## 2 Model of a Single-Oil Wedge Hydrodynamic Bearing-Rotor System  
The photograph of the test platform for this study is shown in Figure 1. A stainless shaft with a 9.52 mm diameter and 490 mm length was supported by a relatively rigid Oilite bronze bushing and a cylindrical, fluid-lubricated bearing. The horizontal shaft was driven by a 90 watt AC induction motor with a variable speed controller. An rpm transducer at the left end was combined with a rotating disk, which has a square slit and mounted on the shaft, and a photo interrupter is used to detect the speed as the motor rotates. A flexible coupling connects the motor shaft and the stainless shaft. A single-oil wedge hydrodynamic bearing was set at the right end. The type of the hydrodynamic bearing was a circle wedge. Two eddy current sensors set on the outside case of the bearing and the other two eddy current sensors installed next to bearing protection are used to measure the displacements of the journal and shaft. An oil tank installed above journal bearing to store ferrofluid. A stand of four radial spring supporting the rotor was used to position the initial position of the rotor journal to the center of the fluid-film bearing during the experiment. In the midspan, one 0.78 kg counterweight plate was attached to the shaft. Two seals made of bronze as bearing protections were located at both sides of the mass disc to protect the rotor system. A centerline adjuster was used to position the rotor journal to the center of the fluid-film bearing during the experiment. Vibration data were obtained from two pairs of XY eddy current transducers located in the fluid-lubricated bearing shell and located near the seal, respectively, and an rpm transducer was installed next to the flexible coupling in order to obtain rpm signals. There were two eddy current sensors set on the outside case of the bearing to measure the displacements of the journal. Also, there were eight slots in the outside case which were used to install the permanent magnets. The permanent magnets are cylindrical magnets with a diameter of 10 mm and a height of 20 mm, and the material is NdFeB N35. The serial numbers of the magnets are shown in Figure 2. In this study, 3 to 8 magnets were inserted into the eight slots around the bearing to form four different types of magnetic fields. Assuming that the fluid is stable and flows as a laminar flow, the pressure distribution of the hydrodynamic bearing can be shown as in Figure 3. In the figure, \(O_{1}\) is the geometric center of the fluid bearing and \(R_{1}\) is the bearing radius, \(O_{2}\) is the center of the journal under a load \(W\) applied at the journal, the acute angle \(\varphi\) of line \(O_{1}O_{2}\) and the load line is called the attitude angle, \(\Omega\) is the rotating speed, \(d\) is the journal diameter, and \(R_{2}\) is the journal radius, \(c\) is the gap of the journal and bearing, eccentricity \(e\) is the distance of \(O_{1}\) and O\({}_{2}\), \(h\) is the oil film thickness, and \(h_{min}\) is the minimum oil film thickness and equal to \(c\) minus \(e\). Figure 4 displays the model of a symmetric rigid rotor system. According the research of [PERSON] [28], the mathematical model which represents the balance of forces in the symmetric rotor can be expressed as  
\[ \begin{array}{c}M_{1}\tilde{z_{1}}+D_{S}\tilde{z_{1}}+(K_{1}+K_{2})z_{1}-K_{ 2}z_{2}=mr\Omega^{2}e^{l\Omega t}\\ M_{2}\tilde{z_{2}}+M_{f}\big{(}\tilde{z_{2}}-2j\lambda\Omega\tilde{z_{2}}- \lambda^{2}\Omega^{2}z_{2}\big{)}+[D+\varphi_{2}(|z_{2}|)]\big{[}\tilde{z_{2}} -j\lambda\Omega z_{2}\big{]}+[K_{0}+\\ \varphi_{1}(|z_{2}|)]z_{2}+K_{3}z_{2}+K_{2}(z_{2}-z_{1})=0,\end{array}