## 3 Friction-Induced Vibration
### Cross-Coupling Stiffness Effects  
This section studies the effects of the system parameters on friction-induced, nonlinear vibrations and bifurcations. The influence of the cross-coupling stiffness on the sliding time length is presented in the parameter windows of \(\tau_{\text{rel}}-\Omega\) and \(\mu-\Omega\).  
Consider the parameters based on the experiments in Ref. [11]:  
\[\beta=0.04,\ \zeta=0.05,\ R_{0}=1.05,\text{ and }R_{d}=20R_{0},\]  
The response of \(\tau_{\text{rel}}-\Omega\) is calculated during \(\Omega\in[0.03,\ 1]\) in Figure 9. It is noted that when the backward whirling oscillation induced by dry friction happens, the sliding time length is always less than one. This phenomenon indicates the existence of rotor-rolling and disk-slipping transitions where grazing-sliding bifurcation occurs. Such rotor/stator-coupled oscillation has been recognized as slipping oscillation [11]. The angular velocity (\(\Omega\)) is larger at the beginning, where the pure rolling motion part becomes smaller and then increases during one oscillation period. When the cross-coupling stiffness is smaller (\(\gamma\leq 0.05\)), the sliding time length (\(\tau_{\text{rel}}\)) reaches the smallest value, around \(\Omega\approx 0.45\). When \(\gamma\) is larger (\(\gamma\geq 0.12\)), the sliding time length (\(\tau_{\text{rel}}\)) becomes less than one at \(\Omega>0.33\). This phenomenon indicates that the angular velocity is very high for the chosen parameters of the rotor/stator system on which grazing-sliding bifurcation occurs. When \(\gamma\) goes up, the sliding time length (\(\tau_{\text{rel}}\)) equal to one happens at a very low angular velocity. Such a phenomenon indicates that the pure rolling oscillation happens at the very low angular velocity, where the backward whirling oscillation induced by dry friction just occurred.  
Figure 8: Experiment of the backward whirling oscillation: (**a**) test platform, (**b**) rotating orbits, and (**c**) frequency–amplitude characteristics.  
The response of \(\Omega-\mu\) to represent the vibration characteristics of stick-slip whirling is presented in Figure 10. It can be observed that the parameter response of \(\Omega-\mu\) consists of two parts: pure rolling and grazing-sliding bifurcation oscillation. Consider the parameter \(\gamma=0\); the black curves of \(\mu_{i}\) and \(\mu_{ii}\) illustrate the coefficients of the dry friction. In such a case, the pure rolling vibration at \(\tau_{\text{rel}}=1\) governs the motion domain. The stick-slip vibrations that happen are associated with grazing-sliding bifurcations for the range in which the coefficients of the dry friction are smaller than \(\mu_{u}\) and larger than \(\mu_{l}\). \(\mu_{u}\) goes to the minimum value, around \(\Omega\approx 0.45\), which is consistent with the phenomenon in Figure 5. When \(\gamma\) increases, the pure rolling vibration at the sliding time length \(\tau_{\text{rel}}=1\) happens for the larger friction coefficient. This phenomenon indicates that the pure rolling oscillation happens at a very low cross-coupling stiffness. So, the condition of the pure rolling motions can be guaranteed, and the conditions can be applied to predict the nonlinear vibrations and bifurcations of the rotor/stator system with no cross-coupling stiffness [13,14,17].  
From the results in Figures 9 and 10, we can conclude the following: (1) In such a rotor/stator-coupled nonlinear system, pure rolling vibrations (\(\tau_{\text{rel}}=1\)), continuous crossing vibrations (\(\tau_{\text{rel}}=0\)), and grazing-sliding-bifurcation-induced vibrations can occur. (2) If the coefficient (\(\mu\)) of the dry friction is very large or the cross-coupling stiffness (\(\gamma\)) is very small, pure rolling vibration may occur. (3) If the angular velocity (\(\Omega\)) is very low, the backward whirling oscillation induced by dry friction can be produced in rotor/stator systems, and pure rolling vibration may also happen. Meanwhile, if he cross-coupling stiffness is in the low range, then pure rolling exists for larger ranges.