Dynamic instability of a beam undergoing periodic motions over supports

This paper investigates the dynamic stability of a beam moving over two bi-lateral supports using finite element analysis, with essential conditions applied via the penalty function approach. Computational advantages of the penalty-based approach compared to that of the Lagrangian multipliers are highlighted in the context of such unique problems in dynamics. Penalty-based numerical formulation for the moving beam results in a system of second-order differential equations with periodic coefficients. The governing equations are reduced to state-space form and Floquet–Lyapunov theory is applied to investigate dynamic stability of the moving beam. The instability characteristics are studied for a range of amplitudes and frequencies for sinusoidal longitudinal motions of the beam. In addition to the predictions using Floquet–Lyapunov theory, further instability regions based on first and higher approximations are identified. The instability results for periodic motion compare well with previous research and new results are presented taking into consideration the effect of damping. The penalty-based finite element formulation is found to be effective when applied to this class of dynamics problems. The avenues for further research are also highlighted.