Transverse impact of a Hertzian body with an infinitely long Euler-Bernoulli beam

We study in detail, using analytical approximations and numerical solution, the transverse impact interaction of a compact body with an infinitely long Euler-Bernoulli beam. The beam is initially stationary, linear and dissipation-free; a Hertzian spring mediates body-beam contact; and the body is otherwise rigid. Impact interaction obeys two nonlinear differential equations with a fractional order derivative. Prior progress on the infinite-beam problem has been limited. Here we completely characterize the possible contact behaviors in terms of one nondimensional number, S, which governs separations and sustained contact regimes. For small S, there is just one contact phase followed by separation. For large S no separation occurs, and sustained contact occurs with decaying oscillations. For intermediate S, separation occurs one or more times, followed eventually by sustained contact. The number of separations can be large over a small range of S. A semi-analytical approximation matches well the smaller-S behavior until first separation. A separate asymptotic approximation matches the long-time sustained contact behavior for higher S, independent of the intervening number of separations. The two approximations work on overlapping ranges of S. Neither approximation captures the multiple separations of intermediate S, where we use full numerics with a published recipe for fractional order systems. The numerics match the abovementioned analytical results.