Flexural-Torsional Dynamics of Locally Resonant Finite Metamaterial

This paper deals with the flexural-torsional dynamics of finite beams with many spring-mass absorbers periodically attached. Especial emphasis is given to the description of the vibration absorption properties and the bandgap formation. As an example of structures presenting dynamic coupling between bending and torsion, a channel beam is considered. Vlasov´s theory is employed to formulate mathematically the problem. An exact analytical solution is given for the free and forced vibration of the metamaterial beam. The attenuation of the dynamic forced response of the beam, by selecting appropriately the absorbers' parameters, is numerically analysed with the help of the obtained analytical solution. Natural frequencies of the system are used for establishing a criterion to determine the bandgap edge. Finally, an approximate simple procedure is proposed to estimate the bandgap frequency range using only the inertial properties of the beam. This procedure is applicable to arbitrary end conditions.