On the FEM modeling of mechanical systems controlled by relative motion of a member: A pendulum–mass interaction test case

A new approach for simulating the dynamics of mechanical systems with relative movements of members by the finite element method (FEM) is proposed. The moving member is modeled by a mass traveling along a guiding beam. The mass is incorporated into the beam element that is being traversed. The nodal inertia forces due to the relative mass’ motion are derived from the rate of changing the mass matrix of this composite element. These inertia forces, which are added to the regular nodal forces, reflect the Coriolis-type effects present in the system. Several ways of calculating and including them into the time-integrating procedure are presented. The approach allows treating the moving member as a controller; with a desired pattern of the member's motion imposed on the system as input to analyze its response as output. The proposed procedure is tested on the problem of controlling pendulum-like oscillations of a beam by a moving mass. The element size and the time-step of numerical integration to secure sufficient accuracy of the approach for different mass motion patterns are discussed in detail. In particular, it is demonstrated that the case of the mass motion pattern attenuating the pendulum oscillations with a constant damping ratio can be precisely simulated.