Propagation Properties of Three-Coupled Periodic Mechanical Systems

General three-coupled periodic systems are dealt with by means of transfer matrices of single units. The solutions of the associated characteristic equation are discussed in terms of invariant quantities by exploiting the well-known reversibility of its coefficients. An exhaustive description of the free wave propagation patterns is given on the invariants’ space where propagation domains with qualitatively different character are identified. Afterwards, two three-coupled periodic mechanical models are considered: pipes and truss beams. A nonlinear mapping from the invariants’ space to the physical parameters plane provides with a concise representation of the pattern of the propagation domains. A mechanical interpretation associated with the boundaries of these regions is given. The analyzed models give rise to equations of motion where the three-coupled nature stems from the coupling between longitudinal (mono-coupled) and transversal (bi-coupled) dynamics. The evolution of the propagation properties when the coupling parameters tend to vanish is eventually discussed.