宽带噪声激励下含分数阶导数的van der Pol-Duffing振子的可靠性  

Relability of van der Pol-Duffing Oscillator with Fractional Derivative under Wide-Band Noise Excitations

作　　者：陈林聪[1] 李海锋[1] 梅真 朱位秋[2] 

机构地区：[1]华侨大学土木工程学院,福建厦门361021 [2]浙江大学力学系,浙江杭州310027

出　　处：《西南交通大学学报》2014年第1期45-51,共7页Journal of Southwest Jiaotong University

基　　金：国家自然科学基金资助项目(10932009;11002059);教育部博士点新教师基金资助项目(20103501120003);福建省自然科学基金资助项目(2010J05006);中央高校基本科研业务费专项资金资助项目(JB-SJ1010)

摘　　要：为了研究宽带噪声激励下含分数阶导数的van der Pol-Duffing振子的首次穿越问题,首先应用广义谐波平衡技术,将分数阶导数表示的回复力分解为等效拟线性阻尼力和拟线性回复力,获得不含分数阶导数的等效非线性随机系统;然后,应用随机平均法将等效非线性随机系统近似为一维扩散过程,再建立和求解相应的后向Kolmogorov方程,获得系统的条件可靠性函数和平均首次穿越时间计算式;最后,通过实验结果表明,所提方法与蒙特卡罗法模拟结果吻合得非常好;系统的可靠性随分数阶数的增加而提高;分数阶导数表示的回复力不能简单地当作一类特殊的阻尼力.To investigate the first-passage time of van der Pol-Duffing oscillator with fractional derivative under wide-band noise excitations, the restoring force described by a fractional derivative was firstly separated into the equivalent quasi-linear dissipative force and quasi-linear restoring force by using the generalized harmonic balance technique, which yields an equivalent nonlinear stochastic system without fractional derivative. Then, the equivalent nonlinear stochastic system was approximated as one-dimensional diffusive process by using the stochastic averaging method, and the backward Kolmogorov equation associated with the averaged equation was then established and solved to yield the conditional reliability function and mean first-passage time of system. Finally, the numerical simulation demonstrates that the analytical results agree well with those derived by the Monte Carlo simulation; the system reliability improves with the fractional order; and the restoring force described by the fractional derivative can not be regarded as a special damping force.

关 键 词：分数阶导数 可靠性 随机平均法 van der Pol—Duffing振子 宽带噪声 

分 类 号：O324[理学—一般力学与力学基础]