矩形薄板在面内随机参数激励下的随机分岔研究

摘要

The stochastic two dimensional dynamical model of a simply supported thin rectangular plate subjected to in-plate stochastic parametrical excitation was proposed based on elastic theory and Galerkin's approach. The model was simplified applying the stochastic average theory of quasi-integral Hamilton system. Then the methods of Lyapunov exponent and boundary classification associated with diffusion process were utilized respectively to analyze the local and global stochastic stability of the trivial solution of system. It was explored how the stochastic Hopf bifurcation of the model is influenced by the qualitative changes in stationary probability density of system response. It is concluded that the stochastic Hopf bifurcation occurs at two critical parametric values. The results of numerical simulation support the theoretical analysis.%建立了四边简支的矩形薄板在受面内随机激励时的振动模型,并用Galerkin法将该系统化简为二自由度常微分非线性动力学方程组.得出系统的广义能量(Hamilton函数)表达式后,又利用拟不可积Hamilton系统平均理论将方程等价为一个一维的Ito随机扩散过程,并通过计算该系统的最大Lyapunov指数来研究系统的局部随机稳定性,同时利用基于随机扩散过程的奇异边界理论研究了模型的全局稳定性,最后通过稳态概率密度函数的形状变化探讨了系统参数变化对系统随机Hopf分岔的影响.数值模拟结果验证了理论分析的正确性.