Computation of response moments of high dimensional MDOF structure excited by stationary non-Gaussian random fields

Output moments of a non-linear dynamical system excited by a non-Gaussian random field can be obtained in practice only by simulation techniques. When the dynamical system can be decomposed in a low dimension non-linear part acting on a high dimension linear part, the original problem reduces to calculate output moments of a high dimension linear system. The proposed method suggests that work should be directed in the frequency domain. Time trajectories are then obtained through Fourier transform. Such a procedure does not introduce any approximation errors due to the time integration numerical scheme nor does it introduce any transient state. Further quasi-static correction terms can be introduced when a truncated modal basis is utilized in order to describe the low frequency dynamic response.     

An extension of the Kazakov relationship for non-Gaussian random variables and its use in the non-linear stochastic dynamics

A generalization for non-Gaussian random variables of the well-known Kazakov relationship is reported in this work. If applied to the stochastic linearization of non-linear systems under non-Gaussian excitations, this relationship allows us to define the significance of the linearized stiffness coefficient. It is the sum of that one known in the literature (the mean of the tangent stiffness) and of terms taking into account the non-Gaussianity of the response. Moreover, the relationship here given is used for finding alternative formulae between the moments and the quasi-moments. Lastly, it is used in the framework of the moment equation approach, coupled with a quasi-moment neglect closure, for solving non-linear systems under Gaussian or non-Gaussian forces. In this way an iterative procedure based on the solution of a linear differential equation system, in which the values of the response mean and variance are those of the precedent iteration, is originated. It reveals a good level of accuracy and a fast convergence.     

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.     

Response of linear systems to stationary bandlimited non-Gaussian processes

A method is developed for calculating statistics of the state of a linear system subjected to an arbitrary stationary bandlimited non-Gaussian process. The method is based on the representations of the input process obtained from a Shanon’s sampling theorem and Monte Carlo simulation. It is shown that the system output at a time t can be approximated by a finite sum of deterministic functions of t with random coefficients given by equally spaced values of the input process over a window of finite width centered on t. The number of terms in the sum depends on both input and system memory. Numerical examples show that the proposed method is simple to implement, efficient, accurate, and can also be applied to input process that are not bandlimited.     

Probability distribution applicable to non-Gaussian random processes

This paper presents a probability density function representing a non-Gaussian random process in closed form. The probability density is based on the Kac-Siegert solution of Volterra's stochastic series expansion of a nonlinear system. A method is developed, however, to obtain the Kac-Siegert solution from knowledge of the time history only of the random process, and the result is expressed as a function of a normal distribution. Then, by applying the change of random variable technique, the asymptotic probability density function applicable to the response of a nonlinear system (which is a non-Gaussian random process) is developed in closed form. A comparison of the presently developed probability density function and the histogram constructed from a record indicating strong non-Gaussian characteristics shows excellent agreement.     

A Monte Carlo simulation model for stationary non-Gaussian processes

A class of stationary non-Gaussian processes, referred to as the class of mixtures of translation processes, is defined by their finite dimensional distributions consisting of mixtures of finite dimensional distributions of translation processes. The class of mixtures of translation processes includes translation processes and is useful for both Monte Carlo simulation and analytical studies. As for translation processes, the mixture of translation processes can have a wide range of marginal distributions and correlation functions. Moreover, these processes can match a broader range of second order correlation functions than translation processes. The paper also develops an algorithm for generating samples of any non-Gaussian process in the class of mixtures of translation processes. The algorithm is based on the sampling representation theorem for stochastic processes and properties of the conditional distributions. Examples are presented to illustrate the proposed Monte Carlo algorithm and compare features of translation processes and mixture of translation processes.     

Existence and construction of translation models for stationary non-Gaussian processes

Translation models are memoryless transformations of Gaussian processes specified by their marginal distribution F and covariance function ξ. Iteration schemes are commonly used to find probability laws of Gaussian images of translation models, although these schemes may not converge since translation models do not exist for arbitrary functions F and ξ. Pairs (F,ξ) for which translation models exist are said to be consistent. Optimization algorithms are developed for constructing translation models that, for consistent pairs (F,ξ), match F and ξ, and, for inconsistent pairs (F,ξ), match F or ξ and approximate ξ or F. The resulting translation models can be used in Monte Carlo simulation studies.     

A translation model for non-stationary, non-Gaussian random processes

A model for simulation of non-stationary, non-Gaussian processes based on non-linear translation of Gaussian random vectors is presented. This method is a generalization of traditional translation processes that includes the capability of simulating samples with spatially or temporally varying marginal probability density functions. A formal development of the properties of the resulting process includes joint probability density function, correlation distortion and lower and upper bounds that depend on the target marginal distributions. Examples indicate the possibility of exactly matching a wide range of marginal pdfs and second order moments through a simple interpolating algorithm. Furthermore, the application of the method in simulating statistically inhomogeneous random media is investigated, using the specific case of binary translation with stationary and non-stationary target correlations.     

Numerical discretization of stationary random processes

The increasing interest of the research community to the probabilistic analysis concerning the civil structures with space-variant properties points out the problem of achieving a reliable discretization of random processes (or random fields in a multi-dimensional domain). Given a discretization method, a continuous random process is approximated by a finite set of random variables. Its dimension affects significantly the accuracy of the approximation, in terms of the relevant properties of the continuous random process under investigation. The paper presents a discretization procedure based on the truncated Karhunen–Loève series expansion and the finite element method. The objective is to link in a rational way the number of random variables involved in the approximation to a quantitative measure of the discretization accuracy. The finite element method is applied to evaluate the terms of the series expansion when a closed form expression is not available. An iterative refinement of the finite element mesh is proposed in this paper, leading to an accurate random process discretization. The technique is tested with respect to the exponential covariance function, that enables a comparison with analytical expressions of the approximated properties of the random process. Then, the procedure is applied to the square exponential covariance functions, which is one of the most used covariance models in the structural engineering field. The comparison of the adaptive refinement of the discretization with a non-adaptive procedure and with the wavelet Galerkin approach allows to demonstrate the computational efficiency of the proposal within the framework of the Karhunen–Loève series expansion. A comparison with the Expansion Optimal Linear Estimation (EOLE) method is performed in terms of efficiency of the discretization strategy.     

Modelling non-Gaussian random fields using transformations and Hermite polynomials

In the context of modelling residual roughness on nominally flat moderately polished metal surfaces, a method is proposed for solving problems related to sample function properties and/or special points such as maxima, minima, saddle points for random fields having non-Gaussian height distributions by recasting them in terms of the corresponding problems for the much more tractable Gaussian random fields by means of transformations. Special reference is made to the expansion of the transformations in series of Hermite polynomials. While the use of Hermite polynomials in connection with transformations of random fields and the useful results they yield with regard to covariance functions are well known, this paper derives the most general explicit formula for the expectation of any product of several Hermite polynomials in correlated Gaussian arguments thereby allowing their application to the higher moments of the transformed random field, in particular, to the third moment, which may be used to measure skewness.     

Probability distributions of peaks and troughs of non-Gaussian random processes

This paper deals with the development of probability density functions applicable for peaks, troughs and peak-to-trough excursions of a non-Gaussian random process where the response of a non-linear system is represented in the form of Volterra's second-order functional series. The density functions of peaks and troughs are derived in closed form and presented separately. It is found that the probability density function applicable to peaks (and troughs) is equivalent to the density function of the envelope of a random process consisting of the sum of a narrow-band Gaussian process and sine wave having the same frequency. Furthermore, for a non-Gaussian random process for which the skewness of the distribution is less than 1.2, the density function of peaks (and troughs) can be approximately presented in the form of a Rayleigh distribution. The parameter of the Rayleigh distribution is given as a function of parameters representing the non-Gaussian characteristics. The results of comparisons between newly derived density functions and histograms of peaks, troughs and peak-to-trough excursions constructed from data with strong non-linear characteristics show that the distributions well represent the histograms for all cases.     

Efficient stationary multivariate non-Gaussian simulation based on a Hermite PDF model

An efficient stationary multivariate non-Gaussian simulation method is developed using spectral representation and third order Hermite polynomial translation. An approximate closed form relationship is employed to identify the Hermite translation parameters based on target skewness and kurtosis. This preserves a high degree of accuracy over the entire admissible range of the Hermite translation, and eliminates the need for iterative solution of the translation parameters. The Hermite PDF model is suitable for a wide range of strongly non-Gaussian stochastic process. In addition, an explicit bidirectional relationship between the target non-Gaussian and Gaussian correlation is developed to eliminate the need for iteration or numerical integration to identify the underlying Gaussian correlation. Examples apply the simulation method to both theoretical targets and experimental wind pressure data.     

Space–time extreme value statistics of non-Gaussian random fields

This paper focuses on two new methods for predicting the extreme values of a non-Gaussian random field in both space and time. Both methods rely on the use of scalar time series expressing spatial extremes. These time series are constructed by sampling the available realizations of the random field over a suitable grid defining the domains in question and extracting the extreme values for each time point. In this way, time series of spatial extremes are produced. The realizations of the random field are obtained from either measurements or Monte Carlo simulations. The obtained time series provide the basis for estimating the extreme value distribution using recently developed techniques for time series, which results in an accurate practical procedure. The proposed prediction methods are applied to two specific cases. One is a second-order random ocean wave field, whose statistics deviate only mildly from the Gaussian, and the other is an example of a random field whose statistics is strongly non-Gaussian.     

多点平稳随机载荷识别方法研究

根据矩阵谱分解的思想,提出多点任意相关的随机载荷识别方法.鉴于响应功率谱矩阵和激励功率谱矩阵具有相同的秩且为非负定,首先推导完全相干的功率谱矩阵的识别方法,将任意相干的激励功率谱矩阵进行谱分解成完全相干的功率谱矩阵之和,利用响应信息识别分解后的完全相干功率谱进而完成对激励功率谱的识别.该方法具有较高的识别精度.针对求解逆问题中的适定性进行了讨论,指出病态的原因并运用条件数权重法,该法能在一定程度上减轻病态,提高识别精度.通过实验和仿真验证上述方法的正确性.最后对提高识别精度提出了建议.     

Simulation of stationary Gaussian/non-Gaussian stochastic processes based on stochastic harmonic functions

A new model is proposed to represent and simulate Gaussian/non-Gaussian stochastic processes. In the proposed model, stochastic harmonic function (SHF) is extended to represent multivariate Gaussian process firstly. Compared with the conventional spectral representation method (SRM), the SHF based model requires much fewer variables and Cholesky decompositions. Then, SHF based model is further extended to univariate/multivariate non-Gaussian stochastic process simulation. The target non-Gaussian process can be obtained from the corresponding underlying Gaussian processes by memoryless nonlinear transformation. For arbitrarily given marginal probability distribution function (PDF), the covariance function of the underlying multivariate Gaussian process can be determined easily by introducing the Mehler’s formula. And when the incompatibility between the target non-Gaussian power spectral density (PSD) or PSD matrix and marginal PDF exists, the calibration of the target non-Gaussian spectrum will be required. Hence, the proposed model can be regarded as SRM to efficiently generate Gaussian/non-Gaussian processes. Finally, several numerical examples are addressed to show the effectiveness of the proposed method.     

一种新的非高斯随机振动数值模拟方法

在振动工程领域,采用蒙特卡洛仿真方法求解复杂随机动力学问题时需要精确模拟各种随机振动激励信号。当随机振动激励具有显著的非高斯特征时,用传统的高斯振动去近似将产生较大的分析误差,需要研究精确的非高斯振动数值模拟技术。现有各种非高斯随机模拟方法一般只能模拟具有高峰值特征的随机振动,即超高斯随机振动,并且算法复杂不够直观,需要进行多次反复迭代,模拟精度和效率都有待提高。本文提出了一种新的基于幅值调制和相位重构的非高斯随机振动数值模拟方法,算法简洁直观,并充分利用快速傅里叶变换算法提高模拟效率,不仅可以模拟具有指定统计特性和频谱特性的超高斯随机振动,还能模拟亚高斯随机振动,具有广泛的适应性。数值仿真实验验证了该方法的有效性和精确性。     

Fatigue damage assessment for a spectral model of non-Gaussian random loads

In this paper, a new model for random loads–the Laplace driven moving average–is presented. The model is second order, non-Gaussian, and strictly stationary. It shares with its Gaussian counterpart the ability to model any spectrum but has additional flexibility to model the skewness and kurtosis of the marginal distribution. Unlike most other non-Gaussian models proposed in the literature, such as the transformed Gaussian or Volterra series models, the new model is no longer derivable from Gaussian processes. In the paper, a summary of the properties of the new model is given and its upcrossing intensities are evaluated. Then it is used to estimate fatigue damage both from simulations and in terms of an upper bound that is of particular use for narrowband spectra.     

Linear models for non-Gaussian processes and applications to linear random vibration

Linear models are finite sums of specified deterministic, continuous functions of time with random coefficients. It is shown that linear models provide (i) accurate approximations for real-valued non-Gaussian processes with continuous samples defined on bounded time intervals, (ii) simple solutions for linear random vibration problems with non-Gaussian input, and (iii) efficient techniques for selecting optimal designs from collections of proposed alternatives. Theoretical arguments and numerical examples are presented to establish properties of linear models, illustrate the construction of linear models, solve linear random vibration with non-Gaussian input, and propose an approach for optimal design of linear dynamic systems. It is shown that the proposed linear model provides an efficient tool for analyzing linear systems in non-Gaussian environment.     

Cycle distribution and fatigue damage assessment in broad-band non-Gaussian random processes

The aim of the paper is to propose a method to assess the cycle distribution and the fatigue damage in stationary broad-band non-Gaussian processes; the method is a further development of an existing procedure proposed for Gaussian processes [Int J Fatigue 2002; 24(11)]. By introducing a suitable transformation, we link a non-Gaussian process to an underlying Gaussian one, for which we can estimate the cumulative distribution of counted cycles; the corresponding joint density for the non-Gaussian process is then derived. The analysis of time histories measured on Mountain-bikes in off-road tracks shows that the new method is able to correctly assess the distribution of ‘rainflow’ counted cycles taking into account the non-normality of the load.     

Numerical and experimental studies of linear systems subjected to non-Gaussian random excitations

This paper presents a numerical simulation scheme for generating symmetric non-Gaussian random processes governed by prescribed kurtosis and spectral density. The generated process is represented as a continuous stationary random signal with occasional spikes superimposed on a Gaussian random background. The generated time history data records are used to simulate random excitations acting on linear single-degree-of-freedom systems. The results of the numerical simulation are compared with those measured experimentally. For a wide-band random excitation with kurtosis close to 3, the response kurtosis is found to be very sensitive to small changes in the excitation kurtosis. This is manifested by the appearance of significant spikes in the time history records when the excitation records do not display any significant spikes. The influence of the system damping is also examined for narrow-band and wide-band random excitations, and some differences are reported in the results.