A clamped beam, images solution

Abstract

The method of images is a well-known idea in theoretical physics, but not often used in structural acoustics. When considering the problem of obtaining Green’s function for a general bounded domain, the reflection is described by an image source, and position and sign of the image is chosen to fulfill the boundary conditions. The Green’s function can then be found as the superposition of the free Green’s function for the two sources. Consider now a finite beam, clamped at the boundaries and driven by a point force. In order to make use of periodicity, expand the region along the x axis. To the original force an infinite number of image sources were added. The image expansion is even, and the clamped boundaries convert to simply supported edges. The reforcing supports are introduced in the equation as infinite sums. The system is now infinite, implying Fourier transforms to be used on the spatial coordinate. The transformed displacement and inverse transform are solved for formally. The support forces are related to the beam displacement, so that the yet-unknown reaction forces are solved for. The resulting infinite sums are given explicitly using the Poisson sum formula and the contour integration.