# Dynamic Simulation of a 5 Degrees of Freedom Rotor Dropping on a Protective Bearing
## 2 Establishment of Dynamic Simulation Model for 5-Dof Rotor-Protective Bearing System
\begin{cases}0&s\leq s_{0}\\ k(s-s_{0})^{\epsilon}+c\dot{s}*STEP(s,s_{0},-d_{m},1,s_{0},0)&s>s_{0}\end{cases}  \tag{5}\]  
The STEP function is used to simulate the non-convergence of the calculation results caused by the sudden change in the contact damping. Due to the large impact in the rotor drop system, the penalty depth \(d_{m}\) is taken as 0.015 mm in the contact setting. For the Impact function [22], the most important thing is to determine the contact stiffness coefficient, damping, and contact coefficient, because the internal contact force of the protective bearing and the contact force between the rotor and the inner ring of the protective bearing are in line with the basic conditions of the Hertz theory: (1) Assuming that the contact object only undergoes elastic deformation and obeys Hooke's law, due to the high hardness of the bearing steel, relative to the radius of the ball, (2) it is assumed that only the maximum surface compressive stress is considered, and the assumption is reasonable because the normal contact force on the contact surface of the rolling element is much greater than the tangential friction force. (3) Assuming that the semi-radial ratio of the contact surface size to the surface curvature of the contact object is very small, many previous experiments have proved that the results calculated according to this assumption are consistent with the experimental results, so the assumption is reasonable. Therefore, the Hertz theory is generally used to calculate the internal contact force of the protective bearing, and this theory is used to calculate the contact stiffness coefficient in this paper. According to the Hertz theory, the contact between the rolling element and the raceway in the protective bearing belongs to point contact. Under the action of load Q, the contact deformation between the ball and the raceway with the same material properties is [23]:  
\[\Delta=e_{\delta}\sqrt[3]{(\sum\rho)Q^{2}} \tag{6}\]  
where \(\rho\) is the principal curvature of the contact surface, \(e_{\delta}\) is the Hertzian contact coefficient related to the principal curvature of the contact surface, which can be calculated from reference [24], and Q is the normal load on the contact surface. Therefore, the contact coefficient between the rolling element and the raceway is \(k_{b}=e_{\delta}{}^{-3/2}(\sum\rho)^{-1/2}\), and the contact coefficient is \(3/2\)[25]. Because of the complexity of damping calculation, there is no reliable calculation method at present, and the experimental method is basically used to determine the damping. According to the experiment, the damping value is 1000 N\(\cdot\)S/m. The contact between the rotor and the inner ring of the protective bearing is also calculated by the Hertz theory. The contact problem between the rotor and the inner ring can be understood as the line-surface contact problem in Hertz contact. Since both the rotor and inner ring of the bearing are cylinders with finite length, the contact deformation under load is [26]:  
\[\Delta=3.83\times 10^{-5}\frac{Q^{0.9}}{l^{0.8}} \tag{7}\]  
where \(l\) is the line contact width, and the contact width when the rotor contacts with the bearing inner ring is the bearing width, taken as \(l=20\) mm. It can be concluded that the contact between the rotor and the inner ring of the protective bearing is \(k_{r}=4.268\times 10^{11}\) N/mm\({}^{-10/9}\), and the contact coefficient is 10/9. The experimental method is also selected for the value of contact damping, and the commonly used contact damping value of \(1000\)N \(\cdot\) m/s is selected [27]. It is also very important to set the contact friction. There are various methods to calculate the friction coefficient in the dynamic simulation of the bearing. In a previous study, it was found that the friction coefficient of 0.007 had a good effect, so the friction coefficient between the rolling body and the raceway was set to 0.007 in the modeling simulation.