## 3 Optimization and Matching Design of Rubber-Based SMA Suspension Systems
### Optimization of Cab Suspension System Based on Adams Vibration  
#### 3.2.1 Adams Design Optimization Variable Setting  
The stiffness and damping adjustments of the commercial vehicle cab suspension system are pivotal for adapting to varying road conditions. By optimally tuning the suspension system's stiffness and damping, the vehicle's stability, comfort, and safety can be significantly enhanced. This section utilizes the established Adams multi-body dynamics model to parameterize the stiffness and damping of the suspension system. Given that the primary driving force in the cab originates from the frame and is predominantly concentrated in the z-direction, the stiffness and damping of the shock absorber in the z-direction are employed as design variables. The initial values and variation ranges for each design variable are presented in Table 7, while additional variables and their respective values are referenced in Table 5.  
2.2 Optimization of Stiffness and Damping of Rubber-Based SMA Suspension Under Different Loading Rates  
Based on the primary excitation frequency range of the driver's cab excitation source, harmonic excitation is applied to the SMA suspension system in the driver's cab at loading rates of 120 mm/s, 240 mm/s, 360 mm/s, 480 mm/s, and 600 mm/s, corresponding to frequencies of 10 Hz, 20 Hz, 30 Hz, 40 Hz, and 50 Hz, respectively. A displacement amplitude of 3 mm and a duration of 1 s are maintained across these tests. The loading rates are approximately calculated by dividing the excitation amplitude by one-quarter of the cycle. Figures 11 and 12 illustrate the time-domain displacement and acceleration responses of the seat in the z-direction at different loading rates. These data were obtained by establishing a reference coordinate system at the seat and measuring the relevant parameters within this reference coordinate system using Adams/View2016 version software. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Initial Value** & **Lower Limit** & **Upper Limit** \\ \hline Suspended z-direction stiffness & & & \\ (Kzd N/mm) & 176.33 & 110 & 260 \\ \hline Suspension z-direction damping & & & \\ (Czd\textbackslash{}N\textbackslash{}s/mm) & 1.4408 & 0.1 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Design variables for cab suspension system. As shown in the above figure, under low-frequency excitation corresponding to a small loading rate, the z-direction displacement amplitude and z-direction acceleration amplitude at the seat are both relatively large. Under high-frequency excitation corresponding to high loading rates, the z-direction displacement amplitude at the seat is small, but it takes a long time to reach a stable state, while the z-direction acceleration amplitude is higher. By utilizing the \(z\)-axis stiffness and damping of the suspension as design variables and imposing the constraint that the displacement amplitude of the driver's seat must be less than 10 mm, the optimization aims to minimize the root mean square (RMS) value of the \(z\)-axis acceleration of the seat. The stiffness and damping parameters listed in Table 7 are optimized under these conditions. To balance reducing calculation time and ensuring robust optimization results, the maximum number of iterations is set to 100, with a tolerance level of 0.1. Under the specified excitation conditions, which include a loading rate of 120 mm/s (corresponding to an excitation frequency of 10 Hz) and a displacement amplitude of 3 mm, the iterative optimization process targeting the root mean square (RMS) value of the \(z\)-axis acceleration of the seat is illustrated in Figure 13. Additionally, the time-frequency response optimization iteration processes for the \(z\)-axis acceleration and \(z\)-axis displacement of the seat are presented in Figures 14 and 15, respectively. As illustrated in Figure 13, the root mean square (RMS) value of the \(z\)-axis acceleration of the seat has decreased from 3.307 m/s\({}^{2}\) to 1.991 m/s\({}^{2}\).