Nonlinear Vibration Solution for an Inclined Timoshenko Beam Under the Action of a Moving Force with Constant/Nonconstant Velocity
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This study is focused on the nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions subjected to a moving force under the influence of three types of motions, including accelerating, decelerating and constant-velocity types of motion. The nonlinear governing coupled partial differential equations (PDEs) of motion for the bending rotation of the warped cross section of the beam and its longitudinal and transverse displacements are derived by using Hamilton’s principle.
To obtain the dynamic response of the beam under the action of a moving force, the derived nonlinear coupled PDEs of motion are solved by applying Galerkin’s method. Then the dynamic response of the beam is obtained using the mode summation technique. Furthermore, the calculated results are verified by the results obtained by finite-element method (FEM) analysis. In the next step, a parametric study of the response of the beam is conducted by changing the magnitude of the traveling concentrated force, its velocity and boundary conditions for the beam. Similarly, their sensitivity to the dynamic response of the beam is also studied. It is observed that the existence of quadratic-cubic nonlinearity in the governing coupled PDEs of motion renders the hardening/softening behavior in the dynamic response of the beam. Moreover, we note that any restriction imposed on stretching of the mid-plane of the beam introduces a nonlinear behavior in the PDEs of motion of the beam.
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