## 4 Results
In addition, due to the nonlinearities from Hertzian contact and bearing clearance, this combination resonance has soft hysteresis characteristics, and the quasi-period response jumps to the VC period-1 motion branch as \(\Omega\) decreases to 388.20 rad/s. Figure 6: For \(\delta_{0}=1.5\) μm, numerical bifurcation diagrams of local maxima (**a**) \(y(\tau)\) and (**b**) \(x(\tau)\) when \(\Omega\) sweeps up (black dots) and down (red dots). For certain applications, such as high-speed or high-precision machines, to increase bearing stiffness and reduce the run-out and noise due to elastic deformation or clearance [11; 39], it is possible to eliminate clearance and introduce small interference (negative clearance) in the bearings. Figure 8 presents frequency-response curves of the VC parametric response for \(\delta_{0}\) = -1.5 \(\upmu\)m. In this case, the equivalent resonant frequencies of the two degrees-of-freedom of System (5) are about \(\omega_{x0}\) = 1959.49 rad/s and \(\omega_{y0}\) = 2105.55 rad/s. Compared to the case of \(\delta_{0}\) = 1.5 \(\upmu\)m, the resonant excitation frequency is higher in the horizontal direction than in the vertical direction, and this frequency exchange, due to the adjusting of bearing clearance, was previously illustrated by [PERSON] [56] and [PERSON] [25], respectively. Herein, the VC primary resonances as \(\Omega\) rotates around \(\omega_{x0}\)/\(N_{b}\) = 244.94 rad/s and \(\omega_{y0}\)/\(N_{b}\) = 263.19 rad/s, 1/2-order subharmonic resonances as \(\Omega\) rotates around 2\(\omega_{x0}\)/\(N_{b}\) = 489.88 rad/s and 2\(\omega_{y0}\)/\(N_{b}\) = 526.38 rad/s, and the case of \(\omega_{x0}\) + \(\omega_{y0}\) = \(\Omega_{\rm VC}\) combination resonance as \(\Omega\) rotates around \(\omega_{x0}\)/\(N_{b}\) + \(\omega_{y0}\)/\(N_{b}\) = 508.13 rad/s are also excited. The difference compared to the case of \(\delta_{0}\) = 1.5 \(\upmu\)m is that, as shown in Figure 9, the primary and 1/2-order subharmonic resonances of the system in vertical directions (i.e., the \(R_{1}\) and \(R_{3}\) ranges) exhibit soft-hard and hard spring characteristics, respectively, and the bifurcations excited by the combination resonance at the points \(C_{1}\) and \(C_{2}\) are all supercritical Hopf types. These bifurcation characteristics are clearly illustrated by the numerical bifurcation diagrams in \begin{table}
\begin{tabular}{c c c c c} \hline \(\Omega\) & **406.9645** & **407.0645** & **407.1645** & **407.2645** \\ \hline \multirow{4}{*}{\(\lambda_{m}\)} & \(-0.018\) + 1.031i & \(-0.019\) + 1.006i & \(-0.019\) + 0.971i & \(-0.066\) + 0.923i \\  & \(-0.018\) \(-1.031i\) & \(-0.019\) \(-1.006i\) & \(-0.019\) \(-0.971i\) & \(-0.066\) \(-0.923i\) \\  & \(-0.014\) + 0.831i & \(-0.016\) + 0.851i & \(-0.017\) + 0.883i & 0.027 + 0.925i \\  & \(-0.014\) \(-0.831i\) & \(-0.016\) \(-0.851i\) & \(-0.017\) \(-0.883i\) & \(0.027\) \(-0.925i\) \\ \hline \end{tabular}
\end{table} 
Table 4: Period-1 motion Floquet multipliers \(\Lambda_{m}\) around \(B_{7}\). Figure 7: For \(\delta_{0}\) = 1.5 \(\upmu\)m, VC quasi-period response characteristics, (**a**) orbit (black line) and its Poincare mapping (red dots), (**b**) frequency spectrum when \(\Omega\) = 400 rad/s. Figure 10. In addition, the VC 2-order superharmonic resonances of the two degrees-of-freedom of the system are strongly excited as \(\Omega\) rotates around 1/2-\(\omega_{x0}\)/\(N_{b}\) = 122.47 rad/s and 1/2-\(\omega_{y0}\)/\(N_{b}\) = 131.60 rad/s (i.e., the \(R_{6}\) and \(R_{7}\) ranges in Figure 8), respectively. Figure 11 presents the response characteristics of the 2-order superharmonic resonance in the \(y\) direction of the system, in which the superharmonic frequency component dominates the period-1 response of \(y(\tau)\). Figure 8: For \(\delta_{0}\) = \(-\)1.5 \(\upmu\)m, stable (solid) and unstable (dashed) VC periodic peak-to-peak frequency–response curves in \(x\) (red line) and \(y\) (black line) directions.