Vibration mitigation using finite graded metamaterials: a rod with attached local resonators

Abstract

The paper concerns the analysis of vibration transmission and mitigation in a finite rod performing axial vibrations with attached multiple damped resonators. The resonators are not assumed to be periodically spaced or to have similar dynamic properties. The behaviour is analysed in terms of waves travelling between the discontinuities. Reflection and transmission coefficients of the rod with one local resonator attached are utilised in the analysis. Values of these coefficients for each of the attached resonators, along with the boundary conditions and field transfer function, are used to derive analytical expressions for the displacement of the finite rod at any point. Once the displacement is determined, the transmission loss is calculated. Moreover, the eigenfrequencies of the graded metamaterial finite rod are identified using the phase closure principle. The influence of the mass, the damping and the location of the resonators on the vibration attenuation are investigated. Numerical examples are presented. Good agreement is noted between the results of the analytical studies and finite element simulations in COMSOL. It is shown that the graded metamaterial rod can provide better vibration attenuation as compared to the conventional periodic metamaterial.