A direct probabilistic approach to solve state equations for nonlinear systems under random excitation

In this paper, a direct probabilistic approach (DPA) is presented to formulate and solve moment equations for nonlinear systems excited by environmental loads that can be either a stationary or nonstationary random process. The proposed method has the advantage of obtaining the response’s moments directly from the initial conditions and statistical characteristics of the corresponding external exci-tations. First, the response’s moment equations are directly derived based on a DPA, which is completely independent of the It?/filtering approach since no specific assumptions regarding the correlation structure of excitation are made. By solving them under Gaussian closure, the response’s moments can be obtained. Subsequently, a multiscale algo-rithm for the numerical solution of moment equations is exploited to improve computational efficiency and avoid much wall-clock time. Finally, a comparison of the results with Monte Carlo (MC) simulation gives good agreement. Furthermore, the advantage of the multiscale algorithm in terms of efficiency is also demonstrated by an engineering example.     

Cumulant-neglect closure for non-linear oscillators under random parametric and external excitations

The statistical moments of a non-linear system responding to random excitations are governed by an infinite hierarchy of equations; therefore, suitable closure schemes are needed to compute the more important lower order moments approximately. One easily implemented and versatile scheme is to set the cumulants of response variables higher than a given order to zero. This is applied to three non-linear oscillators with very different dynamic properties, and with Gaussian white noises acting as external and/or parametric excitations. It is found that the accuracy of computed second moments can be improved greatly by extending from the second order closure (Gaussian closure) to the fourth order closure, and that further refinement is unnecessary for practical purposes. Treatment of nonstationary transient response is also illustrated.     

含随机参数的多体系统动力学分析

基于Lagrange方程建立了含随机参数的多体系统的动力学 模型,利用广义坐标分离法将随机微分代数方程转化为随机纯微分方程,利用Newmark法进行数值解算. 应用随机因子法求解系 统随机响应的数字特征,获得统计意义下的解. 以旋转杆滑块系统为例,考虑系统中载荷、物理和几何参数的随机性,通过与Monte Carlo法结果的对比验证了文中方法的正确性和有效性. 计算结果表明,部分随机参数的分散性对多体系统动力响应的影响不可忽略,利用随机参数的动力学模型将能客观地反映出系统的动力学行为.     

A consistent closure method for non-linear random vibration

—A closure method is proposed for calculating moments of the state vector of a non-linear system satisfying an Itô stochastic differential equation. It is based on a finite set of moment equations and a sequence of probabilities converging to the exact distribution of the state vector that is obtained by the minimization of an objective function. Numerical results for one-dimensional diffusion processes show that the proposed closure technique is robust in the sense that resultant moments are satisfactory even when crude approximations are used for the probability of the state vector.     

Statistical characteristics of the diffusion of a chemically active additive in a turbulent mixing zone

In the article a numerical solution of the connected system of the equations of turbulent transfer for the fields of the velocity and concentration of a chemically active additive is used to calculate a number of the second moments of the concentration field in a flat mixing zone. The system of transfer equations is derived from the equations for a common function of the distribution of the fields of the pulsations of the velocity and the concentration [1] and is simplified in the approximation of the boundary layer. A closed form of the transfer equations is obtained on the level of three moments, using the hypothesis of four moments [2] and its generalized form for mixed moments of the field of the velocity and the field of a passive scalar. The differential operator of the closed system of the equations of turbulent transfer for the fields of the velocity and the concentration is found by a method of closure not of the parabolic type but of a weakly hyperbolic type [3]. An implicit difference scheme proposed in [4] is used for the numerical solution. The results of the numerical solution are compared with the experimental data of [5].     

非Gauss分布随机波浪力对海洋平台的非线性作用

为避免求解决定Maikov过程转移概率密度的Fokker—Planck方程，基于尺度分离的假设导出了一组描述非线性海洋平台受非Gauss分布随机波浪载荷作用所产生响应的矩量的常微分方程组。矩量方程清楚地反映出分别对应随机载荷和结构响应的两种不同统计特性的相互关系。由于矩量方程不依赖载荷的概率分布的具体细节，以它来模拟随机激励作用下的非线性系统将免于Monte Carlo方法所面临的正确模拟载荷概率分布的困难任务。将摄动法用于矩量方程可使线性化不再需要，这样就不会因为线性化而产生不可预料的误差。     

Constructing transient response probability density of non-linear system through complex fractional moments

The probability density function for transient response of non-linear stochastic system is investigated through the stochastic averaging and Mellin transform. The stochastic averaging based on the generalized harmonic functions is adopted to reduce the system dimension and derive the one-dimensional Itô stochastic differential equation with respect to amplitude response. To solve the Fokker–Plank–Kolmogorov equation governing the amplitude response probability density, the Mellin transform is first implemented to obtain the differential relation of complex fractional moments. Combining the expansion form of transient probability density with respect to complex fractional moments and the differential relations at different transform parameters yields a set of closed-form first-order ordinary differential equations. The complex fractional moments which are determined by the solution of the above equations can be used to directly construct the probability density function of system response. Numerical results for a van der Pol oscillator subject to stochastically external and parametric excitations are given to illustrate the application, the convergence and the precision of the proposed procedure.     

Gauss white noise perturbations of nonholonomic mechanical systems

The perturbations of nonholonomic mechanical systems under the Gauss white noises are studied in this paper.It is proved that the differential equations of the first-order moments of the solution process coincide with the corresponding equations in the non-perturbational case,and that there areε~2-terms but noε-terms in the differential equations of the second-order moments.Two propositions are obtained.Finally,an example is given to illustrate the application of the results.     

Higher order linearization in non-linear random vibration

In this paper a higher order linearization method for analyzing non-linear random vibration problems is presented. The non-linear terms of the given equation are replaced by unknown linear terms. These are in turn described by extra non-linear differential equations. The combined system of equations is then linearized to arrive at a higher degree-of-freedom equation for the original system. The method is illustrated by considering the Duffing oscillator under white noise input. The equivalent two d.o.f linear system is derived by the present method. Numerical results on steady state variance and PSD functions are obtained. These are found to be better than the simple linearization results.     

Extended Thermodynamics of dense gases with many moments – The macroscopic approach

To understand the non-equilibrium phenomena beyond the assumption of local equilibrium in rarefied polyatomic gases, a new version of extended thermodynamics has been recently introduced. It proposes two hierarchies of balance equations; the first of these is the usual one, while the second is an “energy block”. In this framework, we investigate here the model with an arbitrary but fixed number of moments and obtain its closure up to every order with respect to equilibrium. This is done in the context of the macroscopic approach. This is not a trivial aspect; in fact, if we impose the conditions up to a given order with respect to equilibrium, there is always the risk to obtain restrictions on the same expressions by imposing the equations at an higher order! An article has been published to prove that this eventuality does not occur. Similarly, if we impose the conditions for a model with a given number of moments, there is always the risk to obtain restrictions on the same expressions by imposing the equations for the model with an higher number of moments! Here this risk is confirmed, but controlled.     

A perturbation method for estimation of dynamic systems

A novel framework called the Perturbed Jth Moment Extended Kalman Filter (PJMEKF), based on a classical perturbation technique is proposed for estimating the states of a nonlinear dynamical system from sensor measurements. This method falls under a class of architectures under investigation primarily to study the interplay of major issues in nonlinear estimation such as nonlinearity, measurement sparsity, and initial condition uncertainty in an environment with low levels of process noise. Taylor series expansion of the departure motion dynamics about the best estimate is used to derive a series representation of the unforced motion. It is found that such series representation evolves as a set of differential equations that force each other in a cascade manner, adding up to give the unforced motion (in a so-called “triangular” structure). This formal perturbation solution for the departure motion dynamics is used in deriving the differential equations governing the time evolution of the high order statistical moments of the estimation error. These tensor differential equations are found to possess a similar high order triangular structure in addition to being symmetric (in N tensorial dimensions and we appropriately term the evolution equations as Tensor Lyapunov Equations of statistical moment perturbations). Elegance of the tensor differential equations thus derived is accompanied by the computational advantages due to symmetry in all tensorial dimensions. A vector matrix representation of tensors is proposed with which the representation and solution of the tensor differential equations can be carried out effectively. Approximations are introduced to incorporate low levels of process noise forcing function in the propagation phase of the moment equations. The statistics thus propagated are used in a filtering framework to estimate the state vector of a nonlinear system from noisy measurements, within the traditional Kalman update paradigm. The Kalman gain thus determined is utilized in updating all high order moments in preparation for the subsequent propagation phase leading to improved estimation accuracy. The filter developed is applied to an orbit estimation problem and comparisons are presented with classical extended Kalman filter.     

基于神经网络的结构可靠性优化设计

结合随机摄动技术和随机模拟方法,提出了可靠性优化设计的一种数值逼近法,将服从任意分布的可靠性概率约束等价转化为确定型约束,可以迅速准确地获得优化设计结果。针对具有多失效模式的结构可靠性优化设计,提出了随机模拟-神经网络方法,模拟得到随机设计变量与系统可靠度之间的显性函数表达式,简化了计算过程,同时可以获得较高的计算精度,具有很好的工程实用价值。     

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 stability of three-dimensional linear systems under parametric random action

The stability of a three-dimensional linear system, driven by a parametric random excitation is considered. The stability of the system is investigated using a combination of stochastic averaging method and a probabilistic approach. For this purpose, it is necessary to find the transient probability density of the components of the vector random process. Thus, the invariant measure of the system may be calculated from the stationary solutions of the associated Fokker–Planck equations. These solutions are obtained numerically, using the sweep method. As a comparison criterion, a digital simulation of It? equations has been carried out using the Monte-Carlo simulation (MCS) method. As an application, the example of instability of a thin-walled bar under the effect of parametric random action is considered     

Numerical modeling of turbulent-transfer processes in a mixing zone

The series of moments of the velocity field in a two-dimensional zone of mixing is calculated in this article by numerically solving a system of turbulent-transfer differential equations derived from an equation for a single-point distribution function of the velocity pulsation field [1] and simplified to an approximation of the boundary layer. The closed form of the transfer equation is obtained at the level of the third moments using the Millionshchikov hypothesis [2]. The differential operator of the system under this closure turns out to be weakly hyperbolic [3], and not parabolic. A difference scheme is proposed that realizes the method of matrix fitting [4]. A comparison is carried out with an experiment [5, 6].     

Sensitivity,probabilistic and stochastic analysis of the thermo-piezoelectric phenomena in solids by the stochastic perturbation technique

The main aim here is to present the application of the generalized stochastic perturbation technique to thermo-piezoelectric analysis of solid continua. The general nth order Taylor series representation for all random input parameters and the state functions is employed to formulate the coupled thermo-electro-elasticity equilibrium equations of the additional order; a determination of any probabilistic moments and characteristics is described; the discretization of the problem in terms of the Stochastic perturbation-based Finite Element Method is also provided. Since this expansion includes the lowest order partial derivatives, the structural sensitivity analysis using direct differentiation is performed at the same time with probabilistic modeling contrasted with the Monte-Carlo simulation results. The probabilistic approach is extended here towards an accounting for the stochastic ageing processes, which appear frequently in aggressive external environments and under dynamic excitation. The two parametric stochastic process with Gaussian initial value and ageing velocity is tested for this purpose. The entire procedure is tested on the example of the thermo-electro-elastic pulsation of the beam modeled analytically using the symbolic software MAPLE, where polynomial approximations, design sensitivities, probabilistic moments and their histories are computed and visualized.     

Estimating Critical Hopf Bifurcation Parameters for a Second-Order Delay Differential Equation with Application to Machine Tool Chatter

Nonlinear time delay differential equations are well known to havearisen in models in physiology, biology and population dynamics. Theyhave also arisen in models of metal cutting processes. Machine toolchatter, from a process called regenerative chatter, has been identifiedas self-sustained oscillations for nonlinear delay differentialequations. The actual chatter occurs when the machine tool shifts from astable fixed point to a limit cycle and has been identified as arealized Hopf bifurcation. This paper demonstrates first that a class ofnonlinear delay differential equations used to model regenerativechatter satisfies the Hopf conditions. It then gives a precisecharacterization of the critical eigenvalues on the stability boundaryand continues with a complete development of the Hopf parameter, theperiod of the bifurcating solution and associated Floquet exponents.Several cases are simulated in order to show the Hopf bifurcationoccurring at the stability boundary. A discussion of a method ofintegrating delay differential equations is also given.     

Experimental investigation of a maximum entropy assumption for acceleration terms within a poly‐disperse moment framework

Moment transport methods are being developed to model poly‐dispersed multiphase flows by transporting statistical moments of the particle size–velocity joint probability density function (JPDF). A common feature of these methods is the requirement to reproduce or approximate the form of the JPDF from the transported moments for calculation of body force terms and other source terms. This paper examines the application of a maximum entropy technique against phase Doppler anemometry data sets from an electrostatically charged kerosene spray and also an automotive pressure swirl atomizer. An assessment of which moments are required to reproduce the JPDFs using a maximum entropy assumption to a sufficient level of accuracy is made. It is found that it is possible to reproduce the JPDFs to a high level of accuracy using a large number of moments; however, this incurs large computational overheads. If the moments to be transported are chosen on the basis of physical reasoning (such as the relationship between size and velocity due to drag) it is possible to reduce the number of moments to those which would be conserved via balance equations. This permits an approximation to the JPDF commensurate with the closure level of the moment transport method and thus the closure model method is naturally scalable with the degree of information from available conservation equations. Copyright © 2008 John Wiley & Sons, Ltd.