Vibration isolation by a periodic beam with embedded acoustic black holes based on a hybrid dynamics method

The acoustic black hole (ABH) has been proved to reduce broadband vibration response in beams and plates. While the traditional analytical and semi-analytical methods can only deal with the response of simple ABH structures; for complex ABH structures, the numerical methods such as the finite element method (FEM) have to be resorted and these methods are too often time-consuming. In this work, the vibration isolation by a beam structure embedded with periodic embedded symmetric ABHs is investigated. The vibration transmission of a single embedded symmetric ABH beam unit is first studied by the Riccati transfer matrix method (RTMM). A comparative analysis of the convergence speed and the computation efficiency of the unit by the RTMM and the FEM demonstrates the computation time using the RTMM increases linearly with the number of segments while that using the FEM increases exponentially and quickly exceeds the former as the number of segments increases. The computation time is consistent with the computational complexity associated with their respective algorithms. A hybrid dynamics method (HDM) is then proposed to derive for the vibration transmission solution of the beam with single and multiple periodically embedded symmetric ABHs. A comparison of the responses with those calculated by the RTMM demonstrates that the proposed HDM provides an efficient tool for solving the vibration response of the finite beam with periodic embedded ABHs, leading to much improved computational efficiency with ensured numerical stability and accuracy. This advantage in computation efficiency becomes even more obvious for larger structures when the number of the ABH units increased considerably. The proposed hybrid dynamic approach provides a basis for solving the vibration transmission/isolation problems of more complex ABH structures.