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考虑几何非线性项和阻尼的影响,给出了四边简支的正交各向异性矩形层合板在两项横向简谐激励作用下的非线性振动微分方程,利用伽辽金法导出了相应的达芬型非线性强迫振动方程。应用多尺度法对组合共振问题进行求解,得到了系统在稳态运动下的幅频响应方程。基于李雅普诺夫稳定性理论,得到了解的稳定性判定条件。通过数值算例,分析了不同参数对系统组合共振及其分岔特性的影响。结果表明,随着调谐参数、板厚度、阻尼系数以及激励力等参数的改变,系统存在多幅值现象、滞后现象和跳跃现象,出现不稳定解,且在某些参数点处具有运动性态发生变化的分岔特性,表现出较为复杂的动力学特性。
Considering the influence of geometrical nonlinearity and damping, the nonlinear vibration differential equations of rectangular anisotropic rectangular laminated plates with four edges on two transverse harmonic stimuli are given. The corresponding Galerkin method is derived Davenpnein nonlinear forced vibration equation. The multi-scale method is used to solve the combined resonance problem, and the amplitude-frequency response equation of the system under steady-state motion is obtained. Based on the Lyapunov stability theory, the conditions for the stability of the solution are obtained. Numerical examples are given to analyze the influence of different parameters on the resonance and bifurcation characteristics of the system. The results show that with the change of parameters such as tuning parameters, plate thickness, damping coefficient and excitation force, the system has multi-amplitude phenomenon, hysteresis phenomenon and jump phenomenon, which results in unstable solution and kinematics at some parameter points The changed bifurcation characteristics show more complex kinetic characteristics.