An exact and explicit formulation for acoustic black hole beam–multibody systems: dynamic stiffness matrices and mode counts

This paper presents an exact and explicit dynamic stiffness (DS) formulation for analyzing the vibration of acoustic black hole (ABH) beam–multibody systems. The proposed formulation first addresses both longitudinal and flexural vibrations for ABH elements, which require merely one DS element to exactly capture their full dynamics. For complex structures, a novel DS element, rigid body–beam–rigid body (R–B–R) element, is developed. For the first time, exact mode counts of ABH beams are derived to enable the implementation of the Wittrick-Williams algorithm, thus facilitating efficient natural frequency determination, enriched by an inverse iterative method with a homotopy algorithm for systems with damping. Free vibration analyses of uniform/ABH beam–multibody systems are performed to showcase the accuracy and the efficiency of the proposed formulation, in comparison with finite element results. Vibration analyses of a beam cell with an ABH core and N-cell compound beam illustrate the capability and the efficacy of the proposed technique in dealing with complex structures. Leveraging the exact and explicit nature of the DS formulation, this work provides benchmark solutions for ABH–multibody systems, while offering a powerful simulation tool for ABH–induced applications such as noise/vibration control, energy harvesting, and metamaterial design.