## 4 Results and Discussion
### Effects of Mass Unbalance  
In this section, the clearance-to-radius ratio \(C/R\) is set at 0.485%. To thoroughly investigate the influence mechanism of base motion on the rotor system under deteriorating unbalanced conditions, two unbalanced values are selected: 252 g mm and 360.04 g mm. Given that base motion may excite higher-order responses under large unbalanced conditions, the harmonic order is set to H = 8 to ensure computational accuracy. Figure 16: Frequency–response curves and rotor trajectories at the SFD node under base yawing motion (\(C/R=0.485\%,\;U=144.02\;\mathrm{g\cdot mm}\)) (a) frequency–response curves; (b) rotor trajectories (red: clearance; dot: center). Figure 15: Frequency–response curves of harmonic components at disk node under base pitching motion (\(C/R=0.485\%,\;U=144.02\;\mathrm{g\cdot mm}\)): (a) \(\omega_{by}=8\;\mathrm{rad/s}\); (b) \(\omega_{by}=9\;\mathrm{rad/s}\) (c) \(\omega_{by}=10\;\mathrm{rad/s}\). The frequency-response characteristics at the SFD node and the disk node under base pitching motion are presented in Figures 17 and 18, where unstable solutions are denoted by dashed lines. With the increase of pitching angular velocity from 6 rad/s to 9 rad/s, a slight shift of the resonance peak toward higher frequencies is observed. Under moderate unbalance (\(U=252\) g \(\cdot\) mm), unstable solutions persist at a pitching angular velocity of 6 rad/s, which are identified as saddle-node bifurcation points through Floquet multiplier analysis. These unstable solutions completely vanish when the pitching angular velocity reaches 7 rad/s. In contrast, under large unbalanced conditions (\(U=360.04\) g \(\cdot\) mm), the bifurcation type remains consistently characterized as saddle-node bifurcation, with only a reduction observed in the rotational speed range exhibiting bistable behavior phenomenon. Figure 19 quantitatively illustrates the variation trends of the instability region (\(\Delta\widetilde{\Omega}\)) with respect to pitching angular velocity under different unbalanced conditions. For \(U=144.02\) g \(\cdot\) mm, the instability region is minimal under fixed base conditions and completely disappears when the pitching angular velocity increases to 4 rad/s. At \(U=252\) g \(\cdot\) mm, the initial instability region exhibits \(\Delta\widetilde{\Omega}=0.041\) and gradually diminishes with increasing pitching angular velocity, ultimately vanishing when the pitching angular velocity reaches 7 rad/s. For \(U=360.04\) g \(\cdot\) mm, the initial instability region demonstrates  
Figure 17: Frequency–response curves and rotor trajectories at the SFD node under base pitching motion (\(C/R=0.485\%\)) (**a**) \(U=252\) g \(\cdot\) mm; (**b**) \(U=360.04\) g \(\cdot\) mm. Figure 18: Frequency–response curves and rotor trajectories at the disk node under base pitching motion (\(C/R=0.485\%\)) (**a**) \(U=252\) g \(\cdot\) mm; (**b**) \(U=360.04\) g \(\cdot\) mm. \(\Delta\widetilde{\Omega}=0.195\) and exhibits an approximately linear decreasing trend with increasing pitching angular velocity, maintaining an unstable region of \(\Delta\widetilde{\Omega}=0.119\) even at 10 rad/s. These observations indicate that base pitching motion generally enhances system stability; however, its suppression effectiveness demonstrates a distinct unbalanced threshold characteristic. At lower unbalanced magnitudes, the bistable behavior can be completely suppressed through moderate base pitching excitation. As the unbalanced magnitude increases, the instability region expands significantly, requiring correspondingly higher pitching angular velocities for effective suppression. Figure 20 presents the frequency-response curves of horizontal displacement at the SFD node under a base pitching angular velocity \(\omega_{by}=9\) rad/s for different unbalanced magnitudes. The three-dimensional spectral plots reveal varying degrees of change in the response amplitudes across different order components as the rotational speed increases.