# A Numerical Analysis of the Nonlinear Dynamics of Shimmed Conical Gas Foil Bearings
## 4 Results and Discussion  
To demonstrate the capability of the present model a few simulation results are presented. These simulations are supposed to capture the static and dynamic, as well as the linear and nonlinear stability behaviour of CGFB. The parameters used for these investigations are shown in Table 2.  
In the first simulation, the static performance of CGFBs is investigated. The equilibrium positions for different masses and external axial forces are calculated by setting \(\mathbf{g}(\tau,\mathbf{s})\) to 0. For any bearing able to support an axial load it is crucial to provide an axial stiffness. Therefore, the axial force must increase the further the conical journal moves into the bearing. Because of the coupling of radial and axial forces in conical bearings, it is important to investigate these axial forces for a constant rotor mass which results in a constant radial force. Figure 5 shows the axial forces for different rotor masses and the corresponding axial displacement of the conical journal for a CGFB with and without the use of shims with a thickness of \(t_{s}=25\times 10^{-6}\) m. It can be seen that, without shims, the axial force only slightly increases with increasing axial displacement. The axial force is mostly dependent on the rotor mass. This results in a very small range of axial force which \begin{table}
\begin{tabular}{l l l} \hline Ambient pressure & \(p_{\text{a}}\) & \(1.01\times 10^{5}\) Pa \\ Viscosity of air & \(\mu\) & \(1.95\times 10^{-5}\) Pa s \\ Nominal clearance & \(c_{0}\) & \(31.8\times 10^{-6}\) m \\ Half cone angle & \(\varphi\) & \(15^{\circ}\) \\ Maximum radius & \(r_{1}\) & \(19.05\times 10^{-3}\) m \\ Bearing length & \(L\) & \(38.1\times 10^{-3}\) m \\ Foil stiffness & \(k_{\text{b}}\) & \(4.739\times 10^{18}\) N/m\({}^{3}\) \\ Loss factor & \(\eta\) & 0.25 \\ Rotational speed & \(n\) & 10,000 rpm \\ Rotor unbalance & \(m_{\tau}u\) & 0 kgm \\ \hline \end{tabular}
\end{table} 
Table 2: Bearing parameters and fluid properties of a CGFB (\(\varphi=15^{\circ}\)). Figure 4: Deviation of the dimensionless y-displacement of the equilibrium position \(\varepsilon_{\text{yeq}}\) of the case shown in Figure 3b for a variation of \(N_{\theta}\) (**a**) and \(N_{z}\) (**b**). can be provided for a constant rotor mass. If the axial force onto the bearing is too low, the bearing clearance would increase until the radial force could no longer be supported. This range of supportable axial force is a key parameter for the integration into a rotor system with two opposing conical bearings. The larger the range of a bearing, the more robust a rotor system can be designed. The strong dependency between rotor mass and axial force results in the problem mentioned in the introduction. The simulations with the mentioned simplifications show that a CGFB with a uniform bearing clearance can support no axial load without a radial force acting on the bearing. This problem can be solved by using shims to modify the bearing clearance into a nonuniform shape. The results for a shimmed CGFB in Figure 5 show that the dependency on the rotor mass can be strongly decreased while the range of supportable axial force increases. This makes a shimmed CGFB much more suitable for the use in rotor systems with two opposing CGFBs. To investigate why the shimmed and unshimmed CGFBs show such different static properties, the calculated pressure distributions in a shimmed and unshimmed CGFB on the unwound lateral surface of the conical air gap are shown in Figure 6. In order to be able to compare the pressure distributions in a meaningful way, the axial displacement of the rotor was fixed to \(\varepsilon_{x}=-0.5\) and the same mass of \(m_{r}=5\,\mathrm{kg}\) was used for both calculations. Without shims, one pressure maximum resulting from decreasing air gap in circumferential direction occurs. At \(\theta\approx 330^{\circ}\), the influence of the Guembel condition is clearly visible as the pressure at this point does not drop below one.