## 1 Introduction
[32] presented the amplitude and stability of VC responses for a rolling bearing-rotor system in a large speed range; however, the mechanism of the resonant amplitude has been little addressed. [PERSON] [33] observed high-order subharmonic vibration responses under low damping and strong nonlinearity conditions in high speed rotating machinery with asymmetric bearing supports, based on the numerical integration of a finite difference formulation. [PERSON] et al. [34] showed the soft and hard hysteretic 1/2-order subharmonic resonances in different degrees of freedom of a ball bearing-rotor system using the multi-scale method. Although this study also found combination resonances between the 1/2-order subharmonic resonance intervals, this has not been analyzed in depth. [PERSON] et al. [35] established a 6 DOF nonlinear ball bearing-rotor model and found that the occurrence of 2-order subharmonic resonance is provoked by the coupling of Hertzian contact and internal clearance nonlinear factors of ball bearings. [PERSON] et al. [36] noted that the subharmonic hysteretic resonance become significant and the resonance amplitude moved to lower frequencies as the assembly gap increased for an angular contact high-speed ball bearing system. However, these studies [33; 34; 35; 36] relate to the case of external excitations from unbalanced rotors, for which a small excitation produces a large response only if the frequency of the external excitation is close to that of the linear natural frequency of the system, and it is essentially different from that of the parametric excitation system. Therefore, a global mechanism analysis of VC parametric resonances in rolling bearings is necessary. Bearing clearance is a basic parameter of rolling bearings, and has a significant effect on bearing life, installation, and bearing stiffness [11; 12]. [PERSON] [37] previously showed that the resonant frequency and amplitude can be reduced to some extent as the clearance adjusts. [PERSON] et al. [38]noted that the bearing life declines gradually with positive clearance and rapidly with increasing negative clearance, and the life can be maximized for a small negative operating clearance. In the field of bearings, it is generally considered that adverse vibration and noise behaviors can be reduced by adjusting the ball bearings to zero clearance [39]. In particular, a clearance-free operation is usually desirable for precision or high-speed machinery, such as machine tools and turbines [11]. Therefore, the study of VC parametric resonances in clearance-free operations for rolling bearing systems is valuable. The harmonic balance and alternating frequency/time domain (HB-AFT) method is an effective means to obtain the steady-state harmonic solutions of a general nonlinear system. It can be used to quickly obtain the frequency domain information of the nonlinear term for the harmonic balance process of the AFT discretization technology [40]. [PERSON] et al. [41] and [PERSON] et al. [42] developed the generalized HB-AFT method to analyze the quasi-periodic response characteristics of multiple degrees-of-freedom and multi-frequency excitation systems. [PERSON] et al. [22] embedded continuation and [PERSON]'s Floquet stability analysis technology into the HB-AFT method, which can automatically track the periodic response and bifurcation characteristics of nonlinear systems. Recently, [PERSON] et al. [43] developed a harmonic balance method combined with the model incremental transfer matrix of the system, which is highly convenient for dealing with systems with multiple degrees-of-freedom. The motivation of the present work is to provide a better understanding of the VC parametric resonances in a ball bearing. The mechanisms of various new cases of parametric resonances are provided mainly using the HB-AFT method with the Floquet theory, and their detailed response characteristics, such as time series, orbit, stability, and hysteresis, are investigated.