## 5 Results Discussion  
### The Results and Analysis of the Fractional Model of ML-ABH Structure  
Previous studies have all been based on analyses conducted using integer-order models. However, the properties of viscoelastic materials are actually closely related to the excitation frequency. To obtain a more accurate understanding of the impact of viscoelastic materials on structural energy transmission characteristics, we have introduced the fractional-order Kelvin model for analysis. The fractional-order parameter \(\alpha\) characterizes the viscoelastic properties of the material, with a higher order indicating stronger viscoelastic performance. Firstly, we present the total reflection parameter \(\mid R_{01}\mid\) of the fractional-order ABH structure, along with the transmission and reflection matrices \(\mathbf{T}^{+}\), \(\mathbf{T}^{-}\), \(\mathbf{R}^{+}\), and \(\mathbf{R}^{-}\) under two sets of parameters. It is noted here that the total reflection parameter \(\mid R_{01}\mid\) is the parameter of the reflection matrix \(R\) generated when no other structure is connected to the right end of the ABH structure. The transmission and reflection matrices \(\mathbf{T}^{+}\), \(\mathbf{T}^{-}\), \(\mathbf{R}^{+}\), and \(\mathbf{R}^{-}\) are produced when other structures are connected to the right end of the ABH structure. The parameters selected for the ML shape function are \(\gamma\) = 0.1, \(\zeta\) = 0.0002, and \(a\) = 5 \(\times\) 10\({}^{-6}\). The rationale for selecting these parameters is to ensure that the ML shape function and the power-law shape functions (quadratic and quartic curves) are distinctly differentiable, such that the three shape function curves yield pronounced differences in the calculated reflection parameter \(\mid R_{01}\mid\). The ABH structure shape function can be obtained from the above parameters, as shown in Figure 11; the blue curve is the ML function, the green curve is a quartic curve, and the red curve is a quadratic curve. When the three ABH-shaped functions are consistent at the beginning and end, compared to quadratic and quartic curves, the ML curve can decay faster and smoothly transition to the final value. Compared to quadratic and quartic curves, the ML curve exhibits faster decay and a smoother transition to its final value when the three ABH-shaped functions are consistent at both ends. The variation curves of the reflection matrix element \(\mid R_{01}\mid\) obtained from the three shape functions with respect to frequency \(\omega\) are shown in Figure 11b when \(\alpha\) is 0.5, \(h_{0}\) is 0.7 mm, and \(x_{0}\) is 0. The \(\mid R_{01}\mid\) curves all exhibit periodic decay, but the values  
Figure 11: (a) Comparison of three kinds of ABH shape function curves; (b) reflection matrix parameter \(\mid R_{01}\mid\) of ABH beam. corresponding to the ML shape function are smaller, and they decay faster, which is more beneficial for structural vibration reduction. In addition, the relationship curve between \(\mid R_{01}\mid\) and \(\omega\) under different film thicknesses and different fractional orders was studied, as shown in Figure 12. Since the truncation thickness is 8.2554 \(\times\) 10\({}^{-4}\) mm, we select two film covering thicknesses \(h_{0}\) of 0.01 mm and 0.7 mm, which are, respectively, smaller and larger than the truncation thickness, as the research parameters. The results show that the fractional order of the model and the thickness of the membrane have significant effects on the structural vibration reduction. The greater the \(\alpha\), the greater the attenuation of \(\mid R_{01}\mid\), which also means that the vibration energy is continuously dissipated in the reflection process. This is consistent with the actual implementation. The fractional order reflects the dissipation performance of the coated material. The greater the \(\alpha\), the stronger the dissipation performance of the material, that is, the stronger the damping performance.