## 4 Results
### Global Response Characteristics of VC Resonances around the Bearing Clearance-Free Operations
The locations of these combination resonances are clearly exhibited by the numerical bifurcation diagrams in Figure 21. Herein, the case of 1/2-(\(\omega_{x0}+\omega_{y0}\)) = \(\Omega_{\text{VC}}\) combination resonance is triggered from the supercritical secondary Hopf bifurcations occurring at points \(E_{3}\) and \(E_{4}\), and results in quasi-period (e.g., see Figure 22) and period-4 resonant responses (e.g., see Figure 23) in the system; the main frequency components of the response satisfy 1/2-(p + q)= 1, which has been indicated in [25]. Figure 24 shows the quasi-period responses excited in the cases of 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, respectively, and there are chaotic motions excited in the case of the 1/2-(\(\omega_{x0}+1/2\cdot\omega_{y0}\)) = \(\Omega_{\text{VC}}\) combination resonance as shown in Figure 25. It has been found in a number of studies [5; 6; 10; 11; 12; 13; 14; 15; 16] that the quasi-period motion is a typical characteristic of VC vibrations in rolling bearing systems. However, to date, its physical mechanism has not been given. From above analysis, it is clear that various cases of combination resonances are internal physical mechanisms of excited quasi-period responses. Figure 21: For \(\delta_{0}=4.0\)\(\upmu\)m, numerical bifurcation diagrams of local maxima (**a**) \(x(\tau)\) and (**b**) \(y(\tau)\) when \(\Omega\) sweeps up (black dots) and down (red dots) around 100–200 rad/s. Figure 20: For \(\delta_{0}=4.0\)\(\upmu\)m, response characteristics of the VC period-8 motion when \(\Omega\) = 355 rad/s, (a) orbit (black line) and its Poincare mapping (red dots), (**b**) frequency spectrum. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\Omega\) & **353** & **355** & **357** & **363** & **368** \\ \hline \(\omega_{x0}\) (rad/s) & 1767.54 & 1775.00 & 1795.20 & 1833.00 & 1867.97 \\ \(\omega_{y0}\) (rad/s) & 1056.46 & 1065.00 & 1060.80 & 1071.00 & 1076.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The equivalent resonant frequencies of the system predicted from Equation (23). Figure 23: For \(\delta_{0}\) = 4.0 \(\upmu\)m, response characteristics of the VC period-4 motion when \(\Omega\) = 180 rad/s, (**a**) orbit (black line) and its Poincare mapping (red dots), (**b**) frequency spectrum. Figure 22: For \(\delta_{0}\) = 4.0 \(\upmu\)m, response characteristics of the VC quasi-period motion when \(\Omega\) = 181.2 rad/s, (**a**) orbit (black line) and its Poincare mapping (red dots), (**b**) frequency spectrum. Comparing Figures 12 and 13, it is clear that not all cases of the predicted VC parametric resonances will be excited; for example, the predicted 1/2-order subharmonic resonance \(R_{3}\) in the interval a-a. For \(\delta_{0}=-1.1\)\(\upmu\)m, the equivalent resonant frequencies of the system are about \(\omega_{x0}=1921.18\) rad/s and \(\omega_{y0}=1915. adjusting the bearing clearance may make it operate in a linear range to some extent, and thus avoid some resonances.