## 2 Methodology
### Projection Matrix  
The CMS methods start by splitting the nodal displacement vector \(\mathbf{u}(t)\) into entries related to master DoFs \(\mathbf{u}_{\text{m}}\in\mathbb{R}^{N_{\text{m}}}\) and slave DoFs \(\mathbf{u}_{\text{s}}\in\mathbb{R}^{N_{\text{s}}}\), where \(N_{\text{m}}+N_{\text{s}}=N\). The master DoFs, denoted by subscript m, correspond to \(N_{\text{m}}\) physical DoFs retained in the generalized coordinates. Other DoFs are considered slave DoFs and are denoted by subscript s. Here the master DoFs include the DoFs that the contact forces are applied to. The equation of motion Equation (1) and coordinate transformation matrix Equation (2) are then partitioned accordingly. Thus, Equation (2) is written as follows:  
\[ \begin{bmatrix}\mathbf{u}_{\text{m}}\\ \mathbf{u}_{\text{s}}\end{bmatrix} =\mathbf{R} \begin{bmatrix}\mathbf{u}_{\text {m}}\\ \boldsymbol{\eta}\end{bmatrix} \text{,where }\mathbf{R}= \begin{bmatrix}\mathbf{I}& \mathbf{0}\\ \mathbf{R}_{\text{sm}}&\mathbf{R}_{\text{ss}}\end{bmatrix}  \tag{4}\]  
where \(\mathbf{I}\in\mathbb{R}^{N_{\text{m}}\times N_{\text{m}}}\) is an identity matrix. \(\boldsymbol{\eta}\in\mathbb{R}^{n_{\text{g}}}\) is the time-dependent vector of the amplitude of normal modes (\(n=N_{\text{m}}+n_{\text{g}}\)). Accordingly, the projection matrices of Rubin (\(\mathbf{R}_{\text{Rubin}}\)) and Craig-Bampton (\(\mathbf{R}_{\text{CB}}\)) model order reduction techniques are computed based on the considered normal modes and static deflections ([13; 14; 15; 32]):  
\[\mathbf{R}_{\text{Rubin}}= \begin{bmatrix}\mathbf{I}&\mathbf{0}\\ \mathbf{\Psi}_{\text{s}}^{\mathsf{T}}(\mathbf{\Psi}_{\text{m}}^{\mathsf{T}})^ {-1}&-\mathbf{\Psi}_{\text{s}}^{\mathsf{T}}(\mathbf{\Psi}_{\text{m}}^{\mathsf{ T}})^{-1}\mathbf{\Phi}_{\text{m}}^{\mathsf{f}}+\mathbf{\Phi}_{\text{s}}^{ \mathsf{f}}\end{bmatrix}  \tag{5}\]  
\[\mathbf{R}_{\text{CB}}= \begin{bmatrix}\mathbf{I}&\mathbf{0}\\ \mathbf{\Psi}^{\text{c}}&\mathbf{\Phi}^{\text{c}}\end{bmatrix}  \tag{6}\]  
Matrices \(\mathbf{\Psi}^{\mathsf{T}}\in\mathbb{R}^{N\times N_{\text{m}}}\) and \(\mathbf{\Phi}^{\mathsf{f}}\in\mathbb{R}^{N\times n_{\text{g}}}\), in Rubin projection matrix \(\mathbf{R}_{\text{Rubin}}\), are, respectively, the matrices of the residual flexibility attachment modes and the selected normal modes with free boundary condition at master DoFs. Here, for the sake of simplicity, it is assumed that the structures are constrained and have no rigid body motion, so the rigid body modes are not considered in the Rubin basis. \(\mathbf{\Psi}^{\text{c}}\in\mathbb{R}^{N_{\text{s}}\times N_{\text{m}}}\) and \(\mathbf{\Phi}^{\text{c}}\in\mathbb{R}^{N_{\text{s}}\times n_{\text{g}}}\) matrices in Craig-Bampton projection matrix \(\mathbf{R}_{\text{CB}}\) include constraint modes and the selected normal modes with fixed boundary condition at master DoFs, respectively. To compute these static deflections and normal modes, first, the tangential stiffness matrix of the structure in static equilibrium (\(t=0\)) is computed:  
\[\mathbf{\tilde{K}}=\frac{\partial\mathbf{f}_{\text{int}}(\mathbf{u}(0))}{ \partial\mathbf{u}} \tag{7}\]  
here, \(\mathbf{f}_{\text{int}}(\mathbf{u}(0))=0\) and \(\mathbf{\tilde{K}}\) comprise the undeformed linear stiffness matrix. Then, to obtain the free interface modes of the Rubin model, the nonlinear normal modes are computed as Equation (8). The attachment modes \(\mathbf{G}^{\text{m}}\) are the static displacements of the structure due to unit force applied at master DoFs.