Frequency graded 1D metamaterials: A study on the attenuation bands

Depending on the frequency, waves can either propagate (transmission band) or be attenuated (attenuation band) while travelling through a one-dimensional spring-mass chain with internal resonators. The literature on wave propagation through a 1D mass-in-mass chain is vast and continues to proliferate because of its versatile applicability in condensed matter physics, optics, chemistry, acoustics, and mechanics. However, in all these areas, a uniformly periodic arrangement of identical linear resonating units is normally used which limits the attenuation band to a narrow frequency range. To counter this limitation of linear uniformly periodic metamaterials, the attenuation bandwidth in a one-dimensional finite chain with frequency graded linear internal resonators are investigated in this paper. The result shows that a properly tuned frequency graded arrangement of resonating units can extend the upper part of the attenuation band of 1D metamaterial theoretically up to infinity and also increases the lower part of the attenuation bandwidth by around 40% of an equivalent uniformly periodic metamaterial without increasing the mass. Therefore, the frequency graded metamaterials can be a potential solution towards low frequency and wideband acoustic or vibration insulation. In addition, this paper provides analytical expressions for the attenuation and transmission frequency limits for a periodic mass-in-mass metamaterial and demonstrates the attenuation band is generated by the high absolute value of the effective mass not only due to the negative effective mass