## 4 Procedures
### Case Studies
#### 4.2.1 Analytical Coupling Approach for Vehicle Vibration Analysis
\]  
There is no connection between substructure C and interface c1 and between substructure A and interface c2 \(\left(\mathrm{H}_{\mathrm{cc1}}^{\mathrm{C}}=\mathrm{H}_{\mathrm{czc1}}^{ \mathrm{C}}=0\right)\) and \(\mathrm{H}_{\mathrm{ac2}}^{\mathrm{A}}=\mathrm{H}_{\mathrm{clc2}}^{\mathrm{A} }=0\)). These relationships simplify the formulation of the coupling matrix. Therefore \([\mathrm  
\[\sum_{\mathrm{s=1}}^{\mathrm{s=N=3}}\left[\psi_{\mathrm{s}}^{\mathrm{ T}}\right]\left[\psi_{\mathrm{s}}\right]\bigotimes \left[\mathrm{H}_{YY}^{\mathrm{s}}\right]\] \[= \begin{bmatrix}1\\ 0\end{bmatrix}\begin{bmatrix}1&0\end{bmatrix} \bigotimes \begin{bmatrix} \mathrm{H}_{\mathrm{c1 cl}}^{\mathrm{A}}&\mathrm{H}_{\mathrm{c1 cl}}^{\mathrm{A} }=0\\ \mathrm{H}_{\mathrm{c2c1}}^{\mathrm{A}}=0&\mathrm{H}_{\mathrm{c2c2}}^{\mathrm{A} }=0\end{bmatrix} + \begin{bmatrix}-1\\ 1\end{bmatrix}\begin{bmatrix}-1&1\end{bmatrix} \bigotimes \begin{bmatrix} \mathrm{H}_{\mathrm{c1 cl}}^{\mathrm{B}}&\mathrm{H}_{\mathrm{c2c2}}^{\mathrm{B} }\\ \mathrm{H}_{\mathrm{c2c1}}^{\mathrm{B}}&\mathrm{H}_{\mathrm{c2c2}}^{\mathrm{B} }\end{bmatrix}  \tag{24}\] \[+ \begin{bmatrix}0&-1\end{bmatrix}\begin{bmatrix}0&-1\end{bmatrix}  \bigotimes \begin{bmatrix}\mathrm{H}_{\mathrm{c1 cl}}^{\mathrm{C}}=0&\mathrm{H} _{\mathrm{c1 cl}}^{\mathrm{C}}=0\\ \mathrm{H}_{\mathrm{c2c1}}^{\mathrm{C}}=0&\mathrm{H}_{\mathrm{c2c2}}^{\mathrm{C} }=0\end{bmatrix} =\begin{pmatrix}\mathrm{H}_{\mathrm{c1 cl}}^{\mathrm{A}}+ \mathrm{H}_{\mathrm{c1 cl}}^{\mathrm{B}}&-\mathrm{H}_{\mathrm{c1 cl}}^{\mathrm{B} }\\ -\mathrm{H}_{\mathrm{c2c1}}^{\mathrm{B}}&\mathrm{(H}_{\mathrm{c2c2}}^{\mathrm{B} }+\mathrm{H}_{\mathrm{c2c2}}^{\mathrm{C}})\end{bmatrix}\]  
The resulting coupled receptance matrix, \(\left[\mathrm{H}\right]_{\mathrm{jettundsen}}^{\mathrm{ABC}}\), is calculated by substituting Equations (20)-(24) into Equation (19). It accounts for dynamic stiffness at the interfaces and the flexible effects of bushings or mounts:  
\[\left[\mathrm{H}\right]_{\mathrm{jettundsen}}^{\mathrm{ABC}}= \begin{bmatrix}\mathrm{H}_{\mathrm{a}a}^{\mathrm{A}}&\mathrm{H}_{ \mathrm{acl}}^{\mathrm{A}}&0&0\\ \mathrm{H}_{\mathrm{c1a}}^{\mathrm{A}}&\mathrm{H}_{\mathrm{c1 cl}}^{\mathrm{A} }&\mathrm{H}_{\mathrm{c1 cl}}^{\mathrm{A}}&0\\ 0&\mathrm{H}_{\  
Use of LM-FBS Coupling Algorithm  
To apply this framework, we start by determining the Boolean matrix that satisfies the compatibility conditions, ensuring that displacements are equal at interface connection points. In this context, [H] is analogous to [H\({}_{\alpha\alpha}\)] from the previous method. The boundary conditions at the interfaces are expressed as \(X_{\text{c1a}}^{\text{ABC}}=X_{\text{c1b}}^{\text{ABC}},X_{\text{c2c}}^{\text{ABC }}=X_{\text{c2b}}^{\text{ABC}}\), ensuring compatibility between the degrees of freedom (DoFs) at the interface coordinates. These conditions align the displacements at connection points, as enforced by the Boolean matrix. \[[\text{B}]= \begin{bmatrix}X_{\text{a}}&X_{\text{c1a}}&X_{\text{b}}&X_{\text{c1b }}&X_{\text{c2b}}&X_{\text{c}}&X_{\text{c2c}}\\ 0&-\text{I}&0&\text{I}&0&0&0\\ 0&0&0&0&-\text{I}&0&\text{I}\\ \end{bmatrix} c1 \tag{26}\]  
and [X] represents the displacement vector of the DoF coordinates corresponding to measurement points of interest: \(\{\text{X}\}=\left\{X_{\text{a}},X_{\text{c1a}}\right\};X_{\text{b}};X_{\text{ c1b}};X_{\text{c2b}};X_{\text{c}};X_{\text{c2c}}\right\}\). It is important to note that Equation (25) is derived by assuming a rigid interface, and it is not valid in the context of a non-rigid connection between substructures A and B. In the case of flexible coupling, the Boolean matrix would be determined from Equation (12). The next step involves calculating the following:  
\[[\text{B}][\text{H}][\text{B}]^{\text{T}}= \begin{bmatrix}0&-\text{I}&0&\text {I}&0&0&0\\ 0&0&0&0&-\text{I}&0&\text{I}\end{bmatrix}