## 4 Results and Discussion
### Effects of SFD Parameters  
To investigate the influence of squeeze film damper parameters on system instability, analyses were conducted with fixed unbalanced magnitude \(U=252\) g \(\cdot\) mm and pitching angular velocity \(\omega_{by}=10\) rad/s, to investigate the effects of varying clearance-to-radius ratios (\(C/R\)) on the system frequency-response characteristics. Figure 21 illustrates the frequency-response curves at the SFD node under \(C/R=0.485\%\) and \(0.61\%\), where circle markers denote saddle-node bifurcation points. The results indicate that at \(C/R=0.485\%\), the system maintains stable motion within the specified speed range. When \(C/R\) increases to \(0.61\%\), although the frequency-response curves in both horizontal and vertical directions exhibit trends similar to those at \(C/R=0.485\%\), enhanced nonlinear effects lead to the emergence of unstable solutions. Floquet multiplier analysis identifies these instability points as saddle-node bifurcations, resulting in the reappearance of bistable characteristics in the system. Figure 22 presents the frequency-response curves at the SFD node when \(C/R\) increases to \(0.728\%\), where circle markers indicate saddle-node bifurcation points and triangular markers denote secondary Hopf bifurcation points. Multiple unstable solution segments appear on the periodic solution curve, with Table 3 detailing the characteristic variations of Floquet multipliers at bifurcation points. Figure 21: Frequency–response curves at the SFD node under base pitching motion (\(U=252\) g \(\cdot\) mm, \(\omega_{by}=10\) rad/s) (**a**) horizontal displacement; (**b**) vertical displacement. Figure 20: Frequency–response curves of harmonic components at SFD node under base pitching motion (\(C/R=0.485\%\), \(\omega_{by}=9\) rad/s): (**a**) \(U=252\) g \(\cdot\) mm; (**b**) \(U=360.04\) g \(\cdot\) mm. For the unstable region A-B, a saddle-node bifurcation occurs at point A, while at point B, Floquet multipliers cross the unit circle as complex conjugate pairs, indicating a secondary Hopf bifurcation. This bifurcation suggests the loss of stability in periodic solutions near this rotational speed, leading to quasi-periodic motion. Regions C-D and E-F exhibit hardening spring characteristics, indicating the potential occurrence of complex jump phenomena near the bifurcation points. Since the harmonic balance method approximates system motion through finite harmonic terms truncation, it can only yield periodic solutions. To verify the existence of non-periodic solutions within specific speed ranges, time-domain analysis was conducted using the Newmark numerical integration method [17] with a time step corresponding to 1/200 of the period. To ensure computational convergence, calculations extended over 2000 periods at each investigated rotational speed, with the final 100 periods used to construct Poincare sections. All simulations were initialized with a uniform nodal displacement of y = 1 \(\upmu\)m.  
Figure 23 illustrates the rotor trajectories, phase diagrams, and Poincare sections at the SFD node for various rotational speeds. At \(\widetilde{\Omega}=1.025\), the system exhibits stable period-1 motion. At \(\widetilde{\Omega}=1.05\), the phase space trajectory demonstrates irregular evolution with scattered points in the Poincare section, indicating the possible chaotic motion. At \(\widetilde{\Omega}=1.075\), the system displays quasi-periodic motion with a complete toroidal structure in the Poincare section. At \(\widetilde{\Omega}=1.10\), despite intersecting rotor trajectory, the Poincare section shows a single point, indicating a return to period-1 motion. The time-domain computational results validate the harmonic balance method's predictions regarding non-periodic solutions.