On the Nature of Symmetrical Contacts of a Beam Impacting a Rigid Wall

Abstract
The impact of a compliant structure against a rigid obstacle is governed by multiple time and spatial scales, ranging from rapid local deformations and stress wave propagation to slower global motions. This interaction presents significant challenges for numerical methods, particularly in resolving the immediate postimpact response and contact dynamics. In this study, we focus on the impact of a beam on a rigid obstacle as a minimal yet representative example of such multiscale phenomena. A local asymptotic solution is derived for the short-time response. At leading order, the analysis reveals two distinct contact patterns governed by the ratio of the impact velocity to the beam curvature at the contact point: a single concentrated force whose intensity decays with the inverse square root of time or a pair of traveling forces that move away from each other at a rate inversely proportional to the square root of time. Notably, in the latter case, the beam segment between the traveling forces remains stationary, posing challenges for numerical methods based on penetration conditions. These findings highlight the utility of asymptotic methods in capturing the singular and bifurcated nature of contact forces in multiscale impact problems. The solution can be used to improve or provide a benchmark for assessing numerical approaches in the impact analysis of compliant beam structures with rigid walls.