Dynamic Behavior of Elastic Bars and Beams Impinging on Ideal Springs

The impacting and rebounding behaviors of straight elastic components are investigated and a unified approach is proposed to analytically predict the whole process of the collision and rebounding of straight elastic bars and beams after each of them impinges on ideal (massless) elastic spring(s). The mathematical problems with definitive solution are formulated, respectively, for both the constrained-motion and free-motion stages, and the method of mode superposition, which is concise and straightforward especially for long-time interaction and multiple collision cases, is successfully utilized by repeatedly altering boundary and initial conditions for these successive stages. These two stages happen alternatively and the collision process terminates when the constrained motion no longer occurs. In particular, three examples are investigated in detail; they are: a straight bar impinges on an ideal elastic spring along its axis, a straight beam vertically impinges on an ideal elastic spring at the beam's midpoint, and a straight beam vertically impinges on two ideal springs with the same stiffness at the beam's two ends. Numerical results show that the coefficient of restitution (COR) and the nondimensional rebounding time (NRT) only depend on the stiffness ratio between the ideal spring(s) and the elastic bar/beam. Collision happens only once for the straight bar impinging on spring, while multiple collisions occur for the straight beam impinging on springs in the cases with large stiffness ratio. Once multiple collisions occur, COR undergoes complicated fluctuation with the increase of stiffness ratio. Approximate analytical solutions (AASs) for COR and NRT under the cases of small stiffness ratio are all derived. Finally, to validate the proposed approach in practical collision problems, the influence of the springs' mass on the collision behavior is demonstrated through numerical simulation.