载流圆板在磁场中的随机稳定性分析

The Stochastic Stability of a Current Carrying Circular Plate in a Magnetic Field

本文根据大挠度板壳力学基础理论和电磁弹性力学理论,建立了载流圆板的非线性磁弹性随机振动力学模型,采用伽辽金变分法将其变换成非线性常微分动力学方程.通过拟不可积哈密顿系统的平均理论将该方程等价为一个一维伊藤随机微分方程.通过计算该方程的最大Lyapunov指数判断该系统的局部随机稳定性,并进一步采用基于随机扩散过程的奇异边界理论判断该系统的全局稳定性.最后通过讨论该系统的稳态概率密度函数图的形状变化讨论了该动力系统的随机Hopf分岔的变化规律,并采用数值模拟对理论分析进行了验证.

According to the theories of the elasticity with large deflection and the magnetic elasticity, the nonlinear magneto-elastic random vibration model of a current carrying circular plate was established. Then the nonlinear ordinary differential dynamic equation was derived using the Galerkin’s variation method. The equation was equivalent to a one dimensional Ito stochastic differential equation through the average theory method of quasi non integrable Hamiltonian systems. The stochastic local stability of the system was determined by calculating the largest Lyapunov exponent of the equation. The global stability of the system was determined using the theory of singular boundary based on stochastic diffusion process. Finally, the stochastic Hopf bifurcation of the system was discussed through the shape change of the steady-state probability density function diagram of the system. And the theoretical analysis is verified by numerical simulation.