The short-time impulse response of Euler-Bernoulli beams

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 roott. The first correction involves small amplitudes and fast oscillations: something like t(3/2) 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 (x(2)). 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 epsilon, the ensuing support forces are of O(epsilon(-1/2)). Finally, the displacement a G as a function of time shows structure at all scales, and is nondifferentiable at infinitely many points.