Mass-spring model for acoustic metamaterials consisting of a compact linear periodic array of dead-end resonators

This paper presents a mass-spring model to predict the normal incidence acoustic response of a metamaterial composed of a compact linear periodic array of dead-end resonators. The dead-end resonators considered are ring-shaped Helmholtz resonators. The model is based on a mass-spring analogy and considers the thermoviscous losses in the metamaterial following an effective fluid approach. A matrix equation of acoustic motion is derived for the finite case of N-periodic arrays. Under external excitation, its direct solution predicts the sound absorption coefficient and transmission loss. Under the homogeneous case, the solution of its associated eigenvalue problem predicts the acoustic eigenfrequencies and mode shapes. The dispersion relation is also solved to predict the beginning of the first stopband, and a low frequency approximation allows development of a formula to estimate the first eigenfrequency. The results show that the system with N degrees of freedom has three stopbands over the frequency range studied, with zero sound absorption and transmission. The model also helps to understand how the acoustic dissipation, at a given resonant frequency, is affected by the position of the acoustic velocity nodes (eigenmodes) in the geometry of the metamaterial. Prototypes are designed, manufactured, and tested in an impedance tube to validate the model.