# A Numerical Analysis of the Nonlinear Dynamics of Shimmed Conical Gas Foil Bearings
## 5 Conclusions  
According to the author's knowledge, this paper presents the first transient model for CGFBs. A well-known transient model for GFJBs and GFTBs is modified and applied to CGFBs.  
The results show a drastically different static behaviour depending on whether the bearing clearance is modified by shims or not. A bearing with a uniform clearance shows a strong coupling between radial load and axial force and a comparatively weak correlation between axial force and axial displacement. A shimmed CGFB shows contrary behaviour. While the axial force is strongly coupled to the axial displacement, the radial load only has a minor impact. This gives a shimmed CGFB a much greater range of supportable axial force for a constant rotor mass compared to an unshimmed bearing. For a robust rotor system, a certain range of supportable axial forces is necessary to compensate for manufacturing inaccuracies and non-constant axial loads in practical applications. This indicates that for practical use a CGFB with non-uniform bearing clearance could be much more suitable.  
The results of a linear stability analysis show that especially the axial load of the bearing determines whether a stable equilibrium position occurs or not. For increasing rotational speeds, a higher axial force is necessary to stabilise the system. The first presented transient simulation shows the decaying vibrations of a perfectly balanced rotor with a stable equilibrium position. The second one captures a stable limit cycle of a perfectly balanced rotor. This limit cycle represents a self-excited sub-synchronous vibration that dominates in the axial direction. Although with much smaller amplitudes, this sub-synchronous vibration can also be seen in the radial directions. The third simulation with an added unbalance shows additional sub- and super-harmonic vibrations which are, contrary to the vibrations in the balanced case, non-integer multiples of a lower frequency component.  
Figure 15: Waterfall diagrams for the x- and z-displacement for a rotor mass \(m_{\mathrm{r}}=1.5\) kg and axial force \(F_{\mathrm{ax}}=52\) N without (**a**,**b**) and with (**c**,**d**) a rotor unbalance \(m_{\mathrm{r}}\mu=10^{-6}\) kgm.  
The presented waterfall diagrams show that above a threshold speed, self-excited vibrations occur with an almost constant frequency at increasing rotational speeds. In contrast, neither a constant frequency nor a constant frequency ratio to the harmonic is observed for the additional sub- and super-harmonic vibrations in the case of an added unbalance. These frequencies seem to change parallel to the harmonic (1 \(\Omega\)) or the negative harmonic (\(-1\)\(\Omega\)).  
These simulations demonstrate the capability of the presented transient model to describe the static and dynamic behaviour of shimmed CGFBs. The findings highlight the significance of a non-uniform bearing clearance and the crucial role of the axial load in maintaining bearing stability. These novel conclusions contribute to a better understanding of the underlying phenomena and can provide guidance for the design of complex rotor systems supported by CGFBs.  
Conceptualization, M.S.; methodology, M.S.; software, M.S.; formal analysis, M.S.; investigation, M.S.; resources, R.L.; data curation, M.S.; writing--original draft preparation, M.S.; writing--review and editing, R.L. and M.S.; visualization, M.S.; supervision, R.L.; project administration, M.S.; funding acquisition, R.L. All authors have read and agreed to the published version of the manuscript.  
We acknowledge support by the German Research Foundation and the Open Access Publication Fund of TU Berlin.  
Not applicable.  
Data Availability Statement: Data sharing does not apply to this article.  
The authors declare no conflict of interest.