## 3 Optimization and Matching Design of Rubber-Based SMA Suspension Systems
### Structural Parameter Matching Design of Rubber-Based SMA Composite Vibration Dampers
The resulting response surface, which illustrates the relationship between the diameter and loading rate of the SMA metal module as independent variables and the z-direction damping of the rubber-based SMA composite damper as the dependent variable, is presented in Figure 17.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**P1—XYPlane.H1** & **P2—XYPlane.V2** & **P3—Plane4.V1** & **P4—Plane4.H2** & **P5—Extrude2.FD1** & **P6—Stiffness** \\ **(m)** & **(m)** & **(m)** & **(m)** & **(N/mm)** \\ \hline
20 & 45 & 80 & 55 & 2 & k1 \\
18 & 45 & 80 & 55 & 2 & k2 \\
22 & 45 & 80 & 55  
Based on the response surface analysis of various design variables concerning the stiffness and damping characteristics of rubber-based SMA composite dampers and guided by the optimization results for suspension stiffness and damping, the structural parameters of the rubber-based SMA composite dampers have been methodically optimized and designed. An array encompassing a range of design variables, along with the corresponding stiffness and damping calculation results, was imported into Matlab. The primary objective was to minimize the absolute percentage error between the calculated stiffness values and the target stiffness values under different loading rates. As additional criteria, the stiffness corresponding to the maximum amplitude was set to be 1.4 times that of the minimum amplitude, and the damping corresponding to the maximum amplitude was set to be 1.65 times that of the minimum amplitude. Consequently, the optimal combination of structural parameters was determined, and the  
Based on the response surface analysis results, several modifications to the optimized SMA metal module have been identified. The diameter of the SMA metal module has been reduced from 45 mm to 33 mm, and its thickness has been decreased from 2 mm to 1.75 mm. Conversely, the thickness of the outer rubber material has increased from 15.5 mm to 22.75 mm, while the width of the composite shock absorber has slightly decreased from 55 mm to 53 mm. These changes are attributed to the insensitivity of rubber stiffness to frequency variations in the low-frequency range, which reduces the amplification of stiffness damping in composite dampers at these frequencies. As the frequency increases, the rubber material undergoes dynamic hardening, which enhances the sensitivity of the inner SMA metal module to frequency changes, thereby gradually increasing the stiffness-damping amplification of the composite vibration damper in the mid-to-high-frequency range. Additionally, by reducing the diameter and thickness of the SMA metal module, the SMA's superelastic strain limit can be reached with smaller excitation amplitudes. When subjected to larger excitation values, the SMA undergoes complete martensitic transformation, thereby improving the overall stiffness of the composite vibration damper. The parameterized model of rubber-based SMA composites were updated with optimized structural parameters and their z-stiffness and damping values recalculated at different loading rates. The results are shown in Table 12.  
Based on the data presented in the table, matching curves for stiffness, damping, target stiffness, and target damping of the optimized composite vibration damper were plotted, as illustrated in Figure 19. The results show significant variation in the stiffness and damping values of the optimized structure at a loading rate of 120 mm/s, whereas better alignment is observed at other loading rates. This discrepancy can be attributed to the characteristics of the rubber-based SMA composite vibration damper, where both stiffness and damping increase with the loading rate, with this increase being more pronounced at lower loading rates. A reduction in the stiffness and damping values at the 120 mm/s loading rate was specifically implemented to enhance overall fitting accuracy and better align with the target stiffness and damping values.