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

The dynamic response of a Timoshenko beam with immovable ends resting on a nonlinear viscoelastic foundation and subjected to motion of a traveling mass moving with a constant velocity is studied. Primarily, the beam’s nonlinear governing coupled PDEs of motion for the lateral and longitudinal displacements as well as the beam’s cross-sectional rotation are derived using Hamilton’s principle. On deriving these nonlinear coupled PDEs the stretching effect of the beam’s neutral axis due to the beam’s fixed end conditions in conjunction with the von-Karman strain-displacement relations is considered. To obtain the dynamic responses of the beam under the act of a moving mass, derived nonlinear coupled PDEs of motion are solved by applying Galerkin’s method. Then the beam’s dynamic responses are obtained using mode summation technique. Furthermore, after verification of our results with other sources in the literature a parametric study on the dynamic response of the beam is conducted by changing the velocity of the moving mass, damping coefficient, and stiffnesses of the foundation including linear and cubic nonlinear parts, respectively. It is observed that the inclusion of geometrical and foundation stiffness nonlinearities into the system in presence of the foundation damping will produce significant effect in the beam’s dynamic response. 1. Introduction The topic of vibration study of structural elements such as strings, beams, plates, and shells under the act of a moving mass is of great interest and importance in the field of structural dynamics. It should be noted that the review of numerous reported studies related to the dynamic behavior of mechanical/structural systems discloses that almost linear behavior of such systems is considered. Indeed, in reality such systems inherently and naturally have nonlinear behavior, for example, due to the geometrical nonlinearity or when they are subjected to external loadings comparatively large enough. As we will see later on in the modeling of the problem, the stretching of the beam’s neutral axis due to fixed ends condition adds another nonlinearity to the dynamical behavior of the system. In addition, there are some other external distinct mechanical elements having nonlinear behavior attached to such structures like shock energy absorbers or viscoelastic foundations which will add further other nonlinearities in the model analysis. From mechanical point of view, any beam structure can be modeled as a thin or thick beam for which different theories usually can be implemented. In extending the issue of the