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 The first correction involves small amplitudes and fast oscillations: something like The initial displacement of points away from G is something like For small t, the deformed shape at points x far from G is oscillatory with decreasing amplitude, something like 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 Finally, the displacement of G as a function of time shows structure at all scales, and is nondifferentiable at infinitely many points.
The Short-Time Impulse Response of Euler-Bernoulli Beams
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, Oct. 17, 2002; final revision, Aug. 18, 2003. Associate Editor: O. O’Reilly. Discussion on the paper should be addressed to the Editor, Prof. Robert M. McMeeking, Journal of Applied Mechanics, Department of Mechanical and Environmental Engineering University of California–Santa Barbara, Santa Barbara, CA 93106-5070, and will be accepted until four months after final publication of the paper itself in the ASME JOURNAL OF APPLIED MECHANICS.
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Chatterjee, A. (May 5, 2004). "The Short-Time Impulse Response of Euler-Bernoulli Beams ." ASME. J. Appl. Mech. March 2004; 71(2): 208–218. https://doi.org/10.1115/1.1667531
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