## 4 Results and Discussion
### Effects of Base Motions
#### 4.1.2 Base Pitching Motion  
Figure 12 illustrates the frequency-response curves at SFD and disk nodes under various base pitching angular velocities. The results demonstrate that with increasing pitching velocity, the deflection at the SFD node shows an overall upward trend. According to Equation (4), the inertial moment term \(\Omega\)\({}_{i}^{P}\)\(\omega_{by}\) introduced by base pitching motion increases with rotational speed, leading to additional displacement in the system. The amplitude at the critical position of the disk node exhibits characteristics of an initial increase followed by a decrease. At \(\omega_{by}=2\) rad/s, the bifurcation points near the resonance peak remain saddle-node bifurcation points. When \(\omega_{by}\) increases to 4 rad/s, the bistable phenomenon in the solution curves disappears, and the system hardening spring characteristics are suppressed. As the pitching angular velocity increases to 9 rad/s, the primary resonance peak at the disk node significantly decreases, but a secondary resonance peak emerges in the high-speed region \(\left(\overset{\sim}{\Omega}>1.3\right)\). This phenomenon indicates that under conditions of relatively small mass unbalance, appropriate base pitching angular velocity can improve the system bifurcation characteristics and enhance dynamic stability. However, excessive base pitching angular velocity can lead to significant static offset in the high-speed region and may even generate secondary resonance phenomena. Figure 11: Frequency–response curves of harmonic components under base rolling motion (\(C/R=0.485\%\), \(U=144.02\) g \(\cdot\) mm, \(\omega_{bx}=10\) rad/s) at nodal points of: (**a**) SFD; (**b**) disk. Figure 13 demonstrates the frequency-response characteristics of various harmonic components when the base pitching angular velocity is \(\omega_{by}=6\) rad/s. Analysis results indicate that the horizontal response at the SFD node contains static displacement components, fundamental frequency component \(\Omega\), and higher harmonic components \(2\Omega\) and \(3\Omega\), with static displacement and fundamental frequency \(\Omega\) components being dominant. The disk node response is primarily governed by the constant term and fundamental frequency component \(\Omega\), with the static displacement component exhibiting a notable peak characteristic near the critical speed. The static displacement component at the SFD node generally increases with rotational speed but shows a slight decrease near the critical speed. This phenomenon is further validated by the journal orbit at the SFD node, shown in Figure 14: when the speed approaches the critical speed (\(\widetilde{\Omega}=1.03\)), despite the significantly enlarged orbit, its center is positioned closer to the bearing center. This dynamic characteristic is similar to the self-centering effect observed in rotor systems with static eccentricity SFD [7]. To better observe the secondary resonance phenomenon at the disk node, Figure 15 presents the frequency-response curves of various harmonic components in the horizontal direction under different pitching angular velocities. At a pitching angular velocity of 8 rad/s, the synchronous frequency-response curve exhibits a single distinct peak at \(\widetilde{\Omega}=1.09\). When the pitching angular velocity increases to 9 rad/s, this primary peak amplitude decreases, and a secondary peak emerges at \(\widetilde{\Omega}=1.40\). Upon further increasing the pitching angular velocity to 10 rad/s, the primary peak continues to decrease while the secondary peak amplitude increases, making the dual-peak phenomenon more pronounced. This phenomenon can be attributed to the large static eccentricity induced at high rotational speeds under base pitching motion. The significant static  
Figure 14: Rotor trajectories at the SFD node under different rotational speeds (\(C/R=0.485\%\), \(U=144.02\) g \(\cdot\)mm, \(\omega_{by}=6\) rad/s).