A wave and Rayleigh-Ritz method to compute complex dispersion curves in periodic lossy acoustic black holes

In this work we first propose a wave and Rayleigh-Ritz method (WRRM) to analyze the vibrations of periodic systems. It is well-known that one of the main difficulties of the Rayleigh- Ritz method (RRM) is to find suitable basis functions that satisfy the problem boundary conditions. In the WRRM, this is done by expanding the system response as a superposition of the basis of the nullspace of the matrix defining the system periodic conditions. Therefore, the WRRM widens the potential of the RRM and constitutes a computationally efficient alternative to the wave and finite element method (WFEM) for structures requiring very fine meshes. Although the method is of general application, in this paper it is presented for an infinite periodic beam consisting of acoustic black holes (ABHs) cells with damping layers. Periodic ABHs exhibit broadband vibration reduction thanks to bandgap formation at low frequencies and to the ABH effect in the mid-high frequency range. The WRRM allows one to obtain the equations of motion of the periodic ABH beam and to define linear and quadratic eigenvalue problems to respectively obtain the system's real and complex dispersion curves. While the former have been studied at extent, the latter have been barely analyzed for periodic ABH beams or plates. And they are of critical importance given that ABHs are strongly damped systems and that damping layer properties may be frequency-dependent. Therefore, the second contribution of this work is to carefully inspect the role played by the imaginary part of the complex dispersion curves on the functioning of the ABH periodic beam. Its value below and above the ABH cut-on frequency is inspected and a parametric analysis considering variations on the ABH profile and damping layer geometry is carried out. The underlying physics is explained and, finally, a transmissibility analysis on a finite ABH periodic beam with five cells is provided to confirm the mechanisms found for infinite periodic damped ABH beams.