## 2 Methodology
### Projection Matrix
The residual flexibility attachment modes \(\mathbf{\
abla}^{\mathrm{r}}\), are computed by enforcing the spectral orthogonality between the attachment modes and the normal modes as Equation (9):  
\[(\mathbf{\mathrm{M}}-\omega_{\mathrm{r}}^{2}\bar{\mathbf{\mathrm{K}}})\mathbf{\mathrm{\Phi}} _{\mathrm{i}}^{\mathrm{f}}=0 \tag{8}\]  
\[\mathbf{\
abla}^{\mathrm{r}}=\mathbf{\mathrm{G}}^{\mathrm{r}}-\sum_{p=1}^{n_{\mathrm{g} }}\frac{\mathbf{\mathrm{\Phi}}_{p}^{\mathrm{f}}\mathbf{\mathrm{\Phi}}_{p}^{\mathrm{f} \,\mathrm{T}}}{\omega_{p}^{2}},\quad\mathbf{\mathrm{G}}^{\mathrm{r}}=\bar{\mathbf{ \mathrm{K}}}^{-1} \begin{bmatrix}\mathbf{\mathrm{I}}\\ \mathbf{\mathrm{0}}\end{bmatrix}  \tag{9}\]  
where \(\omega_{p}\) is the natural frequency of \(p^{\mathrm{th}}\) free interface normal mode \(\mathbf{\mathrm{\Phi}}_{p}^{\mathrm{f}}\). The fixed interface normal modes and constraint modes of the Craig-Bampton method are computed from the partitions of the tangential stiffness matrix as Equations (10) and (11). The constraint modes \(\mathbf{\
abla}^{\mathrm{c}}\) are the static displacements of the structure due to unit displacement applied at master DoFs. \[(\mathbf{\mathrm{M}}_{\mathrm{ss}}-\omega_{\mathrm{f}}^{2}\bar{\mathbf{\mathrm{K}}}_{ \mathrm{ss}})\mathbf{\mathrm{\Phi}}_{\mathrm{i}}^{\mathrm{c}}=0 \tag{10}\]  
\[\mathbf{\
abla}^{\mathrm{c}}=\bar{\mathbf{\mathrm{K}}}_{\mathrm{ss}}^{-1}\bar{\mathbf{ \mathrm{K}}}_{\mathrm{sm}} \tag{11}\]  
In this context, the residual flexibility attachment modes and constraint modes employed in the Rubin and Craig-Bampton bases are referred to as static modes. To account for the dependency of normal modes of a geometrically nonlinear structure on its state, the linear basis could be enhanced by adding Static Modal Derivatives (SMDs). This yields enhanced projection matrices [21,22]:  
\[\mathbf{\mathrm{R}}_{\mathrm{Rubin-MD}}= \begin{bmatrix}\mathbf{\mathrm{I}}&\mathbf{0}&\bm {0}\\ \mathbf{\mathrm{\Psi}}_{\mathrm{s}}^{\mathrm{r}}(\mathbf{\mathrm{\Psi}}_{\mathrm{m}}^{ \mathrm{r}})^{-1}&-\mathbf{\mathrm{\Psi}}_{\mathrm{s}}^{\mathrm{r}}(\mathbf{\mathrm{ \Psi}}_{\mathrm{m}}^{\mathrm{r}})^{-1}\mathbf{\mathrm{\Phi}}_{\mathrm{m}}^{ \mathrm{f}}+\mathbf{\mathrm{\Phi}}_{\mathrm{s}}^{\mathrm{f}}&-\mathbf{\mathrm{\Psi}}_{ \mathrm{s}}^{\mathrm{r}}(\mathbf{\mathrm{\Psi}}_{\mathrm{m}}^{\mathrm{r}})^{-1}\mathbf{ \mathrm{\Theta}}_{\mathrm{m}}^{\mathrm{f}}+\mathbf{\mathrm{\Theta}}_{\mathrm{s}}^{ \mathrm{f}}\end{bmatrix}  \tag{12}\]  
\[\mathbf{\mathrm{R}}_{\mathrm{CB-MD}}= \begin{bmatrix}\mathbf{\mathrm{I}}&\mathbf{0}&\mathbf{0} \\ \mathbf{\mathrm{\Psi}}^{\mathrm{c}}&\mathbf{\mathrm{\Phi}}^{\mathrm{c}}&\mathbf{\mathrm{ \Theta}}^{\mathrm{c}}\end{bmatrix}  \tag{13}\]  
where \(\mathbf{\mathrm{\Theta}}^{\mathrm{f}}\) and \(\mathbf{\mathrm{\Theta}}^{\mathrm{c}}\) are matrices of the selected SMDs obtained by introducing a finite displacement around the free interface and fixed interface modes, respectively. The reduction methods are characterized by the projection matrices of Equations (12) and (13) referred to as Enhanced Rubin and Enhanced Craig-Bampton methods, respectively. In the next section, the computation of SMDs is discussed.