非线性主共振

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

  • 摘要: 探究了不同温度效应对柔性压电薄板非线性主共振变化的影响。依据von-Kármán板大挠度理论,结合弹性力学有限变形基础理论及Bubnov-Galerkin原理,推导出了系统的非线性振动方程与协调方程。通过平均法,获得了柔性压电薄板在主共振条件下的幅频响应方程。根据Routh-Hurwitz稳定判据,对系统解的稳定性进行了分析,并利用仿真软件进行了数值模拟,以探讨不同参数(板厚度、长宽比、外激励和温度差)对薄板主共振响应的影响。结果表明:在(1,1)模态下,解的不稳定区域较为显著,而在(2,1)和(1,2)模态下则有所减小。外激励的增大导致系统幅值相应增加;随着温度、板厚度或长宽比的增加,系统幅值则减小。在主共振状态下,柔性压电薄板存在2个鞍结分岔点。2个跳跃节点之外的部分对应系统的单值响应,而2个跳跃节点之间的部分则对应三值响应,包含2个稳定解和一个不稳定解。2个稳定节点周围的轨迹呈现收敛状态,而另一个鞍点周围的轨迹则呈发散状态。

     

    Abstract: 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.

     

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