## 2 Analysis of Stiffness and Damping Characteristics of Rubber-Based SMA Composite Dampers
The initiation of martensitic transformation depends on the attainment of a critical transformation stress. Under a sufficiently high excitation amplitude, a thin damping layer accumulates the requisite stress through incremental strain, thereby instigating the martensitic transformation. This is reflected in the force-displacement curve as a progressive increase in displacement corresponding to the stiffness inflection point of the overall structure. The results for the stiffness, damping, and energy dissipation of composite dampers with varying SMA metal module thicknesses and diameters are presented in Table 2.  
Figure 2: Stress-displacement curves corresponding to SMA metal modules with different thickness and diameters: (**a**) different thicknesses; (**b**) different diameters. Additionally, the variation curves for each parameter as a function of the SMA metal module's thickness and diameter are illustrated in Figure 3.  
As illustrated in Figure 3, an increase in the thickness of the SMA metal module results in a corresponding rise in both the stiffness and damping of the shock absorber. The amplitude of these increases also escalates with the increased thickness of the SMA metal module. Owing to the substantial disparity in stiffness and damping between SMA materials and rubber materials, variations in the structural parameters of SMA metal modules significantly influence stiffness and damping to a greater extent than those of rubber vibration-damping components. Figure 3 demonstrates that as the diameter of the SMA metal module increases, the stiffness and damping of the shock absorber gradually diminish, with the rate of decrease tapering off progressively. This phenomenon occurs because larger diameters in SMA metal modules correlate with reduced stiffness. Consequently, under identical displacement conditions, the strain in the SMA metal module is reduced, leading to lower energy dissipation within a single loading and unloading cycle. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**SMA Structural** & **Numerical** & **Stiffness** & **Damping** & **Dissipated** \\
**Parameters** & **Value (mm)** & **(N/mm)** & **(N-s/mm)** & **Energy (N-mm)** \\ \hline \multirow{4}{*}{Thickness (mm)} & 1 & 153.2233 & 1.2134 & 534.6851 \\  & 1.5 & 164.4317 & 1.2675 & 581.6425 \\  & 2 & 176 & 1.4408 & 639.8969 \\  & 2.5 & 237.7487 & 2.1188 & 941.7358 \\  & 3 & 323.193 & 3.024 & 1343.6 \\ \hline \multirow{4}{*}{Diameter (mm)} & 30 & 245.5661 & 1.8637 & 825.8955 \\  & 35 & 201.2777 & 1.619 & 717.6486 \\ \cline{1-1}  & 40 & 176 & 1.4408 & 639.8969 \\ \cline{1-1}  & 45 & 159.6214 & 1.3993 & 620.145 \\ \cline{1-1}  & 50 & 145.1112 & 1.3916 & 617.795 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Stiffness, damping, and dissipated energy parameters corresponding to different thicknesses and diameters of SMA metal modules. Figure 3: The variation curves of stiffness and damping with the thickness and diameter of SMA metal modules: (**a**) SMA metal module thickness; (**b**) SMA metal module diameter. The Influence of Structural Parameters of Rubber Vibration Dampers on the Stiffness and Damping Characteristics of Composite Dampers  
To investigate the influence of the outer rubber damping element's thickness on the damper's damping characteristics, the outer diameter D4 of the rubber damping element was varied while keeping all other structural parameters constant. The outer diameters were adjusted to 66 mm, 73 mm, 80 mm, 87 mm, and 94 mm, corresponding to thicknesses of 8.5 mm, 12 mm, 15.5 mm, 19 mm, and 22.5 mm, respectively. A harmonic excitation with an amplitude of 3 mm and a loading rate of 75 mm/s was applied. The resulting force-displacement curves for the various thicknesses of the outer rubber damping element are shown in Figure 4.  
As illustrated in Figure 4, increasing the thickness of the outer rubber damping element results in a gradual increase in the displacement at the stiffness turning point during the loading process.