The Short-Time Impulse Response of Euler-Bernoulli Beams

Abstract

We study an undamped, simply supported, Euler-Bernoulli beam given an instantaneous impulse at a point G, far from its ends. The standard modal solution obscures interesting mathematical features of the initial response, which are studied here using dimensional analysis, an averaging procedure of Zener, a similarity solution for an infinite beam, asymptotics, heuristics, and numerics. Results obtained include short-time asymptotic estimates for various dynamic quantities, as well as a numerical demonstration of fractal behavior in the response. The leading order displacement of G is proportional to t. The first correction involves small amplitudes and fast oscillations: something like t3/2 cos(t−1). The initial displacement of points away from G is something like t cos(t−1). For small t, the deformed shape at points x far from G is oscillatory with decreasing amplitude, something like x−2 cos(x2). The impulse at G does not cause impulsive support reactions, but support forces immediately afterwards have large amplitudes and fast oscillations that depend on inner details of the impulse: for an impulse applied over a time period ε, the ensuing support forces are of O(ε−1/2). Finally, the displacement of G as a function of time shows structure at all scales, and is nondifferentiable at infinitely many points.