Dynamic analysis of an inclined Timoshenko beam traveled by successive moving masses/forces with inclusion of geometric nonlinearities

Abstract

In the first part of this paper, the nonlinear coupled governing partial differential equations of vibrations by including the bending rotation of cross section, longitudinal and transverse displacements of an inclined pinned–pinned 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 are derived. To do this, the energy method (Hamilton’s principle) based on the large deflection theory in conjuncture with the von-Karman strain-displacement relations is used. These equations are solved using the Galerkin’s approach via numerical integration methods to obtain dynamic responses of the beam under act of a moving mass/force. In the second part, the nonlinear coupled vibrations of the beam traveled by an arbitrary number of successive moving masses/forces are investigated. To do a thorough study on the subject at hand, a parametric sensitivity analysis by taking into account the effects of the magnitude of the traveling mass or equivalent concentrated force, the velocity of the traveling mass/force, beam’s inclination angle, length of the beam, height of the beam and spacing between successive moving masses/forces are carried out. Furthermore, the dynamic magnification factor and normalized time histories of the mid-point of the beam are obtained for various load velocity ratios, and the results are illustrated and compared to the results obtained from traditional linear solution. The influence of the large deflections caused by a stretching effect due to the beam’s immovable end supports is captured. It is seen that the existence of quadratic–cubic nonlinear terms in the coupled governing PDEs of motion renders stiffening (hardening) behavior of the dynamic responses of the beam under the action of a moving mass/force.