## 4 Results
### Discussion on the Verification of Global VC Parametric Resonances  
Different from previously works on parametric VC primary resonances, in this paper we extend to study many other parametric VC resonance types such as 1/2-order subharmonic resonances, 2-order superharmonic resonances, and various VC combination resonances. For verification of these new findings, we use our proposed method to reanalyze a classical work from [PERSON] et al. [14], where they presented that sub/super-harmonic, multi-period, quasi-periodic, and chaos-like motions emerge in System (5). As shown in Table 6, this ball bearing model has a larger bearing clearance \(\delta_{0}\) = 20 \(\upmu\)m and a lighter radical load \(W\) = 58.8 N compared to the above-studied model. \begin{table}
\begin{tabular}{c c} \hline \hline
**Item** & **Value** \\ \hline Contact stiffness \(C_{b}\) (N/m\({}^{3/2}\)) & \(1.334\times 10^{10}\) \\ Ball diameter \(D_{b}\) (mm) & 11.9062 \\ Pitch diameter \(D_{h}\) (mm) & 52.0 \\ Number of balls \(N_{b}\) & 8 \\ Bearing clearance \(\delta_{0}\) (\(\upmu\)m) & 20 \\ Equivalent mass \(m\) (kg) & 16 \\ Damping factor \(c\) (Ns/m) & 2940 \\ Radial load \(W\) (N) & 58.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Specifications of the ball bearing in [14]. Figure 27: For \(\delta_{0}=-1.1\)\(\upmu\)m, the response characteristics in 1/2-order subharmonic resonance, (**a**) the frequency–response curves in \(x\) (red line) and y (black line) directions, and (**b**) numerical bifurcation diagrams of local maxima of \(y(\uptau)\) when \(\Omega\) sweeps up (black dots) and down (red dots). Figure 26: For \(\delta_{0}=-1.1\)\(\upmu\)m, stable (solid) and unstable (dashed) VC periodic peak-to-peak frequency–response curves in \(x\) (red line) and \(y\) (black line) directions. Figure 28 presents frequency-response curves of the VC parametric response of the model in Table 6. In this case, the equivalent resonant frequencies of the model are about \(\omega_{x0}=1463.75\) rad/s and \(\omega_{y0}=370.74\) rad/s. Compared to the case of \(\delta_{0}=4.0\)\(\upmu\)m in Section 4.2 (i.e., \(\omega_{x0}=1881.47\) rad/s and \(\omega_{y0}=1047.21\) rad/s), the resonant excitation frequencies fall a lot in [PERSON]'s model. Herein, the VC primary resonances as \(\Omega\) rotates around \(\omega_{x0}/N_{b}=182.97\) rad/s and \(\omega_{y0}/N_{b}=46.34\) rad/s are also excited (i.e., the \(R_{1}\) and \(R_{3}\) ranges). These two primary resonances exhibit soft spring and soft-hard spring characteristics similar to the case of \(\delta_{0}=4.0\)\(\upmu\)m studied. Through supercritical and subcritical period-doubling bifurcations trigger at points \(F_{1}\) and \(F_{2}\), the 1/2-order subharmonic resonance in the horizontal direction is strong excited in a wide range \(R_{4}\) as \(\Omega\) rotates around \(2\cdot\omega_{y0}/N_{b}=92.68\) rad/s, and VC period-2 resonant responses occur. It should be note that 2-order superharmonic resonance in the vertical direction dominates the resonant amplitude \(R_{6}\) in the vertical direction as \(\Omega\) rotates around 1/2\(\cdot\omega_{x0}/N_{b}=91.48\) rad/s, because the main frequency component of \(x(\tau)\) in this range is 2-order harmonics (i.e., the response characteristics at \(\Omega=88\) rad/s presented in Figure 29). Moreover, the period-2 solution branch \(F_{1}\)-\(F_{2}\) undergoes period-doubling bifurcations at \(F_{3}\), \(F_{4}\), \(F_{5}\) and \(F_{6}\), generating period-4 branch \(F_{3}\)-\(F_{4}\) and \(F_{5}\)-\(F_{6}\). Especially, there are various VC motion types arising subsequently on branch \(F_{5}\)-\(F_{6}\), including chaotic responses, and this branch contains the beat/chaos-like range b-b described by [PERSON] et al. (2014) (see Figure 30). Figure 30: The peak-to-peak frequency–response curves of literature [14], in (**a**) \(x\) and (**b**) \(y\) directions. Herein, the blue and red lines are the period-1 and period-2 solution branches of Figure 28, respectively, obtained by our work.