## 1 Introduction
[21] investigated broadband vibration mitigation properties of ABH beams equipped with four types of dampers. [PERSON] et al. [22] designed an ultralight phononic beam composed of a complex lattice of ABH to achieve broad low-frequency bandgaps (BGs). [PERSON] et al. [23] introduced ethylene vinyl acetate material into a composite ABH beam to investigate the low-frequency elastic wave attenuation properties. [PERSON] et al. [24] designed a lightweight and high-stiffness beam insulator embedded in a graded double-leaf ABH for vibration reduction in the low-frequency range. In addition, the application of ABH structure is greatly affected by the spatial distribution position. To avoid the limitation of space size on the application of ABH, scholars have proposed spiral ABH structures, such as the Archimedean spiral configuration [25] and helix-acoustic configuration [26]. Since viscoelastic materials are added to the ABH structure to achieve vibration reduction properties, the fractional calculus theory can describe the properties of viscoelastic materials better than the integer theory [27; 28]. For now, there are many research results of uniform beam and non-uniform beam structures based on the fractional calculus theory. [PERSON] and [PERSON] [29] investigated the linear and nonlinear vibration analysis of viscoelastic Timoshenko nanobeams based on fractional calculus theory. [PERSON] et al. studied the vibration problems of viscoelastic beams [27] and viscoelastic sandwich beams [28] based on the fractional calculus theory. However, to the authors' knowledge, there is limited research on dynamic models of viscoelastic ABH beams described by fractional-order theory. Current research on fractional-order models for ABH structures primarily focuses on incorporating fractional-order boundary conditions and fractional-order acoustic fields [30], rather than introducing fractional calculus theory based on perspective constitutive modeling or ABH composite structure properties. Consequently, a fractional-order vibration model of the ABH composite double-layer beam is studied and established in this paper. Moreover, the vibration analysis methods for the ABH structure mentioned above are incapable of analyzing the vibration characteristics of the fractional-order ABH beam model. To derive the steady-state solution for the vibration of a fractional-order non-uniform beam, the wave method is introduced within the context of this paper. The theoretical framework of the wave method is extended, and a recursive formulation is derived to characterize the vibrational behavior of such structures. Power-law ABHs might exhibit even narrower bandwidths. The power-law profile might be optimized for a specific frequency, leading to poor performance outside that range [14]. Furthermore, from a mathematical perspective, the Mittag-Leffler (ML) function, though lesser-known, exhibits broad variational flexibility and serves as a generalized function. Owing to its parsimonious parameterization, diverse curve morphologies can be generated through parametric optimization, thereby enabling efficient identification of configurations that enhance the vibration attenuation performance of ABH structures. Therefore, a generalized ABH shape function is proposed, namely the two-parameter ML function, to expand the bandwidth range for vibration reduction and enhance the energy dissipation properties of ABH structures with viscoelastic layers. It is noteworthy that the proposed shape function exhibits generalized characteristics, allowing the vibration damping performance of the ABH structure to be optimized through parameter tuning. Furthermore, compared to power-law functions (m = 2 and 4), the ML-shaped function achieves superior vibration attenuation with reduced material consumption. Due to the significant impact of ML parameters on the vibration reduction characteristics of ABH structures, we introduced the Particle Swarm Optimization (PSO) algorithm to obtain the optimal parameters for achieving the best vibration reduction characteristics.