## 4 Procedures
### Case Studies
#### 4.2.2 Numerical Modeling of a Dynamic Structure  
This case study analyzes a simplified dynamic structure consisting of three interconnected substructures, as shown in Figure 7. The model incorporates two connection points, labeled 2 and 3, with each connection point having two DoFs: one translational and one rotational, due to the use of beam elements in a 2D plane. The system is modeled as a free-free structure with three beam substructures, which can serve as a simplified representation of a vehicle structure--comprising the rear, middle, and front sections, as shown in Figure 7a. The analysis is performed using the generalized coupling method, with the results compared to those obtained from numerical finite element analysis (FEA)  
Figure 5: Tesla Model S chassis 2; image retrieved from TurboSquid [32]. Figure 6: Vibrational model of EV suspension assembly using receptance coupling and three substructures. and modal analysis. To ensure simplicity and consistency in the DoFs across all methods, each substructure is modeled as a single beam element with uniform properties. The key assumptions are as follows: (1) the system exhibits linear dynamic behavior, with no non-linearities in stiffness or mass; (2) the analysis is conducted for an undamped system to focus on the coupling method and its comparison with existing undamped models; (3) subsystems A, B, and C are rigidly coupled without flexibility at the connections; (4) no geometric non-linearities or large deformations are considered; (5) the receptance data for each subsystem are assumed to be known, and in the case study presented, the substructure receptances are captured using an analytical method based on the stiffness and mass matrices of beam elements which are representative of subsystems; and (6) the frequency range is assumed to capture primarily flexible modes without higher-order complexities. Generalized Coupling Method Using LM-FBS  
The dynamic system, denoted as ABC, is divided into three substructures, as shown in Figure 7b. To apply the LM-FBS algorithm, the Boolean matrix is constructed to satisfy the compatibility conditions at the interface connection points, ensuring equal displacements at the interface connection points. The boundary conditions at the interfaces can be expressed as follows:  
\[y_{2a}^{A}=y_{2b}^{B},\theta_{2a}^{A}=\theta_{2b}^{B},y_{3b}^{B}=y_{3c}^{C}, \theta_{3b}^{B}=\theta_{3c}^{C} \tag{32}\]  
Using these compatibility conditions, the Boolean matrix [B] is formulated based on the DoFs at the interface coordinates, as shown in Equation (33):  
Figure 7: Dynamic structure consists of three substructures with beam element. \[[\text{B}]= \begin{bmatrix}0&0&\text{I}&0&-\text{I}&0&0&0&0&0&0&0\\ 0&0&0&1&0&-\text{I}&0&0&0&0&0&0\\ 0&0&0&0&0&0&1&0&-\text{I}&0&0&0\\ 0&0&0&0&0&0&0&1&0&-\text{I}&0&0\end{bmatrix}  \tag{33}\]  
The generalized displacement vector \(\{\text{X}\}\), which encompasses the DoFs of internal nodes and connection points, is expressed as follows:  
\[\{\text{X}\}=\left\{y_{1}^{\text{A}};\theta_{1}^{\text{A}};y_{2\text{a}}^{ \text{A}};\theta_{2\text{a}}^{\text{A}};y_{2\text{b}}^{\text{B}};\theta_{2 \text{b}}^{\text{B}};y_{3\text{b}}^{\text{B}};\theta_{3\text{b}}^{\text{B}}; y_{3\text{c}}^{\text{C}};\theta_{3\text{c}}^{\text{C}};y_{4}^{\text{C}}; \theta_{4}^{\text{C}}\right\} \tag{34}\]  
The receptance matrix for the coupled system ABS is assembled as follows:  
\[\left[\text{H}^{\text{ABC}}\right]_{12\times 12}= \begin{bmatrix}\left[\text{H}^{ \text{A}}\right]_{4\times 4}&\left[0\right]_{4\times 4}&\left[0\right]_{4 \times 4}\\ \left[0\right]_{4\times 4}&\left[\text{H}^{\text{B}}\right]_{4\times 4}&\left[0 \right]_{4\times 4}\\ \left[0\right]_{4\times 4}&\left[0\right]_{4\times 4}&\left[\text{H}^{\text{C}} \right]_{4\times 4}\end{bmatrix}