## 4 Results
### Discussion on the Verification of Global VC Parametric Resonances
Figure 29: Response characteristics of the VC period-2 motion when \(\Omega=88\) rad/s, (**a**) orbit (black line) and its Poincare mapping (red dots), (**b**) time series and their frequency spectra. Furthermore, the stable period-1 solution branch undergoes supercritical Hopf bifurcations at points \(F_{7}\) and \(F_{8}\), and the coming unstable solution branch \(F_{7}\)-\(F_{8}\) (i.e., \(R_{5}\) range around \(\omega_{x0}\)/\(N_{b}+\omega_{y0}\)/\(N_{b}\) = 229.31 rad/s) contains various VC responses such as period-5, period-6 and beat/chaos-like motions (see Figure 30). The response characteristics of the quasi-period motion as \(\varOmega\) rotates at 242.2277 (i.e., in literature [14], \(\omega_{8}\) takes 6000 rpm) are presented in Figure 31. Herein, the values of the main incommensurable dimensionless VC frequencies p and q are about 0.7715 and 0.2285, respectively, and p + q = 1, where the value 1 is the dimensionless VC excitation frequency \(\varOmega_{\text{VC}}\) = \(N_{b}\)-\(\varOmega\). Moreover, for the original time scale, the corresponding main frequency components of the response are \(N_{b}\)-\(\varOmega\)\(\cdot\)p \(\approx\) 1495.03 rad/s and \(N_{b}\)-\(\varOmega\)\(\cdot\)q \(\approx\) 442.79 rad/s, which are close to the predicted equivalent resonant frequencies \(\omega_{x0}\) = 1463.75 rad/s and \(\omega_{y0}\) = 370.74 rad/s of the two degrees-of-freedom of System (5). As stated in Section 4.1, this indicates that the case of \(\omega_{x0}+\omega_{y0}=\varOmega_{\text{VC}}\) combination resonance occurs in range \(R_{5}\). Different from the previous analysis (see Figures 7b and 16b), as shown in Figure 31b, in this case the response-frequency spectrum contains component of 2-q, so there may be the case of \(\omega_{x0}+\)1/2-\(\omega_{y0}=\varOmega_{\text{VC}}\) combination resonance also excited in \(R_{5}\), and actually the value \(\omega_{x0}\)/\(N_{b}\) + 1/2-\(\omega_{y0}\)/\(N_{b}\) = 206.14 rad/s (see Figure 28) is in range \(R_{5}\). This can explain why the instability range \(R_{5}\) excited by the combination resonance is so large in [PERSON]'s model. In other words, due to the more asymmetric stiffness characteristics (where \(\omega_{x0}\) = 1463.75 rad/s and \(\omega_{y0}\) = 370.74 rad/s), [PERSON]'s model can contain more types of parametric resonances, and these resonances may overlap (literature [27] contains a detailed discussion of this overlapping phenomenon). Due to the vibration mechanism being more complicated, many researchers [44; 60] have devoted themselves to investigating the dynamic characteristics of asymmetric rolling bearing-rotor systems. The response characteristics of the period-5 motion as \(\varOmega\) rotates at 211.9492 rad/s (i.e., in literature [14], \(\omega_{\text{s}}\) takes 5250 rpm) are presented in Figure 32, where its main frequency components p and q are 4/5 and 1/5, respectively; satisfying p + q = 1, and using the above analysis process, it can be found that this period-5 response is also excited by the case of \(\omega_{x0}+\omega_{y0}=\varOmega_{\text{VC}}\) combination resonance. Therefore, the case of \(\omega_{x0}+\omega_{y0}=\varOmega_{\text{VC}}\) combination resonance is the internal mechanism of excitation of various types of VC responses in \(R_{5}\) range described by [PERSON] et al. [14] (see Figure 30). In addition, it can be easily verified by our analysis process that the resonant amplitude \(R_{12}\) comes from 6-order superharmonic resonance in the vertical direction, because this resonant range emerge as \(\varOmega\) rotates around 1/6-\(\omega_{x0}\)/\(N_{b}\) = 30.50 rad/s (see Figures 28 and 30). As shown in Figure 33, our above analysis results agree well with the numerical bifurcation diagrams. For experimental verification, because the VC vibration response is a weak signal, the vibration is usually measured by non-contact displacement sensor in experiments [24; 34; 61].