Flexural wave band gaps and vibration reduction properties of a locally resonant stiffened plate

A locally resonant stiffened plate is constructed by attaching a two-dimensional periodic array of spring-mass resonators to a traditional periodic stiffened plate. A method based on the finite element method and Bloch theorem is presented for calculating the flexural wave dispersion relation and forced vibration response of the proposed locally resonant stiffened plate. The method is validated by comparing the predictions with simulations by FEM software COMSOL. The effects of the spring-stiffness and mass ratio of local resonators on the flexural wave band gap and vibration reduction performance are analysed, which can facilitate the design of the locally resonant stiffened plate for vibration-reduction applications in engineering. The main findings of this work are as follows.1) The local resonator can have a significant effect on the propagation of flexural wave in stiffened plate. On the one hand, the local resonator is able to create a low-frequency local resonance band gap; on the other hand, it can enhance the high-frequency Bragg band gap. Within the band gap frequency range, the vibration of the locally resonant stiffened plate can be reduced remarkably.2) The spring-stiffness of local resonators shows a notable influence on the band gap and vibration reduction performance of the locally resonant stiffened plate. As the spring-stiffness gradually increases, the nature frequency of local resonator is gradually tuned to higher frequency, and the phenomenon of band-gap transition and band-gap near-coupling may arise. Under the near-coupling condition, the pass band between two band gaps turns narrow, and it seems that these two band gaps form a super-wide pseudo-gap (within which only a very narrow pass band exists). This behaviour is of great interest for the broad band vibration reduction applications. Moreover, the complete band gap will disappear if the nature frequency of local resonator is tuned to a higher value than a threshold frequency, which is dependent on the geometrical and material parameters.3) The influence of the additional mass ratio of local resonator on the band gap behavior is highly relevant to the nature frequency of local resonator. If the nature frequency of resonator is lower than the band-gap near-coupling frequency, both the local resonance band gap and Bragg band gap are broadened with increasing the additional mass ratio of resonator. When the nature frequency of resonator is close to the band-gap near-coupling frequency, the phenomenon of band-gap near coupling and band-gap transition may arise or disappear as the additional mass ratio of resonator gradually changes. When the nature frequency of resonator is higher than the band-gap near-coupling frequency, on the one hand, the lower frequency band gap will disappear rapidly with increasing the mass ratio of resonator. However, it will be present again if the mass ratio of resonator increases up to a large enough value. On the other hand, the higher frequency band gap is broadened with increasing the mass ratio, but if the mass ratio is tuned to a larger value than a specific value, this band gap will transform from local resonance band gap to Bragg band gap, and the normalized gap width of this band gap will be narrowed with increasing the mass ratio.