## 2 Methodology
### Static Modal Derivatives
#### 2.2.1 Dominant Mode Selection  
The dominant modes have a high participation factor during the system dynamics. A frequency-based approach is generally used, selecting those normal modes whose natural frequencies are close to the excitation frequency (\(\Phi=[\Phi^{(1)},\dots,\Phi^{(m)}]\)). This provides the normal modes of the linear system excited by external forces. However, the internally resonant, geometrically coupled modes, and static modes describing the local flexibility of the contact region, can contribute considerably to the behavior of a structure, particularly in strongly or even moderately nonlinear cases. Their contribution can be significant enough that the modes' derivatives are necessary in reduction bases to obtain accurate responses. The approach proposed here enables the selection of these modes. It yields a more robust reduced model that remains valid under higher levels of geometric nonlinearity. Furthermore, in terms of the convergence analysis needed to ensure the proper reduction basis, faster convergence is achieved, resulting in a shorter computing time.  
To achieve this, a geometrically linear transient analysis with frictional contact (GLF analysis) along with a series of geometrically nonlinear static analyses (GNLS analyses) are conducted. The GLF analysis identifies potentially internally resonant modes excited by higher harmonic components of friction forces, while the GNLS analyses reveal geometrically coupled modes that participate significantly in the structure response.  
First, the geometrically linear model with friction contact is analyzed, and the beam deflections during the steady state are maintained at different times \(\mathbf{U}_{t}=[\mathbf{u}_{t}^{1},\mathbf{u}_{t}^{2},\dots,\mathbf{u}_{t}^{n}]\). Then, the static force necessary to obtain a deflection equivalent to the linear dynamic response amplitude is computed as follows:  
\[\mathbf{f}=\mathbf{K}\mathbf{X}\text{sin}(\beta) \tag{18}\]  
when \(\mathbf{X}\) and \(\beta\) are, respectively, the amplitude and the phase of the forced response of the linear system. Then, a set of nonlinear static analyses is performed at different force levels in the range [0 \(\mathbf{f}\)] to activate different levels of geometric nonlinearity, collecting the resulting beam deflections \(\mathbf{U}_{st}=[\mathbf{u}_{st}^{1},\mathbf{u}_{st}^{2},\dots,\mathbf{u}_{st}^ {n}]\). The two sets of displacements are assembled in a matrix \(\mathbf{U}=[\mathbf{U}_{st},\mathbf{U}_{t}]\).  
Figure 1: Selection of modes to obtain projection basis.  
Here, the dominant modes are defined as those with considerable participation in basis \(\mathbf{U}\). In other words, when \(\mathbf{U}\) is projected onto these modes, they exhibit the highest participation factors. So, the participation of a set of modes (\(\mathbf{Y}\)) in displacement vectors of matrix \(\mathbf{U}\) is computed by least square approximation \(\mathbf{U}\approx\mathbf{Y}\mathbf{Q}\). Then, the modes with the highest values in matrix \(\mathbf{Q}\) will be selected as the dominant modes. This set of modes includes the free interface normal modes and residual flexibility attachment modes for the Rubin method and the fixed interface normal modes and constraint modes for the Craig-Bampton method,  
As shown in Figure 1, when the dominant modes \(\mathbf{Y}_{\mathrm{D}}\) are selected, the SMDs of these modes (\(\mathbf{\Theta}\)) are computed. In the next section, the selection of the most relevant SMDs is described.