Pseudo-excitation Method Research Articles

Stochastic Seismic Response Analysis of Base-Isolated High-Rise Buildings

Stochastic seismic responses of a base-isolated high-rise building subjected to non-stationary random seismic earthquake ground motion are investigated by combining the pseudo excitation method and equivalent linearization method, for which the superstructure is modeled as a multi-degree-of–freedom system considering the shear-flexural effect, the hysteretic restoring forces of the isolators are described by the Bouc-Wen differential equation model which has a good performance on the transition from elastic responses to plastic responses. With the linearization for such a Wen's model, a first order differential equation will be obtained, which will couple with the governing equations of the isolated structure and so make up the closed-form expressions for base-isolated high-rise buildings, thus the stochastic seismic responses of the simplified systems will be obtained conveniently. Consequently, the solution of the stochastic seismic responses of a base-isolated structure considering hysteretic nonlinear behavior is transformed to a deterministic step-by-step integration problem. The precision of the pseudo excitation method is verified by Monte-Carlo simulation, the results for a 17-storey frame with height:width ratio=5.1 show that the isolation technique greatly reduces the inter-storey drifts and absolute accelerations of the superstructure for high-rise buildings and that the responses are substantially underestimated if the flexural effect is neglected.

Probability Analysis for the Damage of Gravity Dam

Damage reliability analysis is an emerging field of structural engineering which is very significant in structures of great importance like arch dams, large concrete gravity dams etc. The research objective is to design and construct an improved method for damage reliability analysis for concrete gravity dam. Firstly, pseudo excitation method and Mazar damage model were used to analyze how to calculate damage expected value excited by random seismic loading and deterministic static load on the condition that initial elastic modulus was deterministic. Moreover, response surface method was improved from the aspects of the regression of sample points, the selection of experimental points, the determined method of weight matrix and the calculation method of checking point respectively. Then, the above method was used to analyze guarantee rate of damage expected value excited by random seismic loading and deterministic static load on the condition that initial elastic modulus was random. Finally, a test example was given to verify and analyze the convergence and stability of this method. Compared with other conventional algorithm, this method has some strong points: this algorithm has good convergence and stability and greatly enhances calculation efficiency and the storage efficiency. From what has been analyzed, we find that damage expected value is insensitive to the randomness of initial elastic modulus so we can neglect the randomness of initial elastic modulus in some extent when we calculate damage expected value.

Sensitivity Analysis for Structures Subjected to Stationary Random Excitations

A method for sensitivity analysis of stationary random vibration is proposed within the framework of pseudo-excitation method (PEM). As the stationary random vibration analysis is transformed into harmonic vibration analysis, therefore the sensitivity analysis of stationary random responses is transformed into sensitivity analysis of pseudo harmonic responses of structures subjected to pseudo harmonic excitations. Using the PEM principle once again, the formulae for calculating the sensitivities of power spectrum density (PSD) and variances of random responses are derived. Based on the above work, the study of statistical characteristics of stationary random responses for structures with uncertain parameters is carried out.

Stochastic space vibration analysis of a train-bridge coupling system

The Pseudo-Excitation Method (PEM) is applied to study the stochastic space vibration responses of train-bridge coupling system. Each vehicle is modeled as a four-wheel mass-spring-damper system with two layers of suspension system possessing 15 degrees-of- freedom. The bridge is modeled as a spatial beam element, and the track irregularity is assumed to be a uniform random process. The motion equations of the vehicle system are established based on the d'Alembertian principle, and the motion equations of the bridge system are established based on the Hamilton variational principle. Separate iteration is applied in the solution of equations. Comparisons with the Monte Carlo simulations show the effectiveness and satisfactory accuracy of the proposed method. The PSD of the 3-span simply-supported girder bridge responses, vehicle responses and wheel/rail forces are obtained. Based on the <TEX>$3{\sigma}$</TEX> rule for Gaussian stochastic processes, the maximum responses of the coupling system are suggested.

Sensitivity analysis and optimization of vehicle–bridge systems based on combined PEM–PIM strategy

This paper presents a new optimization method for coupled vehicle-bridge systems subjected to uneven road surface excitation. The vehicle system is simplified as a multiple rigid-body model and the single-span bridge is modeled as a simply supported Bernoulli-Euler beam. The pseudo-excitation method transforms the random surface roughness into the superposition of a series of deterministic pseudo-harmonic excitations, which enables convenient and accurate computation of first and second order sensitivity information. The precise integration method is used to compute the vertical random vibrations for both the vehicle and the bridge. The sensitivities are used to find the optimal solution, with vehicle ride comfort taken as the objective function. Optimization efficiency and computational accuracy are demonstrated numerically.

The Application of Pseudo-Excitation Method to the Ride Comfort Research of Wheel-Loader

The wheel-loader is modeled as a linear 12-DOF system in this paper for researching its ride comfort under typical working situations. The pseudo-excitation method is introduced for vibration analysis, which accuracy has been validated. By using this method and taking a certain type of wheel-loader as a representative example, the spectral density of real response of wheel-loader such as the vertical acceleration of driver, the for-and-aft angular acceleration of bucket and the wheel relative dynamic force due to different carrying capacity, speed and the road surface level are obtained. The results offer the theoretical evidence for the farther wheel-loader vibration reduction research.

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.

Ride-Comfort-Oriented Suspension Optimization Using the Pseudo-Excitation Method

The main purpose of this article is to optimize the stiffness and damping parameters of suspension systems to improve the ride comfort of a coach by using the pseudo-excitation method (PEM) and the method of moving asymptotes (MMA). The finite element model of the coach under investigation has 1778 degrees of freedom. The surface roughness of the highway is regarded as a spatial stationary random process, which is accurately transformed into the superposition of a series of deterministic pseudo-harmonic surface unevennesses by using PEM. By doing so, the proposed method considerably simplifies the solution of the vibration equations, the derivation of sensitivities of various random responses, and the execution of the optimization process. MMA is applied to solve the ride comfort optimization problem. The proposed method is well justified by the numerical example.

Symplectic random vibration analysis of a vehicle moving on an infinitely long periodic track

Based on the pseudo-excitation method (PEM), symplectic mathematical scheme and Schur decomposition, the random responses of coupled vehicle–track systems are analyzed. The vehicle is modeled as a spring–mass–damper system and the track is regarded as an infinitely long substructural chain consisting of three layers, i.e. the rails, sleepers and ballast. The vehicle and track are coupled via linear springs and the “moving-vehicle model” is adopted. The latter assumes that the vehicle moves along a static track for which the rail irregularity is further assumed to be a zero-mean valued stationary Gaussian random process. The problem is then solved efficiently as follows. Initially, PEM is used to transform the rail random excitations into deterministic harmonic excitations. The symplectic mathematical scheme is then applied to establish a low degree of freedom equation of motion with periodic coefficients. In turn these are transformed into a linear equation set whose upper triangular coefficient matrix is established using the Schur decomposition scheme. Finally, the frequency-dependent terms are separated from the load vector to avoid repeated computations for different frequencies associated with the pseudo-excitations. The proposed method is subsequently justified by comparison with a Monte-Carlo simulation; the fixed-vehicle model and the moving-vehicle model are compared and the influences of vehicle velocity and class of track on system responses are also discussed.

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.

PEM Based Sensitivity Analysis for Acoustic Radiation Problems of Random Responses

A new effective method for computing the acoustic radiation and its sensitivity analysis of a structure subjected to stochastic excitation is presented. Previous work in the area of structural and acoustic sensitivity analysis systems was mostly focused on the deterministic excitation. New methods are developed to account for stochastic excitation. The structural-acoustic response is calculated using finite element method and boundary element method combined with stochastic analysis techniques. An accurate and highly efficient algorithm series for structural stationary random response analysis, pseudo-excitation method (PEM), is extended to acoustic random analysis in this paper, which was used to calculate structural random analysis in the past. So the acoustic radiation problems of random responses are transformed to the structural-acoustic harmonic analyses. This is a time-saving progress in comparison with traditional method. Based on the PEM, the acoustic radiation sensitivities of the structure are developed in emphasis that are transformed to harmonic sensitivity analyses. They are validated by comparing with the results of finite difference sensitivity method. Numerical examples are given to demonstrate the effectiveness of the methods and the program.

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.

Design optimization of acoustic radiation from structures under random excitation

Acoustic design optimization of structures subjected to random excitations is investigated in this paper. In this work, a Pseudo excitation method (PEM), an accurate and highly efficient algorithm series for structural stationary random response analysis, is extended to deal with random acoustic radiation problems. A boundary element method (BEM) and a finite element method (FEM) are combined with the PEM to calculate acoustic radiation from random excited structures. A design optimization model is presented, in which the pressure power spectral density (PSD), mass of the structure or eigenvalues of the structure is used as an objective function or constraint function. The thickness of the structure is used as a design variable. Based on acoustic radiation sensitivity analysis and structural sensitivity analysis, the design optimization model is solved with sequential linear programming. Numerical examples of sensitivity analysis and acoustic design optimization are given to demonstrate the effectiveness of the proposed method

Analysis of Stationary Random Responses for Non-Parametric Probabilistic Systems

The move from conceptual design, through fabrication to observation and measurement on the resulting physical structure is fraught with uncertainty. This, together with the necessary simplifications inherent when using the finite element technique, makes the development of a predictive model for the physical structure sufficiently approximate that the use of random structural models is often to be preferred. In this paper, the random uncertainties of the mass, damping and stiffness matrices in a finite element model are replaced by random matrices, and a highly efficient pseudo excitation method for the dynamic response analysis of non-parametric probability systems subjected to stationary random loads is developed. A numerical example shows that the dynamic responses calculated using a conventional (mean) finite element model may be quite different from those based on a random matrix model. For precise fabrication, the uncertainties of models cannot be ignored and the proposed method should be useful in the analysis of such problems.

Analysis of stationary random responses for non-parametric probabilistic systems

The move from conceptual design, through fabrication to observation and measurement on the resulting physical structure is fraught with uncertainty. This, together with the necessary simplifications inherent when using the finite element technique, makes the development of a predictive model for the physical structure sufficiently approximate that the use of random structural models is often to be preferred. In this paper, the random uncertainties of the mass, damping and stiffness matrices in a finite element model are replaced by random matrices, and a highly efficient pseudo excitation method for the dynamic response analysis of non-parametric probability systems subjected to stationary random loads is developed. A numerical example shows that the dynamic responses calculated using a conventional (mean) finite element model may be quite different from those based on a random matrix model. For precise fabrication, the uncertainties of models cannot be ignored and the proposed method should be useful in the analysis of such problems.

A STOCHASTIC EQUIVALENT LINEARIZATION FOR ANALYSIS OF MULTIPLE-DEGREE-OF-FREEDOM FLEXIBLE STRUCTURES

Stochastic equivalent linearization (SEL) method has gained wide popularity because of its versatility in application to multiple-degree-of-freedom (MDOF) nonlinear systems. It is restricted in the random vibration analysis of flexible structures because of the implicit nonlinear system. This paper presents a new method for the equivalent nonlinear system of flexible structures. Using this method, the implicit geometrically nonlinear system can be represented as an explicit equivalent nonlinear system. According to the modal analysis method, the geometrically nonlinear force is replaced by a high-order moment of modal coordinate. The MDOF physical system is translated into a modal system, which could be solved easily. Based on the equivalent nonlinear system, the nonlinear random vibration method is presented by using SEL technology. By using the pseudo-excitation method, the efficiency of the nonlinear random vibration method is increased obviously. The rapid calculation makes it possible to analyze the nonlinear random vibration of MDOF flexible structure. The validation shows that the method has reasonable precision and high efficiency and it could be used in the random vibration analysis of practical flexible structures.

A response surface approach to fatigue reliability of ship structures

Today ship designers are facing ever more stringent safety and environmental requirements as well as economic ones. With the advancement of rules and procedures in the classification societies and the tailwind of modern computing technology, spectral fatigue based on first principles has become commonplace in ship design. So far the centrepiece of such a process is by and large a deterministic acceptance criterion despite the highly uncertain nature of the problem. As a result, the observed damage in a fatigue prone area is often significantly inconsistent with the predicted one, making client-facing a daunting task. As a small step towards a reliability-based solution, this paper attempts to introduce structural uncertainties into established fatigue design assessment process such as the ShipRight FDA3 procedure and the supporting software by Lloyd's Register. Key design parameters are randomised in a spectral fatigue model, where a pseudo-excitation method is used to reflect the non-linear effect of inertial loads and external wave pressure in the splash zone. A stepwise response surface method is used in tandem with t by t fine mesh finite element analysis to obtain the probability of failure. The calculation is demonstrated for an oil tanker example.

Pseudo-excitation-method-based sensitivity analysis and optimization for vehicle ride comfort

In this article, sensitivity analysis formulae for optimizing vehicle suspension systems are derived using the pseudo excitation method (PEM). A spatial finite element model is used to describe the dynamic behaviour of a vehicle running on a randomly uneven road, of which the irregularity is assumed to be a Gaussian random process. Based on the random equations of motion with the right-hand side random acceleration replaced by a pseudo acceleration excitation, various first and second orders of sensitivity formulae are calculated conveniently by differentiating these equations. The optimal solutions when vehicle ride comfort is the objective function are derived by means of these flexibilities. The optimization efficiency and the computational accuracy are numerically justified.

A New Analytical Method for Stochastic Response of Structure-Damper System

Fundamental principles from structural dynamics, pseudo excitation method and perturbation techniques are used to develop a new fast stochastic method for seismic analysis of the combined structuredamper system. In the approach, the mathematical equation of structure-damper system is expressed in the perturbation form, based on which the inverse operation of the matrices is avoided. Moreover, the new method also does not need the solution of any complex eigenvalue problem, in contrast to other methods found in the literature. Finally, the computation efficiency of the method is examined, and numerical comparisons with exact results are carried out to verify the accuracy of the proposed method. In all cases examined, the approach presented here shows excellent agreement with the exact results.