## 2 Model of a Single-Oil Wedge Hydrodynamic Bearing-Rotor System
 \tag{1}\]  
where \(z_{1}\) (\(t\)) and \(z_{2}\) (\(t\)) are the displacements of the right and left node points, \(M_{1}\) and \(M_{2}\) are the masses of the mass disk and the rotor disk of bearing, \(D_{S}\) is the generalized external damping coefficient, \(K_{0}\) is the radial rigidity of the oil film, \(K_{1}\) and \(K_{2}\) are the rigidity coefficients of the shaft, \(K_{3}\) is the radial rigidity provided by the spring outside the journal which balances the gravity of the journal, \(\varphi\) is  
Figure 1: The experimental platform. Figure 2: Magnet position numbers. the nonlinear radial rigidity of the fluid, \(\Omega\) is the rotation speed, and \(\lambda\) is the average speed ratio in the fluid, and \(m\) and \(r\) are mass and radius of modal unbalance, respectively. \(D\) and \(M_{f}\) are damping and inertia coefficients respectively. \(\varphi_{1}\) and \(\varphi_{2}\) are nonlinear functions of the radial displacement, respectively. Equation (1) can be linearized, and the eigenvalue is taken to deduce the following instability threshold speed  
\[\Omega\leq\frac{1}{\lambda}\sqrt{\frac{K_{1}}{M_{1}}+\frac{K_{2}(K_{0}+K_{3})} {M_{1}[K_{2}+(K_{0}+K_{3})]}}.\]  
Expand the value in the root number, the equation can be rewritten as  
\[\Omega\leq\frac{1}{\lambda}\sqrt{\frac{K_{1}[K_{2}+(K_{0}+K_{3})]+K_{2}(K_{0}+K _{3})}{M_{1}[K_{2}+(K_{0}+K_{3})]}}.\]  
\(K_{1}\) is the rigidity coefficient of the flexible coupling to the mass disc. Since one end is fixed by the flexible coupling, the rigidity coefficient \(K_{1}\) is large. \(K_{2}\) is the rigidity coefficient of the mass disc to the fluid bearing, far from the fixed end, and the coefficient \(K_{2}\) is less than \(K_{1}\). \(K_{0}\) is the radial stiffness of the oil film, the rigidity coefficient provided by the liquid is also smaller than the rigidity coefficient provided by the flexible coupling. \(K_{3}\) is the radial stiffness provided by the spring outside the journal. The characteristics of the spring itself provide a small coefficient of rigidity. Thus, \(K_{2}\), \(K_{0}+K_{3}\) are smaller than \(K_{1}\), and the value in the root number can approximate to  
\[\Omega\leq\frac{1}{\lambda}\sqrt{\frac{K_{1}[K_{2}+(K_{0}+K_{3})]}{M_{1}[K_{2 }+(K_{0}+K_{3})]}}\approx\frac{1}{\lambda}\sqrt{\frac{K_{1}}{M_{1}}}\approx \Omega_{th}, \tag{2}\]  
where \(\Omega_{th}\) is distinguished as the instability threshold speed. Figure 4: Rigid rotor system model. Figure 3: Schematic diagram of pressure distribution of hydrodynamic bearing. The above equation can be reduced to \(\Omega\leq\frac{1}{\lambda}\sqrt{\frac{K_{1}}{M_{1}}}\), i.e., the instability threshold speed of the rotor can be increased by reducing \(\lambda\), increasing \(K_{1}\), or reducing \(M_{1}\). However, \(M_{1}\) is rarely changed as the rotor system has been set up. Only \(\lambda\) and \(K_{1}\) can be handled to change the instability threshold speed. In terms of oil bearing, the viscosity of the lubricating fluid is one of the conditions that can be used to change the value of \(K_{1}\), and the higher the lubricating fluid viscosity, the greater the \(K_{1}\), and the less likely the system will cause fluid instability.