# A Numerical Analysis of the Nonlinear Dynamics of Shimmed Conical Gas Foil Bearings
## 4 Results and Discussion
Figure 13 shows the movement of the shaft in x-, y- and z-directions and the corresponding frequency spectrum. The sub- and super-harmonic vibrations observed in the case without an unbalance are still present with the same frequencies but slightly different amplitudes. In addition, the unbalance response with a frequency ratio of 1 can be seen in x- and y-direction and a few more sub- and super-harmonics occur. Two sub-harmonics and one super-harmonic with a frequency ratio of 0.272, 0.455 and 1.73 appear mostly in x-direction and y-direction. Contrary to the  
Figure 4:1 Rotor movement in x-, y- and z-direction in a shimmed CGFB after a drop from the bearing center for a rotor mass \(m_{\mathrm{r}}=1.5\) kg and axial force \(F_{\mathrm{Ax}}=52\) N (**a**,**c**,**e**) and the corresponding frequency spectrum (**b**,**d**,**f**) of the stationary limit cycle after 100 rotations. vibration in the perfectly balanced case, the frequencies of these sub- and super-harmonics are non-integer multiples of a lower frequency component. To further investigate the nonlinear stability behaviour and the characteristics of the appearing limit cycles, Figure 14 shows the limit cycles of the simulations above, with and without a rotor unbalance after the first 500, for an additional 1000 shaft rotations. It could be assumed that the limit cycle of the unbalanced case is just the limit cycle from the balanced case overlain by the harmonic response to the unbalance force. However, this would not explain the additional sub- and super-synchronous vibrations mentioned above. It seems like the unbalance force not only causes a harmonic response but also excites additional sub- and super-synchronous vibrations itself. To classify the nature of these limit cycles a three-dimensional Poincare map shown in Figure 14 can be used. In both cases, the Poincare map shows a three-dimensional closed loop with an uncountable number of points. This classifies both limit cycles despite their different appearances as quasi periodic. Waterfall diagrams for the x- and z-direction with and without a rotor unbalance are presented in Figure 4:2 Sub- and super-harmonic vibrations as a result of instabilities can be observed. The parameters are the same as in Figures 10 and 12. These plots can be used to investigate how the different sub- and super-harmonic vibrations behave at different rotational speeds. They are constructed by calculating the frequency spectrum for different speeds from transient simulations. For every rotational speed, a transient simulation starting from the bearing center with a duration of 3 s is performed. To ensure a steady state vibration, the first second of the simulation is discarded and the remaining 2 s are used for the Fourier transform to obtain the frequency spectrum. It can be seen that in case of a perfectly balanced rotor, no harmonic vibration occurs and that the sub- and super-harmonic vibrations mainly appear after a rotational speed of 8000 rpm. The frequency of these vibrations stays almost constant for rising rotational speeds. Therefore, the frequency ratio to the harmonic changes for different speeds. As seen before in Figure 13, the dynamic behaviour becomes more complex in case a rotor unbalance is added. In the x-direction the harmonic (denoted as 1 \(\Omega\)) is clearly visible. The self-excited vibrations from the case without an unbalance are still present. Additional sub- and super-harmonics appear in the x-direction. These vibrations show a different behaviour. Their frequencies do not stay constant but increase or decrease with an increasing rotational speed. These frequencies seem to change parallel to the harmonic (1 \(\Omega\)) or the negative harmonic (\(-1\ \Omega\)). Figure 4:3 Trajectory after a drop from the bearing center for a rotor mass \(m_{\mathrm{r}}=1.5\) kg and axial force \(F_{\mathrm{ax}}=52\) N with a rotor unbalance \(m_{\mathrm{r}}u=10^{-6}\) kgm in a shimmed CGFB; (**a**) x-y-plane (**b**) y-z-plane.