含末端质量的悬臂梁随机非线性振动的随机平均法

Stochastic averaging method for stochastic nonlinear vibration of cantilever beam with end mass XIE Nana,GE Gen

提出了一种可适用于含末端质量的悬臂梁强非线性振动的随机平均法。该方法将原本只能解决仅含有刚度非线性项振子的随机平均法扩展到了能解决既含有刚度非线性又含有惯性非线性的振子。先对含有末端质量的悬臂梁应用凯恩方法进行了建模,然后再基于哈密尔顿函数将振子化为关于瞬态等效振幅和瞬态相位的两个随机微分方程,随后应用随机平均原理将随机微分方程化简为一个关于等效振幅的伊藤方程。并在此基础上得出了在末端质量取不同值时的等效振幅的稳态概率密度以及位移和速度的联合概率密度。数值模拟很好地证明了该理论方法的正确性。

Here,a stochastic averaging method was proposed for strongly nonlinear vibration of a cantilever beam with end mass.This method extended the random average method being only able to solve an oscillator with stiffness nonlinearity into the one able to solve oscillators with both stiffness nonlinearity and inertia nonlinearity.A cantilever beam with end mass was modeled using Kane method,and then the oscillator was converted into two stochastic differential equations with respect to transient equivalent amplitude and transient phase based on Hamilton function,and the stochastic differential equations were simplified into an ITO equation with respect to equivalent amplitude by using the stochastic average principle.Furthermore,the steady-state probability density of equivalent amplitude and the joint probability density of displacement and velocity were solved when the end mass having different values.The theoretical correctness of the proposed method was verified with numerical simulation.