# A Numerical Analysis of the Nonlinear Dynamics of Shimmed Conical Gas Foil Bearings
## 4 Results and Discussion
Because only one maximum occurs only a small circumferential section is relevant for the resulting forces. Equation (8) yields that, the smaller the relevant section, the more constant the ratio between the resulting radial and axial forces becomes. This explains the strong dependency between mass and axial force in an unshimmed CGFB. Figure 7 shows the section of the pressure distribution in shimmed and unshimmed CGFB through the respective pressure maxima in circumferential and axial direction. For an unshimmed bearing, the pressure maximum is slightly shifted to the larger opening of the bearing. This is expected, due to the higher circumferential speed of the conical journal at this side. In a shimmed CGFB, three local pressure maxima occur at the circumferential locations in each shim. Therefore, the relevant section for the resulting forces is much larger compared to an unshimmed CGFB. When calculating the radial forces, these maxima partially compensate each other, but when calculating the axial force, they add up. This explains why the coupling between rotor mass and axial force is much smaller for a shimmed bearing than for an unshimmed one. It is interesting to note that the pressure is always greater than 1 and thus the Guembel condition has no influence on the pressure distribution at this equilibrium position. The maximum pressure is slightly shifted to the smaller side of the bearing. This can be explained by the higher change of the film height resulting from the assumed modification of the bearing clearance seen in Equation (7). This effect at the  
Figure 5: Axial force for different rotor masses and axial displacements for a CGFB without (**a**) and with (**b**) the use of shims. smaller side of the bearing dominates the effect due to higher circumferential speed at the larger side. Figures 6 and 7 show that the generated pressure inside a shimmed CGFB is much higher than in a unshimmed one. This can be explained by the additional ramps created by the shims, which generate additional pressure. As can also be seen in Figure 5, this leads to a considerably higher axial force for the same axial displacement. The dynamic and linear stability behaviour of a CGFB can be analysed by evaluating the leading eigenvalue of the Jacobian matrix for an equilibrium position. An equilibrium position with a negative real part of the leading eigenvalue can be characterised as stable. If the real part of the eigenvalue is positive the equilibrium position is unstable. The eigenvalue analysis of the matrix \(\mathbf{J}\) of an equilibrium position \(\mathbf{s}_{\text{E}}\) yields \(N_{\lambda}\) eigenvalues. The stability of the equilibrium is governed by the leading eigenvalue \(\lambda_{L}\), which is the one with the highest real part. This yields the stability criterion shown in Equation (13). \[\text{Re}(\lambda_{L})=\text{max}\big{(}\text{Re}(\lambda_{1}),...,\text{Re}( \lambda_{N_{\lambda}})\big{)}<0 \tag{13}\]  
This is a linear approach and only evaluates the stability at the equilibrium position itself and gives no information about possible stable or unstable limit cycles. Figure 8 shows the real part of the leading eigenvalue for the equilibrium positions for different axial forces and rotor masses for a shimmed CGFB. It can be seen that the axial force in particular is extremely important to guarantee a stable equilibrium point. The higher the axial force, the more stable the equilibrium position becomes. This is a interesting result  
Figure 6: Pressure distribution on the unwound lateral surface of the conical air gap without (**a**) and with (**b**) shims for a rotor mass \(m_{t}=5\) kg and axial displacement \(\epsilon_{\text{z}}=-0.5\). Figure 7: Section of the pressure distribution in shimmed and unshimmed CGFB through the respective pressure maxima in circumferential (**a**) and axial direction (**b**) for a rotor mass \(m_{t}=5\) kg and axial displacement \(\epsilon_{\text{z}}=-0.5\). because generally the axial force or the axial direction does not play an important role regarding the rotor stability.