## 6 Conclusions  
In this paper, a more generalized ABH shape function is proposed, which is employed to improve the energy dissipation characteristics of ABH structures incorporating viscoelastic layers. Compared with traditional power-law shape functions, the ML shape function represents a more generalized function whose parameters can be adjusted to enhance the energy dissipation characteristics of ABH structures. Since fractional-order constitutive models can more accurately capture the properties of viscoelastic materials, a fractional-order viscoelastic ABH structure vibration model was established, which was close to the real vibration results. The current literature on the development of fractional-order models for ABH structures predominantly focuses on fractional-order boundary conditions and acoustic fields. In contrast, the fractional-order ABH beam model proposed in this work is established from the perspectives of constitutive models and the mechanical behavior of composite structures,  
Figure 15: (**a**) A 3D surface plot of \(\mid\!R_{01}\!\mid\) with respect to frequency \(\omega\) and fractional order \(\alpha\) when \(\gamma=0.5694\); \(\xi=0.4231\); \(\alpha=0.0012\) and \(h_{0}=0.01\) mm. ; (**b**) a 3D surface plot of \(\mid\!R_{01}\!\mid\) with respect to frequency \(\omega\) and the film thickness \(h_{0}\) when \(\gamma=0.5694\); \(\xi=0.4231\); \(\alpha=0.0012\) and \(\alpha=0.8\); (**c**) a 3D surface plot of \(\mid\!R_{01}\!\mid\) with respect to frequency \(\omega\) and the film length \(L\)-\(x_{0}\) when \(\gamma=0.5694\); \(\xi=0.4231\); \(a=0.0012\); \(\alpha=0.8\) and \(h_{0}=0.01\) mm. resulting in an inhomogeneous beam formulation. To investigate the energy transmission and reflection characteristics of ML-ABH structures under steady-state conditions, the wave method is introduced. Furthermore, we have extended the theoretical framework of the wave method for analyzing inhomogeneous beam structures. The parameters of the proposed shape function exhibit a pronounced influence on the reflection coefficient (\(\,\left|\,R_{01}\,\right|\,\)) of the ABH structure. To achieve optimal vibration reduction performance of the ABH structures, the PSO algorithm is introduced to optimize these parameters. The effectiveness of the wave method was verified by comparing it with \(\,\left|\,R_{01}\,\right|\,\) in the literature [31]. Research on new shape functions and fractional-order models shows that the thickness and length of the coating, as well as the fractional order of the model, have a significant impact on the energy transmission and reflection characteristics of ML-ABH structures. By optimizing algorithms and selecting ML shape function parameters reasonably, the optimal vibration reduction performance can be achieved. Overall, the higher the fractional order, the stronger the damping characteristics of the structure, thereby enhancing its ability to attenuate vibrations. The thicker the coating of the ABH beam, the higher the external excitation frequency, and the stronger the ability to attenuate vibration. In addition, the longer the length of the film, the more significant the suppression of ABH structure vibration. However, when the thickness of the film is large, scale effects may occur, which may lead to a decrease in vibration damping performance. Parameter selection is not universally governed by maximization or minimization; instead, it requires balanced optimization to achieve optimal performance in ABH structures. Therefore, according to different ABH structural vibration models, it is necessary to choose the appropriate coating length and thickness in order to obtain the best vibration reduction performance. It is worth noting that regardless of the material selection or structural design, ABH structures cannot dissipate all energy and must have some energy reflection. Therefore, we can connect a damping device on the right side of an ABH for secondary energy dissipation, thereby further reducing vibration energy.