## 2 Methodology
### Development of General Coupling Formulas with Full DoFs Using GRCFBS  
The receptance coupling method, initially formulated by [PERSON], describes the coupling of two subsystems (A and B) to yield the assembled subsystem receptance matrix at the interface and internal coordinates, as expressed as follows [8]:  
\[ \begin{bmatrix}H_{an}^{AB}&H_{dc}^{AB}&H_{ab}^{AB}\\ H_{dc}^{AB}&H_{dc}^{AB}&H_{cb}^{AB}\\ H_{bu}^{AB}&H_{bc}^{AB}&H_{bb}^{AB}\end{bmatrix} = \begin{bmatrix}H_{aa}^{A}&H_{ ac}^{A}&0\\ H_{aa}^{A}&H_{cc}^{A}&0\\ 0&0&H_{bb}^{AB}\end{bmatrix} - \begin{bmatrix}H_{ac}^{A}\\ H_{cc}^{A}\\ -H_{bc}^{B}\end{bmatrix}\begin{bmatrix}H_{cc}^{A}+H_{cc}^{B}\end{bmatrix} ^{-1} \begin{bmatrix}H_{ac}^{A}&H_{cc}^{A}&-H_{bc}^{B}\end{bmatrix} .\end{bmatrix} \tag{5}\]  
In this formulation, subscripts \(a\) and \(b\) refer to the internal DoFs within substructures \(A\) and \(B\), while subscript \(c\) corresponds to the connection DoFs between subsystems. In this paper, it is assumed that no internal generalized coordinates within the subsystems are involved, meaning that the points of interest are limited to the connection nodes for simplicity. This assumption does not impact the generality of the procedure. Emphasis is placed on determining \(H_{cc}^{AB}\), which captures the direct receptance at the interface based on the individual subsystems' receptance matrices at the connection coordinates. Under various coupling assumptions, \(H_{cc}^{AB}\) can be expanded for two conditions of weak and strong coupling between rotational and translational DoFs at the interfaces. From Equation (5), \(H_{cc}^{AB}\) can be expressed in terms of \(H_{cc}^{A}\) and \(H_{cc}^{B}\) as follows:  
\[H_{cc}^{AB}=H_{cc}^{A}-H_{cc}^{A}\Big{(}H_{cc}^{A}+H_{cc}^{B}\Big{)}^{-1}H_{cc} ^{A}. \tag{6}\]  
Using Equations (4) and (6), the receptance coupling for the assembled system AB can be defined as follows:  
\[ \begin{bmatrix}H_{tt(c)}^{AB}&H_{tr(c)}^{AB}\\ H_{rt(c)}^{AB}&H_{rr(c)}^{AB}\end{bmatrix} = \begin{bmatrix}H_{tt(c)}^{A}&H_{tr(c)} ^{A}\\ H_{rt(c)}^{A}&H_{rr(c)}^{A}\end{bmatrix} - \begin{bmatrix}H_{tt(c)}^{A}&H_{tr(c)} ^{A}\\ H_{rt(c)}^{A}&H_{rr(c)}^{A}\end{bmatrix}\begin{bmatrix}D_{tt}^{AB}&D_{rr}^{AB}\\ D_{rt}^{AB}&D_{rr}^{AB}\end{bmatrix} ^{-1} \begin{bmatrix}H_{tt(c)}^{A}&H_{tr(c)} ^{A}\\ H_{rt(c)}^{A}&H_{rr(c)}^{A}\end{bmatrix} . \tag{7}\]  
where the matrix \(D\) denotes receptance at the connection points and is expressed as follows:  
\[ \begin{bmatrix}D_{tt}^{AB}&D_{tr}^{AB}\\ D_{rt}^{AB}&D_{rr}^{AB}\end{bmatrix} = \begin{bmatrix}H_{tt(c)}^{A}&H_{rr(c)}^{ A}\\ H_{rt(c)}^{A}&H_{rr(c)}^{A}\end{bmatrix} + \begin{bmatrix}H_{tt(c)}^{B}&H_{rr(c)} ^{B}\\ H_{rt(c)}^{B}&H_{rr(c)}^{B}\end{bmatrix} . \tag{8}\]  
Assuming a negligible coupling between the translational and rotational DoFs at the connection points between subsystems A and B--meaning that the off-diagonal terms \(D_{tr}^{AB}\) and \(D_{rt}^{AB}\) in the coupling matrix are approximately zero--the simplified Equation (7) is as follows:  
\[ \begin{bmatrix}H_{tt(c)}^{AB}&H_{tr(c)}^{AB}\\ H_{rt(c)}^{AB}&H_{rr(c)}^{AB}\end{bmatrix} = \begin{bmatrix}H_{tt(c)}^{A}&H_{tr(c)} ^{A}\\ H_{rt(c)}^{A}&H_{rr(c)}^{A}\end{bmatrix} - \begin{bmatrix}H_{tt(c)}^{A}&H_{tr(c)} ^{A}\\ H_{rt(c)}^{A}&H_{rr(c)}^{A}\end{bmatrix}\begin{bmatrix}D_{tt}^{AB}&0\\ 0&D_{rr}^{AB}\end{bmatrix} ^{-1} \begin{bmatrix}H_{tt(c)}^{A}&H_{tr(c)}^{A}\\ H_{rt(c)}^{A}&H_{rr(c)}^{A}\end{bmatrix} . \tag{9}\]  
Due to the principle of reciprocity in dynamic systems, the cross-recceptances are equal (\(H_{rt(c)}^{A}=H_{tr(c)}^{A}\)) to maintain the symmetry of the system. The coupling between translational and rotational DoFs at each node is typically weak or negligible under conditions where either the structural or dynamic properties of the connection points or interfaces inherently limit rotational-translational interactions (e.g., pinned connections or hinges), or the physical layout or symmetry of the nodes reduces the likelihood of significant cross-influence.