Admissible shape functions to characterize acoustic black hole effect based on semi-analytical model

Acoustic Black Holes (ABHs) are wedge-shaped structures with power-law profiles and have been increasingly investigated for vibration control because of its energy concentration effect. In an ideal scenario, the phase velocities of vibration gradually retard to zero and the energy of flexural vibration wave can concentrate in the vicinity of the tip edge thanks to its power-law profile. Variation of thickness, however, brings about difficulties for theoretical analysis in which most existing models need special methods to deal with the diminishing thickness. Recently, a growing number of publications focus on semi-analytical method to solve vibrating response in structures with polynomial profile. Unfortunately, such an admissible shape function to describe the displacement field in polynomial-profiled structures is not easy to find out. The aim of this paper is to explore available shape functions in one-dimensional ABH beams. At the first, the equations of motion are derived based on energy expressions and Euler-Lagrange equation. Then, various shape functions are adopted to characterize the vibrating performance in beams with embedded ABH features, and the displacement fields are decomposed of a set of basis functions analogous to wavelet transform methodology. The results are compared with the Finite Element Method (FEM). Numerical simulations reveal that the smoothness and the decay speed of shape functions affect the complexity of numerical treatment and the accuracy of semi-analytical model greatly. Also, it is shown that the special attention on the two ends of the ABH beam should be addressed when these shape functions are applied. The present work is a supplement to semi-analytical theory allowing the embodiment of vibration control and energy harvesting because of its energy-based feature.