Non-stationary Random Response Research Articles

Experimental and Numerical Studies of Nonstationary Random Vibrations for a High-Pier Bridge under Vehicular Loads

In the current study, the nonstationary random responses of moving wheels induced by road roughness in a time domain were obtained and treated as the nonstationary inputs to study the nonstationary vibration of a high-pier bridge under vehicles with variable speed. A full-scale vehicle model with 12 degrees of freedom was used; vehicle wheels were considered as patch contact instead of point contact with the bridge road surface. The vehicle-bridge coupling equations were established by combining the equations of motion of both the bridge and vehicle using the displacement relationship and the interaction force relationship at the contact patches. The midspan deflections caused by the stationary and nonstationary inputs were compared with the measured responses under different parameters including vehicle acceleration and vehicle deceleration. The verified results showed that the proposed method can accurately simulate the vibration of the bridge under vehicles moving with variable speeds. Using the stationary random process to model the road surface disturbance to vehicles with variable speeds, the dynamic effects can be either underestimated or overestimated. The proposed method was then used to study the ride comfort for vehicles moving with a variable speed on high-pier bridges.

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.

2511 非定常不規則振動応答の自乗平均値の積分値の簡易計算法 : 異なる振幅非定常性に対する検討(一般セッション(3),OS・一般セッション講演)

2511 非定常不規則振動応答の自乗平均値の積分値の簡易計算法 : 異なる振幅非定常性に対する検討(一般セッション(3),OS・一般セッション講演)

Sensitivity and Hessian Matrix Analysis of Evolutionary PSD Functions for Nonstationary Random Seismic Responses

AbstractThis paper describes a numerical method for calculation of the sensitivity and Hessian matrix of the response evolutionary power spectral density (PSD) functions of structures subjected to uniformly modulated evolutionary random seismic excitation. The method is formulated based on the pseudoexcitation method and the Gauss precise time step integration method. The evolutionary nonstationary random response analysis is converted into step-by-step integration computations using the pseudoexcitation method. The formulas of the pseudoresponses and their first and second derivatives with respect to the structural design variables are derived based on the Gauss precise time step integration method. The evolutionary PSD functions, their sensitivity, and the Hessian matrix are calculated using the pseudoresponses and their first and second derivatives, respectively. Finally, the evolutionary PSD functions’ sensitivity and Hessian matrix analysis of a three-story, two-bay planar frame subjected to uniforml...

Sensitivity and Hessian matrix analysis of power spectral density functions for uniformly modulated evolutionary random seismic responses

This paper describes a numerical method for calculation of the sensitivity and Hessian matrix of the response PSD functions of structures subjected to uniformly modulated evolutionary random seismic excitation. The method is formulated based on the pseudo excitation method and Newmark method. The evolutionary non-stationary random response analysis is converted into step-by-step integration computations using the pseudo excitation method. The formulas of the pseudo responses, their first and second derivatives with respect to the structural design variables are derived based on the Newmark method. The PSD functions, their sensitivity and Hessian matrix are calculated using the pseudo responses, their first and second derivatives, respectively. Then the computation procedure of sensitivity and Hessian matrix of PSD functions is given in detail. Finally, the PSD functions’ sensitivity and Hessian matrix analysis of a three-story, two-bay planar frame subjected to the uniformly modulated evolutionary random earthquake ground motion has been studied to elucidate the proposed 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.

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.

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.

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.

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.

20719 非定常不規則振動応答の自乗平均値の積分値の簡易計算法 : 振幅および周波数非定常性を考慮した場合(一般講演 振動・音響)

20719 非定常不規則振動応答の自乗平均値の積分値の簡易計算法 : 振幅および周波数非定常性を考慮した場合(一般講演 振動・音響)

G1000-2-5 非定常不規則振動応答の自乗平均値の積分値の簡易計算法 : 振幅および周波数非定常性を考慮した付加構造物系の応答(機械力学・計測制御部門一般講演(2):振動,社会変革を技術で廻す機械工学)

G1000-2-5 非定常不規則振動応答の自乗平均値の積分値の簡易計算法 : 振幅および周波数非定常性を考慮した付加構造物系の応答(機械力学・計測制御部門一般講演(2):振動,社会変革を技術で廻す機械工学)

2417 非定常不規則振動応答の自乗平均値の積分値の簡易計算法 : 周波数非定常性を考慮した場合(OS24.一般セッション(4),オーガナイズドセッション)

2417 非定常不規則振動応答の自乗平均値の積分値の簡易計算法 : 周波数非定常性を考慮した場合(OS24.一般セッション(4),オーガナイズドセッション)

212 非定常不規則振動応答の自乗平均値の積分値の近似計算法 : 種々の系および入力に対する検討

The response of the structure subjected to nonstationary random vibration such as earthquake excitation is nonstationary random process. Calculating method for statistical characteristics of such a response is complicated. Mean square value is one of the representative statistical values. Then, practical and simplified method to obtain theoretical statistical value is required. 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 of the structure is proposed. Nonstationary random p input excitation is given as product of envelop function and stationary random process. Mean square value of nonstationary random response is assumed to be product of square of envelop function and mean square value of stationary random response. It is found that the proposed method gives exact value of integral of mean square value of the response

A simplified calculation method for non-stationary random seismic response of jacket platform

Jacket platform was simulated by non-uniform cantilever beam subjected to axial loading. Based on the Hamilton theory, the equation of bending motion was developed and solved by the classical Ritz method combined with the pseudo-excitation method (PEM) for non-stationary random response with non-classical damping. Usually, random response of this continuous structure is obtained by orthogonality of modes and some normal modes of the structure are needed, causing inconvenience in the analysis of the non-uniform beam whose normal modes are not easy to be obtained. However, if the PEM is extended to calculate random response by combining it with the classical Ritz method, the responses of non-uniform beam, such as auto-power spectral density (PSD) function, cross-PSD and higher spectral moments can be solved directly avoiding the calculation of normal modes. The numerical results show that the present method is effective and useful in aseismic design of platforms.

402 非定常不規則振動応答の自乗平均値の積分値の簡易計算法 : 付加構造物系の応答に対する計算法

402 非定常不規則振動応答の自乗平均値の積分値の簡易計算法 : 付加構造物系の応答に対する計算法

Dynamic response analysis of stochastic truss structures under non-stationary random excitation using the random factor method

The dynamic characteristics and responses of stochastic truss structures under non-stationary random excitation are investigated using a new method called the random factor method (RFM). Using the RFM, the structural physical parameters and geometry can be considered as random variables. Based on the assumption that the randomness on each element is the same for all elements, the structural stiffness and mass matrices can then respectively be described by the product of two parts corresponding to the random factor and the deterministic matrix. From the expressions of structural random response in the frequency domain, computational expressions for the mean value, standard deviation and variation coefficient of the mean square value of the non-stationary random displacement and stress response are developed by means of the random variable’s functional moment method and the algebra synthesis method. The influences of the randomness of the structural parameters on the dynamic characteristics, structural displacement and stress responses are demonstrated using a truss structure.

2721 非定常不規則振動応答の自乗平均値の積分値の簡易計算法 : 付加構造物系の応答について(OS27.一般セッション(1),ポスターセッションP-4)

2721 非定常不規則振動応答の自乗平均値の積分値の簡易計算法 : 付加構造物系の応答について(OS27.一般セッション(1),ポスターセッションP-4)