Nonlinear dynamic analysis of an axially loaded rotating Timoshenko beam with extensional condition included subjected to general type of force moving along the beam length

In this paper the non-planar nonlinear dynamic responses of an axially loaded rotating Timoshenko beam subjected to a three-directional force traveling with a constant velocity is studied. On deriving the nonlinear coupled partial differential equations (PDEs) of motion the stretching effect of the beam’s neutral axis due to the pinned-pinned ends’ condition in conjunction with the von Karman strain-displacement relation are considered. The beam’s nonlinear governing coupled PDEs of motion for the bending rotations of warped cross-section, longitudinal and lateral displacements are derived using Hamilton’s principle. To obtain the dynamic responses of the beam, derived PDEs of motion are solved by applying Galerkin’s method. Furthermore, subsequent to the verification of obtained results, a parametric study on the dynamic responses of the beam is conducted by changing the value of the concentrated traveling force-bending moment, load velocity, frequency fluctuation of the load velocity and rotational speed of the beam, respectively. It is observed that the existence of nonlinearity in the governing coupled PDEs of motion due to the beam mid-plane stretching introduces a noticeable effect on the size of the beam’s stiffness.