Wave propagation in three-coupled periodic structures

Free wave propagation patterns for general three-coupled periodic structures are investigated by means of the transfer matrix approach. It is shown that an exhaustive description of the propagation domains requires spaces that are stratified in homogeneous regions, whose dimension is given by the number of invariants of the transfer matrix characteristic equation and whose boundaries are represented by codimension-one manifolds. Three types of three-coupled periodic mechanical models characterized by constitutive elastic and/or inertial coupling between mono- and bi-coupled dynamics, namely pipes, thin-walled beams and truss beams, are considered. From the design standpoint, an adequate representation of the propagation domains pattern is obtained through a nonlinear mapping from the space of the invariants to the physical parameters plane. The analyzed models give rise to equations of motion where the three-coupled nature stems from the coupling between transversal (bi-coupled) and longitudinal (mono-coupled) dynamics for the pipes and truss beams, whilst coupling occurs between transversal and torsional (mono-coupled) dynamics when it comes to the thin-walled beams. A mechanical interpretation associated with the bounding frequencies of the propagation regions is given and the evolution of the propagation properties when coupling parameters tend to vanish is discussed.