Nonstationary Random Excitation Research Articles

Random vibration analysis of train–bridge under track irregularities and traveling seismic waves using train–slab track–bridge interaction model

The frequent use of bridges in high-speed railway lines greatly increases the probability that trains are running on bridges when earthquakes occur. This paper investigates the random vibrations of a high-speed train traversing a slab track on a continuous girder bridge subjected to track irregularities and traveling seismic waves by the pseudo-excitation method (PEM). To derive the equations of motion of the train–slab track–bridge interaction system, the multibody dynamics and finite element method models are used for the train and the track and bridge, respectively. By assuming track irregularities to be fully coherent random excitations with time lags between different wheels and seismic accelerations to be uniformly modulated, non-stationary random excitations with time lags between different foundations, the random load vectors of the equations of motion are transformed into a series of deterministic pseudo-excitations based on PEM and the wheel–rail contact relationship. A computer code is developed to obtain the time-dependent random responses of the entire system. As a case study, the random vibration characteristics of an ICE-3 high-speed train traversing a seven-span continuous girder bridge simultaneously excited by track irregularities and traveling seismic waves are analyzed. The influence of train speed and seismic wave propagation velocity on the random vibration characteristics of the bridge and train are discussed.

Girsanov Transformation–Based Reliability Modeling and Testing of Actively Controlled Structures

The problem of estimation of the time-variant reliability of actively controlled structural dynamical systems under stochastic excitations is considered. Monte Carlo simulations, reinforced with Girsanov transformation-based sampling variance reduction, are used to tackle the problem. In this approach, the external excitations are biased by an additional artificial control force. The conflicting objectives of the two control forces-one designed to reduce structural responses and the other to promote limit-state violations (but to reduce sampling variance)-are noted. The control for variance reduction is fashioned after design-point oscillations based on a first-order reliability method. It is shown that for structures that are amenable to laboratory testing, the reliability can be estimated experimentally with reduced testing times by devising a procedure based on the ideas of the Girsanov transformation. Illustrative examples include studies on a building frame with a magnetorheologic damper-based isolation system subject to nonstationary random earthquake excitations. (C) 2014 American Society of Civil Engineers.

Random vibration analysis of structures by a time-domain explicit formulation method

Non-stationary random vibration of linear structures with uncertain parameters is investigated in this paper. A time-domain explicit formulation method is first presented for dynamic response analysis of deterministic structures subjected to non-stationary random excitations. The method is then employed to predict the random responses of a structure with given values of structural parameters, which are used to fit the conditional expectations of responses with relation to the structural random parameters by the response surface technique. Based on the total expectation theorem, the known conditional expectations are averaged to yield the random responses of stochastic structures as the total expectations. A numerical example involving a frame structure is investigated to illustrate the effectiveness of the present approach by comparison with the power spectrum method and the Monte Carlo simulation method. The proposed method is also applied to non-stationary random seismic analysis of a practical arch bridge with structural uncertainties, indicating the feasibility of the present approach for analysis of complex structures.

Efficient MCS for random vibration of hysteretic systems by an explicit iteration approach

A new method is proposed for random vibration anaylsis of hysteretic systems subjected to non-stationary random excitations. With the Bouc-Wen model, motion equations of hysteretic systems are first transformed into quasi-linear equations by applying the concept of equivalent excitations and decoupling of the real and hysteretic displacements, and the derived equation system can be solved by either the precise time integration or the Newmark-<TEX>${\beta}$</TEX> integration method. Combining the numerical solution of the auxiliary differential equation for hysteretic displacements, an explicit iteration algorithm is then developed for the dynamic response analysis of hysteretic systems. Because the computational cost for a large number of deterministic analyses of hysteretic systems can be significantly reduced, Monte-Carlo simulation using the explicit iteration algorithm is now viable, and statistical characteristics of the non-stationary random responses of a hysteretic system can be obtained. Numerical examples are presented to show the accuracy and efficiency of the present approach.

G1010601 ガタを考慮した弾塑性サポート支持配管系の動的信頼性に関する基礎検討([G101-06]機械力学・計測制御一般セッション(6))

In this study, we investigate the change of structural dynamic reliability of piping system supported by elasto-plastic damper with gap subjected to nonstationary random input excitation, when a width of gap is changed. The elasto-plastic damper has a bilinear hysteretic characteristics. The piping system is modeled by L-type piping systems with a weight at one end. Analytical model of piping system is constructed by using finite element method. The support force is given by bilinear model. The inputs are given by narrow range white Gaussian noises. In this study we can consider the structural integrities of both the piping system and the elasto-plastic supporting devices by taking two indexes; dynamic reliability of pipings and accumulated energy of support into account. Furthermore, the effect of parameters such as gap width are searched. Numerical simulations are perfomed using L-type piping system model for random inputs based on probablistic vibration theory. This study is applicable to the seismic design of piping systems supported by elasto-plastic dampers subjected to the actual earthquake inputs.

非定常不規則入力を受ける摩擦振動系の不規則振動応答特性 : 最大応答に対する検討

非定常不規則入力を受ける摩擦振動系の不規則振動応答特性 : 最大応答に対する検討

Analytical Solutions of a Class of Nonlinear Single Degree-of-Freedom Systems to Nonstationary Random Excitations

An analytical method for responses of a class of nonlinear single degree-of-freedom (DOF) systems under nonstationary random excitations (NSREs) is presented. It is based on the transformation of the nonlinear stochastic differential equations that govern the vibrations of nonlinear systems under NSREs into a polynomial of which the exact roots are available, and therefore solutions without approximation of the nonlinear system can be evaluated. For demonstration purposes, the van der Pol-Duffing oscillator under a time-modulated zero mean Gaussian white noise process is considered. Representative solutions are verified by the Monte Carlo simulation (MCS) technique. The main conclusions are as follows. First, a simple and relatively straightforward method has been developed to transform the nonlinear stochastic differential equations of a class of nonlinear single DOF systems into polynomials for which exact solutions are available. The analytical solution developed in this paper is for the deterministic equation of motion for a specific realization of the input excitation function f(t). The statistics of the displacement response are obtained using the expectation operator applied to the exact solutions computed for every realization of f(t). Second, for the first time solutions without approximation are obtained for a nonlinear single DOF system with nonlinearities involving nonlinear velocity as well as displacement and under NSRE. Third, for solutions of the van der Pol-Duffing oscillator the computational time required by the present method is much more efficient than that using the MCS technique.

The Time Domain Simulation of Non-Stationary Road Roughness

The input of road roughness, which affects the ride comfort and the handling stability of vehicle, is the main excitation for the running vehicle. The time history of the road roughness was researched with the random phases, based on the stationary power spectrum density of the road roughness determined by the standards. Through the inverse Fourier transform, the random phases can be used to get the road roughness in time domain, together with the amplitude. Then, the time domain simulation of the non-stationary random excitation when the vehicle ran at the changing speed, would also be studied based on the random phases. It is proved that the random road excitation for the vehicle with the changing speed is stationary modulated evolution random excitation, and its power spectrum density is the stationary modulated evolutionary power spectrum density. And the numerical results for the time history of the non-stationary random inputs were also provided. The time history of the non-stationary random road can be used to evaluate the ride comfort of the vehicle which is running at the changing speed.

Instantaneous Stochastic Optimal Control of Seismically Excited Structures Based on Time Domain Explicit Method

A new instantaneous stochastic optimal control (ISO) for the linear building structures subjected to non-stationary random excitations is proposed. A plane shear structure is taken as the example to illustrate the proposed method. The main advantage of the method is that the control force is easy to be calculated because the expression of the control force is independent of the state of the system.

Time variant reliability model updating in instrumented dynamical systems based on Girsanov’s transformation

The problem of updating the reliability of instrumented structures based on measured response under random dynamic loading is considered. A solution strategy within the framework of Monte Carlo simulation based dynamic state estimation method and Girsanov’s transformation for variance reduction is developed. For linear Gaussian state space models, the solution is developed based on continuous version of the Kalman filter, while, for non-linear and (or) non-Gaussian state space models, bootstrap particle filters are adopted. The controls to implement the Girsanov transformation are developed by solving a constrained non-linear optimization problem. Numerical illustrations include studies on a multi degree of freedom linear system and non-linear systems with geometric and (or) hereditary non-linearities and non-stationary random excitations.

Dynamic Responses of Supercavitating Vehicles under Non-Stationary Random Excitation

When the front end of the supercavitating vehicles subjects to very large axial non-stationary random excitation at high speed motion under water, it is necessary to analyze dynamic responses of supercavitating vehicles under non-stationary random excitation. The dynamical equation of supercavitating vehicles is transformed into the form of state equations. The Simpson integration method is going to calculate the integral term of the general solution of state equation to improve the precise integration method. The explicit expression of dynamic responses of supercavitating vehicles is deduced, the means and variances of structural responses are calculated with operation laws of the first moment and second moment. Under different sailing speeds and different cone-cavitator angles dynamic responses of supercavitating vehicles are given by the examples, and the effectiveness of the method was demonstrated.

Response analysis of nonlinear multi-degree-of-freedom systems with uncertain properties to non-Gaussian random excitations

The investigation reported in this paper is concerned with the development of an approach for response analysis of multi-degree-of-freedom (mdof) nonlinear systems with uncertain properties of large variations and under non-Gaussian nonstationary random excitations. The developed approach makes use of the stochastic central difference (SCD) method, time co-ordinate transformation (TCT), and adaptive time schemes (ATS). It is applicable to geometrically complicated systems idealized by the finite element method (FEM). For demonstration of its use and availability of results for direct comparison, a two-degree-of-freedom (tdof) nonlinear asymmetric system with uncertain natural frequencies and under Gaussian and non-Gaussian nonstationary random excitations is considered. Computed results obtained for the system with and without uncertain natural frequencies, and under Gaussian and non-Gaussian nonstationary random excitations are presented. It is concluded that the approach is relatively simple, accurate and efficient to apply.

Nonlinear stochastic optimal control strategy of hysteretic structures

Referring to the formulation of physical stochastic optimal control of structures and the scheme of optimal polynomial control, a nonlinear stochastic optimal control strategy is developed for a class of structural systems with hysteretic behaviors in the present paper. This control strategy provides an amenable approach to the classical stochastic optimal control strategies, bypasses the dilemma involved in It<TEX>$\hat{o}$</TEX>-type stochastic differential equations and is applicable to the dynamical systems driven by practical non-stationary and non-white random excitations, such as earthquake ground motions, strong winds and sea waves. The newly developed generalized optimal control policy is integrated in the nonlinear stochastic optimal control scheme so as to logically distribute the controllers and design their parameters associated with control gains. For illustrative purposes, the stochastic optimal controls of two base-excited multi-degree-of-freedom structural systems with hysteretic behavior in Clough bilinear model and Bouc-Wen differential model, respectively, are investigated. Numerical results reveal that a linear control with the 1st-order controller suffices even for the hysteretic structural systems when a control criterion in exceedance probability performance function for designing the weighting matrices is employed. This is practically meaningful due to the nonlinear controllers which may be associated with dynamical instabilities being saved. It is also noted that using the generalized optimal control policy, the maximum control effectiveness with the few number of control devices can be achieved, allowing for a desirable structural performance. It is remarked, meanwhile, that the response process and energy-dissipation behavior of the hysteretic structures are controlled to a certain extent.

Seismic behaviour of isolated multi-span continuous bridge to nonstationary random seismic excitation

This paper concentrates on the results of responses of a multi-span continuous bridge isolated with double concave friction pendulum bearings subjected to non-stationary random seismic excitation characterized by the incoherence, the wave-passage, and the site-response effects. The earthquake excitation is modelled as a non-stationary random process as uniformly modulated broad-band excitation. To perform the seismic isolation procedure, the double concave friction pendulum bearings which are sliding devices that utilize two spherical concave surfaces are placed at each of the six support points of the deck. The non-stationary response of the isolated bridge is compared with the corresponding stationary response in order to study the effects of non-stationary characteristics of the earthquake input motion. Solutions obtained from the stationary and non-stationary stochastic analyses for the isolated bridge to spatially varying earthquake ground motions are compared for the special cases of the earthquake ground motion model. The spatially varying earthquake ground motions are described stochastically based on an empirical coherency loss function and a filtered power spectral density function. The site effect is considered by a transfer function derived from one dimensional wave propagation theory. It is observed that the stationary assumption is reasonable for the considered ground motion duration.

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.

Stochastic Floor Response Spectra under Non-Stationary Random Excitation and Model Uncertainty

The purpose of this paper is to propose one analytical method to make stochastic floor response spectra in order to easily estimate the dynamic characteristics of secondary system such as piping, equipment and so on, under the stochastic non-stationary excitation and model uncertainty. Applying the transition matrix method with state space to the secondary system mounted on main structure, the non-stationary standard deviation of absolute acceleration response relating to the secondary system is obtained under the assumption of no structural uncertain properties. Then, finally, in order to obtain the statistical results of response in consideration of model uncertainty, through combining the extreme value theory, FOSM(First Order Second Moment) method and ACO(Ant Colony Optimization) algorithm, the floor response spectra can be stochastically derived. This proposed floor response spectra may be called the Stochastic Floor Response Spectra. The analytical results are compared with those by Monte Carlo simulation, and show the accuracy and efficiency.

G100041 不規則振動論に基づく弾塑性サポート支持配管系の地震応答解析手法に関する基礎検討([G10004]機械力学・計測制御部門一般セッション(4):地震応答・スロッシング)

In this study, we estimate structural dynamic reliability of piping systems supported by elasto-plastic damper subjected to nonstationary random input excitation. Analytical model of elasto-plastic damper has a bilinear hysteretic characteristics. Analytical model of piping systems is constructed by using finite element method. At first, we make nonstationaty random inputs by combining of stationary random processes and envelop functions. At second, we investigate the stresses of the piping and the absorbed energy of damper for nonstationary random input excitations. At third, we evaluate structural dynamic reliability based on probabilistic vibration theory. This structural dynamic reliability takes into account of the integrity of both the piping systems and elasto-plastic dampers. At last, we examine the effects of the supporting conditions such as supporting locations and capacities of damper on structural dynamic reliability, and also study the impronement of structural dynamic reliability reliability.

Studies of the performance of particle dampers under dynamic loads

This paper presents a systematic investigation of the performance of particle dampers (vertical and horizontal) attached to a primary system (single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF)) under different dynamic loads (free vibration, stationary random excitation as well as nonstationary random excitation, with single component or multi-component), and the optimum operating regions are all determined. The amount of dissipated energy due to impact and friction, and the concept of “ Effective Momentum Exchange” are shown to be suitable “global” measures to interpret the physics involved in the behavior of particle dampers. Using the well-established discrete element method, the motion of vertical particle dampers can be analyzed and classified into three different regions, and the associated damping characteristics can be interpreted. The first mode of a MDOF primary system can be effectively controlled by a properly designed particle damper; however, the higher modes are more affected by other parameters. Consequently, extensive parametric studies are presented to evaluate the effects of various system parameters, such as: mass ratio, primary system damping, coefficient of restitution, container dimensions, excitation amplitude and components, input locations and damper locations.

Effect of material uncertainty on stochastic response of dams

The present research considers the effect of uncertain material parameters on the stochastic dynamic response of an asphaltic lining dam–foundation system subjected to both stationary and non-stationary random excitation. The uncertain material parameters of interest are the shear modulus and mass density, which are modelled using a log-normal distribution. The Monte Carlo simulation method determines the stochastic seismic response of the dam–foundation system to spatially varying ground motion with uncertain material parameters. Consideration of the spatial variability of ground motion incorporates incoherence and wave passage effects from both stationary and non-stationary random excitations. A Heaviside modulating function is used throughout the study for the non-stationary responses. The results obtained indicate that variability of the shear modulus should be included in a stochastic dynamic analysis of an asphaltic lining dam–foundation system, but variability of the mass density can be neglected. Also, from the results, stationarity is a reasonable assumption for asphaltic lining dams in response to typical durations of strong shaking.

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.