The effects of mode shapes on the temporal response of flexible closed-loop linkages under the impulse excitation

•The dynamic modeling of the closed-loop mechanisms with flexible links.•Using the Gibbs–Appell formulation, the Newton’s kinematic impact law and the Timoshenko beam theory to model the dynamic motion equations.•Detecting the exact impact moments and solving the relevant differential/algebraic equations.•Comprehensively studying the effects of shape functions on the vibration response of the elastic closed-loop linkages.•Presenting a criterion for validating the simulation results. In this paper, by relying on the Timoshenko beam theory and the assumed modes method, we have presented a dynamic modeling of the closed-loop flexible linkages in the non-impact and impact stages. The dynamic equations for the suspension stage are obtained by employing the effective, but less used, Gibbs-Appell formulation, and the governing equations for the impact stage are derived by means of the Newton’s impact method. Although the motion equations have been extracted for an n-link mechanism in general, the simulations are performed for two closed-loop manipulators consisting of four elastic links. In order to model the flexibility of the links, the two mentioned manipulators are respectively simulated with the mode shapes associated with the clamped–clamped (C–C) and clamped-free (C-F) boundary conditions. In fact, the primary goal of this research is to investigate the effects of the mode shapes on the temporal response of these types of mechanical systems. Lastly, a criterion based on the mechanical energy conservation law is presented for validating the obtained results.