# 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
The following equations can be listed for the force analysis of the four-degree-of-freedom rotor:  
\[\left\{ \begin{array}{l}m\ddot{x}=F_{tt}+F_{tb}\\ m\ddot{y}=F_{na}+F_{nb}-F_{G}\\ J_{x}\ddot{\theta}_{x}-J_{z}\omega\dot{\theta}_{y}=F_{rb}l_{b}-F_{ra}l_{a}\\ J_{y}\ddot{\theta}_{y}+J_{z}\omega\dot{\theta}_{x}=F_{ta}l_{a}-F_{tb}l_{b}\\ \end{array} \right. \tag{1}\]  
The equation of motion of the rotor around the axis is:  
\[J_{z}\ddot{\theta}_{\omega}=-F_{ta}R_{r}-F_{tb}R_{r} \tag{2}\]  
When the rotor rotates by an angle around the center of mass, the following system of equations can be listed based on geometric relationships [18]:  
\[\left\{ \begin{array}{l}x_{b}=x_{G}-l_{b}\tan\theta_{y}\\ x_{a}=x_{G}+l_{a}\tan\theta_{y}\\ y_{b}=y_{G}+l_{b}\tan\theta_{x}\\ y_{a}=y_{G}-l_{a}\tan\theta_{x}\\ \end{array} \right. \tag{3}\]  
In this paper, the five-degree-of-freedom rotor-protective bearing system [19] is simulated by the ADAMS multi-body dynamics software package, and the working environment is selected as follows: (1) Set the coordinate system: in the ADAMS modeling environment, there is a coordinate system that the origin is fixed but can rotate with the model, which is the general coordinate system of the system. In addition, the system will automatically mark a marker point on the center of mass of the rigid body, and use Cartesian coordinates according to the actual modeling needs. (2) Unit system selection: the ADAMS/view unit system has MMKS, MKS, CGS, and IPS, and this paper uses the basic unit MMKS, that is, the unit mm, mass kg, force N, and time s. (3) Set the characteristics of the object: The mass and moment of inertia of the component designed according to the actual size are automatically calculated by the system, but not designed according to the actual size. The mass and moment of inertia can be customized. (4) Solver setting: when solving the contact collision problem, based on the BDF rigid integration program, the GSTIFF algorithm with variable order, variable step size, and variable coefficient is adopted, and the SI2 integrator with stable glue is selected to solve the discontinuous problem. For the whole bearing model, the constraint relationship between each part of the bearing is divided as follows: (1) the outer ring of the bearing is fixed with the base, so the outer ring is set as a fixed pair to fix the bearing on the base; (2) set a contact pair among a rolling body in that protective bearing, an inner ring raceway and an outer ring raceway, wherein the contact set is the key of the whole modeling process, the contact stiffness, the damping and the contact coefficient need to be set in the contact setting, and the friction contact is set at the same time; (3) a contact pair is arranged between the inner ring of the bearing and the rotor, and the contact stiffness, damping, contact coefficient, friction coefficient, and the like also need to be set; and (4) the axial force is ignored, and a plane pair is arranged between the rotor and the ground so as to axially restrain the rotor. Using the ADAMS multi-body dynamics simulation software package to simulate the rotor-protective bearing system, the most important thing is the establishment of the model and the reasonable setting of parameters. In ADAMS, there are two ways to define contact: one is based on the Impact function, and the other is based on the recovery coefficient method. The Impact function is more applicable and accurate. It uses the contact stiffness coefficient and damping coefficient to calculate the impact force. This function is used to calculate the contact force in the model simulation in this paper. When the two cylinders are in contact, use the solver to calculate the contact force between the two cylinders according to the Impact function [20]. When the position relationship of two contact objects is different, the expression of Impact function is different, which is a piecewise function, and the occurrence [21] of contact is determined by calculating the penetration depth \(s_{0}\):  
\[s_{0}=R-r \tag{4}\]  
\[F=