Vibration Analysis of Viscoelastic Acoustic Black Hole Beams with a New Shape Function

In this study, a Mittag-Leffler (ML) function is introduced as a novel shape function for the fully damped acoustic black hole (ABH) beam, which demonstrates superior vibration damping characteristics compared to traditional shape functions. By incorporating viscoelastic materials, a more generalized fractional-order vibration model is established through the application of fractional-order calculus theory. To analyze the vibration damping properties of fully damped ABH structures, the wave method-a semi-analytical approach-was employed to derive the stationary dynamic response of the structural system. To enhance model performance, the Particle Swarm Optimization (PSO) technique was applied for systematic parameter adjustment in ML shape function. The results indicate that the ML-based shape function achieves enhanced vibration suppression through parameter optimization. The damping characteristics of the viscoelastic material are characterized by the fractional-order component within the model, in which a higher fractional order corresponds to improved damping performance and superior vibration attenuation. This work provides a robust theoretical foundation for future research on ABH structures.