Random Vibration Problems Research Articles

A stochastic Functional Model based method for random vibration based robust fault detection under variable non–measurable operating conditions with application to railway vehicle suspensions

The problem of random vibration based robust fault detection under variable and non–measurable Environmental and Operating Conditions (EOCs) is considered, and a novel stochastic Functional Model (FM) based method is postulated. It is a data–driven method, of the Statistical Time Series (STS) type, and aims at overcoming the well known drawbacks of available methods by achieving high detection performance while eliminating their drawbacks, such as the need for measurable EOCs, for measurement of a high number of vibration signals for proper training, for subjective judgement in selecting method parameters, and for high dimensional non–convex optimization procedures. The method is based on representing the system dynamics, under any set of EOCs, in a proper feature space, within which the healthy dynamics are represented by a proper healthy subspace constructed via a Functional Model. Fault detection is then based upon determining, at a certain risk level, whether or not the current dynamics resides within the healthy subspace. The method's assessment is achieved via simulation results with a case study pertaining to fault detection in a railway vehicle suspension under variable payload, with high detection performance, clearly exceeding that of an alternative Principal Component Analysis (PCA) based method.

Some significant improvements for interval process model and non-random vibration analysis method

Recently the authors proposed the interval process model for dynamic uncertainty quantification and based on this further developed a kind of non-probabilistic analysis method called ‘non-random vibration analysis method’ to deal with the important random vibration problems, in which the excitation and response are both given in the form of interval process rather than stochastic process. Since it has some attractive advantages such as easy to understand, convenient to use and small dependence on samples, the non-random vibration analysis method is expected to become an effective supplement to the traditional random vibration theory. In this paper, some significant improvements are made for the interval process model and the non-random vibration analysis method, making them not only more rigorous in theory but also more practical in engineering. Firstly, the definitions and relevant conceptions of interval process model are further standardized and improved, and in addition some new conceptions such as the interval process vector and the cross-covariance function matrix are complemented. Secondly, this paper proposes the important conceptions of limit and continuity of interval process, based on which the differential and integral of interval process are defined. Thirdly, the analytic formulation of dynamic response bounds is deduced for both of the linear single degree of freedom (SDOF) vibration system and the multiple degree of freedom (MDOF) vibration system, providing an important theoretical basis for non-random vibration analysis. Fourthly, this paper also gives the formulation and corresponding numerical methods of structural dynamic response bounds based on finite element method (FEM) for complex continuum problems, effectively enhancing the applicability of non-random vibration analysis in engineering. Finally, four numerical examples are investigated to demonstrate the effectiveness of the proposed method.

Random vibration fatigue life analysis of electronic packages by analytical solutions and Taguchi method

This paper provides an analytical solution to solve for the electronic assembly random vibration problem. Additionally, this paper employs Taguchi method to investigate the fatigue life performance of electronic vehicles subjected to random vibrations loadings. The current work adopted and modified as necessary a previously published analytical solution to solve for the electronic assemblies random vibration problem. This solution is also used to solve for BGA's axial deflections, strains and stresses. In combination with Taguchi method, this was solution employed to investigate the printed circuit board (PCB) and the integrated circuit (IC) component geometries as well as ball grid array (BGA) solder ball dimensions on solder axial stresses and related that to fatigue life performance of electronic packages under random vibration loading. The results of this newly developed analytical solution are in great match with FEA results in terms of the solder joint axial deflections power spectral density (PSD) function. Also, the stress analysis results of Taguchi method showed that the smaller and thicker PCB as well as larger and thicker IC's might reduce solder stresses and hence improve the fatigue performance. Also solder dimensions, compared to PCB and IC, have minor effects on solder stresses and fatigue life behavior. Finally, an optimal design for best fatigue performance was suggested.

Non-random vibration analysis for general viscous damping systems

The authors recently developed a kind of non-probabilistic analysis method, named as ‘non-random vibration analysis’, to deal with the important random vibration problems, in which the excitation and response are both given in the form of interval process rather than stochastic process. Since it has some attractive advantages such as easy to understand, convenient to use and small dependence on samples, the non-random vibration analysis method is expected to be an effective supplement of the traditional random vibration theory. In this paper, we further extend the non-random vibration analysis into the general viscous damping system, and formulate a method to calculate the dynamic response bounds of a viscous damping vibration system under uncertain excitations. Firstly, the unit impulse response matrix of the system is obtained by using a complex mode superposition method. Secondly, an analytic formulation of the system dynamic response middle point and radius under uncertain excitations is derived based on the Duhamel's integral, and thus the upper and lower response bounds of the system can be obtained. Finally, two numerical examples are investigated to demonstrate the effectiveness of the proposed method.

Based on the virtual experiment study of the impact of load sequence on the calculation process of random vibration fatigue damage.

At present, the linear fatigue accumulation theory proposed by Miner is used to calculate the fatigue problem of random vibration. In this process, the loading order of the loads is ignored, so that the results obtained are riskier. The nonlinear fatigue cumulative damage theory takes into account the effects of the load loading sequence, but the calculation becomes more complicated. Many scholars have shown that the fatigue calculation of alternating loads is not strictly linear, and the high and low order will affect the damage index. This phenomenon will be more obvious in the random vibration process. In this paper, the effects of loading sequence on the calculation of non-Gaussian random vibration fatigue damage are studied by virtual experiments. Through the phase selection method, 7 sets of excitations with different kurtosis were loaded onto the notched specimen. Through the calculation results of fatigue damage in the notch sensitive area, it was confirmed that the damage calculated by linear and nonlinear cumulative damage formula was improved with the increase of kurtosis. The results can make a significant difference.

Random vibrations in unmanned aerial vehicles, mathematical analysis and control methodology based on expectation and probability

The analysis of the random vibrations that occur during the flight of the unmanned aerial vehicles is important, as these random vibrations have random characteristic properties and have the ability to decrease the endurance of such systems. The accuracy of data collected from sensors in the unmanned aerial vehicle system is important for the flight control system. The transferring of these data among different sensors such as inertial measurement unit, axis accelerometers, GPS or cameras is usually affected by many factors. One of the important factors is the random vibration, which is usually caused by aerodynamic excitation or air turbulence. The problem of random vibrations has been studied for a long time and explained in many text books. In this study, the author introduces a mathematical analysis for random vibrations that are independent of their sources by considering these vibrations as a random and non-stationary process and designs a control methodology based on expectations and probability theory to reduce the effect of these vibrations. The analysis used in this research is based on the assumptions of practical approximation techniques.

An explicit time-domain approach for sensitivity analysis of non-stationary random vibration problems

This paper presents an explicit time-domain method for sensitivity analysis of structural responses under non-stationary random excitations. Based on time-domain explicit expressions of dynamic responses, a new and more concise time-domain explicit expression of response sensitivity is derived using the direct differentiation method (DDM). Then a more efficient algorithm for direct construction of the explicit expression of response sensitivity is developed based on the physical meanings of the coefficient matrices in the formulation. The adjoint variable method (AVM) is further used to establish the explicit expression of the sensitivity of an arbitrary response. Finally, based on the time-domain explicit expressions for both dynamic response and its sensitivity, an efficient time-domain approach is proposed to calculate the sensitivity of variance responses of a structure subjected to non-stationary random excitations. Numerical examples of different structural systems under non-stationary random excitations are presented to demonstrate the accuracy and efficiency of the proposed method.

Seismic vibration control of bridges with excessive isolator displacement

The effectiveness of base isolation (BI) systems for mitigation of seismic vibration of bridges have been extensively studied in the past. It is well established in those studies that the performance of BI system is largely dependent on the characteristics of isolator yield strength. For optimum design of such systems, normally a standard nonlinear optimization problem is formulated to minimize the maximum response of the structure, referred as Stochastic Structural Optimization (SSO). The SSO of BI system is usually performed with reference to a problem of unconstrained optimization without imposing any restriction on the maximum isolator displacement. In this regard it is important to note that the isolator displacement should not be arbitrarily large to fulfil the serviceability requirements and to avoid the possibility of pounding to the adjacent units. The present study is intended to incorporate the effect of excessive isolator displacement in optimizing BI system to control seismic vibration effect of bridges. In doing so, the necessary stochastic response of the isolated bridge needs to be optimized is obtained in the framework of statistical linearization of the related nonlinear random vibration problem. A simply supported bridge is taken up to elucidate the effect of constraint condition on optimum design and overall performance of the isolated bridge compared to that of obtained by the conventional unconstrained optimization approach.

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.

Global Response Sensitivity Analysis of Randomly Excited Dynamic Structures

The response of structural dynamical systems excited by multiple random excitations is considered. Two new procedures for evaluating global response sensitivity measures with respect to the excitation components are proposed. The first procedure is valid for stationary response of linear systems under stationary random excitations and is based on the notion of Hellinger's metric of distance between two power spectral density functions. The second procedure is more generally valid and is based on the l2 norm based distance measure between two probability density functions. Specific cases which admit exact solutions are presented, and solution procedures based on Monte Carlo simulations for more general class of problems are outlined. Illustrations include studies on a parametrically excited linear system and a nonlinear random vibration problem involving moving oscillator-beam system that considers excitations attributable to random support motions and guide-way unevenness. (C) 2015 American Society of Civil Engineers.

Response analysis of Rayleigh–Van der Pol vibroimpact system with inelastic impact under two parametric white-noise excitations

In the present paper, the random vibration problem of Rayleigh–Van der Pol oscillator is considered by means of a single-degree-of-freedom system with a unilateral rigid barrier at its equilibrium position. The random excitations are two parametric Gaussian white noises. The nonsmooth transformation method is applied to convert the vibroimpact system to an equivalent system without velocity jump. The modified stochastic averaging technique for energy envelope is used to deal with the transformed system, and the averaging Ito stochastic differential equation is obtained. The analytical solution of the response of the original vibroimpact system is derived by solving the corresponding Fokker–Planck equation and using the inverse transformation of nonsmooth transformation. The validity of the analytical results is verified by those from Monte Carlo simulations based on original system. Effects of different system parameters and parametric white noises on the response of the system are examined. In addition, the critical condition of stochastic bifurcations is also explored.

Random vibrations of Rayleigh vibroimpact oscillator under Parametric Poisson white noise

Random vibration problems for a single-degree-of-freedom (SDOF) Rayleigh vibroimpact system with a rigid barrier under parametric Poisson white noise are considered. The averaged generalized Fokker-Planck-Kolmogorov (FPK) equations with parametric Poisson white noise are derived after using the nonsmooth variable transformation and the approximate stationary solutions for the system's response are obtained by perturbation method. The results are validated numerically by using Monte Carlo simulations from original vibroimpact system. Effects on the response for different damping coefficients, restitution coefficients and noise intensities are discussed. Furthermore, stochastic bifurcations are also explored.

Model reduction for dynamical systems with quadratic output

SUMMARYFinite element models for structures and vibrations often lead to second order dynamical systems with large sparse matrices. For large‐scale finite element models, the computation of the frequency response function and the structural response to dynamic loads may present a considerable computational cost. Padé via Krylov methods are widely used and are appreciated projection‐based model reduction techniques for linear dynamical systems with linear output. This paper extends the framework of the Krylov methods to systems with a quadratic output arising in linear quadratic optimal control or random vibration problems. Three different two‐sided model reduction approaches are formulated based on the Krylov methods. For all methods, the control (or right) Krylov space is the same. The difference between the approaches lies, thus, in the choice of the observation (or left) Krylov space. The algorithms and theory are developed for the particularly important case of structural damping. We also give numerical examples for large‐scale systems corresponding to the forced vibration of a simply supported plate and of an existing footbridge. In this case, a block form of the Padé via Krylov method is used. Copyright © 2012 John Wiley & Sons, Ltd.

CONVOLVED ORTHOGONAL EXPANSIONS FOR UNCERTAINTY PROPAGATION: APPLICATION TO RANDOM VIBRATION PROBLEMS

Physical nonlinear systems are typically characterized with n-fold convolution of the Green’s function, e.g., nonlinear oscillators, inhomogeneous media, and scattering theory in continuum and quantum mechanics. A novel stochastic computation method based on orthogonal expansions of random fields has been recently proposed [1]. In this study, the idea of orthogonal expansion is formalized as the so-called nth-order convolved orthogonal expansion (COE) method, especially in dealing with random processes in time. Although the paper is focused on presentation of the properties of the convolved random basis processes, examples are also provided to demonstrate application of the COE method to random vibration problems. In addition, the relation to the classical Volterra-type expansions is discussed

Linear models for non-Gaussian processes and applications to linear random vibration

Linear models are finite sums of specified deterministic, continuous functions of time with random coefficients. It is shown that linear models provide ( i ) accurate approximations for real-valued non-Gaussian processes with continuous samples defined on bounded time intervals, ( i i ) simple solutions for linear random vibration problems with non-Gaussian input, and ( i i i ) efficient techniques for selecting optimal designs from collections of proposed alternatives. Theoretical arguments and numerical examples are presented to establish properties of linear models, illustrate the construction of linear models, solve linear random vibration with non-Gaussian input, and propose an approach for optimal design of linear dynamic systems. It is shown that the proposed linear model provides an efficient tool for analyzing linear systems in non-Gaussian environment.

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.

Linear random vibration by stochastic reduced‐order models

AbstractA practical method is developed for calculating statistics of the states of linear dynamic systems with deterministic properties subjected to non‐Gaussian noise and systems with uncertain properties subjected to Gaussian and non‐Gaussian noise. These classes of problems are relevant as most systems have uncertain properties, physical noise is rarely Gaussian, and the classical theory of linear random vibration applies to deterministic systems and can only deliver the first two moments of a system state if the noise is non‐Gaussian.The method (1) is based on approximate representations of all or some of the random elements in the definition of linear random vibration problems by stochastic reduced‐order models (SROMs), that is, simple random elements having a finite number of outcomes of unequal probabilities, (2) can be used to calculate statistics of a system state beyond its first two moments, and (3) establishes bounds on the discrepancy between exact and SROM‐based solutions of linear random vibration problems. The implementation of the method has required to integrate existing and new numerical algorithms. Examples are presented to illustrate the application of the proposed method and assess its accuracy. Copyright © 2009 John Wiley & Sons, Ltd.

A direct probabilistic method to solve state equations under random excitation

In random vibration problems, the statistical characteristics of the response, such as the mean value, correlation and variance, are often interested in by people. This paper is concerned with the statistic property of structural dynamic response under uncertain loads which are considered random processes. These random processes can either be stationary or non-stationary. A more accurate method named the direct probabilistic approach (DPA) is presented, which is proposed providing an alternative to the conventional time, frequency domain methods and the state space theory in a random vibration field. In this study, the response’s statistical characteristics can be directly obtained from the statistical characteristics and the initial condition of the corresponding external loads. Finally, two examples are given to compare the DPA method with the conventional method. Good agreement is found, which strongly validates the proposed method.

Evaluation of Karhunen–Loève, Spectral, and Sampling Representations for Stochastic Processes

The Karhunen–Loève, spectral, and sampling representations, referred to as the KL, SP, and SA representations, are defined and some features/limitations of KL-, SP-, and SA-based approximations commonly used in applications are stated. Three test applications are used to evaluate these approximate representations. The test applications include (1) models for non-Gaussian processes; (2) Monte Carlo algorithms for generating samples of Gaussian and non-Gaussian processes; and (3) approximate solutions for random vibration problems with deterministic and uncertain system parameters. Conditions are established for the convergence of the solutions of some random vibration problems corresponding to KL, SP, and SA approximate representations of the input to these problems. It is also shown that the KL and SP representations coincide for weakly stationary processes.

Random vibrations of swings

The paper considers a problem of random vibrations of swings—a pendulum with variable length. The goal of the paper is to estimate mean response energy of the system, subjected to external Gaussian white noise. The Energy Balance method is used to derive analytical results.