## 6 Results and Discussion
### Orbit Simulation  
The nonlinear orbit response of the C-GFB is carried out with an imbalance amount of 8.5 g from the rotating speed of 3 krpm up to 90 krpm with a constant acceleration of 500 rad/s\({}^{2}\). The PZT actuators are controlled with the ramp input from 0 to 50, 100, and 150 V at the initiation of unstable motions, to investigate the capability of the C-GFBs to stabilize the unstable motions in real time. This is the major focus in the orbit simulation. Figure 13 shows the predicted transient responses of journal eccentricities (\(\varepsilon_{X}\), \(\varepsilon_{Y}\)) and corresponding rotational speed when the driving voltage is maintained at 0 V. The result reveals that the journal responses become unstable at ~15 krpm with whirl frequencies of 123-147 Hz and disappear at ~19 krpm, considered as oil whirl motions, and finally diverge at 33 krpm because of excessive oil whip motions. Note that the divergent oil whip appears to be stable in the waterfall plot owing to the limited time resolution of the short time Fourier transform of the plot. Figure 12: Predicted journal eccentricity and attitude angle for C-GFB and shimmed GFB: (**a**) eccentricity and attitude angle, and (**b**) equilibrium position. Note that the oil whirl and oil whip are probably the most common causes of the sub-synchronous instability of hydrodynamic journal bearings. Typically, the oil whirl exhibits whirl frequency tracks at approximately one-half of the rotational speed. Meanwhile, the oil whip phenomenon exhibits the locked whirl frequency at a certain frequency related to the system's natural frequency. This is because the oil whip occurs on those machines that are subject to oil whirl when the oil whirl frequency coincides with and becomes locked into the system's natural frequency [6,7]. Figure 14 illustrates the predicted journal transient responses of C-GFBs with respect to various driving voltages of (a) 50 V control, (b) 100 V control, and (c) 150 V control during the speed-up simulation. The driving voltage is controlled with the ramp input from 6 to 6.5 s at the initiation of unstable responses, with a rotational speed of 32.6 krpm. It is evident that after the real-time voltage control, the originally diverging subsynchronous motions are perfectly suppressed after a few seconds. Moreover, the journal equilibrium position moves close to the bearing center immediately. Notably, high voltage control results in relatively fast stabilization and significant variations in the journal equilibrium position. The predicted journal transient responses are analyzed with waterfall plots for the vertical responses. The horizontal responses are omitted for brevity. The journal response in the case of the 50 V control exhibits a large subsynchronous instability with a frequency of approximately 105 Hz. It finally diverges at the rotational speed of 44.4 krpm. Meanwhile, the 100 and 150 V cases display relatively stable responses until the maximum speed of 90 krpm. These display limited oil whip motions, which do not diverge within the analysis speed range. Notably, the whirl frequencies of the oil whip are approximately 105, 127, and 143 Hz when the driving voltage is 50, 100, and 150 V, respectively. A tendency of increasing whirl frequencies with bearing preloads was also reported by Ref. [8] in the rotordynamic experiment of shimmed GFBs, and Ref. [12] in the nonlinear rotordynamic analysis of an active GFB. It is noteworthy that the piezoelectric preload control of C-GFBs causes the gas-film to be stiff or soft depending on the voltage level. This implies that it is capable of controlling the gas-film stiffness, which mainly determines the frequency and shape of the subsynchronous motion modes [24]. Figure 15 illustrates the filtered subsynchronous amplitudes of the journal vertical motions extracted for all the above transient responses, specifically oil whip motions. As the rotational speed increases, the oil whips initiate again. The onset speed varies according to the voltage level.