Exact Analysis Of Bi-periodic Structures

By using the U-transformation method, it is possible to uncouple linear simultaneous equations, either algebraic or differential, with cyclic periodicity. This text presents a procedure for applying the U-transformation technique twice to uncouple the two sets of unknown variables in a doubly periodic structure to achieve an analytical exact solution. Explicit exact solutions for the static and dynamic analyses for certain engineering structures with doubly periodic properties - such as a continuous truss with any number of spans, cable network and grillwork on supports with periodicity, and grillwork with periodic stiffening members or equidistant line supports - can be found in the volume. The availability of these exact solutions not only helps with the checking of the convergence and accuracy of numerical solutions, but also provides a basis for optimization design for these types of structures. The study of the force vibration and mode shape of periodic systems with non-linear disorder is yet another research area which has attained considerable success by the U-transformation method. This work illustrates the analytical approach and procedure for the problems of localization of the mode shape of nearly periodic systems together with the results.