## Appendix A Enhanced Transformation Matrix  
Equation (14) can be explicitly presented as:  
\[\mathbf{q}_{F}^{i}=\mathbf{T}_{0}^{i}\mathbf{p}_{F}^{i}=\begin{bmatrix} \mathbf{\Phi}_{d}^{i}&\mathbf{\Phi}_{r}^{i}&\mathbf{\Psi}^{i}\\ \mathbf{0}&\mathbf{0}&\mathbf{I}\end{bmatrix}\begin{bmatrix}\mathbf{p}_{d}^{ i}\\ \mathbf{p}_{r}^{i}\\ \mathbf{q}_{b}^{i}\end{bmatrix}. \tag{15}\]Using the transformation matrix \(\mathbf{T}_{0}^{i}\) for the projection in Equation (9), we have:  
\[\hat{\mathbf{M}}_{F}^{i}\hat{\mathbf{p}}_{F}^{i}+\hat{\mathbf{K}}_{F}^{i}\mathbf{ p}_{F}^{i}=\mathbf{0}, \tag{10a}\] \[\hat{\mathbf{M}}_{F}^{i}=\mathbf{T}_{0}^{i}\mathbf{M}_{F}^{i}\mathbf{T}_{0}^{i}, \hat{\mathbf{K}}_{F}^{i}=\mathbf{T}_{0}^{i}\mathbf{K}_{F}^{i}\mathbf{T}_{0}^{i},\] (10b) \[\left[\frac{d^{2}}{dt^{2}}\hat{\mathbf{M}}_{F}^{i}+\hat{\mathbf{K}}_{F}^{i} \right]\mathbf{p}_{F}^{i}=\begin{bmatrix}\hat{\mathbf{\Lambda}}_{d}^{i}&\mathbf{ 0}&\frac{d^{2}}{dt^{2}}\hat{\mathbf{M}}_{c_{d}}^{i}\\ \mathbf{0}&\hat{\mathbf{\Lambda}}_{r}^{i}&\frac{d^{2}}{dt^{2}}\hat{\mathbf{M}} _{c_{r}}^{i}\\ \frac{d^{2}}{dt^{2}}[\hat{\mathbf{M}}_{c_{d}}^{i}]^{T}&\frac{d^{2}}{dt^{2}}[ \hat{\mathbf{M}}_{c_{d}}^{i}]^{T}&\hat{\mathbf{K}}_{b}^{i}+\frac{d^{2}}{dt^{2} }\hat{\mathbf{M}}_{b}^{i}\end{bmatrix}\mathbf{p}_{F}^{i}=\mathbf{0}, \tag{10c}\]  
where the component matrices are:  
\[\hat{\mathbf{\Lambda}}_{d}^{i}=\mathbf{\Lambda}_{d}^{i}+\frac{d^{2}}{dt^{2}} \mathbf{I}_{d}^{i},\quad\Lambda_{d}^{i}=\mathbf{\Phi}_{d}^{i}{}^{T}\mathbf{K} _{s}^{i}\mathbf{\Phi}_{d}^{i},\quad\mathbf{I}_{d}^{i}=\mathbf{\Phi}_{d}^{i}{}^ {T}\mathbf{M}_{s}^{i}\mathbf{\Phi}_{d}^{i}, \tag{11a}\] \[\hat{\mathbf{\Lambda}}_{r}^{i}=\mathbf{\Lambda}_{r}^{i}+\frac{d^{2}}{dt^{2}} \mathbf{I}_{r}^{i},\quad\Lambda_{r}^{i}=\mathbf{\[PERSON]}_{r}^{i}{}^{T}\mathbf{K} _{s}^{i}\mathbf{\[PERSON]}_{r}^{i},\quad\mathbf{I}_{r}^{i}=\mathbf{\[PERSON]}_{r}^{i}{}^ {T}\mathbf{M}_{s}^{i}\mathbf{\[PERSON]}_{r}^{i},\] (11b) \[\hat{\mathbf{M}}_{c_{d}}^{i}=\mathbf{\[PERSON]}_{d}^{i}{}^{T}\mathbf{A}^{i},\quad \hat{\mathbf{M}}_{c_{r}}^{i}=\mathbf{\Phi}_{r}^{i}{}^{T}\mathbf{A}^{i},\quad \mathbf{A}^{i}=\mathbf{M}_{s}^{i}\mathbf{\Psi}^{i}+\mathbf{M}_{c_{r}}^{i},\] (11c) \[\hat{\mathbf{K}}_{b}^{i}=\mathbf{K}_{b}^{i}+\mathbf{K}_{c}^{i}{}^{T}\mathbf{ \Psi}^{i},\quad\hat{\mathbf{M}}_{b}^{i}=\mathbf{\Psi}^{i}{}^{T}\mathbf{M}_{s} ^{i}\mathbf{\Psi}^{i}+\mathbf{\Psi}^{i}{}^{T}\mathbf{M}_{c}^{i}+\mathbf{M}_{c }^{i}{}^{T}\mathbf{\Psi}^{i}+\mathbf{M}_{b}^{i}. \tag{11d}\]  
From the second row in Equation (10c), the coordinate vector of the residual terms can be expressed as:  
\[\mathbf{p}_{r}^{i}=-[\hat{\mathbf{\Lambda}}_{r}^{i}]^{-1}\left[\frac{d^{2}}{dt ^{2}}\hat{\mathbf{M}}_{c_{r}}^{i}\right]\mathbf{q}_{b}^{i}. \tag{12}\]  
Substituting Equation (12) into Equation (11), the generalized coordinates \(\mathbf{q}_{s}^{i}\) for the interior containing the residual modal effects can be rewritten as:  
\[\mathbf{q}_{s}^{i}=\mathbf{\Phi}_{d}^{i}\mathbf{p}_{d}^{i}+\mathbf{\Psi}^{i} \mathbf{q}_{b}^{i}-\frac{d^{2}}{dt^{2}}\mathbf{\hat{F}}_{r}^{i}\mathbf{A}^{i} \mathbf{q}_{b}^{i},\quad\hat{\mathbf{F}}_{r}^{i}=\mathbf{\Phi}_{r}^{i}[\hat{ \mathbf{\Lambda}}_{r}^{i}]^{-1}\mathbf{\Phi}_{r}^{i}{}^{T}, \tag{13}\]  
where \(\hat{\mathbf{F}}_{r}^{i}\) means the residual flexibility matrix. Using \(d^{2}/dt^{2}=-\omega^{2}\) and the series expansion, \(\hat{\mathbf{F}}_{r}^{i}\) can be written as:  
\[\hat{\mathbf{F}}_{r}^{i}=\mathbf{\Phi}_{r}^{i}[\mathbf{\Lambda}_{r}^{i}-\omega ^{2}\mathbf{\hat{\Lambda}}_{r}^{i}]^{-1}\mathbf{\Phi}_{r}^{i}{}^{T}=\mathbf{ \[PERSON]}_{r}^{i}[\mathbf{\Lambda}_{r}^{i}]^{-1}\mathbf{\[PERSON]}_{r}^{i}{}^{T}+ \omega^{2}  
This clearly shows that \(\overline{\mathbf{T}}_{1}^{i}\) is a refined transformation matrix with a residual modal effect. Therefore, using \(\overline{\mathbf{T}}_{1}^{i}\), one can expect a better representation of the modal behavior of the original flexible bodies than with the conventional CB transformation matrix \(\overline{\mathbf{T}}_{0}^{i}\).