An Improved Closure Method for Analysis of Nonlinear Stochastic Systems

The probability density function (PDF) and the mean up-crossing rate (MCR) of the stationary responses of nonlinear stochastic systems excited by white noise is analyzed based on the assumption that the PDF of the responses is approximated with an exponential function of a polynomial in the state variables. Based upon the approximate PDF, a new technique is developed for the approximate PDF solution of Fokker–Planck–Kolmogorov equation, and consequently, the MCR of the system responses is analyzed. Numerical results showed that the approximate PDFs and MCRs approach to the exact ones as the degree of the polynomial increases.     

EPC procedure for PDF solution of non-linear oscillators excited by Poisson white noise

The stationary probability density function (PDF) solution of the responses of non-linear stochastic oscillators subjected to Poisson pulses is analyzed. The PDF solutions are obtained by the exponential-polynomial closure (EPC) method. To assess the effectiveness of the solution procedure numerically, non-linear oscillators are analyzed with different impulse arrival rates, degree of oscillator non-linearity and excitation intensity. Numerical results show that the PDFs obtained with the EPC method yield good agreement with those obtained from Monte Carlo simulation when the polynomial order is 4 or 6. It is also observed that the EPC procedure is the same as the equivalent linearization procedure under Gaussian white noise in the case of the polynomial order being 2.     

Random vibration of viscoelastic system under broad-band excitations

An analytical method is proposed to study the response of a viscoelastic system with strongly non-linear stiffness force and under broad-band random excitations. The random excitations can be additive, or multiplicative, or both, and they can be stationary or non-stationary with evolutionary spectra. With the proposed method, contributions of the viscoelastic force to both damping and stiffness are taken into account separately, and then the extended version of the stochastic averaging, called the quasi-conservative averaging, is applied to the system to derive the averaged equation of energy envelope. Probability density functions of system responses, such as the total energy, the amplitude, and the state variables, can then be obtained analytically. The accuracy of the method is substantiated by comparing the analytical results with those from Monte Carlo simulations. Effects of parameters in the viscoelastic force and in the non-linear stiffness force on the system responses are also investigated.     

Stochastic analysis of oscillators with non-linear damping

The response of a class of oscillators with non-linear damping to stochastic excitation is considered. A partial differential equation which describes approximately the probability density function of the response amplitude is derived. The stationary and non-stationary solutions of this equation are examined. The soundness of the method is tested by comparing the solutions generated by its application to problems with known solutions. The Van der Pol and Rayleigh oscillators are included in the example problems studied.     

Non-linear filtering for a dust-perturbed two-body model

In standard textbooks on classical mechanics, the two-body central forcing problem is formulated as a system of the coupled non-linear second-order deterministic differential equations. Uncertainties, introduced by the astronomical ‘dust’, are not assumed in the orbit dynamics. The dust population produces an additional random force on the orbiting particle. This work is a continuation of the paper (Sharma and Parthasarathy, Proc. R. Soc. A: Math. Phys. Eng. Sci. 463:979–1003, [2007]) in which the authors developed and analyzed the dust-perturbed two-body model, which accounts for the dust perturbation felt by the orbiting particle. The theory of the dust-perturbed stochastic system was developed using the Fokker–Planck equation. This paper discusses the problem of realizing non-linear stochastic filters for estimating the states of the dust-perturbed planar two-body stochastic system, especially from noisy observations. This paper utilizes the Kushner’s theory of non-linear filtering, which involves stochastic observation term in the evolution of conditional probability density, for deriving the stochastic evolutions of the conditional mean and conditional covariance. The effectiveness of the non-linear filters of this paper is examined on the basis of their ability to preserve the perturbation effect, less random fluctuations in the mean trajectory and stability characteristics in the mean and variance trajectories. Most notably, this paper reveals the efficacy of the second-order approximate Kushner filter for the estimation procedure in contrast to the first-order approximate filter. Simulation results are introduced to demonstrate the usefulness of an analytic theory developed in this paper.     

线性随机结构在随机激励下动力响应分析

利用虚拟激励法对随机结构正交展开理论进行扩展,并在Ritz向量子空间中对扩阶系统方程进行动力聚缩,提出了一类可以快速高效地进行线性随机结构复合随机振动分析的计算方法．算例分析表明,该法可以方便地分析随机结构在平稳或非平稳随机激励下的复合随机振动问题,且分析结果与 Monte Carlo模拟分析结果符合良好;与均值参数确定性结构传统随机振动分析计算结果相比,随机结构在相同随机激励下响应自谱密度曲线具有峰值降低、谱宽增大的特点．     

Stochastic finite element analysis of non-linear plane trusses

—This study considers the responses of geometrically and materially non-linear plane trusses under random excitations. The stress-strain law in the inelastic range is based on an explicit differential equation model. After a total Lagrangian finite element discretization, the nodal displacements satisfy a system of stochastic non-linear ordinary differential equations with right-hand-sides given by random functions of time. The exact solution of the above stochastic differential equation is generally difficult to obtain. To seek an approximate solution with good accuracy and reasonable computational effort, the stochastic linearization method is used to find the first and second statistical moments (i.e. the mean vector and the one-time covariance matrix) of the nodal displacements. Results of simple structures under Gaussian white-noise excitation indicate that the proposed method has good accuracy (generally underestimates the r.m.s. stationary response by 5–14%) and requires only a small fraction of the computation time of the time-history Monte-Carlo method.     

Stochastic transient analysis of thermal stresses in solids by explicit time-domain method

Stochastic heat conduction and thermal stress analysis of structures has received considerable attention in recent years. The propagation of uncertain thermal environments will lead to stochastic variations in temperature fields and thermal stresses. Therefore, it is reasonable to consider the variability of thermal environments while conducting thermal analysis. However, for ambient thermal excitations, only stationary random processes have been investigated thus far. In this study, the highly efficient explicit time-domain method(ETDM) is proposed for the analysis of non-stationary stochastic transient heat conduction and thermal stress problems. The explicit time-domain expressions of thermal responses are first constructed for a thermoelastic body. Then the statistical moments of thermal displacements and stresses can be directly obtained based on the explicit expressions of thermal responses. A numerical example involving non-stationary stochastic internal heat generation rate is investigated. The accuracy and efficiency of the proposed method are validated by comparison with the Monte-Carlo simulation.     

A domain decomposition method for stochastic analysis of acoustic fields with hybrid and localized uncertainties

An efficient domain decomposition method (DDM) is proposed for the dynamic analysis of stochastic acoustic fields with hybrid and localized uncertainties. The hybrid and localized uncertainties refer to the parameters that are associated with local properties of the acoustic fields and meanwhile are subjected to different kinds of randomness. To take advantage of the locally distributed feature of uncertain parameters, the full acoustic domain is divided into several sub-domains, along with each localized uncertain parameter being assigned to one specific sub-domain. In each sub-domain, the deterministic Helmholtz equation is transformed to a weak integral form and the discretized governing equation is obtained by employing Chebyshev orthogonal polynomials as admissible functions. The random or interval perturbation technique is applied to the individual governing equation according to the respective uncertainty type, whereby the stochastic governing equation is established. The original acoustic field is eventually recovered by the introduction of penalty functions to impose sound pressure continuity on the interfaces of sub-domains, and the (intervals of) sound pressure, together with its expectation and variance, can be subsequently obtained. The accuracy and efficiency of the proposed method are verified in several numerical examples by comparisons with the results given by brute force Monte Carlo simulations, and the DDM-based independent way of modelling and analysis proves to be quite effective and flexible for uncertainty quantification in acoustic fields.     

Non-stationary response of a van der Pol-Duffing oscillator under Gaussian white noise

This paper investigates the probability density evolution process of a van der Pol-Duffing oscillator under Gaussian white noise. A path integration method is employed with the Gauss–Legendre integration scheme. In the path integration method, the short-time Gaussian approximation scheme is used for computing the one-step transition probability density. Two cases are considered with slight nonlinearity or strong nonlinearity in displacement. The stationary and non-stationary responses of the oscillator are studied. Compared with the simulation result, the path integration method can present a satisfactory probability density function (PDF) solution for each case. Different probability density evolution processes are observed correspondingly. In the case of slight nonlinearity, the PDF undergoes a clockwise motion around the origin. The peak region gradually expands and the PDF eventually forms a circle. By contrast, the strong nonlinearity drives the oscillator to oscillate around the limit cycle. In such a case, the PDF rapidly forms a circle. The circle keeps its shape and develops until the oscillator becomes stationary. More complicated phenomena can be studied by the adopted path integration method.