Probabilistic response of nonlinear systems under combined normal and Poisson white noise via path integral method

In this paper the response in terms of probability density function of nonlinear systems under combined normal and Poisson white noise is considered. The problem is handled via a Path Integral Solution (PIS) that may be considered as a step-by-step solution technique in terms of probability density function. A nonlinear system under normal white noise, Poissonian white noise and under the superposition of normal and Poisson white noise is performed through PIS. The spectral counterpart of the PIS, ruling the evolution of the characteristic functions is also derived. It is shown that at the limit when the time step becomes an infinitesimal quantity an equation ruling the evolution of the probability density function of the response process of the nonlinear system in the presence of both normal and Poisson White Noise is provided.     

Multiplicative cases from additive cases: Extension of Kolmogorov–Feller equation to parametric Poisson white noise processes

In this paper the response of nonlinear systems driven by parametric Poissonian white noise is examined.As is well known, the response sample function or the response statistics of a system driven by external white noise processes is completely defined. Starting from the system driven by external white noise processes, when an invertible nonlinear transformation is applied, the transformed system in the new state variable is driven by a parametric type excitation. So this latter artificial system may be used as a tool to find out the proper solution to solve systems driven by parametric white noises. In fact, solving this new system, being the nonlinear transformation invertible, we must pass from the solution of the artificial system (driven by parametric noise) to that of the original one (driven by external noise, that is known). Moreover, introducing this invertible nonlinear transformation into the Itô’s rule for the original system driven by external input, one can derive the Itô’s rule for systems driven by a parametric type excitation, directly. In this latter case one can see how natural is the presence of the Wong–Zakai correction term or the presence of the hierarchy of correction terms in the case of normal and Poissonian white noise, respectively. Direct transformation on the Fokker–Planck and on the Kolmogorov–Feller equation for the case of parametric input are found.     

Higher order statistics of the response of MDOF linear systems under polynomials of filtered normal white noises

This paper exploits the work presented in the companion paper in order to evaluate the higher order statistics of the response of linear systems excited by polynomials of filtered normal processes. In fact, by means of a variable transformation, the original system is replaced by a linear one excited by external and linearly parametric white noise excitations. The transition matrix of the new enlarged system is obtained simply once the transition matrices of the original system and of the filter are evaluated. The method is then applied in order to evaluate the higher order statistics of the approximate response of nonlinear systems to which the pseudo-force method is applied.     

Path integral, functional method, and stochastic dynamical systems

The path-integral approach to dynamical behavior of systems subject to Gaussian white noise is presented in a straightforward manner. Starting from the Chapman-Kolmogorov equation, the transition probability density, and therefore moments and other statistics of the random response are ultimately expressed in terms of functional integrals over the sample-path space. Accordingly, various characteristic functions are replaced by a single generating functional from which moments of all orders are simply calculated through functional differentiation. This generating functional is proven to satisfy a closed system of functional differential equations. These equations are solved in the case of linear systems, their generating functional being obtained in explicit form. Also given in this paper is an integral equation satisfied by the probability densities. Three kinds of approximation method, namely perturbation expansion, Feynman's variational method, and the WKB method, are developed based on the path-integral formalism. They can be used to study the transient as well as stationary behavior of nonlinear systems.     

Poisson white noise parametric input and response by using complex fractional moments

In this paper the solution of the generalization of the Kolmogorov–Feller equation to the case of parametric input is treated. The solution is obtained by using complex Mellin transform and complex fractional moments. Applying an invertible nonlinear transformation, it is possible to convert the original system into an artificial one driven by an external Poisson white noise process. Then, the problem of finding the evolution of the probability density function (PDF) for nonlinear systems driven by parametric non-normal white noise process may be addressed in determining the PDF evolution of a corresponding artificial system with external type of loading.     

Numerical solutions of Fokker-Planck equation of nonlinear systems subjected to random and harmonic excitations

A new approach for an efficient numerical implementation of the path integral (PI) method based on non-Gaussian transition probability density function (PDF) and the Gauss-Legendre integration scheme is developed. This modified PI method is used to solve the Fokker-Planck (FP) equation and to study the nature of the stochastic and chaotic response of the nonlinear systems. The steady state PDF, periodicity, jump phenomenon, noise induced changes in joint PDF of the states are studied by the modified PI method. A computationally efficient higher order, finite difference (FD) technique is derived for the solution of higher-dimensional FP equation. A two degree of freedom nonlinear system having Coulomb damping with a variable friction coefficient subjected to Gaussian white noise excitation is considered as an example which can represent a bladed disk assembly of turbo-machinery blades. Effects of normal force and viscous damping on the mean square response are investigated.     

横浪中船舶的随机混沌运动

采用概率密度函数和数值模拟的方法研究随机横浪中船舶的混沌运动特性和发生混沌运动的临界参数条件。综合考虑非线性阻尼、非线性恢复力矩以及白噪声横浪激励,建立了船舶的横摇非线性随机微分方程。用随机Melnikov均方准则确定混沌运动的系统参数域后,应用路径积分法求解随机微分方程得到了响应的概率密度函数。研究发现:当噪声强度大于混沌临界值时,船舶出现随机混沌运动;对于高的白噪声激励强度,系统响应有两种较大可能的状态并在这两个状态间随机跳跃,这时船舶的运动不稳定并可能发生倾覆。     

A novel mean square formulation of stochastic nonlinear dynamic systems based on Adomian decomposition

An Adomian decomposition based mathematical framework to derive the mean square responses of nonlinear structural systems subjected to stochastic excitation is presented. The exact mean square response estimation of certain class of nonlinear stochastic systems is achieved using Fokker–Planck–Kolmogorov (FPK) equations resulting in analytical expressions or using Monte Carlo simulations. However, for most of the nonlinear systems, the response estimation using Monte Carlo simulations is computationally expensive, and, also, obtaining solution of FPK equation is mathematically exhaustive owing to the requirement to solve a stochastic partial differential equation. In this context, the present work proposes an Adomian decomposition based formalism to derive semi-analytical expressions for the second order response statistics. Further, a derivative matching based moment approximation technique is employed to reduce the higher order moments in nonlinear systems into functions of lower order moments without resorting to any sort of linearization. Three case studies consisting of Duffing oscillator with negative stiffness, Rayleigh Van-der Pol oscillator and a Pendulum tuned mass damper inerter system with linear auxiliary spring–damper arrangement subjected to white noise excitation are undertaken. The accuracy of the closed form expressions derived using the proposed framework is established by comparing the mean square responses of the systems with the exact solutions. The results demonstrate the robustness of the proposed framework for accurate statistical analysis of nonlinear systems under stochastic excitation.     

Stochastic linearization method with random parameters for SDOF nonlinear dynamical systems: prediction and identification procedures

This paper describes firstly, the calculation of the Power Spectral Density Function (PSDF) for the stationary response of SDOF nonlinear second-order dynamical systems excited by a white or a broad-band Gaussian noise, and secondly, the identification of a single-degree-of-freedom (SDOF) nonlinear dynamical second-order dynamical system driven by a broad-band or a colored Gaussian noise. The two aspects are based on the use of a stochastic linearization method with random parameters which is an efficient way of approximating the PSDF. The gain obtained by this method is shown on a SDOF nonlinear dynamical system. In addition, it is shown that the stochastic linearization method with random parameters is an efficient approach for identifying a SDOF nonlinear dynamical system.     

Higher order statistics of the response of MDOF linear systems excited by linearly parametric white noises and external excitations

The aim of this paper is the evaluation of higher order statistics of the response of linear systems subjected to external excitations and to linearly parametric white noise. The external excitations considered are deterministic or filtered white noise processes. The procedure implies the knowledge of the transition matrix connected to the linear system; this, however, has already been evaluated for obtaining the statistics at single times. The method, which avoids making further integrations for the evaluation of the higher order statistics, is very advantageous from a computational point of view.     

Stochastic analysis of time-variant nonlinear dynamic systems. Part 2: Methods of statistical moments, statistical linearization and the FPK equation

In this, the second of a two part paper, the applications of the Fokker-Planck-Kolmogorov (FPK) method to stochastic analysis of time-variant nonlinear systems are considered. A new class of dynamic systems with stochastic nonlinearity and jump parametric exitations is introduced. The comparison of accuracy of different statistical methods such as statistical linearization is discussed.     

Stochastic response analysis of nonlinear systems under Gaussian inputs

In this paper the extension of Itô's rule for the case of vector real valued functions of the response of nonlinear systems excited by zero-mean Gaussian white noise processes is presented. A suitable particularization of the vector function, in order to obtain the statistical moments of every order to the response, is treated, obtaining the differential equations of the response moments in an elegant and compact form. Polynomial expansion and closure schemes are framed in the context outlined here in order to obtain an effective procedure from a computational point of view. An application to a trigonometric nonlinear system, solved in the literature by the stochastic averaging method, is treated here by the moment equation approach using the polynomial expansion of the nonlinear terms in order to evidence the validity of this approach.     

非线性系统响应计算的一种数字模拟方法

目前,对单自由度非线性振子受随机激励的响应计算已经发展到了多种方法。本文从白噪音与泊松过程的联系出发,把系统受白噪音或调制白噪音激励的响应计算转换为对泊松随机脉冲激励的响应计算,从而发展了一种数字模拟方法。此法与其它模拟计算方法所得结果有良好的一致性,并具有较高的计算效率。     

含分数阶导数项的随机Duffing振子的稳态响应分析

本文对一个含有分数阶导数项阻尼的、Gaussian白噪声激励下的Duffing振子进行了稳态响应分析。首先,基于能量平衡理论,运用等效线性化方法,计算等效系统的线性阻尼及自然频率,建立统计意义下的等效线性化系统。然后,利用平均法建立随机Ito方程,得到随机响应的Markovian近似;给出描述振子振幅概率密度函数演化的Fokker-Planck方程,并得到它的稳态解。进一步,对于含有响应振幅的等效线性系统,借助由Laplace变换得到的转换函数,得到原系统的条件功率谱密度,结合振幅的稳态概率密度作为权重函数,给出原系统功率谱密度的估计,以及响应的统计量的估计。数值模拟的结果说明所提出的功率谱密度的近似解析表达式是可靠的,它甚至适用于Duffing振子具有强非线性回复力的情形,因为它可以较好的表现出功率谱密度共振频谱加宽及多峰现象的出现。     

非线性弹性地基上矩形薄板的非线性振动与奇异性分析

研究非线性弹性地基上小挠度矩形薄板的非线性振动，应用弹性力学理论建立非线性弹性地基上小挠度矩形薄板受简谐激励作用的动力学方程，利用Galerkin方法将其转化为非线性振动方程。根据非线性振动的多尺度法求得系统主参数共振-主共振情况的一次近似解，并进行数值计算。分析了阻尼系数、地基系数、激励参数等对系统主参数共振-主共振的影响。系统主参数共振-主共振曲线均具有跳跃现象。随着阻尼、地基系数的改变，系统响应曲线具有“类软刚度特征”。随着参数激励幅值的改变，系统响应曲线具有“类硬刚度特征”。应用奇异性理论得到系统主参数共振-主共振稳态响应的转迁集和分岔图。     

The study of dynamic characteristics in nonlinear systems

A new algorithm is proposed to calculate the Laplace transform of the product of functions, so that the transformation could be used to solve the problem of nonlinear systems. The nonlinear transfer functions and approximate solutions of pulse response in the first- and second-order systems, which are calculated by this method, are the same as those by the Volterra series method, but could be obtained more conveniently. The frequency responses of first- and second-order systems with cubic nonlinearities and their stability problems are studied. A judgement is proposed for second-order nonlinear systems to prevent some special phenomena of nonlinear systems, such as jump, superharmonic resonance, bifurcation, etc     

弹性-粘弹性复合结构随机响应的各阶谱矩的计算方法

提出了一种新的计算弹性-粘弹性复合结构随机响应的各阶谱矩的计算方法，它是一种时域复模态分析方法。利用此方法获得了弹性-粘弹性复合结构在白噪声、滤过白噪声等典型平稳随机激励下随机响应的各阶谱矩的解析表达式，分析了粘弹性对各阶谱矩的影响。此计算方法简便、易用，无论单自由度或多自由度系统均适用，为进一步研究弹性-粘弹性复合结构在随机激励下的可靠性打下良好基础。     

地震作用的相关性对结构瞬态响应的影响

地震作用一般分解为水平运动分量和竖向运动分量,在这两个运动分量的作用下,结构发生大变形时,可能会经历由地震运动分量演变的外部激励和参数激励过程。由于运动分量间的相关性,推导出实际上这两个激励过程也是相关的,而且是完全相关的,但在过去的研究中,为了简化分析,常常假设这两个激励过程是完全独立的。该文以高斯白噪声和过滤高斯白噪声过程模拟地震动过程,以某一单层框架结构为研究对象,采用累积矩截断法,分析高斯白噪声和过滤高斯白噪声这两种地震动激励下单层框架结构的非平稳地震响应。同时考虑地震动分量间的相关性,得到更为精细化的结构随机地震响应,并分析这种相关性对结构响应的影响。结果表明:将地震动作用模拟为更接近实际的过滤高斯白噪声过程时,地震作用相关性对结构响应的影响更为明显,更为不可忽略。     

Probabilistic response of linear structures equipped with nonlinear damper devices (PIS method)

Passive control introducing energy absorbing devices into the structure has received considerable attention in recent years. Unfortunately the constitutive law of viscous fluid dampers is highly nonlinear, and even supposing that the structure behaves linearly, the whole system has inherent nonlinear properties. Usually the analysis is performed by a stochastic linearization technique (SLT) determining a linear system equivalent to the nonlinear one, in a statistical sense. In this paper the effect of the non-Gaussianity of the response due to the inherent nonlinearity of the damper device will be studied in detail via the Path Integral Solution (PIS) method. A systematic study is conducted showing that for a very wide range of parameters the SLT gives satisfactory results in terms of variance of displacement and velocity but not in terms of joint Probability Density Function (PDF). It has also been shown that at steady state the two processes, displacement and velocity, may be considered as independent ones.