Exact Method for Static and Natural Vibration Analyses of Bi-Periodic Structures

Abstract

The U-transformation technique has been applied successfully to the analysis of periodic structures and nearly periodic structures. In this study the technique will be extended to the analysis of bi-periodic structures under static loading or natural vibration, since it is possible to uncouple the governing equation by applying the U-transformation twice. To explain the method used in this paper, a simple cyclic system with bi-periodicity is considered first. It helps to demonstrate the procedures for uncoupling the static equilibrium equation to obtain the closed form solution for displacement and the natural vibration equation to obtain the natural frequencies and modes. Then a continuous beam with equidistant rigid and elastic supports (a structure with bi-periodicity), subjected to a concentrated load, is studied and the generalized analytical solution is derived. Some numerical results are also given. Though not illustrated, it is obvious that the arbitrary static loading condition can be dealt with in the same manner.