Exact Analytical Solutions for Static and Dynamic Analyses of Cyclic Bi-periodic Structures

The U-transformation method can be used to uncouple linear simultaneous equations with cyclic bi-periodic properties. For a system consisting of masses and springs with cyclic bi-periodicity, this paper presents a procedure demonstrating how to apply the U-transformation technique twice in order to uncouple the governing equations of the mass-spring system and obtain exact analytical solutions. The paper also produces the static, natural vibration and forced vibration analyses for the system. Exact explicit solutions are given for these analyses. The availability of these exact solutions not only helps the checking of the convergence and accuracy of numerical solutions, but also provides a basis for the optimum design of these types of structures.