## 3 Case Studies: Application to Continuous and Lumped-Parameters Systems
### Case Study 1: Coupling of Continuous Substructures
#### 3.1.2 Validation of RCFBS Using Numerical FEA
\begin{bmatrix}12\begin{pmatrix}\frac{\mathrm{E} \mathrm{I}_{B}}{\mathrm{L}_{B}^{A}}+\frac{\mathrm{E}\mathrm{I}_{A}}{\mathrm{L}_ {A}^{A}}\end{pmatrix}&6\begin{pmatrix}\frac{\mathrm{E}\mathrm{I}_{B}}{\mathrm{ L}_{B}^{A}}+\frac{\mathrm{E}\mathrm{I}_{A}}{\mathrm{L}_{A}^{A}}\end{pmatrix}&-12 \frac{\mathrm{E}\mathrm{I}_{B}}{\mathrm{L}_{B}^{A}}&6\frac{\mathrm{E}\mathrm{I }_{B}}{\mathrm{L}_{B}^{A}}\\ 6\begin{pmatrix}\frac{\mathrm{E}\mathrm{I}_{B}}{\mathrm{L}_{B}^{A}}+\frac{ \mathrm{E}\mathrm{I}_{A}}{\mathrm{L}_{A}^{A}}\end{pmatrix}&4\begin{pmatrix} \frac{\mathrm{E}\mathrm{I}_{B}}{\mathrm{L}_{B}}+\frac{\mathrm{E}\mathrm{I}_{A}}{ \mathrm{L}_{A}}\end{pmatrix}&-6\frac{\mathrm{E}\mathrm{I}_{B}}{\mathrm{L}_{B}^{A }}&2\frac{\mathrm{E}\mathrm{I}_{B}}{\mathrm{L}_{B}}\\ -12\frac{\mathrm{E}\mathrm{I}_{B}}{\mathrm{L}_{B}^{A}}&-6\frac{\mathrm{E}\mathrm{ I}_{B}}{\mathrm{L}_{B}^{A}}&12\begin{pmatrix}\frac{\mathrm{E}\mathrm{I}_{B}}{ \mathrm{L}_{B}^{A}}+\frac{\mathrm{E}\mathrm{I}_{C}}{\mathrm{L}_{A}^{A}}\end{pmatrix} -6\begin{pmatrix}\frac{\mathrm{E}\mathrm{I}_{B}}{\mathrm{L}_{B}^{A}}+\frac{ \mathrm{E}\mathrm{I}_{C}}{\mathrm{L}_{B}^{A}}\end{pmatrix},\\ 6\frac{\mathrm{E}\mathrm{I}_{B}}{\mathrm{L}_{B}^{A}}&2\frac{\mathrm{E}\mathrm{I }_{B}}{\mathrm{L}_{B}}&-6\begin{pmatrix}\frac{\mathrm{E}\mathrm{I}_{B}}{ \mathrm{L}_{B}^{A}}+\frac{\mathrm{E}\mathrm{I}_{C}}{\mathrm{L}_{c}^{A}}\end{pmatrix} 4\begin{pmatrix}\frac{\mathrm{E}\mathrm{I}_{B}}{\mathrm{L}_{B}}+\frac{ \mathrm{E}\mathrm{I}_{C}}{\mathrm{L}_{c}}\end{pmatrix}\end{bmatrix} ,\\ \left[M_{\mathrm{ABC}}\right]=\frac{1}{420} \begin{bmatrix}156(\mathrm{m}_{B} +\mathrm{m}_{A})&22(\mathrm{L}_{B}\mathrm{m}_{B}-\mathrm{L}_{A}\mathrm{m}_{A})&54& -13\mathrm{L}_{B}\\ 22(\mathrm{L}_{B}\mathrm{m}_{B}-\mathrm{L}_{A}\mathrm{m}_{A})&4\begin{pmatrix} \mathrm{m}_{A}\mathrm{L}_{A}^{2}+\mathrm{m}_{B}\mathrm{L}_{B}^{2}\end{pmatrix} &13\mathrm{L}_{B}&-3\mathrm{L}_{B}^{2}\\ 54&13\mathrm{L}_{B}&156+156&-22(\mathrm{L}_{B}+\mathrm{L}_{c})\\ -13\mathrm{L}_{B}&-3\mathrm{L}_{B}^{2}&-22(\mathrm{L}_{B}+\mathrm{L}_{c})&4 \begin{pmatrix}\mathrm{L}_{B}^{2}+\mathrm{L}_{c}^{2}\end{pmatrix}\end{bmatrix} .\end{split} \tag{13}\]  
Accordingly, the linear equations of motion for the assembly system AB are expressed as follows:  
\[ \begin{split}\begin{cases}\begin{matrix}Y_{A}^{\mathrm{ABC}}\\ \theta_{A}^{\mathrm{ABC}}\\ Y_{A}^{\mathrm{ABC}}\\ \theta_{B}^{\mathrm{ABC}}\end{matrix}=\left[H^{\mathrm{ABC}}\right]\begin{cases} \begin{matrix}\mathrm{F}_{\mathrm{V}_{\mathrm{V}_{\mathrm{V}_{\mathrm{V}_{\mathrm{V}_{ \mathrm{V}_{\mathrm{V}_{\mathrm{V}_{\mathrm{V}_{\mathrm{V}_{\mathrm{V}}}}}}}}}}^{ \mathrm{ABC}}\\ \theta_{A}^{\mathrm{ABC}}\\ \theta_{B}^{\mathrm{ABC}}\end{matrix}.\end{cases}\end{split}  \tag{14}\]Finally, the receptance matrix \([H^{ABC}]\) for the entire system ABC, using the direct method, is given by the following:  
\[[\mathrm{H}^{ABC}]= \begin{bmatrix}\mathrm{H}_{y_{1}y_{1}}^{ABC}&\mathrm{H}_{y_{1} \theta_{1}}^{ABC}&\mathrm{H}_{y_{1}y_{2}}^{ABC}&\mathrm{H}_{y_{1}\theta_{2}}^{ ABC}\\ \mathrm{H}_{y_{1}y_{1}}^{ABC}&\mathrm{H}_{\theta_{1}\theta_{1}}^{ABC}&\mathrm{H }_{\theta_{1}\theta_{2}}^{ABC}&\mathrm{H}_{\theta_{1}\theta_{2}}^{ABC}\\ \mathrm{H}_{y_{2}y_{1}}^{ABC}&\mathrm{H}_{y_{2}\theta_{1}}^{ABC}&\mathrm{H }_{y_{2}\theta_{2}}^{ABC}&\mathrm{H}_{y_{2}\theta_{2}}^{ABC}\\ \end{bmatrix}  \tag{15}\]  
This matrix encompasses both direct and cross-receptance components, which relate translational and rotational displacements to the applied forces and moments at various nodes. For instance, in the first column of the matrix, \(H_{y_{1}y_{1}}^{ABC}\) represents the translational displacement at node 1 due to a force applied at the same node, while \(H_{\theta_{1}y_{1}}^{ABC}\) indicates the rotational displacement at node 1 caused by the force applied in the same node. Similarly, \(H_{y_{2}y_{1}}^{ABC}\) and \(H_{\theta_{2}y_{1}}^{ABC}\) correspond to the translational and rotational displacements at node 2 resulting from a force applied at node 1.