Nonlinear vibration analysis of harmonically excited cracked beams on viscoelastic foundations
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O R I G I N A L PA P E R
Nonlinear vibration analysis of harmonically excited cracked beams on viscoelastic foundations D. Younesian · S.R. Marjani · E. Esmailzadeh
Received: 20 August 2012 / Accepted: 11 October 2012 / Published online: 1 November 2012 © Springer Science+Business Media Dordrecht 2012
Abstract The frequency response of a cracked beam supported by a nonlinear viscoelastic foundation has been investigated in this study. The Galerkin method in conjunction with the multiple scales method (MSM) is employed to solve the nonlinear governing equations of motion. The steady-state solutions are derived for the two different resonant conditions. A parametric sensitivity analysis is carried out and the effects of different parameters, namely the geometry and location of crack, loading position and the linear and nonlinear foundation parameters, on the frequency-response solution are examined. Keywords Cracked beam · Nonlinear vibration · Multiple scales method · Parameter sensitivity · Resonant frequency · Viscoelastic foundation 1 Introduction The presence of surface cracks is quite common in railway tracks as the result of developing high stress D. Younesian · S.R. Marjani Center of Excellence in Railway Transportation, School of Railway Engineering, Iran University of Science and Technology, Tehran 16846-13114, Iran E. Esmailzadeh () Faculty of Engineering and Applied Science, University of Ontario Institute of Technology, Oshawa, Ontario, L1H 7K4, Canada e-mail: [email protected]
levels in the wheel rail contact areas. The cracks normally change the dynamic behavior of the track and sometimes may result in the sudden failure of the railway tracks. There are two main objectives, which could be generally targeted in the vibration analysis of the cracked tracks, namely, (a) the alteration of dynamical behavior and, (b) the crack detection. Vibration response analysis of a beam resting on a foundation, to represent a model of a railway track, has been extensively focused on by railway researchers in recent years. Different linear and nonlinear models have been developed and their analytical results were reported in literature. A comprehensive review on the preliminary studies in this field has been conducted by Fryba [1]. Chen et al. [2] established the dynamic stiffness matrix of an infinite Timoshenko beam, resting on viscoelastic foundation and, subjected to a harmonically moving load. Vostroukhov and Metrikine [3] studied the steady-state response of a railway track when it is traversed by a moving train. They did model the track by two beams rested on periodically positioned supports mounted on a viscoelastic 3-D layer. The dynamic response of infinite Timoshenko beams, resting on either linear or nonlinear viscoelastic foundations, was studied by Kargarnovin et al. [4, 5]. Younesian et al. [6, 7] utilized the Lindsted– Poincare method and analytically studied the vibration of infinite beams on viscoelastic foundations with the randomly distributed parameters. The finite element method was utilized by Ngu
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