# Instability Study of Magnetic Journal Bearing under S-CO\({}_{2}\) Condition
## 3 Experiments with Magnetic Bearings Under S-CO2 Conditions
### Results  
S-CO\({}_{2}\) tests were performed under four different conditions: test 1: 8 MPa and 36 \({}^{\circ}\)C, (350 kg/m\({}^{3}\)); test 2: 9 MPa and 44 \({}^{\circ}\)C (350 kg/m\({}^{3}\)); test 3: 8 MPa and 40 \({}^{\circ}\)C (280 kg/m\({}^{3}\)); and test 4: 9 MPa and 49 \({}^{\circ}\)C, (290 kg/m\({}^{3}\)). These four conditions were selected to test the bearing under two different densities and two different pressure conditions. This is presented in Figure 17.  
Figure 17: CO\({}_{2}\) density map for test condition.  
The shaft trajectory data under S-CO\({}_{2}\) conditions are shown in Figure 18a and show that the shaft does not keep a single revolving center under S-CO\({}_{2}\) conditions when the rpm increases. Therefore, the changing revolution center and shaft's relative position is traced in real time for obtaining the \(\beta\).  
From the data, \(F_{LLIB}\) and \(F_{MB}\) were calculated. The \(F_{LLIB}\) values with respect to the shaft's position are shown in Figure 19, and \(F_{MB}\) values are in Figure 20. From Figure 19, the \(F_{LLIB}\) acts in a direction almost parallel to the shaft velocity. The difference between test 1 and test 4 is that the radial components of \(F_{LLIB}\), \(F_{LLIB,r}\) in tests 1 and 2 are changed more frequently than in tests 3 and 4. This can be explained with Figure 7a, which describes \(F_{r}\) with density change. From Figure 20a, \(F_{MB}\) has constant force to work against the shaft weight, \(F_{gravity}\). \(F_{MB}\) plus \(F_{gravity}\) is plotted in Figure 20b to eliminate the constant force of the AMB. From Figure 20b, the \(F_{MB}\) acts as a second-order system which has stiffness and damping. However, \(F_{LLIB}\) (Figure 19) does not act as a restoring force as \(F_{MB}\) does. In rotor dynamics, the force acting like \(F_{LUB}\) is follower-force from the cross-coupled stiffness, which can increase the energy of the shaft motion with time [23].  
The force magnitudes with various \(\varepsilon\) values during the experiments are summarized in Figure 21. From the fitted line, it is shown that \(F_{LLIB}\) is proportional to the \(\varepsilon\), but the relationship is weak and the \(F_{LLIB}\) has a higher magnitude in test cases 1 and 2 than cases 3 and 4. It is believed to be due to dramatic density change near the pseudo-critical line because the test cases 1 and 2 have more scattering of the data. The \(R^{2}\) values of test 1, test 2, test 3, and test 4 are 0.1624, 0.1708, 0.1950, and 0.2370, respectively. It is inferred that the reason for this difference is due to the frequent change in \(F_{LLIB,r}\) due to density sensitivity which is shown in Figures 7 and 19.  
The \(F_{MB}\) versus \(\varepsilon\) had a similar distribution with \(F_{LLIB}\), as shown in Figure 22. The reason for this wide distribution is inferred to be a response to unstable \(F_{LLIB}\). However, \(F_{MB}\) was not significantly difference among these four tests, unlike \(F_{LLIB}\). It seems that the performance of the AMB used in the experiment reached the limit to  
Figure 18: Shaft trajectory data from the (**a**) air test and (**b**) S-CO\({}_{2}\) test 1 at 30,000 rpm.  
Figure 19: \(F_{LUB}\) distribution vs. shaft position from the 30,000 rpm S-CO\({}_{2}\) test 1 (a); test 2 (b); test 3 (c); and test 4 (d).  
Figure 21: \(F_{LIB}\) distribution vs. \(\epsilon\) from 30,000 rpm S-CO\({}_{2}\) test 1 (a), test 2 (b), test 3 (c), and test 4 (d).  
Figure 22: \(Cont\).