## 5 Results Discussion
### Analysis of PSO Algorithm Optimization Results  
Based on the above two results, it can be observed that the three parameters of the ML shape function have a significant impact on the vibration reduction performance of the structure. Therefore, we introduce the PSO algorithm to optimize these parameters in order to achieve optimal vibration reduction performance. The influence of film coating thickness, coating length, and fractional-order parameter on optimization outcomes is investigated. The objective of this study is to analyze the influence of two coating thicknesses (one greater than and the other less than the ABH truncation thickness) on the minimum reflection element \(\mid\!R_{01}\!\mid\). In addition, the study investigates the effects of three coating lengths (long, medium, and short) and fractional derivative orders (large and small) on the minimum reflection element \(\mid\!R_{01}\!\mid\). Therefore, we have investigated the optimal ML parameters and fitness functions under different coating thicknesses (\(h_{0}\) = 0.01 mm and 0.7 mm), lengths (0.2 L, 0.5 L, and 0.8 L), and fractional orders (\(\alpha\) = 0.2 and 0.8), respectively. Figure 14: Comparison of transmission and reflection matrices under the value of fractional order \(\alpha\) of 0.2, 0.4, 0.6 and 0.8 when the h\({}_{0}\) is 0.7 mm: (**a**) the elements of matrix \(\mathbf{T}^{+}\); (**b**) the elements of matrix \(\mathbf{R}^{+}\). Figure 13: Comparison of transmission and reflection matrices under the value of fractional order \(\alpha\) of 0.2, 0.4, 0.6 and 0.8 when the \(h_{0}\) is 0.01 mm: (**a**) the elements of matrix \(\mathbf{T}^{+}\); (**b**) the elements of matrix \(\mathbf{R}^{+}\). The PSO algorithm parameters are as follows: the population size is 300, the number of iterations is 30, the parameter variation range is [0.00001, 1], and the objective function is \(f=\min(\,\mid\,R_{01}(\alpha)\,\mid\,)\). In this case, the right end of the ABH structure is not connected to any structure. The optimization results of the PSO algorithm are shown in Table 2.  
According to the results in Table 2, it can be concluded that the optimized \(\mid\,R_{01}\,\mid\,\) minimum value is the smallest when the coating length is 0.8 L, indicating that the longer the coating, the more significant its energy dissipation performance. Under the same film thickness and fractional order, the optimized parameters obtained are similar, with the coefficient \(a\) exhibiting a more significant difference when \(h_{0}\) is 0.01 mm. This is mainly due to the long reciprocating motion of the vibration wave in the viscoelastic material layer, which dissipates a large amount of energy. The parameters can achieve the minimum value min = 0.0086 when \(h_{0}\) is 0.01 mm, \(\alpha\) is 0.8, and \(x_{0}\) is 0.2 L, at which point the energy inside the structure is basically dissipated. There is a special case in the table where \(h_{0}\) is 0.7 mm and \(\alpha\) is 0.8, where the minimum value of \(\mid\,R_{01}\,\mid\,\) is large and the energy dissipation result is poor. This may be due to the PSO algorithm becoming stuck in a local optimal solution. Under such parameters, the structural dissipation performance is weak, which is not conducive to structural vibration reduction. With these optimized parameters, the structural dissipation efficiency turns out to be insufficient, impeding effective vibration reduction. This distinction highlights one of the key differences between fractional-order and integer-order models. In subsequent research, we can use other optimization algorithms or PSO improvement algorithms to obtain more accurate optimal ML shape function parameters in order to achieve good structural energy dissipation performance. Based on the results in Table 2, we selected the aforementioned optimal parameters to determine the ABH shape function and analyzed the influence of fractional order, film thickness, and length on the structural vibration reduction performance under these parameters.