## 3 Full Parameter FE-SEA Model Modeling of Heavy Truck Commercial Vehicle
### Obtaining Modeling Parameters of Coupling Loss Factor  
The coupling damping factor is an important parameter used to express the energy exchange between Coupling systems. It is used to represent the power flow or damping effect of one system to another system. The test method testing the coupling loss factor can be used for algorithm modification and model modification. The cab system is divided into 40 subsystems. For \(N\) = 40 subsystems, the PIM power input method can be used to test the coupling loss factor of the system and measure \(\eta_{ij}(i,j=1,2,3,\ldots,N)\) unknowns, which can be expressed as:  
\[\{\eta\}=(\eta_{11}\eta_{12}\ldots\eta_{1N}\eta_{21}\eta_{22}\ldots\eta_{NN} )^{T} \tag{40}\]  
In order to solve the unknown quantity of the system, \(N\) independent tests are carried out. When only the subsystem \(i\) is excited, the power flow balance equation of the system can be obtained:  
\[[E_{1}^{i}B_{1},E_{2}^{i}B_{2},\ldots E_{N}^{i}B_{N}]\{\eta\}=\left\{ \begin{array} []{c}0\\ \vdots\\ 0\\ \frac{p_{i}}{\omega}\\ 0\\ \vdots\\ 0\end{array} \right\} \tag{41}\]  
where Equation (41) is the matrix equation obtained from the \(i\)th test. After completing \(N\) independent tests, the matrix equation can be expressed as [11; 12; 13; 14; 15]:  
\[[\overline{B}]=\{\eta\}=\{p\} \tag{42}\]  
\[[\overline{B}]=\left[ \begin{array}{cccc}E_{1}^{1}B_{1}&E_{2}^{1}B_{1}&\cdots &E_{N}^{1}B_{N}\\ E_{1}^{2}B_{1}&E_{2}^{2}B_{2}&\cdots&E_{N}^{2}B_{N}\\ \vdots&\vdots&\ddots&\vdots\\ E_{1}^{N}B_{1}&E_{2}^{N}B_{2}&\cdots&E_{N}^{N}B_{N}\end{array} \right] \tag{43}\]  
\[B_{i}=\left[ \begin{array}{cccc}-1&0&0&\cdots&0\\ 0&-1&0&\cdots&0\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ 1&1&1&\cdots&1\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ 0&\cdots&0&-1&0\\ 0&\cdots&0&0&-1\end{array} \right] \tag{44}\]  
\[\{P\}=(\frac{p_{1}}{\omega}0...0,\frac{p_{2}}{\omega}0...0,0...0\frac{p_{N}} {\omega})^{T} \tag{45}\]  
In the case that power flow cannot be directly measured, the modal density of the subsystem can be tested. In this case, the reciprocity principle is used to obtain:  
\[\eta_{1}n_{12}=\eta_{2}n_{21} \tag{46}\]  
Combined with Equations (40)-(46) and the system energy and power tested by the PIM method, the coupling loss factor matrix of \(N\) subsystems can be obtained. Due to the small order of magnitude of coupling loss factors in the testing process, some of the coupling loss factors may lose their physical significance in the final results, which need to be modified mathematically to achieve acceptable test accuracy. Using the reciprocityprinciple in Equation (46), the power flow balance equation of the system composed of \(N\) subsystems in the cab can be expressed as:  
\[ \begin{array}{c}\frac{p_{i}}{\omega}=\sum\limits_{k=1}^{N}\eta_{ik}E_{i}-\sum \limits_{\begin{array}{c}j=1\\ j\
eq k\end{array}}^{N}\eta_{ji}E_{j}\\ =\sum\limits_{k=1}^{N}\eta_{ik}n_{i}\frac{E_{i}}{n_{i}}-\sum\limits_{ \begin{array}{c}j=1\\ j\
eq k\end{array}}^{N}\eta_{ji}n_{j}\frac{E_{j}}{n_{j}}\\ =\epsilon_{i}\sum\limits_{k=1}^{N}C_{ik}-\sum\limits_{\begin{array}{c}j=1\\ j\
eq k\end{array}}^{N}C_{ji}\epsilon_{j}(i=1,2,\ldots N)\end{array}  \tag{47}\]  
where the expressions of \(\epsilon_{i}\) and \(C_{ji}\) are as follows:  
\[\left\{ \begin{array}{c}\epsilon_{i}=\frac{E_{i}}{n_{i}}(i=1,2,\ldots N)\\ C_{ji}=n_{i}\eta_{ij}=n_{j}\eta_{ji}=C_{ij}\end{array} \right. \tag{48}\]  
It is assumed that the excitation sources among different subsystems are independent of each other, so the system responses can be superimposed, and the power flow balance equation of the whole system can be expressed as:  
\[[B]\{\epsilon\}=\{p\} \tag{49}\]  
where, \(\{\epsilon\}\) and \([B]\) can be expressed as:  
\[\left\{ \begin{array}{c}\{\epsilon\}=(\epsilon_{1},\epsilon_{2},\ldots, \epsilon_{N})^{T}\\ B_{ij}=\left\{\begin{array}{cc}C_{ij}=\sum\limits_{k=1}^{N}C_{ik}&i=j\\ -C_{ij}&i\
eq j\end{array}\right.\end{array} \right.