Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect

Flexural waves in beams and plates slow down if their thickness decreases. Such property was used in the past for establishing the theory of acoustic black holes (ABH). The aim of the present paper is to establish reliable numerical and experimental approaches for designing, modelling and manufacturing an effective passive vibration damper using the ABH effect. The effectiveness of such vibration absorbers increases with frequency. Initially, the dynamic behaviour of an Euler–Bernoulli beam is expressed using the Impedance Method, which in turn leads to a Riccati equation for the beam impedance. This equation is numerically integrated using an adaptive Runge–Kutta–Fehlberg method, yielding the frequency- and spatially-dependent impedance matrix of the beam, from which the reflection matrix is obtained. Moreover, the mathematical model can be extended to incorporate an absorbing film that assists for reducing reflected waves from the truncated edge. Therefore, the influence of the geometrical and material characteristics of the absorbing film is then studied and an optimal configuration of these parameters is proposed. An experiment consisting of an elliptical plate with a pit of power-law profile placed in one of its foci is presented. The elliptical shape of the plate induces a complete focalisation of the waves towards ABH in case they are generated in the other focus. Consequently, the derived 1-D method for an Euler–Bernoulli beam can be used as a phenomenological model assisting for better understanding the complex processes in 2-D elliptical structure. Finally, both, numerical simulations and experimental measurements show significant reduction of vibration levels.