## 5 Experimental Results
### Oil Whirl and Whip Phenomena  
Figure 6 shows the spectrum plots and full spectrum cascade plots of the oil bearings using ferrofluid as the lubricating oil without any magnetic field. The rotating speed is increasing at 1 Hz/s. If the signals of the two eddy current sensors installed next to bearing protection do not show the shaft impacts to the protective bearing, the motor will drive the shaft to 100 Hz. When the speed of the rotor system reaches the mechanical first balance resonance point, the vibration increases, thus, generating self-excited vibration. In Figure 6, since the rotation speed is under 3024 rpm, the rotor-bearing system is actuated stably. However, when the rotation speed is 3024 rpm, self-excited vibration occurs. The vibration amount is higher than before 3024 rpm obviously. A whirl is generating and the rotor-bearing system is unstable. At the same time, the frequency of the whirl can be measured as 23 Hz by the machine 108 DAI. Thus, the precession speed can be calculated as 1380 rpm. The whip occurs when the rotation speed is 3184 rpm, and the lubricating fluid sub-synchronous frequency and vibration amplitude are proportional to the rotational speed. \begin{table}
\begin{tabular}{c c} \hline \hline
**Oil–based Magnetic Fluid** & **Magnetic Fluid Solution** \\ \hline Base Carrier Oil & SAE 10W/40 \\ Base Carrier Oil Density \(\rho_{f}^{oil}\) & 0.866 g/cm\({}^{3}\) \\ Magnetic Fluid Density after Stabilization & 0.879 g/cm\({}^{3}\) \\ Oleic Acid Volume Fraction \(\phi_{a}^{oil}\) & 1.02\% \\ Solid Phase Particle Volume Fraction \(\phi_{s}^{oil}\) & 0.27\% \\ Colloidal Particle Volume Fraction \(\phi^{oil}\) & 1.29\% \\
100 \({}^{\circ}\)C/cSt Viscosity & 15.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Oil–based magnetic fluid propertiesFrom Figure 6, we see that \(\Omega_{th}\) is 3024 rpm revolutions. Then, \(\lambda\) is obtained as  
\[\lambda=\frac{1380}{3024}=0.456, \tag{15}\]  
where 1380 is the precession speed. By Formula (2), the original fluid stiffness \(K_{B}\) without external magnetic field can be calculated as  
\[\frac{2\pi\times 3024}{60}=\frac{1}{0.456}\sqrt{\frac{K_{B}}{1.136}}. \tag{16}\]  
Thus, the value of the original fluid stiffness \(K_{B}\) is 23,625 N/m. Regarding different magnetic field layouts, the magnetic fluid oil bearing is tested multiple times, and the magnets are placed in different positions under the bearing. According to the slots' positions shown in Figure 2, the recorded experimental data are (a) three magnets are placed in positions No. 3, 5, and 7; (b) five magnets are placed in positions 3-7; (c) seven magnets are placed in positions 2-8; and (d) eight magnets are inserted into the slots. Magnets 1, 3, 5, and 7 are set as north poles toward the center of the bearing and magnets 2, 4, 6, and 8 are set as south poles toward the center of the bearing. The threshold of instability is distinguished according to the full spectrum cascade plots of Figure 7. The whirl occurs when the rotational speeds are 3168, 3184, 3824, and 4480 rpm, and the whip speeds are 3520, 3520, 4480, and 5268 rpm for the four cases. The experimental results show that if the magnetic field is just applied under the horizontal line of the bearing, the oil spin instability threshold risings are few with cases (a) and (b), which increase to about 144 rpm and 164 rpm; if the magnetic field is set around the bearing, the threshold risings are obvious with cases (c) and (d), which increase to about 800 rpm and 1456 rpm. The threshold risings expressed as percentages from cases (a) to (d) are 4.8%, 5.4%, 26.5%, and 48.1%, respectively. The final step is to determine the stiffness increment of the magnetic fluid under magnetic field \(K_{m}\).