Nonlinear Dynamic Analysis of an Inclined Timoshenko Beam Subjected to a Moving Mass/Force with Beam’s Weight Included

Abstract

Abstract. In this study, the nonlinear vibrations analysis of an inclined pinned-pinned self-weight Timoshenko beam made of linear, homogenous and isotropic material with a constant cross section and finite length subjected to a traveling mass/force with constant velocity is investigated. The nonlinear coupled partial differential equations of motion for the rotation of warped cross-section, longitudinal and transverse displacements are derived using the Hamilton's principle. These nonlinear coupled PDEs are solved by applying the Galerkin's method to obtain dynamic responses of the beam. The dynamic magnification factor and normalized time histories of mid-point of the beam are obtained for various load velocity ratios and the outcome results have been compared to the results with those obtained from linear solution. The influence of the large deflections caused by a stretching effect due to the beam's fixed ends is captured. It was seen that existence of quadratic-cubic nonlinear terms in the nonlinear governing coupled PDEs of motion causes stiffening (hardening) behavior of the dynamic responses of the self-weight beam under the act of a traveling mass as well as equivalent concentrated moving force. Furthermore, in a case where the object leaves the beam, its planar motion path is derived and the targeting accuracy is investigated and compared with those from the rigid solution assumption.

Nonlinear Dynamic Analysis of a Timoshenko Beam Resting on a Viscoelastic Foundation and Traveled by a Moving Mass

“…In this study Galerkin's method is chosen as a powerful mathematical tool to analyze the vibrations of a Timoshenko beam. Based on the separation of variables technique, the response of the Timoshenko beam in terms of the linear freeoscillation modes can be assumed as follows [9,[19][20][21]:…”

“…According to von-Karman's theory, the kinematics relations for the beam shown in Figure 1(b) are [19][20][21]…”

“…Note that the total virtual work done by the nonlinear viscoelastic foundation and the traveling mass acting on the beam at the location = ( ) is [9,20,21]…”

“…After doing some mathematics, one would get the nonlinear governing coupled PDEs of motion (EOMs) as follows [9,20,21].…”

An Analytical Study on Dynamic Response of Multiple Simply Supported Beam System Subjected to Moving Loads

“…For decades, the dynamic response of a single beam has been extensively investigated. Many previous studies have provided better solutions to this problem [2][3][4][5][6], and there are also many papers on beam vibration excited by a moving mass [7][8][9][10]. In the recent decades, much effort has been devoted to the analysis of dynamic response of a double-beam system, and the results are significant [11][12][13][14][15][16].…”