Effects of geometric nonlinearity on the response of a long beam on viscoelastic foundation to a moving mass

Abstract

We investigate the effect of the nonlinear terms arising from exact geometry on the dynamic response of the mass-beam-foundation system. In particular, we are interested in the case when the moving speed of the point mass exceeds the critical speed. We replace the infinitely long beam with a sufficiently long finite beam and use a harmonic expansion method to discretize the partial differential equation of motion. The feasibility of this technique is verified by comparing our numerical results for the linear stationary case under a point force against existing analytical solutions. By solving the eigenvalues of the linear problem, one can find the linear critical speed for a specified mass. When the moving speed of the mass is greater than the critical speed, the dynamic response of the nonlinear system eventually settles to a steady state of periodic motion as seen by an observer travelling along with the mass. This is because the beam behaves like a hardening spring, i.e., the magnitude of the resisting bending moment is always larger than its linear counterpart. In order to maintain the uniform speed of the mass during the periodic vibration, the pushing force must change with time. The amplitude of the periodic vibration increases with the moving speed of the mass in the super-critical speed range. The periodic vibration undergoes a Hopf super-critical bifurcation at a nonlinear critical speed, which is slightly higher than its linear counterpart.