Stochastic bifurcations in a nonlinear tri-stable energy harvester under colored noise

In this paper, the stochastic bifurcations and the performance analysis of a strongly nonlinear tri-stable energy harvesting system with colored noise are investigated. Using the stochastic averaging method, the averaged Fokker–Plank–Kolmogorov equation and the stationary probability density (SPD) of the amplitude are obtained, respectively. Meanwhile, the Monte Carlo simulations are performed to verify the effectiveness of the theoretical results. D-bifurcation is studied through the largest Lyapunov exponent calculations, which implies the system undergoes D-bifurcation twice with increasing the nonlinear stiffness coefficients. The effects of the nonlinear stiffness coefficients, noise intensity and correlation time on P-bifurcation are discussed by the qualitative changes of the SPD. Moreover, the relationship between D- and P-bifurcation is explored. If the strength of stochastic jump has obvious gap with respect to the two statuses before and after the occurrence of P-bifurcation, the D-bifurcation will happen, too. Finally, the performance and the capability of harvesting energy from ambient random excitation are analyzed.     

Stationary response of strongly non-linear oscillator with fractional derivative damping under bounded noise excitation

The stochastic averaging method for strongly non-linear oscillators with lightly fractional derivative damping of order α (0<α≤1) subject to bounded noise excitations is proposed by using the generalized harmonic function. The system state is approximated by a two-dimensional time-homogeneous diffusion Markov process of amplitude and phase difference using the proposed stochastic averaging method. The approximate stationary probability density of response is obtained by solving the reduced Fokker–Planck–Kolmogorov (FPK) equation using the finite difference method and successive over relaxation method. A Duffing oscillator is taken as an example to show the application and validity of the method. In the case of primary resonance, the stochastic jump of the Duffing oscillator with fractional derivative damping and its P-bifurcation as the system parameters change are examined for the first time using the stationary probability density of amplitude.     

Stochastic response analysis of noisy system with non-negative real-power restoring force by generalized cell mapping method

The stochastic response of a noisy system with non-negative real-power restoring force is investigated. The generalized cell mapping (GCM) method is used to compute the transient and stationary probability density functions (PDFs). Combined with the global properties of the noise-free system, the evolutionary process of the transient PDFs is revealed. The results show that stochastic P-bifurcation occurs when the system parameter varies in the response analysis and the stationary PDF evolves from bimodal to unimodal along the unstable manifold during the bifurcation.     

Stochastic P-bifurcation analysis of a fractional smooth and discontinuous oscillator via the generalized cell mapping method

The smooth and discontinuous oscillator with fractional derivative damping under combined harmonic and random excitations is investigated in this paper. The short memory principle is introduced so that the evolution process of the oscillator with fractional derivative damping can be described by the Markov chain. Then the stochastic generalized cell mapping method is used to obtain the steady-state probability density functions of the response. The stochastic response and bifurcation of the oscillator with fractional derivative damping are discussed in detail. We found that both the smoothness parameter, the noise intensity, the amplitude and frequency of the harmonic force can induce the occurrence of stochastic P-bifurcation in the system. Monte Carlo simulation verifies the effectiveness of the method we adopt in the paper.     

非均布随机参数结构非线性响应的概率密度演化简

首先考察了概率密度演化理论中的点演化和群演化与概率空间剖分的关系. 继而,讨论了点集筛选的基本准则. 在此基础上推广了点集偏差的概念,对非均匀、非正态的一般多维分布,提出了广义F 偏差（GF 偏差）的概念,避免了偏差计算的NP 难解问题. 探索了GF 偏差与EF 偏差的关系. 以GF 偏差最小化为准则,建议了概率空间最优剖分与点集重整的新策略. 结果表明,上述方法能够处理包含多达数10 个随机变量的结构动力响应概率密度演化分析问题. 最后,指出了需要进一步研究的问题.     

随机结构动力反应分析的概率密度演化方法

提出了随机结构动力反应分析的概率密度演化方法．基于有限单元法基本原理，导出了含有随机参数的结构反应状态方程，进而，通过引入扩展状态向量，建立了随机结构反应的概率密度演化方程．将精细时程积分方法与Lax-Wendroff差分格式相结合，探讨了求解概率密度演化方程的数值方法．对一个8层层间剪切型随机结构进行了算例分析，并与Monte Carlo方法的结果进行了比较．研究表明，随机结构反应的概率密度具有演化特征，且概率密度曲线与正态分布差异甚大，甚至可能出现双峰曲线．     

Complex response analysis of a non-smooth oscillator under harmonic and random excitations

It is well-known that practical vibro-impact systems are often influenced by random perturbations and external excitation forces, making it challenging to carry out the research of this category of complex systems with non-smooth characteristics. To address this problem, by adequately utilizing the stochastic response analysis approach and performing the stochastic response for the considered non-smooth system with the external excitation force and white noise excitation, a modified conducting process has proposed. Taking the multiple nonlinear parameters, the non-smooth parameters, and the external excitation frequency into consideration, the steady-state stochastic P-bifurcation phenomena of an elastic impact oscillator are discussed. It can be found that the system parameters can make the system stability topology change. The effectiveness of the proposed method is verified and demonstrated by the Monte Carlo(MC) simulation.Consequently, the conclusions show that the process can be applied to stochastic non-autonomous and non-smooth systems.     

基于概率密度演化方法的随机结构可靠度分析

随机结构反应的概率密度演化方法能够给出随机荷载作用下随机结构反应的概率密度函数。在此基础上，根据给定的正常使用位移限值要求，直接进行积分给出了随机结构的正常使用可靠度及其失效概率。在实例分析中，与一类情况下的精确解答及基于反应正态分布假定的二阶矩方法分析结果进行了比较。研究表明：基于密度演化方法的随机结构可靠度分析具有很高的精度，而二阶矩方法的可靠度分析结果则往往具有一定的偏差，在失效概率较低时可能给出虚假的失效概率。     

Response analysis of nonlinear vibro-impact system coupled with viscoelastic force under colored noise excitations

This paper is mainly dealing with the stochastic responses of nonlinear vibro-impact (VI) system coupled with viscoelastic force excited by colored noise. By the aid of approximate conversion for the viscoelastic force, the original stochastic VI system is transformed into an equivalent stochastic system without viscoelastic term. Then, the equations of the converted system are simplified by non-smooth transformation, and the stochastic averaging method is employed to solve the above simplified system. A Van der Pol VI oscillator coupled with viscoelastic force is worked out in detail to illustrate the application of the mentioned method, and therewith the analytical solutions fit the numerical simulation results based on the original system. Therefore, the present analytical means of investigating this system is proved to be feasible. Additionally, the exploration of stochastic P-bifurcation by two different ways is also demonstrated in this paper through varying the value of the certain system parameters. Besides, it shows a noteworthy fact that assigning zero or a positive value to the magnitude of viscoelastic force can also lead to the bimodal shape of different degrees in the process of stochastic bifurcations.     

非线性随机结构动力可靠度的密度演化方法

建议了一类新的非线性随机结构动力可靠度分析方法。基于非线性随机结构反应分析的概率密度演化方法,根据首次超越破坏准则对概率密度演化方程施加相应的边界条件,求解带有初、边值条件的概率密度演化方程,可以给出非线性随机结构的动力可靠度。研究了数值计算技术,建议了具有自适应功能的TVD差分格式。以具有双线型恢复力性质的8层框架结构为例进行了地震作用下的动力可靠度分析,与随机模拟结果的比较表明,所建议的方法具有较高的精度和效率。     

一类Markov过程的最大绝对值过程概率密度求解的新方法

随机过程或随机系统响应的最大绝对值概率分布往往是科学与工程中关心的重要挑战性问题.本文从理论与数值上进行了Markov过程的时变最大绝对值过程及其概率分布研究.文中,通过引入扩展状态向量,构造了最大绝对值$\!$-$\!$-$\!$状态量联合向量过程,由此将不具有Markov性的最大值过程转化为具有Markov性的向量随机过程.在此基础上,通过最大绝对值$\!$-$\!$-$\!$状态量之间的关系,建立了联合向量过程的转移概率密度函数.进而,结合Chapman-Kolmogorov方程和路径积分方法,提出了最大绝对值概率密度函数求解的数值方法.由此,可以得到Markov过程最大绝对值过程的时变概率密度函数,可进一步用于结构动力可靠度分析等.通过数值算例,验证了本文所提方法的有效性. 该方法有望推广到更一般随机系统的极值分布估计之中.      

Exact stationary probability density functions for non-linear systems under Poisson white noise excitation

The paper presents exact stationary probability density functions for systems under Poisson white noise excitation. Two different solution methods are outlined. In the first one, a class of non-linear systems is determined whose state vector is a memoryless transformation of the state vector of a linear system. The second method considers the generalized Fokker-Planck (Kolmogorov-forward) equation. Non-linear system functions are identified such that the stationary solution of the system admits a prescribed stationary probability density function. Both methods make use of the stochastic integro-differential equations approach. This approach seems to have some computational advantages for the determination of exact stationary probability density functions when compared to the stochastic differential equations approach.     

Stochastic P-bifurcations of a noisy nonlinear system with fractional derivative element

Acta Mechanica Sinica - This paper investigates the stochastic P-bifurcation (SPB) of a fractionally damped oscillator subjected to additive and multiplicative Gaussian white noise. Variable...     

三类随机系统广义概率密度演化方程的解析解

近年来逐步发展的概率密度演化方法理论为随机动力系统的分析与控制研究提供了新的途径.过去若干年来,已经发展了一系列数值方法如有限差分法、无网格法用于求解广义概率密度演化方程.但是,针对典型随机系统,关于这一方程解析解尚比较缺乏.本文以李群方法为工具,研究给出了Van der Pol振子、Riccati方程和Helmholtz振子3类典型随机非线性系统的广义概率密度演化方程解析解.这些结果,不仅可以作为检验求解广义概率密度演化方程的数值方法结果正确性的判别依据,也为概率密度演化理论的进一步深入研究提供了若干分析实例.      

基于首次穿越时间概率密度的时变可靠性分析

基于时变可靠性性能函数首次穿越时间的概率密度F-PTPD(first-passage time probability density)模型,提出了一种求解机械产品全寿命周期可靠性累计概率密度函数的方法(简称F-PTPD方法),为产品在全寿命周期内可靠性分析和设计提供了工具。首先,采用稀疏网络随机配置方法进行时变可靠性性能函数均值的估计,选取性能函数均值为零的第一个时刻点作为首次穿越点;其次,基于均值的首次穿越点将时变可靠性性能函数进行二阶泰勒展开,利用二次函数的性质求解性能函数首次穿越时间关于随机输入变量的函数;再次,针对首次穿越点函数,采用稀疏网络随机配置方法进行首次穿越时间的四阶原点矩估计;最后,基于四阶原点矩利用最大熵概率密度函数估计方法,推导出首次穿越点的概率分布,获得产品寿命周期内时变可靠性的累计概率密度函数。本文方法可获得产品整个寿命周期失效概率的变化趋势,极大地提高了评估效率,对复杂产品的可靠性评估设计有一定的工程指导意义。     

隧洞围岩力学性态预报的随机介质模型

本文基于系统学观点,利用随机过程以及概率与数理统计方法,视岩体为一随机不确定性系统,建立了描述变形概率密度函数演化所服从的方程,详细讨论了变形发展的概率预报,最后给出了一些有意义的结论和意见。     

广义密度演化方程的δ函数序列解法

随机动力系统响应或状态向量的概率密度函数一般遵循概率密度演化方程,如Liouville方程、FPK方程和Dostupov-Pugachev方程,但是上述方程均属于高维偏微分方程,求解相当困难. 基于概率守恒原理的随机事件描述导出的广义密度演化方程,其维数与系统自由度无关,为随机动力系统分析提供了可能的途径. 从广义密度演化方程的形式解出发,引入δ函数的渐近序列,获得了广义密度演化方程的一种新的数值解法------广义密度演化方程的δ序列解法. 将建议方法与非参数密度估计进行了对比,指出非参数密度估计是该方法的一个特例. 最后,分别采用重构实例和演化实例验证了该方法在一维和多维情形下的有效性.      

RESEARCH FOR ATTRACTING REGION AND EXIT PROBLEM OF A PIECEWISE LINEAR SYSTEM UNDER PERIODIC AND WHITE NOISE EXCITATIONS

离出行为是随机非线性系统的重要现象之一,而离出问题是除随机动力系统理论以外考察随机非线性系统随机稳定性的另一种重要的方法.分段线性系统是一个经典的非线性动力学模型,受随机激励后成为随机系统,但并不是严格的随机动力系统,因而此时随机动力系统理论也不适用.为了研究同时受周期和白噪声激励的分段线性系统,首先使用Poincaré截面模拟其在无噪声时确定性的动力学行为,然后使用Monte Carlo模拟对其在白噪声激励下的离出行为进行了数值仿真分析.其次,为了考察离出问题中的重要参数,系统的平均首次通过时间（mean first-passage time,MFPT）,使用van der Pol变换,随机平均法,奇异摄动法和射线方法进行了量化计算.通过对理论结果与模拟结果的对比分析,得到结论：当系统吸引子对应的吸引域边界出现碎片化时,理论结果与模拟结果的误差极大;而当吸引域边界足够光滑的以后,理论结果与模拟结果才会相当吻合.     

Uncertainty quantification for stochastic dynamical systems using time-dependent stochastic bases

A novel method based on time-dependent stochastic orthogonal bases for stochastic response surface approximation is proposed to overcome the problem of significant errors in the utilization of the generalized polynomial chaos(GPC) method that approximates the stochastic response by orthogonal polynomials. The accuracy and effectiveness of the method are illustrated by different numerical examples including both linear and nonlinear problems. The results indicate that the proposed method modifies the stochastic bases adaptively, and has a better approximation for the probability density function in contrast to the GPC method.     

随机结构静力反应概率密度演化方程的差分方法

随机结构分析的概率密度演化方法是分析随机结构静力反应的一种具有良好前景的方法。本文研究了求解随机结构静力反应概率密度演化方程的差分方法，分别探讨了单边差分格式和Lax-Wendroff格式的计算性态。二者均能满足概率相容性条件并且能够保证均值线性增长。以八层框架结构的静力随机反应为例，对两种差分格式的结果及精确解答进行了具体的比较分析。研究表明，两种差分格式均是收敛和稳定的，在不连续点处存在角点效应．单边差分格式能够保证概率非负性，而Lax-Wendroff格式具有往往更快的收敛速度。就变异系数而言，通常单边差分格式的变异系数随着区间离散数的增长而趋于稳定值，Lax-Wendroff格式则一开始就可得到恒定的值。