## 3 Numerical Analysis
### Slip-Litoff Configuration  
The dynamic behavior of the beam at the steady state when excited at the frequency of 3.2 Hz under a normal preload of 40 N is depicted in Figure 5. Figure 5a shows the beam deflections at different time instants, Figure 5b,c show the contact normal and tangential displacements, respectively, that are indeed the axial and transverse displacements of the beam tip. Figure 5b highlights that the frequency of the axial vibration is twice the frequency of excitation. Figure 4: Frequency response function of the beam tip with four values of contact normal preload. In this configuration, the amplitude of vibration is 21% smaller than the maximum amplitude of the reference system (open contact). The reason for this smaller amplitude can be seen in Figure 6. The contact node state is shown in Figure 6, where slip, stick, and liftoff are depicted by 0, 1, and \(-1\), respectively. It shows that the contact nodes alternate liftoff and slip. As shown in Figure 6, the contact tangential and normal forces are zero during liftoff. The energy dissipated by friction during slip has led to an increase in the damping ratio and a decrease in the vibration amplitude. A comparison is also made between the beam responses obtained by geometrically linear (GL) and nonlinear (GNL) models under the same preload and excitation frequency. The beam tip vibration amplitudes in transverse (\(U_{y}\)) and axial (\(U_{x}\)) directions are presented in Table 1, clearly demonstrating the inaccuracy of the GL model. As shown in Figure 7, the GL model is not capable of predicting the contact liftoff state under the current load and frequency. Consequently, a longer slip state (a higher damping) and a longer stick state (a higher stiffness) are predicted, leading to a smaller amplitude of vibration. This is due to the lack of axial-bending coupling in the GL model that underestimates the axial vibration, which plays a critical role in defining the contact state. \begin{table}
\begin{tabular}{l c c c c} \hline \hline  & \begin{tabular}{c} **Slip–Liftoff** \\ \(\mathbf{U_{y}}\) (mm) \\ \end{tabular} & \begin{tabular}{c} **Open Contact** \\ \(\mathbf{U_{x}}\) (mm) \\ \end{tabular} & \
\begin{tabular}{c} **Open Contact** \\ \(\mathbf{U_{y}}\) (mm) \\ \end{tabular} &
\begin{tabular}{c} **U\(\mathbf{x}\) (mm)** \\ \end{tabular} \\ \hline Geometrically & 123.5 & 0.00039 & 200.8 & 0.0 \\ Linear model & 157.6 & 2.835 & 200.5 & 4.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Vibration amplitude obtained by GL and GNL models in slip–liftoff configuration (\(N_{0}=40\) N, \(\textit{Freq}=3.2\) Hz). Figure 5: (a) Beam deformations, (b) contact normal displacement, and (c) the beam tip displacement in slip–liftoff configuration. Figure 6: (a) Contact node states, (b) contact tangential, and (c) contact normal forces in slip-liftoff configuration (\(N_{0}=40\) N, \(\textit{Freq}=3.2\) Hz). Finally, the performance of the reduction methods is evaluated in this beam configuration. First, the SMDs selection approaches, described in Section 2.2, are investigated. Both the original MMI criterion, based on linear transient analysis, and the modified MMI criterion (MMI-NLS), based on nonlinear static analysis and geometrically linear run with friction contact, are used for the sake of comparison. The steady-state beam response is computed by the enhanced Rubin method. Due to the importance of accurate prediction of the blade tip displacements, the error is considered to be the percentage error in the blade tip transverse vibration amplitude. This error (left axis) and the size of the reduced model (right axis) are shown in Figure 8 (left), versus the number of the normal modes used to determine the dominant modes and the SMDs. The reduced model size refers to the total number of static modes, the dominant normal modes, and the selected SMDs. Since the beam tip transverse and axial displacements (contact DoFs) are only kept as master DoFs in the reduced coordinates, there are two static modes.