Non-Gaussian solution for random rocking of slender rigid block

This paper adopts a random vibration approach to study the response of the slender rigid block to seismic action. The problem is strongly non-linear because of (i) the restoring term and (ii) the quadratic dissipation of energy due to the inelastic impacts, modeled as an impulsive process. The excitation process is firstly assumed to be a Gaussian white noise; secondly, a non-stationary filtered Gaussian white noise is assumed to simulate seismic shaking more accurately. The solution of the associated Fokker-Planck equation in terms of moments of the response is obtained by means of a non-Gaussian closure technique, that enables the complete statistical definition of the approximated transient response process to be achieved. The mean upcrossing rates and the response spectra in terms of displacement are evaluated. The reliability of the solutions derived is assessed by comparing them with Monte Carlo simulations.     

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.     

Finite element solution of Fokker–Planck equation of nonlinear oscillators subjected to colored non-Gaussian noise

Nonlinear oscillators subjected to colored Gaussian/non-Gaussian excitations are modelled through a set of three coupled first-order stochastic differential equations by representing the excitation as a first-order filtered white noise. A 3-D finite element (FE) formulation is developed to solve the corresponding 3-D Fokker Planck (FP) equations. The joint probability density functions of the state variables, obtained as a solution of the FP equation, are typically non-Gaussian and are used for computing the crossing statistics of the response – an essential metric for time variant reliability analysis. The method is illustrated through a noisy Lorenz attractor and a Duffing oscillator subjected to additive colored noise. The increase in state-space dimension when the Duffing oscillator is additionally excited with a parametric Gaussian noise is effectively handled by using stochastic averaging to reduce the state-space dimension. Investigations are carried out to examine the accuracy of the FE method vis-a-vis Monte Carlo simulations. The proposed method is observed to be computationally significantly cheaper for these three problems.     

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．     

A physical approach to structural stochastic optimal controls

The generalized density evolution equation proposed in recent years profoundly reveals the intrinsic connection between deterministic systems and stochastic systems by introducing physical relationships into stochastic systems. On this basis, a physical stochastic optimal control scheme of structures is developed in this paper, which extends the classical stochastic optimal control methods, and can govern the evolution details of system performance, while the classical stochastic optimal control schemes, such as the LQG control, essentially hold the system statistics since there is still a lack of efficient methods to solve the response process of the stochastic systems with strong nonlinearities in the context of classical random mechanics. It is practically useful to general nonlinear systems driven by non-stationary and non-Gaussian stochastic processes. The celebrated Pontryagin’s maximum principles is employed to conduct the physical solutions of the state vector and the control force vector of stochastic optimal controls of closed-loop systems by synthesizing deterministic optimal control solutions of a collection of representative excitation driven systems using the generalized density evolution equation. Further, the selection strategy of weighting matrices of stochastic optimal controls is discussed to construct optimal control policies based on a control criterion of system second-order statistics assessment. The stochastic optimal control of an active tension control system is investigated, subjected to the random ground motion represented by a physical stochastic earthquake model. The investigation reveals that the structural seismic performance is significantly improved when the optimal control strategy is applied. A comparative study, meanwhile, between the advocated method and the LQG control is carried out, indicating that the LQG control using nominal Gaussian white noise as the external excitation cannot be used to design a reasonable control system for civil engineering structures, while the advocated method can reach the desirable objective performance. The optimal control strategy is then further employed in the investigation of the stochastic optimal control of an eight-storey shear frame. Numerical examples elucidate the validity and applicability of the developed physical stochastic optimal control methodology.     

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。     

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.     

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.     

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。     

泊松白噪声激励下斜拉索的面内随机振动

目前针对斜拉索非线性随机振动的研究已广泛开展,但仅限于高斯随机激励情形。然而,现实中大部分的随机扰动都是非高斯的。若使用高斯激励模型将产生较大误差。假设拉索所受非高斯激励为泊松白噪声,研究了泊松白噪声激励下斜拉索面内随机振动。推导了受泊松白噪声激励的斜拉索面内振动的随机微分方程,建立了支配系统平稳响应概率密度函数的广义FPK方程。提出迭代加权残值法求解了四阶广义FPK方程,得到了系统响应概率密度函数的近似稳态闭合解。考察了垂跨比、阻尼系数以及脉冲到达率对拉索面内随机振动响应的影响。结果表明:拉索的响应随着垂跨比的增大,响应呈现不对称现象愈加明显;随阻尼比增加,系统响应得到显著抑制;当脉冲到达率增大,拉索的响应也随之增大,并逐渐接近于高斯白噪声激励的情形。另外,获得的理论结果与蒙特卡罗模拟的结果吻合地非常好。     

Closed-form solutions for the time-variant spectral characteristics of non-stationary random processes

Spectral characteristics are important quantities in describing stationary and non-stationary random processes. In this paper, the spectral characteristics for complex-valued random processes are evaluated and closed-form solutions for the time-variant statistics of the response of linear single-degree-of-freedom (SDOF) and both classically and non-classically damped multi-degree-of-freedom (MDOF) systems subjected to modulated Gaussian colored noise are obtained. The time-variant central frequency and bandwidth parameter of the response processes of linear SDOF and MDOF elastic systems subjected to Gaussian colored noise excitation are computed exactly in closed-form. These quantities are useful in problems which require the use of complex modal analysis, such as random vibrations of non-classically damped MDOF linear structures, and in structural reliability applications. Monte Carlo simulation has been used to confirm the validity of the proposed solutions.     

Design-point excitation for non-linear random vibrations

It has been shown in recent years that certain non-linear random vibration problems can be solved by well established methods of time-invariant structural reliability, such as FORM and importance sampling. A key step in this approach is finding the design-point excitation, which is that realization of the input process that is most likely to give rise to the event of interest. It is shown in this paper that for a non-linear, elastic single-degree-of-freedom oscillator subjected to white noise, the design-point excitation is identical to the excitation that generates the mirror image of the free-vibration response when the oscillator is released from a target threshold. This allows determining the design-point excitation with a single non-linear dynamic analysis. With a slight modification, this idea is extended to non-white and non-stationary excitations and to hysteretic oscillators. In these cases, an approximate solution of the design-point excitation is obtained, which, if necessary, can be used as a ‘warm’ starting point to find the exact design point using an iterative optimization algorithm. The paper also offers a simple method for computing the mean out-crossing rate of a response process. Several examples are provided to demonstrate the application and accuracy of the proposed methods. The methods proposed in this paper enhance the feasibility of approximately solving non-linear random vibration problems by use of time-invariant structural reliability techniques.     

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.     

斜拉索在轴向白噪声随机激励下的响应

建立了在考虑垂度以及拉索张力沿索长变化时拉索的参激随机微分方程,给出了在白噪声随机激励下预测拉索响应的近似理论解--用统计矩截断法求解矩方程,获得高斯闭合解和一阶非高斯闭合解.并采用数值模拟方法进行验证.随后以南京长江二桥A20拉索为研究对象,采用MonteCarlo数值方法对近似理论解进行验证.最后分析了白噪声激励强度、拉索阻尼及拉索张力对响应的影响.结果表明,斜拉索在零均值随机位移激励下的响应为非零均值;在高斯白噪声激励下的一阶位移响应随激励强度的增大而增大.随拉索阻尼和张力增大而减小.     

非平稳随机地震作用的结构整体可靠度分析

应用随机过程的正交展开-随机函数方法,建立了非平稳地震动过程的概率模型,实现了用一个基本随机变量来表达地震动过程的目的。通过选取基本随机变量的代表性离散点集,可以直接获取地震动过程的代表性样本集合。结合概率密度演化理论,进行了多自由度Duffing系统的随机地震反应分析与抗震可靠度计算。研究表明,非平稳地震动过程的概率模型与概率密度演化理论有机结合,可以实现复杂工程结构整体抗震可靠度的精确计算。     

An efficient simulation algorithm for non-Gaussian nonstationary processes

A novel algorithm is proposed for simulating univariate non-Gaussian nonstationary processes (NNP) with the specified evolutionary power spectral density (EPSD)/nonstationary auto-correlation function (NACF) and first four-order time-varying marginal moments (TVMMs). The sample realizations of the target NNP are generated as the outputs from a specific time-varying auto-regressive (TVAR) model via filtering the non-Gaussian and nonstationary white noise inputs. These white noise inputs are also non-Gaussian and nonstationary, and their first four-order TVMMs are predetermined using an approach developed herein according to the specified EPSD/NACF and first four-order TVMMs of the outputs. The conventional Johnson transformation is updated to accommodate the nonstationary cases for producing desired white noise inputs. This algorithm is developed from the linear filtering method (LFM), and inherits the simplicity and high efficiency from LFM. It fills the gaps in LFM-based algorithms for simulating NNP. Two numerical examples, i.e., a ground motion acceleration and a downburst velocity, are presented to fully demonstrate the capabilities of the proposed algorithm by comparing the simulation statistics with the targets.     

多自由度Duffing系统受演变随机激励的非平稳响应

用等效线性化法将受演变随机激励的非线性多自由度Ｄｕｆｆｉｎｇ系统等效为线性时变系统,用虚拟激励法来求解该时变系统的非平稳响应。具有计算简单、精确、效率高的特点。该演变随机激励的谱密度不局限于白噪声和过滤白噪声,各激励之间可以存在相位差。该方法的计算精度由Ｍｏｎｔｅ－Ｃａｒｌｏ法得以验证。     

Random response of a rotating composite blade with flexure–torsion coupling effect by the finite element method

A finite element model is employed to investigate the mean square response of a damped rotating composite blade with flexure–torsion interaction under stationary or non-stationary random excitation. The effects of transverse shear deformation and rotary inertia are considered. The finite element model can satisfy all the geometric and natural boundary conditions of a thick blade. The blade is considered to be subjected to white noise, band-limited white noise or filtered white noise excitation. The numerical results indicate that the increment of rotational speed will reduce the mean square response. It is also found that the mean square response decreases when the low natural frequency of base decreases. Inversely, the mean square response increases when the high natural frequency of base decreases. It is also shown that the fiber orientations have a significant effect on the mean square response of an orthotropic blade under random excitations. Moreover, the flexure–torsion coupling effect on the mean square response is changed by different fiber orientations.     

Parametric identification of systems with non-Gaussian excitation using measured response spectra

The problem of estimating parameters in dynamic systems excited by stochastic processes is addressed. Attention is focused on situations where the response processes are measurable but the excitation processes are non-Gaussian, unmeasurable and known only in terms of parameterised stochastic process models. General techniques for simultaneously estimating system and excitation process parameters are developed, based on the use of both normal, second order spectra and higher order, trispectra. The method is validated through application to some simulated data, relating to an oscillator driven by two specific kinds of non-Gaussian stochastic excitation.     

Response probability estimation for randomly excited quasi-linear systems using a neural network approach

A quasi-linear system is referred to as a system linear in properties and subjected to multiplicative random excitations appearing also in the linear terms. It is known that exact solutions for the stationary moments can be obtained analytically for such a quasi-linear system if the excitations are Gaussian white noises. However, the exact response probability, which is non-Gaussian, is not obtainable analytically. In this paper, a neural network approach is proposed to evaluate the stationary response probability for quasi-linear systems under both additive and multiplicative excitations of Gaussian white noises based on the obtained exact statistical moments. Numerical examples show that the procedure yields accurate results if an appropriate form is assumed for the probability density function. The accuracy of the results is substantiated by comparing them with those obtained from Monte Carlo simulations.