Real wave vectors for dynamic analysis of periodic structures

A modified version of the traditional wave vector computational scheme for the dynamic analysis of long undamped periodic structures is presented. First, the consistency of the complex wave vector mathematical formulation is discussed, placing particular emphasis on the real or complex nature of the resulting characteristic equation from which the natural frequencies are derived. It is shown that the rearrangement in terms of complex waves entering the domain, devised to overcome ill-conditioning arising in the transfer matrix formulation, entails as a side effect an ill-posed problem, as it leads to a complex characteristic equation in a real unknown. Next, the proposed approach is described. It provides for transformation of frequency-dependent real transfer matrices for state vectors to real transfer matrices for wave vectors, thus avoiding an ill-conditioned and ill-posed problem. Finally, applications to mono-coupled and bi-coupled structures are illustrated, aiming at comparing the proposed method with the transfer matrix and complex wave vector approaches.