Journal of Mechanical Science and Technology 24 (10) (2010) 1957~1961 www.springerlink.com/content/1738-494x DOI 10.1007/s12206-010-0704-x Dynamic response of a finite length euler-bernoulli beam on linear and nonlinear viscoelastic foundations to a concentrated moving force Alkim Deniz Senalp 1 , Aytac Arikoglu 1 , Ibrahim Ozkol 1,* and Vedat Ziya Dogan 1 1 Department of Aeronautical Engineering, Istanbul Technical University, Istanbul, 34469,Turkey (Manuscript Received February 23, 2009; Revised March 18, 2010; Accepted June 28, 2010) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract In this paper the dynamic response of a simply-supported, finite length Euler-Bernoulli beam with uniform cross-section resting on a linear and nonlinear viscoelastic foundation acted upon by a moving concentrated force is studied. The Galerkin method is utilized in order to solve the governing equations of motion. Results are compared with the finite element solution for the linear foundation model in order to validate the accuracy of the solution technique. A good agreement between the two solution techniques is observed. The effect of the nonlinearity of foundation stiffness on beam displacement is analyzed for different damping ratios and different speeds of the moving load. The results for the time response of the midpoint of the beam are presented graphically. Keywords: Euler-Bernoulli beam; FEM; Galerkin method; Moving force; Vibration; Viscoelastic foundation ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction Recently, the investigation of the dynamic response of beams on viscoelastic foundations subjected to moving loads has been of great significance in railway engineering. Zheng et. al [1] gave a general solution dynamical problem of an infinite beam on an elastic foundation. Lee [2] investigated the dy- namic response of a Timoshenko beam on a Winkler founda- tion subjected to a moving mass. Thambiratnam and Zhuge [3] studied the dynamics of beams on an elastic foundation subjected to moving loads by using the finite element method. They investigated the effect of the foundation stiffness, travel- ing speed and length of the beam on the dynamic magnifica- tion factor, which is defined as the ratio of the maximum dis- placement in the time history of the mid-point to the static mid-point displacement. Kim [4] investigated the vibration and stability of an infinite Euler-Bernoulli beam resting on a Winkler foundation when the system is subjected to a static axial force and a moving load with either constant or har- monic amplitude variations. The effects of load speed, load frequency, damping on the deflected shape, maximum dis- placement and critical values of the velocity, frequency and axial force are also studied. Kargarnovin and Younesian [5] studied the response of a Timoshenko beam with uniform cross-section and infinite length supported by a generalized Pasternak viscoelastic foundation subjected to an arbitrarily distributed harmonic moving load. Kargarnovin and Youne- sian [6] also analyzed the dynamic response of infinite Ti- moshenko and Euler-Bernoulli beams on nonlinear viscoelas- tic foundations to harmonic moving loads. In this study, the dynamic response of a simply-supported, finite length, uniform cross-section Euler-Bernoulli beam resting on a linear and nonlinear viscoelastic foundation acted upon by a moving concentrated force is studied. In existing literature, research based on the response of beams on founda- tions assumes that the beam is infinite. In this study, an infi- nite track is replaced by a finite track. Since the beam is placed over a very stiff foundation, the moving load has a local effect and it is sufficient to analyze a small portion of the beam. The Galerkin method is used to solve the initial bound- ary value problem that governs the transverse vibration of the beam. Time response histories of the beam are graphically presented for various speeds of force. The effect of non- linearity of the foundation stiffness is also investigated. 2. Theory and formulation In Figs. 1 and 2, simply-supported, homogeneous, isotropic and constant cross-section beams of length L over viscoelastic foundations are shown. Linear and nonlinear foundation mod- els are used. Viscoelastic foundations consist of dashpots and springs. In the literature, the railway track is usually assumed to be linear in order to simplify the track model, although the rail pad and ballast are actually non-linear. The beam is ini- † This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin * Corresponding author. Tel.: +90 2122853111, Fax: +90 2122852926 E-mail address: ozkol@itu.edu.tr © KSME & Springer 2010