Shock and Vibration

Volume 2017 (2017), Article ID 3809415, 13 pages

https://doi.org/10.1155/2017/3809415

## Dynamic Analysis of an Infinitely Long Beam Resting on a Kelvin Foundation under Moving Random Loads

^{1}State Key Laboratory of Structural Analysis for Industrial Equipment, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116023, China^{2}School of Engineering, University of Liverpool, The Quadrangle, Liverpool L69 3GH, UK

Correspondence should be addressed to Y. Zhao; nc.ude.tuld@oahzy

Received 27 December 2016; Accepted 27 February 2017; Published 16 March 2017

Academic Editor: Laurent Mevel

Copyright © 2017 Y. Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Nonstationary random vibration analysis of an infinitely long beam resting on a Kelvin foundation subjected to moving random loads is studied in this paper. Based on the pseudo excitation method (PEM) combined with the Fourier transform (FT), a closed-form solution of the power spectral responses of the nonstationary random vibration of the system is derived in the frequency-wavenumber domain. On the numerical integration scheme a fast Fourier transform is developed for moving load problems through a parameter substitution, which is found to be superior to Simpson’s rule. The results obtained by using the PEM-FT method are verified using Monte Carlo method and good agreement between these two sets of results is achieved. Special attention is paid to investigation of the effects of the moving load velocity, a few key system parameters, and coherence of loads on the random vibration responses. The relationship between the critical speed and resonance is also explored.

#### 1. Introduction

Dynamic analysis of a structure subjected to moving loads has always been an important subject for researchers, which has an extensive engineering background [1–5]. To date, various moving load dynamic problems have been studied, for example, as reviewed in [6], from relatively simple problems of vibration of a beam excited by a moving mass and vibration of a rotating shaft under a moving surface load to complicated vehicle-track-bridge interaction problems.

For a moving load problem, even if the load is modelled as a stationary process, the response at an observed point in a structure is nonstationary because of the load’s motion in space. Such a problem is obviously different from the nonstationary random vibration which has been widely studied in earthquake engineering. In earthquake engineering the nonstationary random vibration response of a structure is induced by the nonstationary random earthquake input. To date, nonstationary earthquake motion and structural analysis of earthquake response have been extensively investigated [7, 8]. However, research into nonstationary random vibration induced by moving random loads is fairly uncommon. In particular, the efficiency in computing dynamic responses of structures under moving random loads is low and thus must be enhanced, which is the motivation of this paper.

In railway engineering, the dynamic interaction between trains and rails can be represented as nonstationary random vibration of an infinitely long beams resting on a Kelvin foundation. The moving load model is more suitable to explore several dynamic aspects of a train-track interaction problem, such as critical speed and resonance characteristics.

For steady-state response of a structure subjected to moving loads, its damping, stiffness, and load velocity are considered important parameters. Achenbach and Sun [9] investigated the responses of a Timoshenko beam of infinite length subjected to a force moving at constant velocity and studied the influence of the damping coefficient and the load velocity on the system responses. Jones et al. [10] presented a model of a track on a layered ground under a moving oscillating load and discussed the effect of the relative speeds of the load, the effect of the ground type, and the effect of the embankment on the vertical displacement of the ground. Lin and Trethewey [11] proposed a method for the dynamic analysis of elastic beams subjected to dynamic loads induced by the arbitrary movement of a spring-mass-damper system. In addition to the steady-state response of a structure subjected to moving loads, the critical state and stability are another major aspect of a moving load problem. Chen et al. [12] established the dynamic stiffness matrix of an infinite Timoshenko beam on viscoelastic foundation under a harmonic moving load and determined the critical velocities and the resonant frequencies. Dieterman and Metrikine [13, 14] derived expressions of the equivalent stiffness of an elastic half space interacting with a beam with finite width and determined the critical velocities of a constant load moving. Suiker et al. [15] investigated the problem of a moving load on a Timoshenko beam-half plane system and analyzed the subcritical/supercritical states.

The nonlinear dynamics of a structure subjected to moving loads is also studied. Yoshimura et al. [16, 17] investigated the deflection of a beam, including the effects of geometric nonlinearity, subjected to moving vehicle loads. Şimşek [18] proposed a method for nonlinear dynamic analysis of a functionally graded beam with pinned-pinned supports subjected to a moving harmonic load on Timoshenko beam theory combined with the von-Kármán’s nonlinear strain-displacement relationship. Castro Jorge et al. [19] studied the dynamic response of a beam on nonlinear elastic foundations under moving constant load and carried out parametric analysis of the load intensity and velocity and the foundation’s stiffness.

The above-mentioned works focus on deterministic moving load problems. In fact, for the very important problems of dynamic interactions between vehicles and structures, the irregularity of a rail or road surface should be considered in structural design and analysis. There are a number of published papers that cover moving random loads. Bryja and Śniady [20] and Śniady [21] assumed that the force arriving at a bridge as a Poisson process of events and developed an analytical technique to determine the bridge’s random vibration response. Chatterjee et al. [22] carried out a continuum analysis for determining the flexural-torsional vibration of a suspension bridge under vehicular movement and the surface irregularity of the bridge pavement was generated from a power spectral density function. Zibdeh [23] studied the random vibration of a simply supported elastic beam under moving random loads with time-varying velocity. Huang and Wang [24] modelled bridges as grillage beam systems and vehicle as a linear multibody model and analyzed the effect of longitudinal grade on the vibration of bridges considering surface roughness. Lombaert et al. [25] built a numerical model of free field traffic-induced vibration during the passage of a vehicle on an uneven road and derived a transfer function between the source and the receiver. Kim et al. [26] proposed a three-dimensional analysis for bridge-vehicle interaction in which measured roadway roughness profiles were used. Chang and Liu [27] made deterministic and random vibration analysis of a nonlinear beam on an elastic foundation subjected to a moving load.

In order to evaluate the random vibration responses of a system induced by moving random load the Monte Carlo method is adopted usually in the above-mentioned studies. The samples of irregular profiles, as input excitation of the system, can be generated according to Fourier series with random phases and the statistical responses of the system can be obtained by using a time-domain integration method. The nonlinear behaviour of structures can be also considered by using Monte Carlo method, which is a significant advantage [28]. However, the time-domain method does not yield the frequency-domain characteristics of the nonstationary random vibration of the system in a straightforward and intuitive manner and it is inconvenient to investigate the resonance phenomenon. Moreover, in order to achieve more reliable predictions a large number of samples must be used in Monte Carlo method; thus a more time-consuming analysis has to be performed.

The power spectral method representing the energy distribution with respect to frequencies is a widely accepted method in random vibration field [29, 30]. For a moving random load problem the frequency-domain method based on the spectral analysis theory of random vibration is extremely attractive for evaluation of the response statistics. However, there are also some difficulties encountered in the practical application of the random vibration theory to moving load problems, such as the following: (1) for a random load moving at constant speed, even though the input load is a stationary random process, the responses of the structure will be a nonstationary process with evolutionary feature due to the load movement. (2) The critical speed of a system is closely related to the structural properties, velocity, and frequency components of the moving load. It is a nearly impossible task to explore the responses under a moving random load using analytical method.

For random vibration analysis of linear systems the PEM is an efficient and accurate algorithm, which has been widely used in the field of earthquake engineering [31–34]. To overcome the above difficulties a hybrid pseudo excitation method-Fourier transform (PEM-FT) is established in this paper. Using PEM-FT the closed-form solution of the evolutionary spectrum (frequency/time) of the systems response is derived, which has a concise expression. Moreover, the computing time of PEM-FT is very short compared with the time-domain Monte Carlo method. Another advantage is that the system’s characteristics in frequency domain can be understood intuitively.

The organization of this paper is as follows. Section 2 introduces the dynamic model and governing differential equation for an infinitely long beam resting on a Kelvin foundation subjected to moving random load. Section 3 presents the equation of motion described in the frequency-wave number domain and then Green function obtained by using integral-transform method. In Section 4 the PEM-FT is developed to evaluate the nonstationary random vibration responses of the beam. Introducing the substitution of an integration parameter the discrete fast Fourier transform is proposed for the numerical integration, which has a high computational efficiency. Further, the critical velocity of moving load is discussed. In Section 5 the proposed method is verified using Monte Carlo method and the influence of the structural damping, stiffness, and load velocity on random vibration responses is investigated. Finally, the main results of this paper are summarized in Section 6. Although a relatively simple mechanical model is studied in this paper, the theory and the numerical analysis method are applicable to the study of nonstationary vibration of more complex coupled train-track systems.

#### 2. Infinitely Long Beam Resting on a Kelvin Foundation under Moving Random Loads

An Euler-Bernoulli beam resting on a viscoelastic foundation of Kelvin type subjected to a sequence of moving random loads, as shown in Figure 1, is meant to represent a train running on a rail track in an approximate way. In the following derivations and discussions, the initial system is considered as quiescent.