Nonstationary probability densities of system response of strongly nonlinear single-degree-of-freedom system subject to modulated white noise excitation

The nonstationary probability densities of system response of a single-degree-of -freedom system with lightly nonlinear damping and strongly nonlinear stiffness subject to modulated white noise excitation are studied.Using the stochastic averaging method based on the generalized harmonic functions,the averaged Fokker-Planck-Kolmogorov equation governing the nonstationary probability density of the amplitude is derived. The solution of the equation is approximated by the series expansion in terms of a set...     

介电弹性球膜的动力稳定性分析

摘要：介电弹性体结构具有卓越的力电性能,然而由于其大变形特性,在动态工作模式下极易出现各类失效问题,这极大阻碍了其工程应用.论文研究与力电失稳行为直接相关的理想介电弹性球膜动力稳定性问题.首先据虚功原理建立电压及压力共同作用下关于伸长比的动力学方程,系统自由能由弹性应变能与静电能组成,而前者基于Mooney-Rivlin模型表出.通过系统首次积分解析给出稳态响应幅值与阶跃电压/阶跃压力的关系曲线,其与静态平衡曲线的交点决定了临界电压/临界压力.研究表明：给定任意电压,材料参数存在某阈值,当超过该值后系统始终保持稳定;对于任意非零压力值,存在类似材料参数阈值;而当压力恰为零时,则始终存在临界电压值,超过该值则系统动力不稳定.     

调制白噪声激励下的单自由度强非线性系统的近似瞬态响应概率密度

研究调制白噪声激励下,包含弱非线性阻尼及强非线性刚度的单自由度系统的近似瞬态响应概率密度．应用基于广义谐和函数的随机平均法,导出关于幅值瞬态概率密度的平均Fokker－Planck－Kolmogorov 方程．该方程的解可近似表示为适当的正交基函数的级数和,其中系数是随时间变化的．应用Galerkin方法,这些系数可由一阶线性微分方程组解得,从而可得幅值响应的瞬态概率密度的半解析表达式及系统状态响应的瞬态概率密度和幅值的统计矩．以受调制白噪声激励的van der Pol-Duffing振子为例验证其求解过程,并讨论了线性阻尼系数及非线性刚度系数等系统参数对系统响应的影响．     

多自由度非线性随机系统的响应与稳定性

响应与稳定性分析一直是随机动力学研究的热点, 发展预测随机响应及判定系统响应性态的方法具有重要的科学意义与广阔的应用前景. 本文综述了有关多自由度非线性随机系统的响应与稳定性的研究. 首先简介用于随机系统响应预测的Fokker-Planck-Kolmogorov方程法、随机平均法、等效线性化法、等效非线性系统法和Monte Carlo模拟法, 评述其优缺点, 进而讨论了多自由度非线性随机系统响应的精确平稳解、近似瞬态解的研究现状. 然后介绍了随机系统稳定性分析的两类方法, 即Lyapunov函数法及Lyapunov指数法,并综述了多自由度非线性随机系统稳定性分析的研究现状. 最后给出几点发展建议.     

Stability Analysis of an Inverted Pendulum Subjected to Combined High Frequency Harmonics and Stochastic Excitations

Stability of vertical upright position of an inverted pendulum with its suspension point subjected to high frequency harmonics and stochastic excitations is investigated. Two classes of excitations, i.e., combined high frequency harmonic excitation and Gaussian white noise excitation, and high frequency bounded noise excitation, respectively, are considered. Firstly, the terms of high frequency harmonic excitations in the equation of motion of the system can be set equivalent to nonlinear stiffness terms by using the method of direct separation of motions. Then the stochastic averaging method of energy envelope is used to derive the averaged Ito stochastic differential equation for system energy. Finally, the stability with probability 1 of the system is studied by using the largest Lyapunov exponent obtained from the averaged Ito stochastic differential equation. The effects of system parameters on the stability of the system are discussed, and some examples are given to illustrate the efficiency of the proposed procedure.