arXiv:2611.16798 · v1

Transverse impact of a Hertzian sphere with an infinite Rayleigh beam

首次提交 2026-11-12 21:41

版本更新 2026-11-12 21:41

雷达观测 2026-11-13 14:15

摘要

We study the impact of a Hertzian sphere on an infinite Rayleigh beam, which presents both interesting possibilities for numerical approximation as well as new insights in the multiple-contact-event regime. Here, Galerkin projection using an abstract continuous variable replaces the history-dependent convolution in the impact equations by a system of local-in-time ordinary differential equations. The corresponding approximate solutions are validated against direct numerical solution of the convolution-based formulation and show agreement. Unlike the corresponding Euler-Bernoulli beam impact problem, which is governed by a single nondimensional parameter, the Rayleigh beam impact dynamics is governed by two nondimensional parameters. A parametric study reveals three distinct impact regimes: one regime with a single contact followed by permanent separation, a second regime with a single contact that is sustained thereafter, and an intervening regime with initial separation followed by multiple contact events with diminishing amplitudes, leading eventually to sustained contact. Short- and long-time asymptotic analyses are used to characterize the boundaries between the permanent separation and the re-contact regimes in different parameter regimes. The three significant new aspects of this paper are the system itself (Rayleigh beam instead of Euler Bernoulli beam), the model order reduction achieved through Galerkin projection in an abstract domain, and the detailed parametric study leading to a simple qualitative picture of possible behaviours.