# A Numerical Analysis of the Nonlinear Dynamics of Shimmed Conical Gas Foil Bearings
## 4 Results and Discussion
A GfJB normally becomes more stable the higher the rotor mass gets. This cannot be said for shimmed CGFBs. If the axial force is high enough to ensure a stable operation, a lower rotor mass can stabilise the rotor even more. Figure 8 also shows the stability borders for different rotational speeds. The higher the rotational speed becomes, the more axial force is needed to stabilise the bearing. It is interesting to note that for higher rotational speeds, a lower rotor mass can be beneficial for the bearing stability. Due to the small range of supportable axial forces, such stability maps cannot be calculated for unshimmed CGFBs. To further investigate the stability behaviour of shimmed CGFBs, three transient simulations for different operating points are presented. Figure 9 shows the trajectory for a bearing with a rotor mass \(m_{\mathrm{r}}=1.25\) kg, and axial load \(F_{\mathrm{ax}}=83\) N at a rotational speed \(n=10,000\) rpm after a drop from the bearing center. This operating point is characterized as stable in Figure 8. The transient simulation supports this statement. The radial and axial movements of the rotor decay within the first 30 journal rotations. After that, the journal remains at an equilibrium position for the rest of the simulated time. Transient simulations are particularly interesting for the assessment of unstable operation. In contrast to the linear stability analysis using the Jacobian matrix at the equilibrium position, a transient simulation can further characterize the nonlinear stability behaviour. Figure 10 shows the transient simulation results for an unstable operating point. The journal does not end up in a stable equilibrium position. Instead, the trajectory approaches a stable limit cycle. After around 100 rotations, the limit cycle does not change significantly for the remaining 400 simulated rotations. Figure 11 shows the journal movement in x-, y- and z-direction and the corresponding frequency spectrum. Because of the assumption of a perfectly balanced rotor, no harmonic vibration occurs. A sub-synchronous vibration with a frequency ratio of 0.725 can be seen in all directions. Such sub-synchronous vibrations are common in nonlinear dynamic systems and are well known to occur in rotor systems supported by GFB. Furthermore, super-harmonic vibrations appear in all directions. The frequencies of these super-harmonic vibrations appear to be integer multiples of the frequency of the mentioned sub-harmonic vibration. This behaviour is common for nonlinear systems. It is noticeable that the sub-harmonic dominates in the axial direction while the first super-harmonic dominates in the x-direction. In the y-direction, the sub-harmonic and the first super-harmonic occur with almost the same amplitudes. Figure 8: Real part of the leading eigenvalue of a shimmed CGFB depending on the rotor mass \(m_{\mathrm{r}}\) and axial force \(F_{\mathrm{ax}}\) of a shimmed CGFB with a rotational speed \(n=10,000\) rpm (**a**) and stability borders for different speeds (**b**). Figure 10: Trajectory after a drop from the bearing center for a rotor mass \(m_{\rm{r}}=1.5\) kg and axial force \(F_{\rm{ax}}=52\) N in a shimmed CGFB; (**a**) x-y-plane (**b**) y-z-plane. Figure 4:0 Trajectory and rotor movement in radial (**a**) and axial direction (**b**) after a drop from the bearing center for a rotor mass \(m_{\rm{r}}=1.25\) kg and axial force \(F_{\rm{ax}}=83\) N in a shimmed CGFB; (**c**) x-y-plane (**d**) three-dimensional trajectory. In addition to the shown simulation for a perfectly balanced rotor, the presented model is also capable of capturing the behaviour of an unbalanced system. Figure 12 shows the trajectory of a rotor with an unbalance of \(m_{\mathrm{r}}u=10^{-6}\) kgm. In comparison to the balanced case, a more complex trajectory occurs. After around 50 shaft rotations the maximum amplitude of the displacement does not grow further. However, no approach to a simple limit cycle, e.g., as seen in Figure 10, can be observed.