Nonlinear systems driven by polynomials of filtered Poisson processes

The perturbation method is applied to determine approximately the mean, variance, skewness and kurtosis of the transient and stationary response of nonlinear systems driven by polynomials of filtered Poisson processes. The analysis is based on the classical perturbation method, the Itô differentiation formula, and properties of the response of linear systems subjected to polynomials of filtered Poisson processes. Two examples are presented to demostrate the efficiency and accuracy of this approximate analysis.     

Response cumulants of nonlinear systems subject to external and multiplicative excitations

A general framework is presented for deriving the differential equations governing the evolution of the response cumulants of linear and nonlinear dynamical systems subjected to external and multiplicative non-Gaussian delta-correlated processes. Significant simplifications of these equations are given based on using appropriate recursive relationships for joint cumulants involving products of one or more variables. A compact form of the equations for the response cumulants is presented which provides insight into the structure of the cumulant equations for specific types of dynamical systems. The procedure developed can easily be implemented in computer software to derive symbolic cumulant equations and to estimate numerically the response cumulants of systems with power-law nonlinearities using approximate cumulant-neglect closure schemes. Comparison between the equations for cumulants and the equations for moments are also presented, with particular emphasis on the advantages and disadvantages of each formulation. Suggestions are given regarding the choice to use cumulant or moment equations for analysing the stochastic response of dynamical systems. The preferred formulation is shown to depend on the type of system analysed (linear or nonlinear), the type of system nonlinearity (polynomial or non-polynomial), and the type of excitation (external or multiplicative, delta-correlated or filtered).     

A method for estimating intensity and impulse response functions offiltered Poisson processes

This paper presents a method for estimating the intensity and impulse response (IR) function of filtered Poisson processes. The case is considered where the filtered Poisson process is modeled as an output of the linear constant-coefficient ordinary differential equations having poles and zeros driven by Poisson impulse processes. It is shown that an explicit formula for estimating the intensity is derived by combining second-and third-order cumulants of the residual time series generated from the discretized filtered Poisson process. It is also shown that the IR function can be estimated from the parameters of the discretized filtered Poisson process. Then, Monte Carlo simulations demonstrate the validity of the proposed method in some specific examples. It is concluded that the proposed method can be extensively applied to actual phenomena appearing in engineering and science     

Characteristic function equations for the state of dynamic systems with Gaussian, Poisson, and Lévy white noise

The Itô formula for semimartingales is applied to develop equations for the characteristic function of the state of linear and non-linear dynamic systems with Gaussian, Poisson, and Lévy white noise, viewed as the formal derivatives of Brownian, compound Poisson, and Lévy processes, respectively. These equations can be obtained if the drift and diffusion coefficient of a dynamic system are polynomials of the system state and the driving noise is Gaussian or Poisson. It was not possible to derive equations for the characteristic function for the state of systems driven by Lévy white noise. Numerical results are presented for dynamic systems with real-valued states driven by Gaussian, Poisson, and Lévy white noise processes.     

Higher order statistics of the response of linear systems excited by polynomials of filtered Poisson pulses

The higher order statistics of the response of linear systems excited by polynomials of filtered Poisson pulses are evaluated by means of knowledge of the first order statistics and without any further integration. This is made possible by a coordinate transformation which replaces the original system by a quasi-linear one with parametric Poisson delta-correlated input; and, for these systems, a simple relationship between first order and higher order statistics is found in which the transition matrix of the dynamical new system, incremented by the correction terms necessary to apply the Itô calculus, appears.     

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.     

Response of linear dynamic systems to non-Erlang renewal impulses: Stochastic equations approach

Linear dynamical systems under random trains of impulses driven by a class of non-Erlang renewal processes are considered. The class considered is the one where the renewal events are selected from an Erlang renewal process. The original train of impulses is recast, with the aid of an auxiliary stochastic variable, in terms of two independent Poisson processes. Thus, by augmenting the state vector of the dynamic system with the auxiliary stochastic variables, the original non-Markov problem is converted to a Markov one.

The differential equations for the response statistical moments can then be derived from the generalised Ito's differential rule.

Numerical results obtained for a few different models and various sets of parameters, show that the present approach allows to account for a variety of inter arrival time's probability distributions. Transient mean value and variance of the response of a linear oscillator have been obtained from the equations for moments.     

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。     

Linear systems with fractional Brownian motion and Gaussian noise

Methods are presented for calculating the evolution in time of the second moment properties of the output of linear systems subjected to fractional Brownian motion and fractional Gaussian noise, defined as the formal derivative of fractional Brownian motion. The study also examines whether the output of linear systems to fractional Brownian motion and fractional Gaussian noise exhibits long range dependence. Numerical examples are presented to illustrate the calculation of output statistics for some linear systems with fractional Brownian motion and fractional Gaussian noise input, and show that output of linear systems to these input processes may not have long memory.     

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.     

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.     

Delay-independent stability of moments of a linear oscillator with delayed state feedback and parametric white noise

The stability of a linear oscillator with delayed state feedback driven by parametric Gaussian white noise is studied in this paper. The first and second order moment equations of the system response are derived by using moment method and Itô differential rule. Based on the moment equations, the delay-independent stable conditions of both moments are proposed: For the first order moment, the sufficient and necessary condition that guarantee delay-independent stability is identified to that of the deterministic system; for the second order moment, the sufficient condition that ensure delay-independent stability depends on noise intensity. The theoretical results are also illustrated with numerical simulations.     

Stochastic control of hysteretic structural systems

A method of stochastic optimal control of hysteretic structural systems under earthquake excitations is presented. Stochastic estimation and control problems are formulated in the form of Itô stochastic differential equations on the basis of the theory of continuous Markov processes. The conditional moment equations given observation data are derived for nonlinear filtering, and are closed by introducing appropriate analytical form of the conditional probability density functions of the state variables. Under the assumption that the admissible controls are expressed as functions of the conditional moment functions the Bellman equation is derived. If the spatial variables of the Bellman equation are defined by a part of the full set of conditional moment functions appearing in the closed moment equations, the resulting Bellman equation is coupled with conditional moment equations both for filtering and for prediction. The Gaussian and non-Gaussian stochastic linearization techniques combined with simple solution techniques to the Bellman equation are examined to solve the Bellman equation or extended Riccati equations without prediction procedures.     

A survey of quantitative and qualitative methods of sensitivity analysis for stochastic dynamic systems

In this paper a brief literature survey concerning sensitivity analysis of stochastic dynamic systems described by Ito equations is presented. In the first part of the paper we review the quantitative methods in the time and frequency domain. We quote the definitions of output and moment sensitivity measures and we present applications for linear systems and nonlinear oscillators with stochastic coefficients under stochastic excitations. The cases of white and coloured noise are considered. This is followed by a discussion of the application of the output sensitivity process to response approximation of nonlinear oscillators. We discuss also the application of sensitivity methods to the approximation of characteristics of the solution of stochastic differential equations. We present another approach to the study of response sensitivity of stochastic systems in the frequency-domain with applications to secondary structural systems. In the second part of the paper we review the qualitative methods. The concept of exponential and practical insensitivity for linear and a class of nonlinear stochastic dynamic systems is presented. We quote the definitions and criteria of the exponential and practical insensitivity which are very close to the definitions of stability concerning selected response variables.     

Small noise expansion of moment lyapunov exponents for two-dimensional systems

We construct an approximation for the moment Lyapunov exponent, the asymptotic growth rate of the moments of the response of a two-dimensional linear system driven by real or white noise. A perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noise. As an example, we study the moment stability of the stationary solution of nonlinear structural and mechanical systems subjected to real noise excitation. The usefulness of the moment Lyapunov exponent in predicting parameter values at which qualitative changes in the probability density function occur (stochastic bifurcation) is also illustrated.     

弯扭组合梁载荷测量试验研究

对金属结构进行载荷测量,应变法是常用方法之一。在弯矩、剪力和扭矩等单独作用下,利用应变法测得数据的准确性以及线性相关性都很好。而实际情况中,结构受力情况复杂,经常受到弯矩、剪力和扭矩等耦合作用,因此利用应变法测得的数据和输入载荷值生成的多元线性回归方程的线性相关性有待进一步验证。本文设计了一种典型的弯扭组合梁结构,在弯矩和扭矩的耦合作用下,利用应变法进行载荷测量,测量数据准确性很好,并且输入载荷值、弯矩和扭矩等多个变量生成的多元线性回归方程的线性相关性很强。     

隔震结构非线性随机地震响应分析的复模态法

本对多自由度双线性滞变隔震结构在过滤白噪声地震激励下的随机响应问题进行了系统研究。首先建立了结构非线性运动方程；然后根据非线性随机振动理论对运动方程进行等效线性化；最后用复模态法获得了等效线性方程的解析解，将复杂的非线性随机响应问题简化为求解一元非线性代数方程问题，并给出了算例。从而建立了多自由度隔震结构非线性随机响应复模态分析的一整套计算方法。     

New partial differential equations governing the joint, response–excitation, probability distributions of nonlinear systems, under general stochastic excitation

In the present work the problem of determining the probabilistic structure of the dynamical response of nonlinear systems subjected to general, external, stochastic excitation is considered. The starting point of our approach is a Hopf-type equation, governing the evolution of the joint, response–excitation, characteristic functional. Exploiting this equation, we derive new linear partial differential equations governing the joint, response–excitation, characteristic (or probability density) function, which can be considered as an extension of the well-known Fokker–Planck–Kolmogorov equation to the case of a general, correlated excitation and, thus, non-Markovian response character. These new equations are supplemented by initial conditions and a marginal compatibility condition (with respect to the known probability distribution of the excitation), which is of non-local character. The validity of this new equation is also checked by showing its equivalence with the infinite system of moment equations. The method is applicable to any differential system, in state-space form, exhibiting polynomial nonlinearities. In this paper the method is illustrated through a detailed analysis of a simple, first-order, scalar equation, with a cubic nonlinearity. It is also shown that various versions of Fokker–Planck–Kolmogorov equation, corresponding to the case of independent-increment excitations, can be derived by using the same approach.

A numerical method for the solution of these new equations is introduced and illustrated through its application to the simple model problem. It is based on the representation of the joint probability density (or characteristic) function by means of a convex superposition of kernel functions, which permits us to satisfy a priori the non-local marginal compatibility condition. On the basis of this representation, the partial differential equation is eventually transformed to a system of ordinary differential equations for the kernel parameters. Extension to general, multidimensional, dynamical systems exhibiting any polynomial nonlinearity will be presented in a forthcoming paper.     

Non-linear stochastic response of marine vehicles

The dynamic behaviour of marine vehicles in extreme sea states is a matter of great concern following some recent and dramatic mishaps. The complex problem of its prediction can be approached from the study, yet of broader scope, of non-linear dynamic systems subjected to stochastic excitations. However, a general non-linear stochastic dynamic theory is not yet available. A new technique, the so-called linearize-and-match method, for predicting the response statistics of non-linear systems, is presented. Essentially, the technique involves the construction of an infinite series of linear systems aimed at the prediction of the response statistical moments of a given order. The linear systems are successively defined by linearizing the original, non-linear system and matching the Volterra functional model response statistics to the desired order. The linear system for predicting second order statistics is shown to coincide with the one obtained using the method of equivalent linearization. Response probability distributions can be constructed from the knowledge of such statistics. Particular attention is devoted to the distribution of maximum entropy and its justification in such underdetermined moment problems. Finally, applications to the roll motion of ships serve to exemplify as well as to assess the accuracy and the versatility of the overall method. Response distributions of maxima so predicted compare very well with digital simulation estimates.