## 4 New Shape Function of ABH Structures  
The ABH structure featuring quadratic and quartic functions has inherent drawbacks, and its energy dissipation and aggregation properties need to be enhanced. Therefore, we introduced a more generalized function, known as the two-parameter ML function, to describe the shape of ABH structure:  
\[ \begin{split} E_{\gamma,\xi}(x)&=\sum_{k=0}^{\infty} \frac{x^{k}}{\Gamma(\gamma k+\xi)}\\ h(x)&=a\frac{1}{E_{\gamma,\xi}(x)}\end{split}  \tag{16}\]  
where \(\gamma\) and \(\xi\) are the two covariates of the ML function; \(a\) is the scaling factor. The ML function can decay quickly first and then slowly reach a stable value, making it easier to truncate. The ML function can be adjusted to simulate other functions such as quadratic and quartic curves by adjusting the parameters, which is a more generalized ABH shape function. By employing a two-parameter ML function, the trend of the curve can be altered through these two characteristic parameters, with the aim of improving the energy dissipation and aggregation performance of the current ABH structure. The influence of the two parameters \(\gamma\) and \(\xi\) on the curve results is shown in Figure 8 when the length of the ABH part \(x=0.2\) m. The parameter \(\gamma\) alters the position of the curve's endpoint, while the parameter \(\xi\) changes the position of the starting point. Figure 8c illustrates the ML-shaped function curves derived from three parameter sets. The results demonstrate that parameter tuning can modulate the variation trend of the curves, enabling them to approximate distinct functions. Figure 7: The reflection matrix parameter \(\mid\!R_{01}\!\mid\) of the ABH beam with 0.7 mm coated layer obtained by the wave method and the approach from literature [31], with two significant discrepancies in the comparative results indicated by green dashed lines. Figure 9 illustrates the relationship between \(h(x)\) and the ML function parameters of \(\gamma\) and \(\zeta\), as well as the length of the ABH part \(x\) when \(a=1\). It can be seen from the results that the smaller the \(\zeta\) and the larger the \(\gamma\), the more drastic the change in \(h(x)\). Figure 8: The variation curve of ML functions with respect to parameters: (**a**) \(\gamma\) = 0.2, 0.4, 0.6 and 0.8 with \(\zeta\) = 0.005; (**b**) \(\zeta\) = 0.002, 0.004, 0.006 and 0.008 with \(\gamma\) = 0.1; (**c**) ML-shaped function tuning chart. Figure 9: (**a**) The surface of \(h(x)\) changing with \(x\) and \(\gamma\) with \(\zeta\) = 0.1, 0.5 and 0.8. (**b**) The surface of \(h(x)\) changing with \(x\) and \(\gamma\) with \(\zeta\) = 0.1, 0.5 and 0.8. The effects of the two parameters \(\gamma\) and \(\xi\) are investigated on the \(\mid\!R_{01}\!\mid\) on the basis of the integer solution model, where \(\alpha\) is 1 and the thickness of the cladding film \(h_{0}\) is 0.7 mm. The rationale for setting \(\alpha=1\) is to eliminate the influence of fractional-order dynamics on the results. Figure 10 illustrates the relationship between \(\mid\!R_{01}\!\mid\) and the ML function parameters of \(\gamma\) and \(\xi\), as well as the external excitation frequency \(\omega\). In Figure 10a,b, the external excitation frequency is set to the range [0, 2000] Hz, with parameter \(\gamma\) varying within [0.01, 1] and parameter \(\xi\) spanning [0.001, 0.5]. The results in Figure 10a indicate that the surface plot exhibits a large concave area. When \(\gamma\) is fixed, \(\mid\!R_{01}\!\mid\) exhibits a trend of first decreasing and then increasing with the rise in \(\omega\). At frequencies below approximately 820 Hz, \(\mid\!R_{01}\!\mid\) decreases as \(\gamma\) decreases. However, at frequencies above 820 Hz, \(\mid\!R_{01}\!\mid\) initially decreases and subsequently increases with decreasing \(\gamma\). When the frequency exceeds approximately 4150 Hz, \(\mid\!R_{01}\!\mid\) first increases and then decreases as \(\gamma\) decreases. Notably, \(\mid\!R_{01}\!\mid\) reaches its minimum value at \(\omega\approx 880\) Hz and \(\gamma\approx 0.5\).