Computing the complex wave and dynamic behavior of one-dimensional phononic systems using a state-space formulation

Inspired by advances in photonics, band structures have been used to investigate the wave behavior along phononic structures as well as their singular properties such as band gaps, defects and topological modes. The most commonly used method for computing band structures is the plane wave expansion, but it cannot be used to compute the forced response of a finite structure with boundary conditions and applied loads. To obtain such dynamic response, it is usually necessary to employ a full finite element model of the whole structure or spectral models that take advantage of the periodicity. This paper proposes a new spectral approach, based on a Riccati differential equation with the impedance as variable, to compute the transfer matrix of one-dimensional phononic systems with arbitrary geometric and material profiles. With this formulation, not only the complex band structure can be computed, but also the forced response. Results for ducts, rods and beams are presented and validated with the extended plane wave expansion and scaled spectral element methods. The proposed approach can also be applied to any one-dimensional periodic structures such as Levy plates, cylindrical shells and photonic systems. It can also be used to investigate with high accuracy and efficient computational cost the physical behavior of generic structures with non-slowly varying properties, gradient-index systems and cloaking devices.