## 3 Optimization and Matching Design of Rubber-Based SMA Suspension Systems  
### Modeling and Analysis of Rubber-Based SMA Suspension System  
#### 3.1.1 Theoretical Analysis of Vibration in the Rubber-Based SMA Suspension System of the Cab  
This portion of the study focuses on the suspension system of a commercial vehicle, which features a symmetrical design on both the left and right sides and employs sleeve-type suspension at both the front and rear. An initial theoretical analysis of vibration isolation is conducted for the cab suspension system, with the simplified model illustrated in Figure 8. Due to the relatively softer suspension stiffness compared to the cab's inherent stiffness, the cab is modeled as a rigid body with six degrees of freedom. In this analysis, the suspension system is represented as a spring exhibiting elastic properties along three orthogonal spatial axes [37]. The suspension stiffness is assumed to be linearly elastic, and damping effects are neglected. To facilitate targeted analysis and simulation of actual conditions, external excitations such as frame vibration and power system vibration are simplified into a fixed-amplitude and -frequency sine load form. In Figure 8, the center-of-mass coordinate system O-XYZ is defined, with the coordinate axes aligned with those of the vehicle coordinate system. Due to the high sensitivity of passengers to vertical vibrations, this study primarily focuses on vibrations in the z-direction. Consequently, the motion equation of the cab suspension can be described as follows:  
\[\mathrm{M\ddot{X}}=\mathrm{M_{R}} \tag{1}\]  
In the formula, \(M_{R}\) is the matrix of suspension dynamic reaction force and dynamic reaction torque, and the cab mass matrix can be expressed as follows:  
\[\mathrm{M}= \begin{bmatrix}\mathrm{m}&0&0&0&0&0\\ 0&\mathrm{m}&0&0&0&0\\ 0&0&\mathrm{m}&0&0&0\\ 0&0&0&\mathrm{J_{x}}&-\mathrm{J_{xy}}&-\mathrm{J_{xz}}\\ 0&0&0&-\mathrm{J_{xy}}&\mathrm{J_{y}}&-\mathrm{J_{yz}}\\ 0&0&0&-\mathrm{J_{xz}}&-\mathrm{J_{yz}}&\mathrm{J_{z}}\\ \end{bmatrix}  \tag{2}\]  
Figure 8: Vibration model of the cab vibration subsystem. The relationship between the displacement \(D_{ci}\) of the \(i\)-th suspension point in the local coordinate system, the displacement \(D_{ri}\) in the center-of-mass coordinate system O-XYZ, and the center-of-mass displacement X is as follows:  
\[D_{ri}=B_{i}D_{ci}=B_{i}E_{i}X \tag{3}\]  
\[B_{i}= \begin{bmatrix}\cos\alpha_{i1}&\cos\beta_{i1}&\cos\gamma_{i1}\\ \cos\alpha_{i2}&\cos\beta_{i2}&\cos\gamma_{i2}\\ \cos\alpha_{i3}&\cos\beta_{i3}&\cos\gamma_{i3}\end{bmatrix}  \tag{4}\]  
\[E_{i}= \begin{bmatrix}1&0&0&0&z_{i}&-y_{i}\\ 0&1&0&-z_{i}&0&x_{i}\\ 0&0&1&y_{i}&-x_{i}&0\end{bmatrix} =(I\,r_{i}) \tag{5}\]  
In the formula, \(B_{i}\) represents the directional cosine matrix in the O-XYZ coordinate system of the \(i\)-th suspension's elastic main axis center of mass, while \(x_{i}\), \(y_{i}\), and \(z_{i}\) represent the suspension coordinates. When there is a vertical displacement excitation on the frame, the displacement \(D_{cbi}\) of the suspension point in its elastic principal axis coordinate system O-XYZ can be expressed as follows:  
\[D_{\rm{rbi}}=B_{i}D_{\rm{cbi}} \tag{6}\]  
The force and deformation of the \(i\)-th suspension in its O\({}_{i}\)-uvvw coordinate system can be calculated by the following equation:  
\[F_{ci}=B_{i}^{T}f_{ri}= \begin{bmatrix}B_{i}^{T}k_{i}B_{i}&B_{i}^{T}k_{i}B_{i} r_{i}\end{bmatrix} X-B_{i}^{T}k_{i}B_{i}D_{\rm{cbi}} \tag{7}\]  
The dynamic reaction force and dynamic reaction moment of the \(i\)-th suspension on the cab can be expressed as the total reaction force \(M_{Ri}\):  
\[M_{Ri}= \begin{bmatrix}-B_{i}^{T}k_{i}B_{i}&-B_{i}^{T}k_{i}B_{i}r_{i}\\ -r_{i}^{T}B_{i}^{T}k_{i}B_{i}&-r_{i}^{T}B_{i}^{T}k_{i}B_{i}r_{i}\end{bmatrix}  X+ \begin{bmatrix}B_{i}^{T}k_{i}B_{i}D_{\rm{cbi}}\\ r_{i}^{T}B_{i}^{T}k_{i}B_{i}D_{\rm{cbi}}\end{bmatrix}  \tag{8}\]  
The motion equation of the cab can be written as  
\[M\ddot{X}=M_{R}=\sum_{i=1}^{n}