Spectral Analysis and Numerical Simulation of 1-d Networks

Abstract

In this and in the following section we shall consider the problem of determining the frequency spectrum and the corresponding modes of a given network of elastic strings or elastic beams. This problem is of great importance for purposes of obtaining Fourier series expansions (Galerkin approximations) of solutions to the dynamic equations of motion and also for certain approaches to the theories of control and optimal design. There is a huge literature devoted to this problem, within which we can identify two groups of papers. The first group, comprising the majority of these papers, is concerned with a finite element approximation of the underlying system and with the generalized eigenvalue problem associated with that approximation. The latter problem is then solved using a suitable numerical procedure, such as the shifted Q-R algorithm or Jacobi’s method, if the dimension of the approximating space is comparatively small, or by other methods such as the Lancos algorithm, the Raleigh-Ritz method, etc., if one is interested in obtaining a particular subset of eigenvalues. For repetitive structures, which are typical in network problems, yet another method has been employed with great success, namely, the method of slicing the spectrum, which is based on Sylvester’s theorem of invariance of the number of positive, vanishing and negative eigenvalues under similarity transformations. This method has been developed for repetitive structures by Wittrick and Williams [110], [111] and others [103]. The underlying idea is that of substructuring, that is, of dividing the structure into smaller identical pieces. The analysis is then performed on the level of the substructure and the results are read into the master structure in an appropriate way.