Non-stationary Random Vibration Research Articles

Non-stationary random vibration of a coupled vehicle–slab track system using a parallel algorithm based on the pseudo excitation method

A parallel algorithm based on the pseudo excitation method (PEM) is applied to investigate the non-stationary random response of a vertically coupled vehicle–slab track system subjected to random excitation induced by track irregularity. The vehicle is simplified as a multibody system with 10 degrees of freedom and the slab track is represented by a three layer Bernoulli–Euler beam model which includes the rail, slab and roadbed. Linear wheel–rail contact provides interaction between the vehicle and slab track models. The track irregularity is assumed to be a Gaussian random process with zero mean value and the non-stationary random response of the coupled vehicle–slab track system is solved using a combination of PEM and step-by-step integration. A parallel algorithm based on PEM is used to accelerate the computation and the effectiveness of the proposed method is verified by the Monte Carlo method. The basic random performance of the coupled vehicle–slab track system is analysed and the effects of the vehicle speed and slab track parameters on the random responses of the vehicle and slab track are investigated. The results indicate that the vehicle speed has a significant effect on the random response of the coupled vehicle–slab track system and that the rail pad parameters are the most important of the slab track parameters discussed.

The Dynamic Analysis and Optimization Technique of Non-Stationary Random Vibration

This paper introduces a new non-stationary random vibration technique for the analysis of time-dependent random vibration systems. It is based on the pseudo-excitation method (PEM) and precise integration method (PIM), which is extended herein and can improve significantly the analysis efficiency. Based on the random vibration equations with the right-hand side random replaced by a pseudo excitation, the first and second orders of sensitivity formulae are calculated conveniently by differentiating these equations. By use of sensitivity information, the optimization of complicated random vibration can be performed easily. Some numerical examples of vehicle-bridge time-dependent coupled system show the effectiveness and accuracy of present method.

Random vibration of a train traversing a bridge subjected to traveling seismic waves

Non-stationary random vibration of 3D time-dependent train–bridge systems subjected to multi-point earthquake excitations, including wave passage effect, is investigated using the pseudo-excitation method (PEM). The motion equation of such a system is established by coupling the train and bridge through wheel–rail contact relationships and accounting for the phase-lags between pier excitations. The horizontal and vertical earthquake excitations are both assumed to be uniformly modulated, fully coherent random excitations with different phases, while the excitation due to track irregularities is assumed to be a 3D, fully coherent random excitation with velocity-dependent time lags. PEM is first proven to be applicable to such time-dependent systems, and is then used to transform the random excitations into a series of deterministic pseudo excitations. By solving for the corresponding deterministic pseudo responses, various non-stationary random responses, including the time-dependent power spectral density functions (PSD) and standard deviations (SD), can be obtained easily. A case study is then presented in which the China-Star high-speed train traverses a seven-span continuous bridge that is being excited by an earthquake. The results show the effectiveness and accuracy of the proposed method by comparison with a Monte Carlo simulation. Additionally, the influences of seismic apparent wave velocity and train speed on the system random responses are discussed.

On the Statistical Distribution of Road Vehicle Vibrations

SUMMARYThis paper presents an alternative method for characterizing the random vibrations produced by transport vehicles. The paper discusses the significance and limitations of the average power spectral density and explains why it is not always adequate as the sole descriptor of road vehicle vibrations as the process generally tends to be non‐stationary and non‐Gaussian. The paper adopts an alternative analysis method, based on the statistical distribution of the moving root‐mean‐square (rms) vibrations, as a supplementary indicator of overall ride quality. A variety of sample vibration records, collected from various vehicle types and routes in Spain and Australia, were used to investigate the suitability of various mathematical models, based on the Weibull distribution. It shows that the model can also effectively describe the statistical parameters of the process, namely the mean, median, standard deviation, skewness and kurtosis. The paper proposes a single mathematical model that can accurately describe the statistical character of the non‐stationary random vibrations generated by road vehicles in general. The proposed generic distribution model, based on the Weibull distribution, was developed to afford additional control over various aspects of the shape of the distribution function. The model was found to be general enough to be able to produce a range of well‐known distributions. Curve‐fitting results using the sum‐of‐squared error (least squares) optimization were found to produce non‐convergent results, which required inclusion of the mean, median, standard deviation, skewness and kurtosis in the optimization algorithm. The paper also shows how the model is capable of accurately describing the statistical parameters of the process, namely the mean, median, standard deviation, skewness and kurtosis. This result is relevant not only for the characterization of ride quality but also for the accurate synthesis of road vehicle vibrations in the laboratory. The results can be used to assist in developing a novel method for simulating non‐stationary (modulated) vibration in the laboratory. The rms distribution function can be used to create an rms level schedule that will enable the synthesis of random vibrations with varying rms levels to better represent the road transport vibration process. Copyright © 2011 John Wiley & Sons, Ltd.

Non-Stationary Random Vibration of the Bridge-Vehicle Coupled System

The non-stationary random responses of the moving tires induced by the road roughness in time domain were obtained, and then those non-stationary responses were treated as the non-stationary inputs to study the non-stationary vibration of a high-pier bridge under vehicles with variable speed. The effect of the two inputs on the mid-span deflections was compared with the measured responses under different parameters including vehicle acceleration and vehicle deceleration. The verified results showed that the proposed method can simulate accurately the vibration of bridge under vehicle moving with variable speeds.

An efficient method for non-stationary random vibration analysis of beams

An efficient method is presented to investigate the non-stationary random vibration response of structures. This method has the advantage of the accuracy of theoretical method in dealing with random loads and the versatility of the finite element method (FEM) in dealing with structures. In this paper, the Euler beam is adopted in the derivation of the governing equation. The uncoupled approach of the frequency-dependent system matrices is presented for solving the motion equation of forced vibration. The time-variance random dynamic response of the beam is analyzed by the precise integral method, meanwhile, the pseudo-excitation is applied to transform the non-stationary random excitation into deterministic pseudo one to simplify the solution of the dynamic equation. Solutions calculated by the FEM with different time step and theoretical analysis are also obtained for comparison. Numerical examples demonstrate the accuracy and high efficiency of the proposed method.

Pseudo Excitation Method and Some Recent Developments

Pseudo Excitation Method (PEM) as a highly efficient and accurate algorithm series of stationary and nonstationary random vibration of structures has been widely used in China. Some well- concerned problems such as the seismic analysis of long-span bridges subjected to differential ground excitations, wind-induced flutter-buffet analysis of slender structures based on 3D finite element modeling have been solved and applied in some important projects taking place in China. Previously, bridges with the main span longer than 150 m were excluded in the 〈〈Specifications of Earthquake Resistant Design for Highway Engineering〉〉 (1989). Since 2008, this official document was replaced by 〈〈Guidelines for Seismic Design of Highway Bridges〉〉, in which PEM is recommended as a basic tool for seismic analysis of long-span bridges. In the design of high speed trains, random vibration can not be effectively analyzed by the traditional approach. The PEM has recently been used instead and some important progresses have been obtained.

Structural Reliability Applications of Nonstationary Spectral Characteristics

This paper presents new closed-form analytical approximations to the first-passage problem in structural reliability by using the exact closed-form solutions for the spectral characteristics of nonstationary random processes. The first-passage problem applied to a structural system possibly with random parameters and subjected to stochastic loading consists of computing the probability of a response quantity exceeding a deterministic threshold in a given exposure time. This paper also investigates, on the basis of benchmark problems, the absolute and relative accuracy of analytical approximations of the time-variant failure probability, such as Poisson, classical Vanmarcke, and modified Vanmarcke approximations, in the case of nonstationary random vibration. The classical and modified Vanmarcke approximations are expressed as time integrals of the closed forms of the corresponding hazard functions. These closed forms refer to linear elastic systems subjected to stationary and nonstationary base excitation...

Non-stationary random vibration analysis for train–bridge systems subjected to horizontal earthquakes

This paper investigates the use of PEM (the pseudo-excitation method) and PIM (the precise integration method), to compute the non-stationary random responses of three-dimensional train–bridge systems subjected to lateral horizontal earthquakes. A method for estimating their maxima is also proposed and evaluated. Each vehicle of the train is described by 27 degrees of freedom and the bridge deck is modeled by three-dimensional Euler beam elements. It is assumed that the lateral horizontal earthquake is a uniformly modulated non-stationary random process, while the excitations due to track irregularity are multi-phase stationary random ones. PEM is first proved to be still valid for such time-dependent systems, and then applied to transform the random excitations of the lateral horizontal earthquakes and track irregularities into a series of deterministic pseudo-excitations, respectively. Furthermore, PIM is extended to solve the corresponding pseudo-responses efficiently and precisely. Hence the time-dependent power spectral density functions and standard deviations of the system responses can be obtained conveniently. Finally, a new formula for estimating the maxima of such non-stationary random responses is suggested, based on the first-passage failure criterion. As a case study, the China-Star high-speed train running on a five-span bridge subjected to lateral horizontal earthquakes is investigated. The results show the effectiveness and accuracy of the proposed method and the reliability of the maximum estimation formula, by comparison with the time-history method. The influences of site soils and train speeds on system random responses are discussed.

Nonstationary random vibration analysis of coupled vehicle‐bridge systems

Purpose — The purpose of this paper is to present a new nonstationary, random vibration method for the analysis of coupled vehicle‐bridge systems with vertical track irregularity. Design/methodology/approach — The vehicle is modeled using a two‐layer suspension system and hence possesses ten degrees of freedom. The bridge is simulated using a Bernoulli‐Euler beam and the longitudinal track irregularity is taken as a uniformly modulated, evolutionary random process that includes phase lags between successive wheels. The pseudo‐excitation method (PEM) is extended to include time‐dependent systems for the first time, thus making it possible to compute the nonstationary random vibration of coupled vehicle‐bridge systems. Additionally, the precise integration method (PIM) is adapted to simulate continuous vehicle force variations in both time and space. Findings — The accuracy and effectiveness of the proposed PEM‐PIM method are confirmed by comparisons with Monte Carlo simulations. The influence of vehicle speed and track irregularity on system random responses are evaluated, and it is shown that the first and second derivatives of the track irregularity should not be arbitrarily ignored, as is usually the case. Originality/value — PEM and PIM are relatively new tools for the numerical solution of complicated random vibration problems and direct dynamic analyses. Until now, they have only been applied to time‐independent systems. However, it is shown herein that the proposed PEM‐PIM method performs nonstationary random vibration analysis of time‐dependent coupled vehicle‐bridge systems efficiently and accurately.

A Pseudo Excitation Method for Vehicle Vibration Analysis in Uneven Speeds

The vehicle riding comfort has become one of important factors of vehicle quality and receives more increasing attention. The driving smoothness of vehicles has immediate influence on the riding comfort of vehicles. In this study, the mechanical model and mathematics model of the 1/2 five degrees of freedom of a vehicle were established, as well as the pseudo- excitation model of road conditions for the front wheel and the rear wheel. By the pseudo- excitation method, the equations of the transient response and the power spectral density were established. After numerical simulation to vehicle vibration response of uneven driving, the results show that the pseudo- excitation method is more convenient than the traditional method and effectively solves the smoothness computation of vehicles while the pseudo- excitation method is used to analyze vehicle vibration under non-stationary random vibration.

Non-stationary random vibration of bridges under vehicles with variable speed

Most researchers assumed the road surface inputs as stationary random process in time domain when studying vehicles with variable speed. This assumption was made to avoid the complexity of the non-stationary random processes. This paper presented a new method of analyzing the non-stationary random response of bridges. Using the covariance equivalence technique, the non-stationary random responses of wheels in time domain were obtained, and thus a new method of analyzing the non-stationary random response of bridges was developed. A two-axle vehicle model and three bridge models were analyzed, namely, a single-span uniform Bernoulli–Euler beam, a three-span stepped beam, and a three-span continuous non-uniform bridge deck. The bridge–vehicle coupling equations were established by combining the equations of motion of both the bridge and the vehicle using the displacement relationship and the interaction force relationship at the contact points. The numerical results indicated that the responses of the tires induced by the road roughness are the non-stationary process, and the amplitudes of responses change as the vehicle velocity varies. Using the responses of the tires as the inputs to study the non-stationary vibration of bridge–vehicle system with variable speed one can obtain more accurate solutions.

Non-stationary random vibration analysis of three-dimensional train–bridge systems

This paper presents a new non-stationary random vibration technique for the analysis of three-dimensional, time-dependent, train–bridge systems that are subjected to excitations caused by track irregularities. It is based on the pseudo-excitation method (PEM), which was previously applicable only to time-invariant systems, but is extended herein and the result is strictly proven to be applicable to time-dependent systems. The analysis proceeds by taking time lags between the wheel excitations into account, in order that the effects of track irregularity can be regarded as a set of three-dimensional, uniformly modulated, multi-point, different-phase, non-stationary random excitations. This enables the random surface roughness of the track to be transformed using PEM into the superposition of a series of deterministic pseudo-harmonic surface irregularities. The precise integration method is then extended to compute the corresponding pseudo responses. By using these pseudo-responses, various non-stationary random responses, including the time-dependent PSD and RMS of the system, can be obtained conveniently and efficiently. Numerical examples show the effectiveness and accuracy of the present method by comparison with Monte Carlo simulation. Additionally, the characteristics of such non-stationary random responses are discussed.

Approximate calculation method for integral of mean square value of nonstationary response

The response of the structure subjected to nonstationary random vibration such as earthquake excitation is nonstationary random vibration. Calculating method for statistical characteristics of such a response is complicated. Mean square value of the response is usually used to evaluate random response. Integral of mean square value of the response corresponds to total energy of the response. In this paper, a simplified calculation method to obtain integral of mean square value of the response is proposed. As input excitation, nonstationary white noise and nonstationary filtered white noise are used. Integrals of mean square value of the response are calculated for various values of parameters. It is found that the proposed method gives exact value of integral of mean square value of the response.

Simplified Calculation Method for Integral of Mean Square Value of Nonstationary Seismic Response

The seismic response of the structure is nonstationary random vibration because earthquake excitation is nonstationary random vibration. Calculating method for the statistical characteristics of such a nonstationary response is complicated. The mean square value of the response is usually used to evaluate the statistical characteristics of random response. Integral of the mean square value of the response corresponds to total energy of the response. In this paper, a simplified calculation method to obtain integral of the mean square value of the nonsatationary response is proposed. In this method, the mean square value of the stationary response is used. As an earthquake excitation, nonstationary filtered white noise is used considering the dynamic characteristics of the ground. Integrals of the mean square value of the nonstationary random response are calculated for various values of parameters. It is found that the proposed method gives exact values of integral of the mean square value of the nonstationary random response.

Non-stationary random vibration analysis of coupled vehicle-bridge systems

Non-stationary random vibration analysis of coupled vehicle-bridge systems

Vehicle Vibration Analysis in Changeable Speeds Solved by Pseudoexcitation Method

The vehicle driving comfort has become one of the important factors of vehicle quality and receives increasing attention. In this paper, the mechanical and mathematical models of the half‐car, five degrees of freedom (DOF) of a vehicle were established, as well as the pseudoexcitation model of road conditions for the front wheel and the rear wheel. By the pseudoexcitation method, the equations of transient response and power spectrum density were established. After numerical simulation to vehicle vibration response of changeable driving, the results show that the pseudoexcitation method is more convenient than the traditional method and effectively solves the smoothness computation problems of vehicles while the pseudoexcitation method is used to analyze vehicle vibration under nonstationary random vibration environments.

Application of a new parametric model-based filter to knock intensity measurement

In this paper, a new parametric model-based filter is proposed for impact-induced fault intensity measurement. The filter designing steps are based on advanced parametric modeling and decomposition of the non-stationary random vibration signals, extracted from a mechanical system. One of the important innovations of this work throughout filter designing is proposing a method for identifying the most proper latent component of the signal in the undamaged state using instant modules of the signal. By considering this fact that knock is one of the important defects in the spark ignition engines, the proposed filter is applied for knock intensity measurement as a case study. For evaluation purpose, the proposed method is compared with other previous methods in literatures.

Non-Stationary Random Vibration of FE Structures Subjected to Moving Loads

An efficient and accurate FEM based method is proposed for studying non-stationary random vibration of structures subjected to moving loads. The loads are assumed to be a stationary process with constant mean value. The non-stationary power spectral densities (PSD) and the time dependent standard deviations of dynamic response are derived by using the pseudo excitation method (PEM) to transform this random excitation problem into a deterministic harmonic excitation one. The precise integration method (PIM) is extended to solve the equation of motion of beams under moving harmonic loads by enhancing the very recent consistent decomposition procedure, in order to simulate the movement of the loads. Six numerical examples are given to show the very high efficiency and accuracy of the method and also to deduce some useful preliminary conclusions from investigation of the dynamic statistical characteristics of a simply supported beam and of a symmetrical three span beam with its centre span unequal to the outer ones.

Non-Stationary Random Vibration of FE Structures Subjected to Moving Loads

An efficient and accurate FEM based method is proposed for studying non-stationary random vibration of structures subjected to moving loads. The loads are assumed to be a stationary process with constant mean value. The non-stationary power spectral densities (PSD) and the time dependent standard deviations of dynamic response are derived by using the pseudo excitation method (PEM) to transform this random excitation problem into a deterministic harmonic excitation one. The precise integration method (PIM) is extended to solve the equation of motion of beams under moving harmonic loads by enhancing the very recent consistent decomposition procedure, in order to simulate the movement of the loads. Six numerical examples are given to show the very high efficiency and accuracy of the method and also to deduce some useful preliminary conclusions from investigation of the dynamic statistical characteristics of a simply supported beam and of a symmetrical three span beam with its centre span unequal to the outer ones.