Dynamic stiffness method for exact modelling of acoustics black hole beams using Meijer G-functions

The acoustics black hole (ABH) effect shows promising potential for wave manipulation and vibration control. An ABH structure features a gradual reduction of the phase velocity of flexural waves alongside wave compression and energy accumulation when entering the tapered ABH portion where the thickness is tailored according to a power-law (with power index m no less than 2). The corresponding non-uniform wavelength distribution over the ABH structure poses great challenges to conventional modelling methods. To alleviate the problem, this paper proposes an exact dynamic stiffness method for modelling ABH beams with arbitrary exponent equal to or greater than 2 under the framework of Euler-Bernoulli beam theory. For ABH with m > 2, a change of variable and the Mellin integral transformation are conducted to derive the integral representations of the exact solution using Meijer G-functions. The solution for the case with m = 2 is also derived for completeness. Then the dynamic stiffness matrix is formulated through symbolic operation. The Wittrick-Williams (WW) algorithm is revamped to cope with the ABH-specific requirement. Numerical examples are given to validate the solution in integral form, the dynamic stiffness matrix, and the efficacy of the improved WW algorithm. The clear advantage of the accurate integral representations over series representations is justified in the higher frequency range. Covering all ABH-relevant scenarios (with m ≥ 2), the exact modelling framework established in this work offers a powerful tool for the modelling and investigation of more complex structures which are built upon ABH beam elements.