# Optimizing Vehicle Body Cross-Sections Using a Parametric Mathematical Model
## 3 Fully Parametric Mathematical Model of Body Structure Based on the Transfer Matrix Method  
The transfer matrix method is used to transform the mechanical analysis problem of an overall structure into the mechanical analysis problem of several units or subsystems [9]. [PERSON] decomposes the sensitivity eigenvalue analysis of a linear mechanical system into an elemental transfer matrix thereby solving for the unknown parameters in the system [10]. [PERSON] ed al. applies the transfer matrix method to a simplified multibody system to solve for the static equilibrium position of the multibody system [11]. This paper applies the theory and method of the transfer matrix method to the structural analysis of the body, deriving a fully parametric mathematical model of the body. The transfer equation for any chain beam structure, as shown in Figure 2, can be obtained using the transfer matrix method. \[S_{n}^{R}=\prod_{i}^{n}\lambda_{i}H_{i}R_{i}S_{i}^{L} \tag{1}\]  
\[S_{i}^{L}=[u_{i}\quad v_{i}\quad\theta_{i}\quad N_{i}\quad Q_{i}\quad M_{i} \quad 1]_{l}^{T} \tag{2}\]  
\[S_{i}^{R}=[u_{i}\quad v_{i}\quad\theta_{i}\quad N_{i}\quad Q_{i}\quad M_{i} \quad 1]_{r}^{T} \tag{3}\]  
Figure 2: Chain beam structure. Equations (2) and (3) describe the state vectors of the left and right nodes of the with section of the beam unit, denoted by \(S_{i}^{L}\) and \(S_{i}^{R}\), respectively. The axial and radial deformations of node \(i\) are represented by \(u_{i}\) and \(u_{i}\), respectively, while \(\theta_{i}\) denotes the angle of turn of node \(i\). The axial, radial, and concentrated force couples at node \(i\) are represented by \(N_{i}\), \(M_{i}\), and \(Q_{i}\), respectively. \[\lambda_{i}= \begin{bmatrix}\cos\alpha_{i}&-\sin\!\alpha_{i}&0&0&0&0&0\\ \sin\alpha_{i}&\cos\alpha_{i}&0&0&0&0&0\\ 0&0&1&0&0&0&0\\ 0&0&0&\cos\alpha_{i}&-\sin\!\alpha_{i}&0&0\\ 0&0&0&\sin\alpha_{i}&\cos\alpha_{i}&0&0\\ 0&0&0&0&0&1&0\\ 0&0&0&0&0&0&1\end{bmatrix}  \tag{4}\]  
In (4), \(\lambda_{i}\) represents the coordinate transformation matrix of cell \(i\), while \(\alpha_{i}\) denotes the angle between the local coordinate system of unit i and the global coordinates. \[H_{i}= \begin{bmatrix}1&0&0&L_{i}/EA_{i}&0&0&0\\ 0&1&L_{i}&0&{L_{i}}^{2}/(2 EI_{i})&{L_{i}}^{3}/(6 EI_{i})&0\\ 0&0&1&0&{L_{i}}/(EI_{i})&{L_{i}}^{2}/(2 EI_{i})&0\\ 0&0&0&1&0&0&0\\ 0&0&0&0&1&0&0\\ 0&0&0&0&L_{i}&1&0\\ 0&0&0&0&0&0&1\end{bmatrix}  \tag{5}\]  
In Equation (5), \(H_{i}\) represents the field matrix of unit \(i\), \(L_{i}\) represents the length of unit \(i\), \(A_{i}\) represents the cross-sectional area of unit \(i\), and \(l_{i}\) represents the cross-sectional moment of inertia of unit i. \[R_{i}= \begin{bmatrix}1&0&0&0&0&0&0\\ 0&1&0&0&0&0&0\\ 0&0&1&0&0&0&0\\ 0&0&0&1&0&0&F_{xi}\\ 0&0&0&0&1&0&F_{yi}\\ 0&0&0&0&0&1&M_{i}\\ 0&0&0&0&0&0&1\end{bmatrix}  \tag{6}\]  
In Equation (6) defines \(R_{i}\) as the pointwise transfer matrix of the state vectors around node \(i\), which includes \(F_{xi}\), \(F_{yi}\), and the external load at node \(i\) of \(M_{i}\). The mathematical model of the chain beam structure can be obtained based on the boundary conditions:  
\[f\Big{(}\{C\}_{(1)},\cdots,\{C\}_{(i)},\cdots\{C\}_{(n)},\Delta Z\Big{)}=0 \tag{7}\]  
In Equation (7), \(\{C\}_{(i)}\) is the set of mechanical properties of the section of the first beam unit, \(\{C\}_{(i)}=\big{\{}A,I_{y},I_{x}\big{\}}_{(i)}\); \(\Delta Z\) is the deformation of the loading point in the loading direction. The structure of the body's skeleton can be viewed as a series of chain beam structures, which are broken down from the overall structure to the local structure using the principle of simplification. This process is illustrated in Figure 3.  
After decomposing the structure, we obtain several chained beam structures with simple forms. We then establish a mathematical model for each chained beam structure separately and iterate upward layer by layer, considering the coupling relationship of each chained structure and the corresponding boundary conditions.