non-Gaussian Random Excitation Research Articles

Stationary response analysis for a linear system under non-Gaussian random excitation by the equivalent non-Gaussian excitation method and the Hermite moment model

An approximate analytical method is developed to obtain the stationary response probability density and mean upcrossing rate for a linear system subjected to non-Gaussian random excitation. The random excitation is a stationary stochastic process prescribed by an arbitrary first-order probability density and a class of power spectral density with a bandwidth parameter, and is described by an Itô stochastic differential equation. In the proposed method, the equivalent non-Gaussian excitation method is first utilized to derive a closed set of moment equations for the system response. The exact solutions of response moments up to the fourth order are calculated from the moment equations. Next, using the response moments, the Hermite moment model is built for the stationary displacement response, and its probability density function is obtained through the model. Finally, the mean upcrossing rate of the displacement is determined from the response probability distribution by making two approximations that the displacement and velocity of the system are independent and the velocity is Gaussian. The proposed method can be used for any excitation probability density and bandwidth parameter value as long as the skewness and kurtosis of the response are within the applicable range of the Hermite moment model. In illustrative examples, the proposed method is applied to linear systems under random excitations with two-types of non-Gaussian probability densities. The effectiveness of the method is demonstrated by comparing the analytical results with the pertinent Monte Carlo simulation results while widely varying the bandwidth ratio between the excitation spectral density and the frequency response function of the system. Moreover, the effect of excitation non-Gaussianity on the mean upcrossing rate is discussed. It is shown that for small bandwidth ratio, the crossing rate differs considerably depending on the excitation probability density. The difference due to the excitation distribution decreases monotonically as the bandwidth ratio increases, and the crossing rate becomes almost the same as that in the case of Gaussian excitation when the bandwidth ratio is large.

Non-stationary non-Gaussian random vibration analysis of Duffing systems based on explicit time-domain method

Non-stationary non-Gaussian random vibration problems of structures are challenging and drawing increasing attention. In the present study, firstly, an explicit time-domain method (ETDM) is proposed to determine the higher-order response statistics of linear systems subjected to non-stationary non-Gaussian random excitations, in which the first four orders of cumulants of dynamic responses are directly formulated through the cumulant operation rule based on the explicit expressions of responses. Secondly, an equivalent linearization – explicit time-domain method (EL-ETDM) is further developed to solve the non-stationary non-Gaussian random vibration problems of Duffing systems, in which the equivalent linear system is derived discarding the assumption of Gaussian response, and the corresponding higher-order cumulant analyses of the linearized system are accomplished by the efficient ETDM. The present approach can account for non-Gaussian random excitations with arbitrary forms, and two specific applications to the Poisson white noise and the square form of Gaussian random process are investigated. Four numerical examples are presented to demonstrate the effectiveness of the proposed methods.

Non-Gaussianity of joint response distribution of a linear system subjected to non-Gaussian random excitation

Non-Gaussianity of joint response distribution of a linear system subjected to non-Gaussian random excitation

Importance sampling for reliability assessment of dynamic systems under general random process excitation

This paper develops a reliability assessment method for dynamic systems subjected to a general random process excitation. Safety assessment using direct Monte Carlo simulation is computationally expensive, particularly when estimating low probabilities of failure. The Girsanov transformation-based reliability assessment method is a computationally efficient approach intended for dynamic systems driven by Gaussian white noise, and this approach can be extended to random process inputs that can be represented as transformations of Gaussian white noise. In practice, dynamic systems may be subjected to inputs that may be better modeled as non-Gaussian and/or non-stationary random processes, which are not easily transformable to Gaussian white noise. We propose a computationally efficient scheme, based on importance sampling, which can be implemented directly on a general class of random processes — both Gaussian and non-Gaussian, and stationary and non-stationary. We demonstrate that this approach is in fact equivalent to Girsanov transformation when the uncertain inputs are Gaussian white noise processes. The proposed approach is demonstrated on a linear dynamic system driven by Gaussian white noise and Brownian bridge processes, a multi-physics aero-thermo-elastic model of a flexible panel subjected to hypersonic flow, and a nonlinear building frame subjected to non-stationary non-Gaussian random process excitation.

Approximate analytical methods for stationary response probability density of a SDOF system with a nonlinear spring under non-Gaussian random excitation

Approximate analytical methods for finding the stationary response probability density of a SDOF system with a wide class of nonlinear springs under non-Gaussian random excitation are proposed. The non-Gaussian excitation is a stationary stochastic process prescribed by an arbitrary probability density with finite variance and a power spectral density with a bandwidth parameter. In order to obtain the response probability distribution of the system, two approximate methods are developed by theoretically extending the Monte Carlo simulation observations on the properties of the response distribution obtained in the previous study. One of the methods is applied when the bandwidth of the excitation power spectrum is narrower than that of the primary resonance peak of the system. In this case, the system becomes quasi-static, and its equation of motion reduces to the equilibrium equation for the stiffness and excitation terms. This equilibrium equation is used to derive the approximate solution of the response distribution. The only requirement of this method is that the nonlinear spring of the system is continuous and monotonically increasing. The other method is proposed for the case of the wide excitation bandwidth compared to the system bandwidth. In this method, the exact solution of the response distribution under Gaussian white noise is utilized. The value of the white noise spectral density parameter in the solution is determined by the equivalent linearization. The second method is applicable to systems with arbitrary integrable nonlinear springs. In numerical examples, we consider three nonlinear systems whose spring characteristics are quite different. Besides, two types of non-Gaussian excitation probability densities with differences in shape are used. It is demonstrated that the proposed methods can reproduce the unique shape of the response probability density caused by the spring nonlinearity and the excitation non-Gaussianity. From the results, the range of the bandwidth ratio between the excitation and the system where the response distribution is accurately obtainable is also examined for each method.

An Analysis of Transient Response Moments of a Linear System Subjected to Non-Gaussian Random Excitation Using Higher-Order Autocorrelation Functions of Excitation

Abstract We present the analytical solutions of the second-, third-, and fourth-order response moments of a single-degree-of-freedom linear system subjected to a class of non-Gaussian random excitation. The non-Gaussian excitation is a zero-mean stationary stochastic process prescribed by an arbitrary probability density and a power spectrum whose peak is located at zero frequency. The excitation is described by an Itô stochastic differential equation in which the drift and diffusion coefficients are determined from the probability density and spectral density of the excitation. In order to obtain the analytical solutions of the response moments, first, we derive the third- and fourth-order autocorrelation functions of the non-Gaussian excitation using its Markov and detailed balance properties. The third-order correlation function is given by the same expression regardless of the difference in the probability density function of the excitation. On the other hand, the fourth-order correlation function is derived under the assumption that the excitation probability density belongs to the Pearson distribution family, which is one of the widest classes of probability distributions. Then, combining the autocorrelation functions of the excitation and the convolution representation of the response, we obtain the exact solutions of the response moments, and it is shown what kind of components the response moments are composed of. Finally, we investigate the dominant time-varying components of the response moments for several different excitation bandwidths.

Joint response distributions of a linear system subjected to non-Gaussian random excitation with a wide range of bandwidth

We examine the joint response probability distributions of a linear system subjected to non-Gaussian random excitation with a wide range of bandwidth. In the previous studies, the transient response characteristics of linear systems subjected to non-Gaussian random excitation were investigated from the view point of the displacement response distribution and velocity response distribution. It was revealed that the probability distribution of the response varies significantly depending upon the bandwidth of the excitation power spectrum. In this paper, firstly, we calculate the joint response distribution of the displacement and velocity by Monte Carlo simulation. The non-Gaussian excitation is prescribed by both a probability density function and a power spectrum with bandwidth parameter. In order to find the response characteristics that appear commonly for different non-Gaussian excitations, we consider two kinds of non-Gaussian distributions for the probability distribution of the excitation. Their shapes are much different from a Gaussian distribution and also distinct from each other. For such non-Gaussian random excitations, the joint response distribution is obtained, and the relationship between the response distribution and the bandwidth of the excitation power spectrum is examined in detail. In addition, we also focus on the property of the distribution to concentrate on a straight line, which is important in discussing the non-Gaussianity in the response. When the bandwidth of the excitation is zero, the excitation can be considered as a step input. In this case, we can derive the solution of the straight line analytically. We illustrate the way of deriving the line for the zero bandwidth. Furthermore, it is shown that the line derived for the zero bandwidth can be used as the line even for the non-zero bandwidth. We also demonstrate that the duration for which the joint distribution concentrates on the straight line becomes shorter as the bandwidth becomes wider.

Effects of non-Gaussian random excitation on displacement and velocity responses of a linear system by using bispectrum and cross-bispectrum

Effects of non-Gaussian random excitation on displacement and velocity responses of a linear system by using bispectrum and cross-bispectrum

Research on the transmission law of kurtosis of SDOF system under nonstationary and non-Gaussian random excitations

Vibration fatigue failure of structures under complex random excitations is a common problem in the field of engineering, and it seriously endangers the reliability and safety of major equipments and structures. It is particularly important to accurately predict the vibration fatigue life of structures under random excitations. Generally, it is assumed that the vibration excitation of engineering structures follows a stationary Gaussian distribution, but the actual excitation usually follows a nonstationary and non-Gaussian distribution, especially under harsh or variable working conditions. The vibration fatigue life of structures is closely related to the structural dynamic response characteristics, and the kurtosis of the structural response has a particularly obvious influence on the vibration fatigue life. In this paper, taking the single-degree-of-freedom system as an example and by decomposing complex random excitations in frequency segments, the dynamic response law of the system under nonstationary and non-Gaussian random excitations is studied in depth, and a mathematical model describing the transmission law of the kurtosis between the decomposed signal of excitation and response signal is established. The results show that there is a close linear relationship between the kurtosis of the structural dynamic response and the kurtosis of the decomposed signal obtained by frequency decomposition of the excitation signal by 3.2 times the half power bandwidth at the damped natural frequency of the structure. It is possible to quantitatively estimate the kurtosis of the structural response according to this relationship, which lays the foundation for accurately predicting the vibration fatigue life of structures under nonstationary and non-Gaussian random excitations.

An analysis of transient response statistics of a linear system subjected to non-Gaussian random excitation by using higher-order autocorrelation functions of excitation

An analysis of transient response statistics of a linear system subjected to non-Gaussian random excitation by using higher-order autocorrelation functions of excitation

Transient response characteristics of linear systems subjected to non-Gaussian random excitation with a wide range of bandwidth

Transient response characteristics of linear systems subjected to non-Gaussian random excitation with a wide range of bandwidth

Transient Response Characteristics of a Linear System Subjected to Non-Gaussian Random Excitation with an Asymmetric Probability Distribution

Transient Response Characteristics of a Linear System Subjected to Non-Gaussian Random Excitation with an Asymmetric Probability Distribution

Transient Moment Analysis of a Linear SDOF System Subjected to Non-Gaussian Random Excitation by Using the Equivalent Non-Gaussian Random Excitation Method

Transient Moment Analysis of a Linear SDOF System Subjected to Non-Gaussian Random Excitation by Using the Equivalent Non-Gaussian Random Excitation Method

Transient response analysis of a linear system subjected to non-Gaussian random excitation by the equivalent non-Gaussian excitation method

Transient response analysis of a linear system subjected to non-Gaussian random excitation by the equivalent non-Gaussian excitation method

Analysis of stationary response distribution of a SDOF nonlinear stiffness system under non-Gaussian random excitation

Analysis of stationary response distribution of a SDOF nonlinear stiffness system under non-Gaussian random excitation

Transient Response Characteristics of Linear Systems Subjected to Non-Gaussian Random Excitation with a Wide Range of Bandwidth

Transient Response Characteristics of Linear Systems Subjected to Non-Gaussian Random Excitation with a Wide Range of Bandwidth

Moment Analysis of a Duffing Oscillator Subjected to Non-Gaussian Random Excitation by Using the Equivalent Non-Gaussian Excitation Method and the Equivalent Linearization

Moment Analysis of a Duffing Oscillator Subjected to Non-Gaussian Random Excitation by Using the Equivalent Non-Gaussian Excitation Method and the Equivalent Linearization

The importance of dynamic behaviour of vibrating systems on the response in case of non-Gaussian random excitations

Dynamic response of vibrating system subjected to non-Gaussian random loads was investigated through a set of numerical simulation on several lumped systems aimed to determine whether and in what form the dynamic behaviour of a vibrating system transfers or masks non-Gaussianity features of the input to the output response. Indeed, in several numerical and experimental activities performed on a Y-shaped specimen it was observed how the system response, both in terms of displacement or stress, changed according to an input variation (stationary and non-stationary Gaussian and non-Gaussian load time histories) and according to a change of the system frequency response function. Moreover, it was observed that even if the system was excited in its frequency range, the response remains unchanged and similar to the input in case of non-stationary and non-Gaussian load, removing preliminarily the possibility to use spectral methods for damage evaluation, going necessarily back to a more “expensive” time-domain analysis. Since the system response characteristics may change significantly according to the input excitation features and to the dynamic system parameters allowing, in some cases, the use of spectral techniques for fatigue damage evaluation also in case of non-Gaussian input loads, the aim of this paper is to understand whether and how the dynamic behaviour of a generic mechanical system transforms the non-Gaussian input excitations into a Gaussian response. To this aim several numerical displacement responses of 1-dof lumped systems characterized by different frequency response functions (resonance frequency position and damping) were analysed and investigated for different stationary and non-stationary Gaussian and non-Gaussian excitations. In such a way, it was possible to a-priori establish under what circumstances the frequency-domain approaches can be adopted to compute the fatigue damage of real mechanical systems.

パワースペクトルの形状差に着目した非ガウス性不規則励振を受ける1自由度線形系の応答分布

パワースペクトルの形状差に着目した非ガウス性不規則励振を受ける1自由度線形系の応答分布

An analysis of response characteristics of dynamical systems under narrow-band non-Gaussian random excitation using equivalent non-Gaussian excitation method

An analysis of response characteristics of dynamical systems under narrow-band non-Gaussian random excitation using equivalent non-Gaussian excitation method