## 2 Modeling of Revolute Joint with Clearances
### Contact Conditions of Journal and Bearing
Its expression is shown in Equation (5):  
\[\delta_{a}=|MA_{i}|-|\mathbf{d_{i}}|\cos\theta-R_{j}\sin\theta \tag{5}\]  
In the case of the two axes being parallel:  
Figure 5 shows the fit condition when the two axes are parallel to each other. The specific values of \(\delta_{r}\) and \(\delta_{a}\) are easier to analyze than when the two axes intersect. When the two axes are parallel, point \(N_{i}\) and point \(N_{j}\) are the midpoints of the axis of the bearing  
Figure 4: The axis of the bearing and journal is not parallel. and the axis of the journal, respectively. The angle of \(\left|\,N_{i}N_{j}\,\right|\) in the vertical direction is \(\alpha\). The projection of \(\left|\,N_{i}N_{j}\,\right|\) in the radial direction is \(e_{1}\), then \(e_{1}=\cos\alpha\). The projection in the axis direction is \(e_{2}\), thus \(e_{2}=\sin\alpha\). According to the geometric model, \(\delta_{a}\) and \(\delta_{r}\) when the two axes are parallel can be calculated as follows:  
\[\delta_{a}=\frac{L_{i}}{2}-\frac{L_{j}}{2}-\left|\,N_{i}N_{j}\,\right|\sin\alpha \tag{6}\]  
\[\delta_{r}=R_{i}-R_{j}-\left|\,N_{i}N_{j}\,\right|\cos\alpha \tag{7}\]  
After obtaining the equations for \(\delta_{a}\) and \(\delta_{r}\), it is easy to distinguish the contact conditions of the bearing and journal. The range of \(\delta_{r}\) and \(\delta_{a}\) values can describe the contact mode, as shown in Inequalities (8) and (9):  
\[\left\{ \begin{array}{l}\delta_{r}>0\text{ free flight mode}\\ \delta_{r}=0\text{ continuous contact mode}\\ \delta_{r}<0\text{ pseudo penetration mode}\\ \end{array} \right. \tag{8}\]  
\[\left\{ \begin{array}{l}\delta_{a}>0\text{ free flight mode}\\ \delta_{a}=0\text{ continuous contact mode}\\ \delta_{a}<0\text{ pseudo penetration mode}\\ \end{array} \right. \tag{9}\]  
It can be seen from Inequalities (8) and (9) that the values of \(\delta_{r}\) and \(\delta_{a}\) are very critical when discussing the contact mode. If these two formulas are combined, that is to say, radial minimum distance and axial minimum distance of the bearing and journal are considered at the same time, the result is as shown in Table 1.  
As can be seen from Table 1, when the joint is in the first mode, both \(\delta_{a}\) and \(\delta_{r}\) are required to be positive. When the contact mode of the bearing and journal is continuous \begin{table}
\begin{tabular}{c c c} \hline
**Contact Mode** & **Radial Minimum Distance** & **Axial Minimum Distance** \\ \hline free flight mode & \(\delta_{r}>0\) & \(\delta_{a}>0\) \\  & \(\delta_{r}=0\) & \(\delta_{a}>0\) \\ continuous contact mode & \(\delta_{r}=0\) & \(\delta_{a}=0\) \\  & \(\delta_{r}>0\) & \(\delta_{a}=0\) \\  & \(\delta_{r}<0\) & \(\delta_{a}>0\) \\ pseudo penetration mode & \(\delta_{r}<0\) & \(\delta_{a}<0\) \\  & \(\delta_{r}>0\) & \(\delta_{a}<0\) \\  & \(\delta_{r}=0\) & \(\delta_{a}<0\) \\ \hline \end{tabular}
\end{table} 
Table 1: Contact mode. Figure 5: Bearing and journal axes in parallel. contact mode, it is necessary to ensure that \(\delta_{a}\) and \(\delta_{r}\) are not less than zero at the same time. When the contact mode is the pseudo penetration mode, at least one of \(\delta_{a}\) and \(\delta_{r}\) must be negative. When the bearing and journal are in continuous contact mode or pseudo penetration mode, there will be impact forces between them, which will affect the motion state of the system to varying degrees. Therefore, it is crucial to analyze the impact force for the movement performance of the whole mechanical system.