## 4 Results
### Global Response Characteristics of VC Resonances around the Bearing Clearance-Free Operations
Based on the above analysis, it is clear that the VC parametric combination resonance can result in various response types to the system for a bigger bearing clearance \(\delta_{0}\). One reason is that, due to the coupling nonlinearities from the bearing clearance and the Hertzian contact of System (5), the amplitude of the combination resonance strongly bends to the left with a large interval of \(\Omega\), and this causes the main frequency components p and q of the response to be modulated significantly, because:  
\[N_{b}\cdot\Omega\cdot\left[ \begin{array}{c}p\\ q\end{array} \right]\approx\left[ \begin{array}{c}\omega_{x0}\\ \omega_{y0}\end{array} \right]. \tag{23}\]  
Secondly, considering the soft spring characteristics of the combination resonance tail, the equivalent resonant frequencies [\(\omega_{x0}\), \(\omega_{y0}\)] in Equation (23) reduce as \(\Omega\) sweeps down; for example, as the values of the main frequency components [p q] are [0.6345, 0.3655], [0.6312, 0.3688], [0.6259, 0.3741], [22/35, 13/35], and [5/8, 3/8] when \(\Omega\) takes 368 rad/s, 363 rad/s, 357 rad/s, 355 rad/s and 353 rad/s (see Figures 16-20), respectively. In addition, [\(\omega_{x0}\), \(\omega_{y0}\)] predicted from Equation (23) becomes smaller gradually, except \(\omega_{y0}\) at \(\Omega=355\) rad/s (see Table 5). Thirdly, in essence the dynamic stiffness (7) is time-varying and influenced by the response characteristics, so the equivalent resonant frequencies \(\omega_{x0}\) and \(\omega_{y0}\) might also fluctuate to some extent. Furthermore, due to the dangerous bifurcation [58] shown at points \(E_{7}\) and \(E_{8}\), there are also other complex hysteresis and jumping behaviors in the case of \(\omega_{x0}+\omega_{y0}=\Omega_{\rm VC}\) combination resonance. Figure 14: For \(\delta_{0}\) = 4.0 \(\upmu\)m, stable (solid) and unstable (dashed) VC periodic peak-to-peak frequency–response curves in \(x\) (red line) and \(y\) (black line) directions. Figure 16: For \(\delta_{0}=4.0\) μm, response characteristics of the VC quasi-period motion when \(\Omega=368\) rad/s, (**a**) orbit (black line) and its Poincare mapping (red dots), (**b**) frequency spectrum. Figure 17: For \(\delta_{0}=4.0\) μm, response characteristics of the VC quasi-period motion (the tori-doubling case) when \(\Omega=363\) rad/s, (**a**) orbit (black line) and its Poincare mapping (red dots), (**b**) frequency spectrum. Figure 15: For \(\delta_{0}=4.0\) μm, numerical bifurcation diagrams of local maxima (**a**) \(x(\tau)\) and (**b**) \(y(\tau)\) when \(\Omega\) sweeps up (black dots) and down (red dots) around 345–375 rad/s. Figure 19: For \(\delta_{0}=4.0\) μm, response characteristics of the VC period-35 motion when \(\Omega=357\) rad/s, (**a**) orbit (black line) and its Poincare mapping (red dots), (**b**) frequency spectrum. Figure 18: For \(\delta_{0}=4.0\) μm, response characteristics of the VC chaotic motion when \(\Omega=353\) rad/s, (**a**) orbit (black line) and its Poincare mapping (red dots), (**b**) frequency spectrum. For \(\delta_{0}=4.0\)\(\upmu\)m, the cases of 1/2-(\(\omega_{x0}+\omega_{y0}\)) = \(\Omega_{\text{VC}}\), 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 are excited as \(\Omega\) is close to 1/2-(\(\omega_{x0}+\omega_{y0}\))/\(N_{b}\) = 183.04 rad/s, 1/2-(1/2-\(\omega_{x0}+\omega_{y0}\))/\(N_{b}\) = 124.25 rad/s, and 1/2-(1/2-\(\omega_{x0}+\omega_{y0}\))/\(N_{b}\) = 150.32 rad/s (i.e., the \(R_{8}\), \(R_{10}\), and \(R_{11}\) ranges in Figure 14), respectively.