## 3 Case Studies
### Case Study 1: Coupling of Continuous Substructures
#### 3.1.2 Numerical FEA for Validation  
To validate the receptance matrix obtained using FBS, numerical FEA has been applied. For subsystem A, the stiffness matrix \(\left[K_{A}\right]\) and mass matrix \(\left[M_{A}\right]\) are as follows [30; 32]:  
\[\begin{split}\left[K_{A}\right]=\frac{EI_{A}}{L_{A}^{3}}\begin{bmatrix} 12&-6L_{A}\\ -6L_{A}&4L_{A}^{2}\end{bmatrix},\\ \left[M_{A}\right]=\frac{m_{A}}{420}\begin{bmatrix}156&-22L_{A}\\ -22L_{A}&4L_{A}^{2}\end{bmatrix}.\end{split} \tag{30}\]  
It is assumed that both substructures A and B have identical masses, i.e., \(m_{A}=m_{B}=m\).  
Similarly, subsystem B's stiffness \(\left[K_{B}\right]\) and mass matrix \(\left[M_{B}\right]\) are derived as follows:  
\[\begin{split}\left[K_{B}\right]=\frac{EI_{B}}{L_{B}^{3}}\begin{bmatrix} 12&6L_{B}&-12&6L_{B}\\ 6L_{B}&4L_{B}^{2}&-6L_{B}&2L_{B}^{2}\\ -12&-6L_{B}&12&-6L_{B}\\ 6L_{B}&2L_{B}^{2}&-6L_{B}&4L_{B}^{2}\end{bmatrix},\\ \left[M_{B}\right]=\frac{m_{B}}{420}\begin{bmatrix}156&22L_{B}&54&-13L_{B}\\ 22L_{B}&4L_{B}^{2}&13L_{B}&-3L_{B}^{2}\\ 54&13L_{B}&156&-22L_{B}\\ -13L_{B}&-3L_{B}^{2}&-22L_{B}&4L_{B}^{2}\end{bmatrix}.\end{split} \tag{31}\]  
The global mass and stiffness matrix for the assembly AB can then be determined from the following:  
\[\begin{split}\left[K_{AB}\right]=\begin{bmatrix}12(\frac{EI_{B}} {L_{B}^{3}}+\frac{EI_{A}}{L_{A}^{3}})&6(\frac{EI_{B}}{L_{B}^{3}}+\frac{EI_{A} }{L_{A}^{3}})&-12\frac{EI_{B}}{L_{B}^{3}}&6\frac{EI_{B}}{L_{B}^{3}}\\ 6(\frac{EI_{B}}{L_{B}^{3}}+\frac{EI_{A}}{L_{A}^{3}})&4(\frac{EI_{B}}{L_{B}}+ \frac{EI_{A}}{L_{A}})&-6\frac{EI_{B}}{L_{B}^{3}}&2\frac{EI_{B}}{L_{B}^{3}}\\ -12\frac{EI_{B}}{L_{B}^{3}}&-6\frac{EI_{B}}{L_{B}^{3}}&12\frac{EI_{B}}{L_{A}^{3 }}-6\frac{EI_{B}}{L_{B}^{3}}\\ 6\frac{EI_{B}}{L_{B}^{3}}&2\frac{EI_{B}}{L_{B}}&-6\frac{EI_{B}}{L_{B}^{3}}&4 \frac{EI_{B}}{L_{B}^{3}}\end{bmatrix},\\ \left[M_{AB}\right]=\frac{1}{420}\begin{bmatrix}156(m_{B}+m_{A})&22(L_{B}m_{B}-L _{A}m_{A})&54&-13L_{B}\\ 22(L_{B}m_{B}-L_{A}m_{A})&4\left(m_{A}L_{A}^{2}+m_{B}L_{B}^{2}\right)&13L_{B}& -3L_{B}^{2}\\ 54&13L_{B}&156&-22L_{B}\\ -13L_{B}&-3L_{B}^{2}&-22L_{B}&4L_{B}^{2}\end{bmatrix}.\end{split} \tag{32}\]  
The linear equations of motion for the assembly system AB are expressed as follows:  
\[\begin{split}\begin{cases}y_{2}^{AB}\\ \theta_{2}^{AB}\\ y_{1}^{AB}\\ \theta_{1}^{AB}\end{cases}=\begin{bmatrix}H^{AB}\\ \frac{F_{\theta_{2}}^{AB}}{F_{\theta_{1}}^{AB}}\\ \frac{F_{\theta_{1}}^{AB}}{F_{\theta_{1}}^{AB}}\end{cases}.\end{split} \tag{33}\]  
where receptance matrix \(\left[H^{AB}\right]\) of the assembly system AB is determined as follows:  
\[\begin{split}\left[H^{AB}\right]=\begin{bmatrix}H^{AB}_{yz_{2}}&H^{AB}_{yz_{2}}&H^{AB}_{yz_{1}}&H^{AB}_{yz_{ 2}}\\ H^{AB}_{\theta_{2}}&H^{AB}_{\theta_{2}}&H^{AB}_{\theta_{1}}&H^{AB}_{\theta_{ 1}}\\ H^{AB}_{y_{1}^{AB}}&H^{AB}_{\theta_{1}\theta_{2}}&H^{AB}_{\theta_{1}\theta_{1}} \end{bmatrix}=\begin{bmatrix}-\omega^{2}[M_{AB}]+[K_{AB}]]^{-1}.\end{split} \tag{34}\]  
It is worth noting that in the FBS method, the system is decomposed into two substructures, each with two generalized coordinates and two DoFs. To maintain consistency in the FEA model, each substructure is also modeled as a single beam element with twoactive nodes and two DoFs. For higher accuracy in FEA, further discretization of each substructure could improve the results.