Response of Periodic Structures by the 2-Transform Method

Periodic structures are defined as structures consisting of identical substructures connected to each other in identical manner. The response of periodic structures to harmonic excitation can be described by a matrix difference equation. The solution of the matrix difference equation can be obtained by the Z-transform method and it yields the response to both end conditions and external excitations. The method developed necessitates the eigenvalues of the transfer matrix for a typical substructure, so that the procedure is capable of analyzing a periodic structure with the same computational effort necessary to analyze a single substructure. Added ad- vantage is derived from the fact that the method does not require the eigenvectors of the transfer matrix. PERIODIC structure is a structure consisting of iden- tical substructures connected to each other in an identical manner. Many physical systems in natural state possess properties that are periodic in space. A typical example is the single crystal in which identically arranged atoms form an infinite or semi-infinite lattice. Many other physical systems are built in the form of periodic structures by design, the object being to reduce cost or to save time or both. For example, a segment of an aircraft fuselage is often made of identical bays connected by identical circumferential frames. Even the circumferential frames in an aircraft fuselage can be looked upon as periodic structures. Monorail tracks, or pontoon bridges can be regarded as examples of periodic structures. Continuous systems with periodic properties can often be treated as discrete periodic systems. The mathematical formulation for discrete periodic systems can be effected by means of the finite element method. The for- mulation consists of a set of simultaneous ordinary dif- ferential equations. There is no single approach to the mathematical analysis of periodic structures. Indeed, the approach depends very much on the nature of the substructure. In particular, it is possible to distinguish between a scalar approach and a matrix ap- proach to the problem formulation. This is related directly to the order of the substructure. More specifically, the scalar approach is suitable for the case in which the substructure represents a single-degree- of-freedom system, whereas the matrix approach is recommended for the case in which the substructure represents a multi-degree-o f-freedom system. The scalar analysis of periodic structures is in a relatively advanced stage of development. Much progress has been made in the matrix analysis of periodic structures but much progress remains to be made. In the following we present a survey of selected works on the subject. A classical example of a simple periodic structure is the one- dimensional array of identical linear springs. Hence, the substructure is the simple harmonic oscillator. Using such a mathematical model, Brillouin' has studied the propagation of harmonic waves in discrete crystal lattices. The same problem is discussed in the text by Morse and Ingard2 who present solutions to both sinusoidal wave motion and tran-