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.     

Non-parametric system identification from non-linear stochastic response

An estimation method is proposed for identification of non-linear stiffness and damping of single-degree-of-freedom systems under stationary white noise excitation. Non-parametric estimates of the stiffness and damping along with an estimate of the white noise intensity are obtained by suitable processing of records of the stochastic response. The stiffness estimation is based on a local iterative procedure, which compares the elastic energy at mean-level crossings with the kinetic energy at the extremes. The damping estimation is based on a generic expression for the probability density of the energy at mean-level crossings, which yields the damping relative to white noise intensity. Finally, an estimate of the noise intensity is extracted by estimating the absolute damping from the autocovariance functions of a set of modified phase plane variables at different energy levels. The method is demonstrated using records obtained by numerical simulation.     

Response spectral density of linear systems with external and parametric non-Gaussian, delta-correlated excitation

An exact, closed form, analytical expressions for a response spectral density of a certain type of systems, subjected to non-Gaussian, stationary, delta-correlated noise are derived. A new, extended mean square stability conditions are derived for such systems.     

Spectral density of oscillator with bilinear stiffness and white noise excitation

The power spectral density of an oscillator with bilinear stiffness excited by Gaussian white noise is considered. A method originally proposed by Krenk and Roberts [J Appl Mech 66 (1999) 225] relying on slowly changing energy for lightly damped systems is applied. In this method an approximate solution for the power spectral density at a given energy level is obtained by considering local similarity with the free undamped response. The total spectrum is obtained by integrating over all energy levels weighting each with the stationary probability density of the energy. The accuracy of the approximate analytical solution is demonstrated by comparing with results obtained by stochastic simulation. It is shown how the method successfully captures the broadening of the resonance peak and the presence of higher harmonics in the power spectral density of strongly non-linear systems.     

Nonlinear stochastic optimal control of Preisach hysteretic systems

The nonlinear stochastic optimal control of Preisach hysteretic systems is studied, and the control procedure is illustrated with an example of the single-degree-of-freedom Preisach system. The Preisach hysteretic system subjected to a stochastic excitation is first replaced by an equivalent non-hysteretic nonlinear stochastic system with displacement-amplitude-dependent damping and stiffness, by using the generalized harmonic balance technique. Then, the relationship between the displacement amplitude and total system energy is established, and the equivalent damping and stiffness coefficients are expressed as functions of the system energy. The averaged Itô stochastic differential equation for the system energy as one-dimensional controlled diffusion process, is derived by using the stochastic averaging method of energy envelope. For the semi-infinite time-interval ergodic control, the dynamical programming equation is obtained based on the stochastic dynamical programming principle, and is solved to yield the optimal control force. Finally, the Fokker–Planck–Kolmogorov equation associated with the averaged Itô equation is established, and the stationary probability density of the system energy is obtained, from which the variances of the controlled system response and the optimal control force are predicted and the control efficacy is evaluated. Numerical results show that the proposed control strategy for Preisach hysteretic systems is very effective and efficient.     

宽带激励下非线性振动系统响应的最小化控制

将一种基于广义谐和函数的随机平均法和随机动态规划原理相结合，提出了一种非线性随机最优控制方法，可以为受宽带激励的单自由度强非线性振动系统设计最优控制规律，以使得系统的稳态响应最小化。方法中的随机平均法用来得到受控系统位移幅值的Ito随机微分方程；用随机动态规划原理为系统稳态响应最小化建立动态规划方程；在控制力为有界的条件下，从动态规划方程中可以导出最优控制规律；通过求解FPK方程得到受控系统的响应。本文用一个具体的例子阐述了这一控制方法的实施过程。     

Random response of Duffing oscillator to periodic excitation with random disturbance

This paper discusses the random response of a non-linear Duffing oscillator subjected to a periodic excitation with random phase modulation. Effects of uncertainty in the periodic excitation and level of the system non-linearity on the response moments and non-Gaussian nature of the response caused by both the system non-linearity and the non-Gaussian loading are investigated. Results are presented in terms of the second- and the fourth-order moments as well as the excess factor of the response and some results are compared with those from the Monte Carlo simulation. An iterated linearisation technique is proposed to improve the accuracy of the numerical results for strongly non-linear systems.     

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.     

非高斯噪声激励下非线性漂移Fokker-Planck方程的非稳态解及其应用

非高斯噪声广泛存在于各种非线性系统，对非高斯噪声所驱动系统的非稳态演化行为进行研究可以更为深入的了解其内在的演化机理．本文对非高斯噪声和高斯白噪声共同驱动的非线性动力学系统的非稳态演化问题进行研究．首先应用格林函数的 $\Omega$ 展开理论在初始区域对非线性动力学系统进行线性化，然后结合本征值和本征矢理论推导出了该系统 Fokker-Planck 方程的近似非稳态解的表达式，最后以 Logistic 系统模型为例分析了非高斯噪声强度，关联时间及非高斯噪声偏离参数对非稳态解以及一阶矩的影响．研究结果表明，用 Logistic 模型描述产品产量增长时，其非稳态解可更好地反映产品产量在不稳定点附近的演化行为．     

The geometry of random vibrations and solutions by FORM and SORM

The geometry of random vibration problems in the space of standard normal random variables obtained from discretization of the input process is investigated. For linear systems subjected to Gaussian excitation, the problems of interest are characterized by simple geometric forms, such as vectors, planes, half spaces, wedges and ellipsoids. For non-Gaussian responses, the problems of interest are generally characterized by non-linear geometric forms. Approximate solutions for such problems are obtained by use of the first- and second-order reliability methods (FORM and SORM). This article offers a new outlook to random vibration problems and an approximate method for their solution. Examples involving response to non-Gaussian excitation and out-crossing of a vector process from a non-linear domain are used to demonstrate the approach.     

Dynamic response and reliability analysis of non-linear stochastic structures

A new probability density evolution method is proposed for dynamic response analysis and reliability assessment of non-linear stochastic structures. In the method, a completely uncoupled one-dimensional governing partial differential equation is derived first with regard to evolutionary probability density function (PDF) of the stochastic structural responses. This equation holds for any response or index of the structure. The solution will put out the instantaneous PDF. From the standpoint of the probability transition process, the reliability of the structure is evaluated in a straightforward way by imposing an absorbing boundary condition on the governing PDF equation. However, this does not induce additional computational efforts compared with the dynamic response analysis. The computational algorithm to solve the PDF equation is studied. A deterministic dynamic response analysis procedure is embedded to compute coefficient of the evolutionary PDF equation, which is then numerically solved by the finite difference method with total variation diminishing scheme. It is found that the proposed hybrid algorithm may deal with non-linear stochastic response analysis problem with high accuracy. Numerical examples are investigated. Parts of the results are illustrated. Some features of the probabilistic information of the response and the reliability are observed and discussed. The comparisons with the Monte Carlo simulations demonstrate the accuracy and efficiency of the proposed method.     

Non-Gaussian models for stochastic mechanics

Memoryless transformations of Gaussian processes and transformations with memory of the Brownian and Lévy processes are used to represent general non-Gaussian processes. The transformations with memory are solutions of stochastic differential equations driven by Gaussian and Lévy white noises. The processes obtained by these transformations are referred to as non-Gaussian models. Methods are developed for calibrating these models to records or partial probabilistic characteristics of non-Gaussian processes. The solution of the model calibration problem is not unique. There are different non-Gaussian models that are equivalent in the sense that they are consistent with the available information on a non-Gaussian process. The response analysis of linear and non-linear oscillators subjected to equivalent non-Gaussian models shows that some response statistics are sensitive to the particular equivalent non-Gaussian model used to represent the input. This observation is relevant for applications because the choice of a particular non-Gaussian input model can result in inaccurate predictions of system performance.     

Stochastic response of linear and non-linear systems to α-stable Lévy white noises

The stochastic response of linear and non-linear systems to external α-stable Lévy white noises is investigated. In the literature, a differential equation in the characteristic function (CF) of the response has been recently derived for scalar systems only, within the theory of the so-called fractional Einstein–Smoluchowsky equations (FESEs). Herein, it is shown that the same equation may be built by rules of stochastic differential calculus, previously applied by one of the authors to systems driven by arbitrary delta-correlated processes. In this context, a straightforward formulation for multi-degree-of-freedom (MDOF) systems is also developed.Approximate CF solutions to the derived equation are sought for polynomial non-linearities, in stationary conditions. To this aim a wavelet representation is used, in conjunction with a weighted residual method. Numerical results prove in excellent agreement with exact solutions, when available, and digital simulation data.     

A copula-based Gaussian mixture closure method for stochastic response of nonlinear dynamic systems

Gaussian closure method is commonly used in the analysis of nonlinear stochastic systems. However, Gaussian closure may lead to unacceptable errors when system response is very much different from being Gaussian, and accuracy of the method decreases as the nonlinearity of the system increases. The need for better accuracy in strongly non-linear problems has caused the development of non-Gaussian closure schemes. In this paper, we develop a new copula-based Gaussian mixture closure method for randomly excited nonlinear systems. Our method relies on the assumption of marginal PDF of response in terms of finite Gaussian mixture model, and the derivation of joint PDF with aid of dependence modeling of Gaussian copula. By substituting the non-Gaussian PDF representation into moment equations of nonlinear system, we further develop an optimization-based closure scheme for the solution of the unknown parameters in joint PDF. In this way, PDF and thus, moments of response of highly nonlinear system can be described in a more flexible and robust way. Effectiveness of the new closure method is demonstrated by a nonlinear and a Duffing oscillator that are subjected to Gaussian white noise. The results are compared with the Gaussian closure and exact solution. It has been shown that Gaussian closure is a special case of the new closure method, and accuracy of Gaussian closure is the lower bound of that of the new closure method.     

A penalty finite element analysis for nonlinear mechanics of biphasic hydrated soft tissue under large deformation

The non-linear response of soft hydrated tissues under physiologically relevant levels of mechanical loading can be represented by a two-phase continuum model based on the theory of mixtures. The governing equations for a biphasic soft tissue, consisting of an incompressible solid and an incompressible, inviscid fluid, under finite deformation are presented and a finite element formulation of this highly non-linear problem is developed. The solid phase is assumed to be hyperelastic, and the stress-strain relations for the solid phase are defined in terms of the free energy function. A finite element model is formulated via the Galerkin weighted residual method coupled with a penalty treatment of the continuity equation for the mixture. Using a total Lagrangian formulation, the non-linear weighted residual statement, expressed with respect to the reference configuration, leads to a coupled non-linear system of first order differential equations. The non-linear constitutive equation for the solid phase elasticity is incrementally linearized in terms of the second Piola-Kirchhoff stress and the corresponding Lagrangian strain. A tangent stiffness matrix is defined in terms of the free energy function; this matrix definition can be applied to any free energy function, and will yield a symmetric matrix when the free energy function is convex. An unconditionally stable implicit predictor-corrector algorithm is used to obtain the temporal response histories. The confined compression mechanical test of soft tissue in stress relaxation is used as an example problem. Results are presented for moderate and rapid rates of loading, as well as small and large applied strains. Comparison of the finite element solution with an independent finite difference solution demonstrates the accuracy of the formulation.     

Response analysis of nonlinear multi-degree-of-freedom systems with uncertain properties to non-Gaussian random excitations

The investigation reported in this paper is concerned with the development of an approach for response analysis of multi-degree-of-freedom (mdof) nonlinear systems with uncertain properties of large variations and under non-Gaussian nonstationary random excitations. The developed approach makes use of the stochastic central difference (SCD) method, time co-ordinate transformation (TCT), and adaptive time schemes (ATS). It is applicable to geometrically complicated systems idealized by the finite element method (FEM). For demonstration of its use and availability of results for direct comparison, a two-degree-of-freedom (tdof) nonlinear asymmetric system with uncertain natural frequencies and under Gaussian and non-Gaussian nonstationary random excitations is considered. Computed results obtained for the system with and without uncertain natural frequencies, and under Gaussian and non-Gaussian nonstationary random excitations are presented. It is concluded that the approach is relatively simple, accurate and efficient to apply.     

Optimal bounded control of harmonically and stochastically excited strongly nonlinear oscillators

A procedure for designing optimal bounded control to minimize the response of harmonically and stochastically excited strongly nonlinear oscillators is proposed. First, the stochastic averaging method for controlled strongly nonlinear oscillators under combined harmonic and white noise excitations using generalized harmonic functions is introduced. Then, the dynamical programming equation for the control problem of minimizing response of the systems is formulated from the partially completed averaged Itô equations by using the dynamical programming principle. The optimal control law is derived from the dynamical programming equation and control constraint without solving the dynamical programming equation. Finally, the stationary probability density of the amplitude and mean amplitude of the optimally controlled systems are obtained from solving the reduced Fokker–Planck–Kolmogorov equation associated with fully completed averaged Itô equations. An example is given to illustrate the proposed procedure and the results obtained are verified by using those from digital simulation.     

Control and dynamics of a SDOF system with piecewise linear stiffness and combined external excitations

The paper considers a problem of stochastic control and dynamics of a single-degree-of-freedom system with piecewise linear stiffness subjected to combined periodic and white noise external excitations. To minimize the system response energy a bounded in magnitude control force is applied to the systems. The stochastic optimal control problem is handled through the dynamic programming approach. Based on the solution to the Hamilton–Jacobi–Bellman equation it is proposed to use the dry friction control law in the non-resonant case. In the resonant case the stochastic averaging procedure has been used to derive stochastic differential equations for system response amplitude and phase. The joint PDF of response amplitude and phase is derived analytically and numerically using the Path Integration approach.     

A progressive damage simulation algorithm for GFRP composites under cyclic loading. Part I: Material constitutive model

A life prediction algorithm and its implementation for a thick-shell finite element formulation for GFRP composites under constant or variable amplitude loading is introduced in this work. It is a distributed damage model in the sense that constitutive material response is defined in terms of meso-mechanics for the unidirectional ply. The algorithm modules for non-linear material behaviour, pseudo-static loading-unloading-reloading response, Constant Life Diagrams and strength and stiffness degradation due to cyclic loading were implemented on a robust and comprehensive experimental database for a unidirectional glass/epoxy ply. The model, based on property definition in the principal coordinate system of the constitutive ply, can be used, besides life prediction, to assess strength and stiffness of any multidirectional laminate after arbitrary, constant or variable amplitude multi-axial cyclic loading. Numerical predictions were corroborated satisfactorily by test data from constant amplitude fatigue of glass/epoxy laminates of various stacking sequences.     

A new model for non-Gaussian random excitations

Very often one is called upon to model time series data which are clearly non-Gaussian, but which retain some aspects of a Gaussian process. In the present paper, a novel methodology which helps in modelling such data is presented. The method is essentially to express the process as a series with finite number of terms, wherein the first term is a Gaussian process with zero mean and unit standard deviation. Non-Gaussian higher order correction terms are added to this such that each succeeding term is orthogonal or uncorrelated with all the previous terms. The unknown coefficients in the series representation can be expressed in terms of the estimated moments of the data. Further the autocorrelation or PSD of the data can be exactly reproduced by the non-Gaussian model. The use of the proposed model is illustrated by considering the unevenness data of railway tracks. Application to response of systems under non-Gaussian excitation is also briefly discussed.