Response of periodic structures

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Virginia Polytechnic Institute and State University

Periodic structures are defined as structures consisting of identical substructures connected to each other in identical manner. In chapter I, the response of periodic structures to harmonic excitation is 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 advantage is derived from the fact that the method does not require the eigenvectors of the transfer matrix.

In chapter II, infinite periodic structures are considered for different types of loading. Furthermore, it is demonstrated that additional savings are possible when the substructure is symmetric.

Chapter III, considers the problem of almost periodic structures. If the system parameters differ slightly from one substructure to another, then the structure becomes almost periodic. An efficient method using a perturbation technique to derive the response of an almost periodic structure is presented. The procedure reduces the solution to a sequential application of the basic algorithm for periodic structures as developed in chapter I.

In chapter IV, we consider the undamped response of a periodic structure using a modal analysis technique. The method allows for arbitrary loads and takes full advantage of the periodic properties of the structure.

In chapter V, an attempt is made to simulate certain continuous systems by periodic structures. The algorithm as developed in chapter I is now adapted to the treatment of continuous systems. The method is capable of deriving the response of damped and undamped systems subject to harmonic distributed loads. The length of the substructure can be made arbitrarily small without increasing the computational effort.