导电微梁的随机Hopf分岔与首次穿越

Hopf Bifurcation and First Passage on Conductive Microbeams

本文研究了受电磁、机械耦合场影响下导电微梁的随机Hopf分岔和首次穿越.基于修正的偶应力理论及磁弹性理论建立了导电微梁在受电磁、机械耦合场影响下的运动方程,并导出微梁的磁弹性随机振动方程.然后根据磁弹性理论和伽辽金变分法将方程化简为非线性微分动力学方程.本文通过拟不可积Hamilton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.然后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.通过讨论该系统的稳态概率密度函数图的形状变化得到了该动力系统的随机分岔的变化规律.然后用数学软件对计算结果进行验算,得到了此模型的Hopf分岔现象,并讨论了系统产生Hopf分岔的条件.最后利用数值模拟法模拟了系统在发生首次穿越的条件,分析了本征长度对系统可靠性函数的影响.结果表明初始能量值、分岔值、材料内禀系数都会对微梁的稳定性产生影响.

We studied stochastic Hopf bifurcations and first passage of conductive microbeams under the influence of electromagnetic and mechanical coupling fields. Based on the modified couple stress theory and magneto-elasticity theory, the motion equation of the conductive microbeam under the influence of electromagnetic and mechanical coupling fields was built, and derived the magneto-elastic random vibration equation of the microbeam. The equations were simplified to nonlinear diffrential dynamics equations according to magnetoelasticity theory and Galerkin variational method. We equivalent the equation to a onedimensional Ito stochastic differential equation through the stochastic average theory of no-integrable Hamliton system. Then the local stochastic stability of the system was judged by calculating the maximum Lyapunov exponent of the equation. And the global stability of the system was judged by the singular boundary theory which based on stochastic diffusion process. By discussing the graphical changes of these steady-state probability density function graphs of the system, we got the variation law of the stochastic bifurcation of the dynamical system. And then the calculation results were checked by mathematical software, the Hopf bifurcation phenomenon of the model is obtained, and we discussed the conditions for the system to generate Hopf bifurcation. Finally, the numerical simulation method was used to simulate the condition of the system in the first passage, and the influence of the materials intrinsic factor on the reliability function of the system was analyzed. The results show that the initial energy value,bifurcation value, and material intrinsic coefficient all affect the stability of the microbeam.