## 2 Methodology
### Development of General Coupling Formulas with Full DoFs Using GRCFBS
\tag{14}\]  
For subsystem B, with two generalized coordinates \(\mathcal{U}_{1}^{B}\) and \(\mathcal{U}_{2}^{B}\), the equation of motion can be expressed in terms of the generalized receptance matrices \(\mathcal{H}_{11}^{B}\), \(\mathcal{H}_{12}^{B}\), \(\mathcal{H}_{21}^{B}\), and \(\mathcal{H}_{22}^{B}\) and the generalized excitation forces \(F_{1}^{B}\) and \(F_{2}^{B}\):  
\[ \begin{split}\left\{\mathcal{U}_{1}^{B}\right\}=\left[\mathcal{H}_ {11}^{B}\right]\left\{F_{1}^{B}\right\}+\left[\mathcal{H}_{12}^{B}\right] \left\{F_{2}^{B}\right\}\ \Rightarrow\ \begin{cases}y_{1}^{B}\\ \theta_{1}^{B}\end{cases}=\begin{bmatrix}H_{y_{1}y_{1}}^{B}&H_{y_{1}\theta_{1}}^{B} \\ H_{y_{1}y_{1}}^{B}&H_{y_{1}\theta_{1}}^{B}\end{bmatrix}\begin{cases}F_{y_{1}}^{B} \\ F_{\theta_{1}}^{B}\end{cases}+\begin{bmatrix}H_{y_{1}y_{2}}^{B}&H_{y_{1}\theta_{2}} ^{B}\\ H_{y_{1}y_{2}}^{B}&H_{y_{1}\theta_{2}}^{B}\end{cases}\text{,}\\ \left\{\mathcal{U}_{2}^{B}\right\}=\left[\mathcal{H}_{22}^{B}\right]\left\{F_{2}^{ B}\right\}+\left[\mathcal{H}_{21}^{B}\right]\left\{F_{1}^{B}\right\}\ \Rightarrow\ \begin{cases}x_{2}^{B}\\ \theta_{2}^{B}\end{cases}=\begin{bmatrix}H_{y_{2}y_{2}}^{B}&H_{y_{2}\theta_{2}}^{B} \\ H_{y_{2}y_{2}}^{B}&H_{y_{2}\theta_{2}}^{B}\end{bmatrix}\begin{cases}F_{y_{2}}^{B} \\ F_{\theta_{2}}^{B}\end{cases}+\begin{bmatrix}H_{y_{2}y_{1}}^{B}&H_{y_{2}\theta_{1}}^{B} \\ H_{y_{2}y_{1}}^{B}&H_{y_{2}\theta_{1}}^{B}\end{cases}\begin{cases}F_{y_{1}}^{B} \\ F_{\theta_{1}}^{B}\end{cases}\text{.}\end{split}  \tag{15}\]Equation (15) can be expressed in the matrix form as follows:  
\[ \begin{cases}\mathcal{U}_{1}^{B}\\ \mathcal{U}_{2}^{B}\end{cases} = \begin{bmatrix}\mathcal{H}_{11}^{B}&\mathcal{H}_{ 12}^{B}\\ \mathcal{H}_{21}^{B}&\mathcal{H}_{22}^{B}\end{bmatrix}\begin{cases}Q_{1}^{B}\\ Q_{2}^{B}\end{cases} . \tag{16}\]  
where \(Q_{1}^{B}\) and \(Q_{2}^{B}\) are the generalized excitation forces, expressed as follows:  
\[Q_{1}^{B}= \begin{cases}F_{U_{1}}^{B}\\ F_{\theta_{1}}^{B}\end{cases} ,\quad Q_{2}^{B}= \begin{cases}F_{V_{2}}^{B}\\ F_{\theta_{2}}^{B}\end{cases} . \tag{17}\]  
For the assembled system AB, the equation of motion determines the generalized coordinates \(\mathcal{U}_{1}^{AB}\) and \(\mathcal{U}_{2}^{AB}\) in terms of the generalized receptance matrices \(\mathcal{H}_{11}^{AB},\mathcal{H}_{12}^{AB},\mathcal{H}_{21}^{AB}\), and \(\mathcal{H}_{22}^{AB}\), and the generalized excitation forces \(Q_{1}^{AB}\) and \(Q_{2}^{AB}\) can be expressed as follows:  
\[ \begin{cases}\mathcal{U}_{1}^{AB}=\mathcal{U}_{1}^{B}\\ \mathcal{U}_{2}^{AB}=\mathcal{U}_{2}^{B}\end{cases} = \begin{bmatrix}\mathcal{H} _{11}^{AB}&\mathcal{H}_{12}^{AB}\\ \mathcal{H}_{21}^{AB}&\mathcal{H}_{22}^{AB}\end{bmatrix}\begin{cases}Q_{1}^{ AB}=Q_{1}^{B}\\ Q_{2}^{AB}=Q_{2}^{B}\end{cases} . \tag{18}\]  
Assuming rigid coupling between subsystems A and B, the compatibility condition \(\mathcal{U}_{2}^{A}=\mathcal{U}_{2}^{B}\), and the equilibrium condition at the interface \(F_{2}^{A}+F_{2}^{B}=0\). Using these conditions, the assembly receptance AB can be expressed in terms of the receptance matrices of subsystems A and B as follows:  
\[ \begin{cases}&\mathcal{H}_{11}^{AB}=\frac{\mathcal{U}_{11}^{AB}}{F_{2}^{AB}}= \frac{\mathcal{U}_{11}^{B}}{F_{1}^{B}}=\mathcal{H}_{11}^{B}-\mathcal{H}_{11} ^{B}\big{(}\mathcal{H}_{22}^{A}+\mathcal{H}_{22}^{B}\big{)}^{-1}\mathcal{H}_ {21}^{B},\\ &\mathcal{H}_{21}^{AB}=\frac{\mathcal{U}_{21}^{AB}}{F_{2}^{AB}}=\frac{\mathcal{ U}_{21}^{B}}{F_{1}^{B}}=\mathcal{H}_{11}^{B}-\mathcal{H}_{11}^{B}\big{(} \mathcal{H}_{22}^{A}+\mathcal{H}_{22}^{B}\big{)}^{-1}\mathcal{H}_{21}^{B},\\ &\mathcal{H}_{22}^{AB}=\frac{\mathcal{U}_{21}^{AB}}{F_{2}^{AB}}=\frac{\mathcal{ U}_{22}^{B}}{F_{2}^{B}}=\mathcal{H}_{22}^{A}\big{(}\mathcal{H}_{22}^{A}+\mathcal{H}_{22} ^{B}\big{)}^{-1}\mathcal{H}_{22}^{B}=\mathcal{H}_{22}^{A}-\mathcal{H}_{22}^{A} \big{(}\mathcal{H}_{22}^{A}+\mathcal{H}_{22}^{B}\big{)}^{-1}\mathcal{H}_{22}^{ A},\\ &\mathcal{H}_{12}^{AB}=\mathcal{H}_{21}^{AB}.\end{cases}