## 5 Conclusions  
In this paper, we investigated the response characteristics of the global VC parametric resonances in a ball bearing system around the bearing clearance-free operations, considering the nonlinearities of bearing clearance and the Hertzian contact between the rolling elements and raceway. The VC period-\(M\) motions and their stabilities were traced using the HB-AFT method and Floquet theory. It was shown that the system can contain primary resonances, 1/2-order subharmonic resonances, 2-order superharmonic resonances, and various cases of combination resonances. The 1/2-order VC subharmonic resonances are excited by supercritical/subcritical period-doubling bifurcations, leading to the emergence of VC period-2 motions. Moreover, it was found that the cases of \(\omega_{x0}+\omega_{y0}=\Omega_{\text{VC}}\), 1/2-(\(\omega_{x0}+\omega_{y0}\)) = \(\Omega_{\text{VC}}\), 1/2-(1/4-\(\omega_{x0}+\omega_{y0}\)) = \(\Omega_{\text{VC}}\), 1/2-(1/2-\(\omega_{x0}+\omega_{y0}\)) = \(\Omega_{\text{VC}}\), and 1/2-(\(\omega_{x0}+1/2\)-\(\omega_{y0}\)) = \(\Omega_{\text{VC}}\) combination resonances can be excited by supercritical/subcritical secondary Hopf bifurcations. Due to the frequency modulation from hysteresis and dynamic stiffness characteristics, the VC parametric combination resonance can result in various VC motion types, such as period-4, period-8, period-35, different cases of quasi-period, and even chaotic responses in the system. Furthermore, it was indicated that the parametric combination resonance is an internal physical mechanism of VC  
Figure 32: Response characteristics of the VC quasi-period motion when \(\Omega\) = 211.9492 rad/s (i.e., in literature [14], \(\omega_{\text{s}}\) takes 5250 rpm), (**a**) orbit (black line) and its Poincare mapping (red dots), (**b**) frequency spectrum.  
Figure 33: Numerical bifurcation diagrams of local maxima of \(y(\tau)\) when \(\Omega\) sweeps up (black dots) and down (red dots).  
quasi-period responses trigged in the rolling bearing system. The above results indicate that adjusting the bearing clearance can control the resonant amplitude and its location. Finally, the classical study from [PERSON] et al. [14] was reanalyzed by our proposed method and the experimental verification of our findings was also discussed. This paper helps clarify the mechanism of complex nonlinear vibration behaviors in rolling bearings.  
Conceptualization, Z.Z.; investigation, Z.Z., Y.Z. and X.L.; methodology, Z.Z.; validation, Z.Z., Y.Z. and X.L.; supervision, T.S. and X.R.; writing--original draft preparation, Z.Z., X.L. and Y.D. All authors have read and agreed to the published version of the manuscript.  
This research was funded by China Postdoctoral Science Foundation (Grant No. 2019M661849), Jiangsu Planned Projects for Postdoctoral Research Funds (Grant No. 2018K164C), Fundamental Research Funds for the Central Universities (Grant No. 30920021155), China Scholarship Council (ID: 201906845022).  
Z.Z. gratefully acknowledges [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON] and [PERSON] for considerable help and valuable suggestions during his academic visit as an eingeladener gastwissenschaftler at Technische Universitat Ilmenau, Germany.  
The authors declare no conflict of interest.