Non-stationary Random Vibration Analysis Research Articles

Nonstationary random vibration analysis of hysteretic systems with fractional derivatives by FFT-based frequency domain method

Nonstationary random vibration problems of nonlinear fractional systems are challenging and drawing increasing attention. This study presents an efficient numerical method for the random vibration analysis of large-scale nonlinear fractional systems subjected to fully nonstationary excitations. Firstly, the fast Fourier transform (FFT)-based frequency domain method originally proposed for the nonstationary random vibration analysis of linear integer-order systems is extended to linear fractional systems, in which the explicit expression of dynamic responses of fractional systems is derived based on the load superposition principle and Newmark method. Secondly, an equivalent linearization analysis framework is constructed to solve the nonstationary responses of hysteretic systems with fractional derivatives, in which the nonstationary response analyses of equivalent linear systems are accomplished by the FFT-based frequency domain method. The presented method can account for nonstationary excitations with arbitrary forms of evolutionary power spectrum, even of the non-separable one. Two numerical examples are used to confirm its superior performance in terms of accuracy and computational efficiency.

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.

An Asymptotic-Homogenization Explicit Time-Domain Method for Random Multiscale Vibration Analysis of Porous Material Structures

Porous material structures have been widely used in civil engineering, mechanical engineering, aerospace engineering and other fields due to their high specific strength and specific stiffness. The stochastic response analysis of porous material structures under random excitations deserves more attention. The multiscale differential governing equations of porous material structures are derived based on the multiscale asymptotic homogenization method (AHM), and the macroscale and microscale explicit time-domain expressions of structural responses are further established. On this basis, the statistical moments of dynamic responses of porous material structures under non-stationary random excitations can be achieved with the explicit time-domain method (ETDM). The purposed method combines the advantages of AHM for high-efficient explicit formulation of macroscale and microscale dynamic responses of porous material structures and the advantages of ETDM for fast analysis of non-stationary random vibration problems. A numerical example is presented to validate the computational accuracy and efficiency of the present approach for non-stationary random vibration analysis of porous material structures.

Nonstationary random vibration analysis of structures with uncertain parameters based on a polynomial dimensional decomposition

AbstractBased on polynomial dimensional decomposition (PDD), a novel method for analysing nonstationary random vibration of structures with uncertain parameters is proposed in this paper. The uncertain structural parameters are assumed to be independent random variables, with certain types of probability distributions; thus, the nonstationary random vibration responses of the structures become stochastic functions of these uncertain parameters. A dimensional decomposition and Fourier expansion were applied to calculate the statistical characteristics of the nonstationary random vibration responses of the structures. To calculate the conditional power spectral density (CPSD) and conditional standard deviation (CSD) of the nonstationary random vibration responses, CPSD and CSD were explicitly expressed using expansion coefficients and polynomial basis. Subsequently, the probability distributions of the structural responses can be effectively calculated in terms of these explicit expressions. Dimension‐reduction integration and Gauss numerical integration were applied to calculate expansion coefficients. The validity of the proposed method was verified by numerical examples, including a single‐degree‐of‐freedom system and a multi‐degree‐of‐freedom system (reinforced concrete shear wall). The results indicate that the proposed method is as accurate as Monte Carlo simulation (MCS), whereas the proposed method requires considerably smaller number of nonstationary random vibration analyses than MCS. Moreover, the nonstationary random vibration analysis of a long‐span cable‐stayed bridge demonstrates that the proposed method is feasible for quantifying the uncertainties of the nonstationary random vibration responses of complex engineering structures with uncertain parameters.

Auxiliary harmonic excitation generalized method for random vibration analysis of linear structures under non-stationary Gaussian excitation

Frequency-domain methods are the most fundamental and effective methods for non-stationary random vibration analysis. However, the existing frequency-domain methods often involve truncation for degree of mode or decomposition of the power spectrum in multi-correlation conditions, which may have impact on the computational accuracy and efficiency of the methods. To avoid the truncation for degree of mode and the decomposition of the power spectrum, an accurate and efficient auxiliary harmonic excitation generalized method is proposed for random vibration analysis of linear structures under non-stationary Gaussian excitation. First, several existing frequency-domain analysis methods are investigated and discussed. Second, the generalized impulse response function, the generalized frequency response function, and the evolutionary generalized frequency response function are proposed. Meanwhile, the physical meaning of the evolutionary frequency response function is re-investigation, in which the response for the evolutionary harmonic excitation and the evolutionary frequency response function proves equivalent. Subsequently, according to the physical meaning, a novel and easy-to-implement algorithm of the response PSD is proposed. The response for the evolutionary auxiliary harmonic excitation can be quickly evaluated based on the time-domain explicit formulation method. Finally, two examples are investigated to verify the accuracy and efficiency of the proposed method.

Nonstationary random vibration analysis of fractionally-damped systems by numerical explicit time-domain method

Random vibration analysis of fractionally-damped systems has received considerable attentions over the past two decades. In this paper, an efficient explicit time-domain approach is developed to determine the response statistics of multi-degrees-of-freedom linear systems endowed with fractional derivative terms subjected to nonstationary random excitations. The explicit time-domain expression of the state vector is constructed for the fractionally-damped system, and on this basis, the statistical moments of responses are directly formulated with the moment operation rule or efficiently calculated with the Monte-Carlo simulation. Dimension-reduced statistical analysis can be easily conducted just focusing on certain responses of interest with the use of the explicit expressions of dynamic responses. The present approach can account for nonstationary random excitations with arbitrary evolutionary power spectrum forms, even of the non-separable kind. A 20-degrees-of-freedom shear-type fractionally-damped system subjected to fully nonstationary random excitation is analysed to illustrate the accuracy and efficiency of the proposed method.

Efficient non-stationary random vibration analysis of vehicle-bridge system based on an improved explicit time-domain method

Based on the recently developed explicit time-domain method (ETDM), this paper presents a new efficient computational framework for non-stationary random vibration analysis of the vehicle-bridge system (VBS) subjected to random track irregularity excitation. In this framework, the time-varying VBS equation is discretized using the time-stepping integration method, leading to an explicit expression for the stochastic system response in terms of the track irregularity excitation. Based on this expression, the response moments of the VBS can be easily evaluated in a direct way; the response probability density can also be obtained by using the Gaussian characteristic of the irregularity excitation. To further improve the computational efficiency, an efficient spectral representation-based formulation is proposed to calculate the excitation covariance matrix required in the analysis, which avoids tedious Fourier integrals required in the traditional ETDM. In the numerical application, the accuracy and efficiency of the proposed method are validated by the solutions obtained using the Monte Carlo and pseudo-excitation methods; the effects of several parameters on the stochastic responses of the VBS are investigated.

An exact and efficient time-domain method for random vibration analysis of linear structures subjected to uniformly modulated or fully non-stationary excitations

This paper presents a novel time-domain method for non-stationary random vibration analysis of linear structures. Specifically, using the well-known spectral representation method, the stochastic excitation process is first decomposed into a set of trigonometric basis functions (TBFs) modulated by a series of orthogonal basis random variables (BRVs). Next, each TBF is used as a deterministic excitation input for the structural model, and the induced structural responses are derived analytically using the modal superposition method. After superposition of each TBF-induced deterministic responses, an explicit closed-form expression between the structural stochastic responses and the BRVs is derived, based on which the response statistics of structures at any time instant can be evaluated easily. Furthermore, the accuracy of the solution is independent of the time step used. Finally, several numerical examples in civil engineering, including a Euler beam under stochastic moving load and a shear frame structure subjected to uniformly modulated or fully non-stationary seismic motions, are studied to illustrate the performance of the proposed method. The obtained structural response statistics are compared with the solutions obtained by the Monte Carlo and evolutionary spectral methods. The results verified the high accuracy and efficiency of the proposed method.

A Non-stationary Random Vibration Analysis of Vehicle-bridge Coupling under Road Surface Coherence Excitation

Based on the pseudo excitation method and the uniform excitation input model of road roughness, the 3D vehicle-bridge coupling non-stationary random vibration model of vehicle variable speed driving is established. The effects of vehicle initial velocity and acceleration on the vertical displacement, root mean square value of acceleration and spectrum characteristics of bridge mid-span are studied. Based on the road roughness power spectrum suggested by national standard GB7031, the pseudo excitation method is used to construct a 3D vehicle-bridge coupling random vibration model to study the influence of the spatial effect of the spectrum input excitation on vehicle-bridge coupling vibration, vehicle vibration and the spectrum characteristics of vehicle-bridge system.

Nonstationary Random Vibration Analysis of Wing with Geometric Nonlinearity Under Correlated Excitation

An algorithm that integrates Karhunen-Loeve expansion (KLE), nonlinear finite element method (NFEM), and a sampling technique to quantify the uncertainty is proposed to carry out random vibration analysis of a structure with geometric nonlinearity under correlated nonstationary random excitations. In KLE, the eigenvalues and eigenfunctions of the autocovariance are obtained by using orthogonal basis functions, and the KLE for correlated random excitations relies on expansions in terms of correlated sets of random variables. The autocovariance functions of excitation are discretized into a series of correlated excitations, and then the structural response is carried out by using NFEM and sampling techniques. The proposed algorithm is applied to both rigid and flexible aircraft wings. Two different types of the boundary condition are studied for the flexible wing: fixed and large mass method (LMM). Results show that the geometric nonlinearity has a stiffening effect on the behavior of the aircraft wing, resulting in an oscillatory response with a lower amplitude, and changes the distribution of the random responses. The response due to LMM boundary condition that is closer to the actual conditions is smaller than the response obtained using fixed boundary condition.

Non-Stationary Random Vibration Analysis of Railway Bridges Under Moving Heavy-Haul Trains

This paper presents a non-stationary random vibration analysis of railway bridges under moving heavy-haul trains by the pseudo-excitation method (PEM) considering the train-track-bridge coupling dynamics. The train and the ballasted track-bridge are modeled by the multibody dynamics and finite element (FE) method, respectively. Based on the linearized wheel-rail interaction model, the equations of motion of the train-ballasted track-bridge coupling system are then derived. Meanwhile, the excitations between the rails and wheels caused by the random track irregularity are transformed into a series of deterministic pseudo-harmonic excitation vectors by the PEM. Then, the random vibration responses of the coupling system are obtained using a step-by-step integration method and the maximum responses are estimated using the 3[Formula: see text] rule for the Gaussian stochastic process. The proposed method is validated by the field measurement data collected from a simply-supported girder bridge (SSB) for heavy-haul trains in China. Finally, the effects of train speed, grade of track irregularity, and train type on the random dynamic behavior of six girder bridges for heavy-haul railways are investigated. The results show that the vertical acceleration and dynamic amplification factor (DAF) of the midspan of the SSB girders are influenced significantly by the train speed and track irregularity. With the increase in the vehicle axle-load, the vertical deflection-to-span ratio ([Formula: see text]) of the girders increases approximately linearly, but the DAF and vertical acceleration fail to show clear trend.

Exact nonstationary responses of rectangular thin plate on Pasternak foundation excited by stochastic moving loads

This paper develops an efficient method termed as PE-PIM to address the exact nonstationary responses of pavement structure, which is modeled as a rectangular thin plate resting on bi-parametric Pasternak elastic foundation subjected to stochastic moving loads with constant acceleration. Firstly, analytical power spectral density (PSD) functions of random responses for thin plate are derived by integrating pseudo excitation method (PEM) with Duhamel's integral. Based on PEM, the new equivalent von Mises stress (NEVMS) is proposed, whose PSD function contains all cross-PSD functions between stress components. Then, the PE-PIM that combines the PEM with precise integration method (PIM) is presented to achieve efficiently stochastic responses of the plate by replacing Duhamel's integral with the PIM. Moreover, the semi-analytical Monte Carlo simulation is employed to verify the computational results of the developed PE-PIM. Finally, numerical examples demonstrate the high accuracy and efficiency of PE-PIM for nonstationary random vibration analysis. The effects of velocity and acceleration of moving load, boundary conditions of the plate and foundation stiffness on the deflection and NEVMS responses are scrutinized.

Non-stationary random vibration analysis of structures under multiple correlated normal random excitations

An algorithm that integrates Karhunen-Loeve expansion (KLE) and the finite element method (FEM) is proposed to perform non-stationary random vibration analysis of structures under excitations, represented by multiple random processes that are correlated in both time and spatial domains. In KLE, the auto-covariance functions of random excitations are discretized using orthogonal basis functions. The KLE for multiple correlated random excitations relies on expansions in terms of correlated sets of random variables reflecting the cross-covariance of the random processes. During the response calculations, the eigenfunctions of KLE used to represent excitations are applied as forcing functions to the structure. The proposed algorithm is applied to a 2DOF system, a 2D cantilever beam and a 3D aircraft wing under both stationary and non-stationary correlated random excitations. Two methods are adopted to obtain the structural responses: a) the modal method and b) the direct method. Both the methods provide the statistics of the dynamic response with sufficient accuracy. The structural responses under the same type of correlated random excitations are bounded by the response obtained by perfectly correlated and uncorrelated random excitations. The structural response increases with a decrease in the correlation length and with an increase in the correlation magnitude. The proposed methodology can be applied for the analysis of any complex structure under any type of random excitation.

Dynamic Analysis of an Infinitely Long Beam Resting on a Kelvin Foundation under Moving Random Loads

Nonstationary random vibration analysis of an infinitely long beam resting on a Kelvin foundation subjected to moving random loads is studied in this paper. Based on the pseudo excitation method (PEM) combined with the Fourier transform (FT), a closed-form solution of the power spectral responses of the nonstationary random vibration of the system is derived in the frequency-wavenumber domain. On the numerical integration scheme a fast Fourier transform is developed for moving load problems through a parameter substitution, which is found to be superior to Simpson’s rule. The results obtained by using the PEM-FT method are verified using Monte Carlo method and good agreement between these two sets of results is achieved. Special attention is paid to investigation of the effects of the moving load velocity, a few key system parameters, and coherence of loads on the random vibration responses. The relationship between the critical speed and resonance is also explored.

A hybrid approach to analyse a beam-soil structure under a moving random load

To study the stochastic response of a beam-soil structure under a moving random load, a hybrid approach based on the pseudo-excitation method and the wavelet method is proposed. Using the pseudo-excitation method, the non-stationary random vibration analysis is transformed into a conventional moving harmonic load problem. Analytical solutions of the power spectral density and standard deviation of vertical displacement are derived in an integral form. However, the integrand is singular and highly oscillatory, and the computational time is an important consideration because a large number of frequency points must be computed. To calculate the response accurately and efficiently, a wavelet approach is introduced. Numerical results show that the frequency band which brings the most significant response is dependent on the load velocity. The hybrid method provides a useful tool to estimate the ground vibration caused by traffic loads.

Fast Equivalent Linearization Method for Nonlinear Structures under Nonstationary Random Excitations

AbstractAn efficient analysis procedure has been developed for the random vibration analysis of nonlinear structures subjected to nonstationary random excitations based on the equivalent linearization method and the time-domain explicit formulation method. When the equivalent linearization method is used for the nonstationary random vibration analysis of a nonlinear structural system, equivalent linear systems are constructed for different time instants and the analysis problem is transformed to a series of analyses of these linear systems. Because a large number of linear analyses must be conducted to get reliable results for the original nonlinear system, efficiency of the analysis algorithm for the linear analysis is crucial for its successful application. The time-domain explicit formulation method, recently proposed for the nonstationary random vibration analysis of linear systems, has much higher computational efficiency than other methods owing to the use of explicit expressions of dynamic response...

Non-stationary random vibration analysis of multi degree systems using auto-covariance orthogonal decomposition

An algorithm that integrates Karhunen–Loeve expansion (KLE) and finite element method (FEM) is proposed to carry out random vibration analysis of complex dynamic systems excited by stationary or non-stationary random processes. In KLE, the auto-covariance function of random process is discretized using orthogonal basis functions. During the response calculations, the eigenvectors of KLE are applied as forcing functions. Three methods are proposed to carry out the random vibration analysis termed as, Method 1A, Method 1B and Method 2. In Method 1A and Method 1B, the basis functions are chosen such that they include multiples of complete or half-cosine and sine functions over the selected time. In Method 2, the basis functions are chosen to be simple piecewise constants. The proposed algorithm is applied to a 2DOF system, a cantilever beam and a stiffened panel for both stationary and non-stationary excitations. Results show that three methods can describe the statistics of the dynamic response with sufficient accuracy. However, Method 1A results have a relatively larger error than that for Method 1B and Method 2 during initial transient time. The Method 2 results have an excellent agreement with analytical results. Moreover, the runtime of Method 2 algorithm is significantly less than both Method 1A and Method 1B algorithms even though its usage results in an increase in the number of KLE terms. Furthermore, Method 2, unlike Method 1A and 1B, neither yields negative and/or infinite eigenvalues for the auto-covariance function nor large inaccuracies.

Random vibration of an elastic half-space subjected to a moving stochastic load

In this paper, a non-stationary random vibration problem of an elastic half-space under a moving stochastic load modelling the dynamic characteristic of the train load is studied. An analytical method combining the pseudo-excitation method (PEM) and Fourier transformation method (FTM) is proposed. Because the load is moving and random, the responses at fixed locations in the half-space have characteristics of evolutionary non-stationary randomness. By means of the FEM, the non-stationary random vibration analysis is transformed into a conventional moving pseudo harmonic load problem. A closed-form solution of random vibration responses is derived in an integral form, which avoids step-by-step integration in time domain and thus leads to great saving in computational time. An adaptive quadrature method is adopted in order to obtain the desired numerical results of the singular integrand with non-stationary power spectral density. Through numerical examples, the proposed method is first validated, and then the dynamics characteristics of the non-stationary random responses of the half-space in all subsonic, transonic and supersonic cases are analysed.

Non-stationary random vibration analysis of a 3D train–bridge system using the probability density evolution method

Rail irregularity is one of the main sources causing train–bridge random vibration. A new random vibration theory for the coupled train–bridge systems is proposed in this paper. First, number theory method (NTM) with 2N-dimensional vectors for the stochastic harmonic function (SHF) of rail irregularity power spectrum density was adopted to determine the representative points of spatial frequencies and phases to generate the random rail irregularity samples, and the non-stationary rail irregularity samples were modulated with the slowly varying function. Second, the probability density evolution method (PDEM) was employed to calculate the random dynamic vibration of the three-dimensional (3D) train–bridge system by a program compiled on the MATLAB® software platform. Eventually, the Newmark-β integration method and double edge difference method of total variation diminishing (TVD) format were adopted to obtain the mean value curve, the standard deviation curve and the time–history probability density information of responses. A case study was presented in which the ICE-3 train travels on a three-span simply-supported high-speed railway bridge with excitation of random rail irregularity. The results showed that compared to the Monte Carlo simulation, the PDEM has higher computational efficiency for the same accuracy, i.e., an improvement by 1–2 orders of magnitude. Additionally, the influences of rail irregularity and train speed on the random vibration of the coupled train–bridge system were discussed.

Stationary and non-stationary random vibration modelling and analysis for an operating wind turbine

The problem of stationary and non-stationary random vibration modelling and analysis for an operating wind turbine is considered by using an acceleration vibration signal measured on a NegMicon NM52/900 fixed speed wind a turbine tower. Three stationary methods, namely Welch spectral estimation, autoregressive, and autoregressive moving average modelling, are considered, along with five non-stationary methods, namely non-parametric Wigner–Ville and spectral correlation, and parametric modelling by means of smoothness priors (SPs) time-dependent autoregressive modelling, functional series (FS) time-dependent autoregressive modelling, and adaptable functional series (AFS) time-dependent autoregressive modelling. The need for non-stationary modelling is mainly dictated by blade rotation and the wind dynamics. The results of the study confirm the cyclo-stationary and broader non-stationary nature of the random vibration, dictating the need for corresponding methods for accurate modelling and analysis. The capabilities and facets of the various methods in terms of modelling and analysing operating wind turbine random vibration are presented.