A semi-analytical method for acoustic black hole cantilever beams

In this paper, a semi-analytical method for acoustic black hole cantilever beams is presented. Based on Euler-Bernoulli beam theory, the governing equation of the free vibration of the variable section beam is derived. The finite difference method (FDM) is used to solve the equation. The mode shape and nature frequency of the one-dimensional acoustic black hole (ABH) cantilever beam is obtained, and the error between the result and the finite element result is within 1%, which proves that the method has high precision. The effects of various parameters of ABH beam on natural frequency and mode shape are studied. The results show that as the exponent m increases, the natural frequency decreases. The higher the order, the lower the percentage. The thickness of the free end hn increases, the natural frequency increases, and the higher the order, the greater the increase. Moreover, m and hn have a certain influence on the node and amplitude of the mode. The semi-analytical method in this paper is simple and efficient, with good accuracy and convergence. The parameterization is convenient, and its good programmability is very convenient, which means that it provides a simple and efficient new method for parameterization and optimization design of the acoustic black hole beams.