## 2 Methodology
### Development of General Coupling Formulas with Full DoFs Using GRCFBS
This simplification applies in cases where the stiffness or mass matrix contributions from rotational-translational coupling are minimal, such as in structures with highly localized DoF constraints or when modeling low-frequency dynamics in certain rigid body approximations. Rearranging Equation (9) results in the following:  
\[H_{tt(c)}^{AB}=H_{tt(c)}^{A}-H_{tt(c)}^{A}D_{tt}^{AB-1}H_{tt(c)}^{A}-H_{tr(c)} ^{A}D_{rr}^{AB-1}H_{rt(c)}^{A}. \tag{10}\]In Equation (10), the third term, \(H_{tr(c)}^{A}D_{rr}^{AB-1}H_{rt(c)}^{A}\), represents the effects of the rotational DoFs of the connection points. If only translational DoFs are considered, the equation simplifies to the following:  
\[H_{tt(c)}^{AB}=H_{tt(c)}^{A}-H_{tt(c)}^{A}D_{tt}^{AB-1}H_{tt(c)}^{A}. \tag{11}\]  
where \(D_{tt}^{AB-1}\) and \(D_{rr}^{AB-1}\) denote the translational and rotational components of dynamic stiffness at the connection area, respectively. For more general scenarios where coupling between rotational and translational DoFs at the interface is significant, the cross-receptance terms, \(D_{tt}^{AB}\) and \(D_{tr}^{AB}\), of matrix D at the connection zone must be considered, resulting in a more complex form of the equation. Using the definition of \([D^{AB}]^{-1}[D^{AB}]=[I]\), the inverse matrix \([D^{AB}]^{-1}= \begin{bmatrix}D_{tt}^{AB}&D_{tr}^{AB}\\ D_{rt}^{AB}&D_{rr}^{AB}\end{bmatrix} ^{-1}\) can be determined as follows:  
\[[D^{AB}]^{-1}= \begin{bmatrix}\left(D_{tt}^{AB}-D_{tr}^{AB}D_{rr}^{AB-1}D_{tr}^ {AB}\right)^{-1}&\left(D_{rt}^{AB}-D_{rr}^{AB}D_{tt}^{AB-1}D_{tt}^{AB}\right)^{ -1}\\ D_{rr}^{AB-1}D_{rt}^{AB}\left(D_{tt}^{AB}-D_{tr}^{AB}D_{rr}^{AB-1}D_{tr}^{AB} \right)^{-1}&D_{tr}^{AB-1}D_{tt}^{AB}\left(D_{rt}^{AB}-D_{rr}^{AB}D_{tr}^{AB-1} D_{tt}^{AB}\right)^{-1}\end{bmatrix} . \tag{12}\]  
The receptance matrices \(H_{tt(c)}^{AB}\), \(H_{tr(c)}^{AB}\), \(H_{rt(c)}^{AB}\), and \(H_{rr(c)}^{AB}\) for the coupled system AB are determined for a single node c representing the connection point:  
\[ \begin{bmatrix}H_{tt(c)}\end{bmatrix} = \begin{bmatrix}H_{xx}&H_{xy}&H_{xz}\\ H_{yx}&H_{yy}&H_{yz}\\ H_{zx}&H_{zy}&H_{zz}\end{bmatrix} \text{, }\left[H_{rr(c)}\right]= \begin{bmatrix}H_{ r_{x}r_{x}}&H_{r_{x}r_{y}}&H_{r_{y}r_{z}}\\ H_{r_{y}r_{x}}&H_{r_{y}r_{y}}&H_{r_{y}r_{z}}\\ H_{r_{z}r_{x}}&H_{r_{z}r_{y}}&H_{r_{z}r_{z}}\end{bmatrix} \text{, }\left[H_{ rr(c)}\right]= \begin{bmatrix}H_{r_{x}x}&H_{r_{x}y}&H_{r_{x}r_{z}}\\ H_{r_{y}x}&H_{r_{y}y}&H_{r_{y}z}\\ H_{r_{x}x}&H_{r_{y}y}&H_{r_{z}z}\end{bmatrix} \text{.} \tag{13}\]  
where \(H_{xx}\), \(H_{xy}\), \(H_{yx}\), and \(H_{yy}\) represent the coupling parameters for translational DoFs (where the excitation load is a force and the measured displacement is translational), while \(H_{r_{x}r_{x}}\), \(H_{r_{x}r_{y}}\), \(H_{r_{y}r_{y}}\), and \(H_{r_{z}r_{z}}\) represent the coupling parameters for rotational DoFs (where the external excitation is a moment). The terms like \(H_{xx}\), \(H_{xy}\), \(H_{yx}\), and \(H_{yy}\) represent the coupling parameters between translational displacement and rotational excitation load (moment) and vice versa. For simplicity, the coupling of two subsystems A and B, each with two DoFs (one translational and one rotational) has been considered. For subsystem A, the equation of motion determining the displacement vector of the generalized coordinate \(\mathcal{U}_{2}^{A}\) in terms of the generalized receptance matrix \(\mathcal{H}_{22}^{A}\) and the generalized excitation forces \(F_{2}^{A}\) (including both translational force \(F_{y2}^{A}\) and moment \(F_{\theta_{2}}^{A}\)) can be expressed as follows:  
\[\left\{\mathcal{U}_{2}^{A}\right\}=\left[\mathcal{H}_{22}^{A}\right]\left\{F_ {2}^{A}\right\}\ \Rightarrow\ \begin{bmatrix}y_{2}^{A}\\ \theta_{2}^{A}\end{bmatrix} = \begin{bmatrix}H_{y_{y}y_{2}}^{A}&H_{y_{y}\theta_ {2}}^{A}\\ H_{y_{y}y_{1}}^{A}&H_{y_{x}\theta_{1}}^{A}\end{bmatrix}\begin{cases}F_{y_{y}}^{A} \\ F_{\theta_{1}}^{A}\end{cases} \text{.}