Propagation of decaying waves in periodic and piecewise periodic structures of finite length

Abstract

A new method is described for the analysis of long and complicated structures which are composed of spatially periodic units or sections of spatially periodic units. The response of such a structure to external excitations is treated as a superposition of wave motions, with account taken of the effects of wave reflection due to change in the construction pattern along the structure and boundary conditions. Since damping usually exists in a real structure, a wave motion always decays as it propagates along the structure. However, this can cause numerical difficulties if the structure is rather long, because a decaying wave appears as a growing wave in the opposite direction and, since propagations in both directions must be accounted for in the analysis, computational errors associated with the apparent growing waves also grow and the numerical results become meaningless. In the proposed new scheme, use is made of wave reflection and transmission matrices, making it possible for all computations to proceed in the direction of wave propagation so that the results are always stable numerically. Furthermore, the required computing time is greatly reduced in comparison with that of an all-encompassing finite-element analysis for the entire structure. Application of the new method is illustrated by examples.