## 4 Procedures
### Case Studies
#### 4.2.1 Analytical Coupling Approach for Vehicle Vibration Analysis  
In this case study, an analytical framework is developed for modeling vehicle vibration, using a simplified quarter car model. The model consists of three interconnected substructures: the vehicle's trimmed body (substructure A), the suspension system (substructure B), and the wheel assembly (substructure C), as illustrated in Figure 3. These substructures are connected through two interfaces \(\mathrm{c_{1}}\), connecting the body and suspension, and \(\mathrm{c_{2}}\), connecting the suspension and wheel. The system is defined by the following DoFs: substructure A, the trimmed body subsystem, is modeled with two DoFs: one internal DoF \(\left\{\mathrm{X_{a}^{A}}\right\}\), representing the reference measurement point within the cabin, and an interface DoF \(\mathrm{X_{c1}^{A}}\), corresponding to the connection point with the suspension. Substructure B, the suspension subsystem, is modeled with three DoFs: one internal DoF \(\mathrm{X_{b}^{B}}\) and two interface DoFs, \(\mathrm{X_{c1}^{B}}\) and \(\mathrm{X_{c2}^{B}}\), connecting to the body and wheel, respectively. The wheel assembly, substructure C, has two DoFs: \(\mathrm{X_{c}^{C}}\) at the hub center, treated as an internal DoF, and \(\mathrm{X_{c2}^{C}}\) at the spindle attachment. The objective is to derive the governing equations for predicting the vehicle body's vibrational response to external excitations, such as road and electric motor operations. Both the Jettmundsen and [PERSON] algorithms are employed, utilizing the receptance matrices of the individual substructures. The resulting coupled system response is obtained by combining the individual substructure responses via the interface boundary conditions. Use of Jettmundsen Coupling Algorithm  
The Jettmundsen method, expressed in Equation (18), partitions the receptance matrices of the substructures and uses graph theory for system assembly [30]. The corresponding matrices \([\mathrm{H_{\alpha\alpha}}]\), \([\mathrm{b}]\), \([\mathrm{H_{\alpha Y}}]\), and \([\mathrm{\psi_{s}}]\), which are derived based on the substructures' DoFs and their connections, are outlined in Equations (20) and (21):  
Figure 3: Coupling of three substructures for road noise prediction. \[[\mathrm{H}_{\alpha\alpha}]= \begin{bmatrix}\mathrm{H}_{\mathrm{aa}}^{\mathrm{A}}&0&0 \\ 0&\mathrm{H}_{\mathrm{aa}}^{\mathrm{B}}&0\\ 0&0&\mathrm{H}_{\mathrm{aa}}^{\mathrm{C}}\end{bmatrix} = \begin{bmatrix} \mathrm{H}_{\mathrm{aa}}^{\mathrm{A}}&\mathrm{H}_{\mathrm{acl}}^{\mathrm{A}} &0&0&0\\ \mathrm{H}_{\mathrm{cla}}^{\mathrm{A}}&\mathrm{H}_{\mathrm{clcl}}^{\mathrm{A}} &0&\mathrm{H}_{\mathrm{cl}}^{\mathrm{B}}\\ 0&\mathrm{H}_{\mathrm{bb}}^{\mathrm{B}}&\mathrm{H}_{\mathrm{bc1}}^{\mathrm{B} }&\mathrm{H}_{\mathrm{bc2}}^{\mathrm{B}}\\ 0&\mathrm{H}_{\mathrm{cb}}^{\mathrm{B}}&\mathrm{H}_{\mathrm{clcl}}^{\mathrm{B} }&\mathrm{H}_{\mathrm{clcl}2}^{\mathrm{B}}\\ 0&\mathrm{H}_{\mathrm{czb}}^{\mathrm{C}}&\mathrm{H}_{\mathrm{czc1}}^{\mathrm{C} }&\mathrm{H}_{\mathrm{czc2}}^{\mathrm{C}}\\ 0&0&\mathrm{H}_{\mathrm{czc}}^{\mathrm{C}}&\mathrm{H}_{\mathrm{czc2}}^{\mathrm{C }}\end{bmatrix}  \tag{20}\]  
where the DoFs \(\alpha\) for the substructures are \(\alpha^{\mathrm{A}}=\left\{\mathrm{a},\mathrm{c1}_{(\mathrm{A})}\right\}, \alpha^{\mathrm{B}}=\left\{\mathrm{b},\mathrm{c1}_{(\mathrm{B})},\mathrm{c2}_ {(\mathrm{B})}\right\}\), \(\alpha^{\mathrm{C}}=\left\{\mathrm{c},\mathrm{c2}_{(\mathrm{C})}\right\}, \gamma=\left\{\mathrm{c1},\mathrm{c2}\right\}\). The mapping matrix \([\psi]\), describing the interconnections between substructures, is defined as follows:  
\[[\psi]= \begin{bmatrix}+1&0\\ -1&+1\\ 0&-1\end{bmatrix} \mathrm{A} \tag{21}\]  
Partitioned matrices correspond to the connections between substructures:  
\[\psi_{\mathrm{A}}= \begin{bmatrix}+1&0\end{bmatrix} ,\quad\psi_{\mathrm{B}}= \begin{bmatrix}-1&1\end{bmatrix} ,\quad\psi_{\mathrm{C}}= \begin{bmatrix}0&-1 \end{bmatrix}