## 3 Friction-Induced Vibration
### Sliding Vibrations  
In this section, we find three categories of sliding oscillations in the friction involved in the rotor/stator system. The parameters considered in the nonlinear rotor system are as follows:  
\[\beta=0.04,\ \gamma=0.05,\ \zeta=0.05,\ R_{0}=1.05,\ R_{d}=20R_{0} \tag{12}\]  
and \(\mu\in[0,0.4]\) and \(\Omega\in[0,1]\). For the study of the sliding motions, we characterize the duration of sliding as follows:  
\[\tau_{rel}=\frac{\tau_{\text{sliding}}}{T} \tag{13}\]  
where \(\tau_{\text{sliding}}\) is the absolute value of the sliding length in one period (\(T\)). When \(\tau_{rel}=1\), the disk performs a pure roll along the stator in one whole period of the oscillation. Figures 2 and 3 present the results of the pure rolling oscillation. Consider the parameters of \(\Omega=0.2\) and \(\mu=0.2\); the rotor/stator system in Equation (3) is characterized with the pure rolling motion of backward whirling oscillation induced by dry friction, as shown in Figure 2a. The green response represents the pure rolling orbit, and the black is the rotor/stator clearance. In Figure 2b, the time history normal components of two vector fields, namely, \(F_{1}(x)\) and \(F_{2}(x)\), and the velocity (\(V_{rel}\)) are presented. The \(V_{rel}\) does not change much during this type of motion and with the value close to zero. It is noted that, in the sliding domain, the condition (5) can always be satisfied in this case, as shown in Figure 2b. Namely, \(\mathcal{L}_{F_{1}}H(x)\cdot\mathcal{L}_{F_{2}}H(x)<0\) is always satisfied. So, as for this type of motion, the relative sliding time length can be concluded as \(\tau_{rel}=1\). Similarly, the pure rolling oscillation in terms of \(\Omega=0.2\) and \(\mu=0.2\) is presented in Figure 3.  
Figure 3: Backward whirling oscillation from pure rolling. (**a**) Disk orbit and (**b**) time history velocity and normal components of vector fields. (Green: continuous pure rolling; black: the stator clearance. \(\Omega=0.8\) and \(\mu=0.3\)). Figure 2: Backward whirling oscillation from pure rolling. (**a**) Disk orbit and (**b**) time history velocity and normal components of vector fields. (Green: continuous pure rolling; black: the stator clearance. \(\Omega=0.2\) and \(\mu=0.2\)). Figures 4 and 5 present the scenario of continuous crossing oscillation. Consider the parameters of \(\Omega=0.9\) and \(\mu=0.16\); the backward whirling oscillation induced by dry friction in the rotor/stator system exhibits the continuous crossing oscillation, as shown in Figure 4a. The blue response represents the continuous crossing oscillation orbits, and the black is the rotor/stator clearance. In Figure 4b, the time history normal components of two vector fields, \(F_{1}(x)\) and \(F_{2}(x)\), and the velocity (\(V_{rel}\)) are presented. The velocity (\(V_{rel}\)) remains zero when it goes through the switching manifold. In this case of the sliding domain, the condition (5) in Figure 2b cannot be satisfied. Namely, \(\mathcal{L}_{F_{1}}H(x)\cdot\mathcal{L}_{F_{2}}H(x)>0\) is always satisfied. So, for this continuous crossing oscillation, the relative sliding time length can be concluded as \(\tau_{rel}=1\). Similarly, the continuous crossing oscillation in terms of \(\Omega=0.8\) and \(\mu=0.16\) is presented in Figure 5.  
When choosing the parameters of \(\Omega=0.4\) and \(\mu=0.2\), the rotor/stator system behaviors a backward whirling oscillation induced by dry friction with the characteristics of sliding oscillations and slipping oscillations. Figures 6 and 7 present the scenario of grazing-sliding bifurcation. The green response represents the pure rolling oscillations, and the blue is the slipping motions. In Figure 6b, the velocity (\(V_{rel}\)) and Lie derivatives are presented. For pure rolling oscillations, the \(V_{rel}\) is near zero, while, for slipping oscillations, the \(V_{rel}\) is over zero in such a rotor/stator nonlinear system with dry friction.