## 4 Results
### Global Response Characteristics of VC Resonances around the Bearing Clearance-Free Operations  
Figure 12 shows the power spectra of the dynamic displacement parameter (\(DDP\)) [9]: \(DDP=x_{\text{peak-to-peak}}+y_{\text{peak-to-peak}}\), where \(x_{\text{peak-to-peak}}=x(t)_{\text{max}}-x(t)_{\text{min}}\), and \(y_{\text{peak-to-peak}}=y(t)_{\text{max}}-y(t)_{\text{min}}\). To reflect the hysteresis characteristics, here the larger \(DDP\) values are selected from the two \(DDP\) values obtained as the controlled parameter \(\Omega\) runs up and down. It is clear that the system contains many types of VC parametric resonances. From Section 4.1, some resonance types of the two degrees-of-freedom of the system have been confirmed, such as the primary resonances (i.e., \(R_{1}\) and \(R_{2}\)), 1/2-order subharmonic resonances (i.e., \(R_{3}\) and \(R_{4}\)), 2-order superharmonic resonances (i.e., \(R_{6}\) and \(R_{7}\)), and the case of \(\omega_{x0}+\omega_{y0}=\Omega_{\text{VC}}\) combination resonance (i.e., \(R_{5}\)). Moreover, the cases of 1/2-(\(\omega_{x0}+\omega_{y0})=\Omega_{\text{VC}}\) combination resonance (i.e., the 2-order summed type of combination resonance [39; 57]) and 1/2-(1/4\(\omega_{x0}+\omega_{y0})=\Omega_{\text{VC}}\) combination resonance (labelled as \(R_{8}\) and \(R_{9}\), respectively) were found in our recent work [25]. As shown in Figure 13, these resonant locations can be predicted well by the equivalent resonant frequencies \(\omega_{x0}\) and \(\omega_{y0}\) from the arithmetic mean of Equation (8) at \(\Omega=700\) rad/s, where we assume that the system is far from any resonance when \(\Omega\) takes 700 rad/s to achieve an almost linearized dynamic stiffness (7). In addition, there are the other 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 (labelled as \(R_{10}\) and \(R_{11}\), respectively) excited as the bearing clearance \(\delta_{0}\) increases beyond 2.0 \(\upmu\)m.  
As shown in Figures 12 and 13, the equivalent resonant frequency \(\omega_{y0}\) quickly falls relative to \(\omega_{x0}\). This results in the resonant amplitudes \(R_{2}\) and \(R_{4}\) shifting to a significantly lower frequency, and the locations of primary resonance \(R_{1}\) and the 1/2-order subharmonic resonance \(R_{4}\) to be closer. For \(\delta_{0}=4.0\)\(\upmu\)m, the equivalent resonant frequencies of the two degrees-of-freedom of System (5) are about \(\omega_{x0}=1881.47\) rad/s and \(\omega_{y0}=1047.21\) rad/s, and the amplitudes of \(R_{1}\) and \(R_{2}\) as \(\Omega\) rotates around \(\omega_{x0}\)/\(N_{b}=235.18\) rad/s and \(\omega_{y0}\)/\(N_{b}=130.90\) rad/s in Figure 14 are the two VC primary resonances,  
Figure 12: Power spectra of response amplitude around the bearing clearance-free operations. Figure 13: The locations of primary resonances (blue lines), 1/2-order subharmonic resonances (black lines), 2-order superharmonic resonances (green lines), and combination resonances (red lines) predicted by the equivalent resonant frequencies \(\omega_{x0}\) and \(\omega_{y0}\). respectively. Herein, the primary resonance \(R_{1}\) and the 1/2-order subharmonic resonance \(R_{4}\) are close to a slight superposition. The case of \(\omega_{x0}+\omega_{y0}=\Omega_{\rm VC}\) combination resonance is triggered from the subcritical and supercritical secondary Hopf bifurcations occurring at points \(E_{5}\) and \(E_{6}\), and bends to the left, exhibiting strong soft spring characteristics (see Figure 15). There are abundant quasi-period responses (e.g., see Figure 16), tori doubling (e.g., see Figure 17) [22,51], and even chaotic motion (e.g., see Figure 18) excitements. Moreover, VC period responses such as period-35 and period-8 motions are also excited by this combination resonance as shown in Figures 19 and 20, respectively.