Fast in-plane dynamics of a beam with unilateral constraints

A computationally efficient technique to simulate the dynamic response of a beam colliding with rigid obstacles is described in this paper. The proposed method merges three key concepts. First, a low-order discretization scheme that maximizes the number of nodes of the discrete model (where impacts are detected) at the expense of the degree of continuity of the constructed displacement field is used. Second, the constrained problem is transformed into an unconstrained one by formulating the impact by using a Signorini complementarity law involving the impulse generated by the collision and the preimpact and postimpact velocity linked through a coefficient of restitution. Third, Moreau's midpoint time-stepping scheme developed in the context of colliding rigid bodies is used to advance the solution. The algorithm is first validated on the nonimpact problem of a cantilever Rayleigh beam subjected to an impulsive discrete load. Then the problem of a cantilever beam vibrating between two (symmetrically located) stops is analyzed. Both cases of discrete and continuous obstacles are considered, and the numerical predictions are compared with published results or those obtained with a commercial code.