## 3 Numerical Analysis
### Slip-Litoff Configuration
The analysis is also performed with another load case (\(N_{0}=200\) N, \(F_{\text{e}}=333.3\) N, and \(Freq=3.2\) Hz). In this case, assuming that the structure remains in the elastic region of the stress-strain curve, the contact again undergoes only slip and liftoff states and the maximum vibration amplitude is 418 mm. The geometric nonlinearity is activated to a greater extent in this case, leading to a higher contribution of geometrically coupled modes on the beam behavior. The results are shown in Figure 8 (right). The performances of Enhanced Rubin (Rubin-MD) and Enhanced Craig-Bampton (CB-MD) methods are also compared to Rubin and Craig-Bampton approaches. In particular, both SMD selection strategies are applied to the Enhanced Craig-Bampton ROM, as was previously the case for the Enhanced Rubin ROM. The results are shown in Figure 9.  
Figure 8: Response error and the size of reduction basis vs. the number of normal modes in slip-liftoff configuration (two load cases). Figure 7: (**a**) Contact node states, (**b**) contact tangential, and (**c**) contact normal forces with a geometrically linear model (\(N_{0}=40\) N, \(Freq=3.2\) Hz). The 8 th and 13 th free interface normal modes and the 9 th and 13 th fixed interface normal modes are axial modes that participate in the response due to their geometric coupling with transverse modes. As seen in Figure 9 (left), adding them to the Rubin and Craig-Bampton bases considerably improved these models' accuracy. However, these reduced models with a size of 16 still exhibit a 3% error. When the SMDs are added to the Rubin basis, the predicted response's accuracy improves significantly. Even when only the first normal mode and the two static modes (there are two contact DoFs) are considered and SMDs are computed, better accuracy is achieved than when employing large ROMs without SMDs. Figures 8 and 9 also showcase the superior performance of the MMI-NLS approach, highlighting the relevant contribution of the selected SMDs. This is more obvious in the case of higher load (Figure 8 (right)), where the response error of Enhance Rubin ROM with three normal modes is about 0.25 % using the MMI-NLS approach and 36% using the MMI approach. The SMD of the first attachment mode (an axial mode) is the mode in which its addition to the reduction basis leads to the priority of MMI-NLS approach. This SMD is not selected by the MMI selection criterion. To better explain the reason for the effect of the modal derivative selection on the accuracy, Table 2 shows the beam tip response error with different modes kept in the reduced space. 1V refers to the first linear normal mode and 1S refers to the first static mode. In the same way, the modal derivative 1 VIS is the derivative of the 1 st normal mode to the first static mode. This table again shows the importance of the selection criteria. The derivative of static modes will not be selected by MMI, since these modes have no considerable participation factors during a linear run. Indeed, the SMDs of the modes that are not excited in a linear run will never be selected by MMI criterion. The resulting error cannot be ignored when geometrically coupled modes or internally resonant modes have considerable participation in the response (stronger nonlinearities). The deformation of modal derivatives of the free interface vibration modes and residual flexibility attachment modes are shown in Figure 10, when the first free interface mode (1V) is the first bending mode. To better exhibition of axial and transverse modes, the axial displacement \(u_{\mathrm{x}}\) and transverse displacement \(u_{\mathrm{y}}\) of each node are depicted separately in the right and left plots, respectively.