## 2 Analysis of Stiffness and Damping Characteristics of Rubber-Based SMA Composite Dampers  
The Influence of SMA Structural Parameters on the Stiffness and Damping Characteristics of Composite Dampers  
The rubber-based SMA composite shock absorber [12] discussed in this article features a sleeve-type suspension structure comprising an outer rubber shock absorber, an inner rubber shock absorber, and a circular SMA metal module, as shown in Figure 1. When a force is applied, the excitation force initially transmits to the outer rubber damping element, which then engages the SMA metal module and induces strain. During the phase transition, the vibration energy dissipates through grain boundary friction. The remaining force then transmits to the cab connecting bolt via the inner rubber element. For the middle-layer SMA metal module to operate effectively, the outer rubber must transmit enough force to generate the necessary stress and strain within the module, thereby achieving the desired damping effect. For the convenience of subsequent simulation research, the DesignModeler module of ANSYS was used to parameterize the rubber-based SMA composite vibration damper. The structural parameters of the rubber-based SMA composite vibration damper are shown in Table 1, where d is the thickness of the circular SMA metal module, H is the overall width of the vibration damper, D1 is the inner diameter of the inner rubber vibration damper, D2 and D3 are the inner and outer diameters of the SMA metal module, and D4 is the outer diameter of the outer rubber vibration damper. \begin{table}
\begin{tabular}{c c c c c} \hline
**D1** & **D2** & **D3** & **D4** & **d** & **H** \\ \hline
20 & 45 & 49 & 80 & 2 & 55 \\ \hline \end{tabular}
\end{table} 
Table 1: Structural parameters of circular ring rubber-based SMA composite vibration dampers. Figure 1: Cross-section view of rubber-based SMA composite vibration damper [12]. To investigate the effect of the thickness and diameter of circular SMA metal modules on the stiffness and damping characteristics of composite vibration dampers, SMA metal modules with varying thicknesses of 1 mm, 1.5 mm, 2 mm, 2.5 mm, and 3 mm were studied under the condition that other structural parameters remained unchanged. A harmonic excitation with an amplitude of 3 mm was applied at a loading rate of 75 mm/s while maintaining a constant thickness of 2 mm for the SMA metal modules. Subsequently, the effect of module diameter on the stiffness and damping properties was examined with inner diameters of 30 mm, 35 mm, 40 mm, 45 mm, and 50 mm. The same harmonic excitation parameters (amplitude of 3 mm and loading rate of 75 mm/s) were applied. The force-displacement curves corresponding to SMA metal modules of different thicknesses and diameters are presented in Figure 2.  
As illustrated in Figure 2a, the enclosed area of the force-displacement curve increases significantly with the thickness of the SMA metal module. This observation can be attributed to the higher proportion of SMA material present in the rubber-based SMA composite vibration damper as the module thickness increases. As a result, the force-displacement characteristics of the overall structure increasingly resemble those inherent to the SMA material. When the thickness of the SMA metal module is equal to or below 1.5 mm, the force-displacement curve exhibits characteristics similar to those of a rubber hysteresis loop, a behavior attributed to the diminutive proportion of SMA material. As depicted in Figure 2b, a reduction in the diameter of the SMA metal module leads to an expansion in the area circumscribed by the force-displacement curve. This also results in an increase in the displacement associated with the stiffness inflection point during loading, thereby enhancing overall structural stiffness. This effect arises because a smaller diameter allows more SMA structures to undergo martensitic and austenitic phase transformations at equivalent displacement load amplitudes, thus augmenting energy dissipation.