Short Time Angular Impulse Response of Rayleigh Beams

Abstract

In the dynamics of linear structures, the impulse response function is of fundamental interest. In some cases one examines the short term response wherein the disturbance is still local and the boundaries have not yet come into play, and for such short-time analysis the geometrical extent of the structure may be taken as unbounded. Here we examine the response of slender beams to angular impulses. The Euler-Bernoulli model, which does not include rotary inertia of cross sections, predicts an unphysical and unbounded initial rotation at the point of application. A finite length Euler-Bernoulli beam, when modelled using finite elements, predicts a mesh-dependent response that shows fast large-amplitude oscillations setting in very quickly. The simplest introduction of rotary inertia yields the Rayleigh beam model, which has more reasonable behaviour including a finite wave speed at all frequencies. If a Rayleigh beam is given an impulsive moment at a location away from its boundaries, then the predicted behaviour has an instantaneous finite jump in local slope or rotation, followed by smooth evolution of the slope for a finite time interval until reflections arrive from the boundary, causing subsequent slope discontinuities in time. We present a detailed study of the angular impulse response of a simply supported Rayleigh beam, starting with dimensional analysis, followed by modal expansion including all natural frequencies, culminating with an asymptotic formula for the short-time response. The asymptotic formula is obtained by breaking the series solution into two parts to be treated independently term by term, and leads to a polynomial in time. The polynomial matches the response from refined finite element (FE) simulations.