中文摘要：

根据板壳力学与磁弹性力学理论,建立了在横向磁场与机械载荷共同作用下的四边简支载流矩形薄板的非线性随机振动模型,利用Galerkin变分法将其化简为非线性微分动力学方程。其次使用拟不可积Hamilton系统的随机平均理论将方程等价为一个一维的It？随机微分方程,并通过计算该系统的最大Lyapunov指数来判断该系统的局部稳定性,同时利用奇异边界理论判断其系统的全局稳定性。最后通过稳态概率密度函数的变化研究了系统参数对发生的随机Hopf分岔的影响。并采用数值模拟对理论分析进行了验证。

英文摘要：

A nonlinear random vibration model of a current carrying thin rectangular plate simply supported at each edge was established when the plate was applied mechanical load in a magnetic field. The model was proposed based on the theories of plates and shells and the magnetic elastic mechanics. It was simplified as a nonlinear dynamics differential equation by using Galerkin variation method. Then the equation was equivalent to be a one-dimensional ItO stochastic differential equation by applying the stochastic average theory of a quasi non-integrable Hamilton system. The local stochastic stability of the system was judged using the maximum Lyapunov index. Its global stability of the system was also judged using the singular boundary theory. Finally the influences of the system parameters on the stochastic Hopf bifurcation were researched through the steady probability density function. The numerical simulation results were shown in the paper.