## 2 Modeling of Revolute Joint with Clearances
### Contact Conditions of Journal and Bearing  
As shown in Figure 3, there are three modes of matching between the revolute joint journal and the bearing during rotation. They are free flight mode, continuous contact mode, and pseudo penetration mode. The first mode refers to the mode in which the journal and bearing do not make contact completely during the rotation, as shown in Figure 3a. The continuous contact mode is the mode in which the bearing and journal make contact, but do not deform during the rotation, as shown in Figure 3b. The pseudo penetration mode refers to the mode wherein the bearing and journal deform to a certain extent as a result of the contact force during the rotation, as shown in Figure 3c. When the revolute joint is in the first mode, the contact force between the two is nonexistent. When the revolute joint is in the other two modes, they will be affected by the contact force. The state of the revolute joint can be resolved by the contact force. It can be  
Figure 3: Contact modes. Figure 2: Possible contact forms between the bearing and journal. (**a–d**) Two-point contact forms; (**e,f**) single-point contact forms; (**g,h**) three-point contact forms; (**i**) four-point contact form; (**j**) surface contact form; (**k**) line contact form; and (**l**) both surface contact and line contact forms. seen that the contact force is crucial. Before judging the magnitude of the contact force, the geometric state of their mutual cooperation can be analyzed first. Figure 4 shows the geometric model of the bearing and journal in the revolute joint. In Figure 4, \(\theta\) represents the included angle between the axis of bearing and the axis of journal and point \(M\) represents the intersection of the two axes. When \(\theta\) is zero, the fit is divided into two states, namely, axis parallel and coincidence. When \(\theta\) is not zero, we consider the two axes to intersect at point \(M\). Here, the matching state of the journal and bearing can be divided into two cases: the intersection state of the two axes and the parallel state of the two axes. In the first case:  
\(A_{i}\) and \(B_{i}\) represent the centers of two end-facing circles of the cylinder inside the bearing. \(A_{j}\) and \(B_{j}\) represent the centers of the two end-facing circles of the journal. \(\mathbf{d_{1}}\) is the vector from \(M\) to \(A_{j}\). \(a\) is the projection of the mode of vector \(\mathbf{d_{1}}\) on the right end face of the inner cylinder of the bearing. \(b\) is the projection of journal radius \(A_{j}P_{r}\) on the right end face of the inner cylinder of the bearing. \(Q\) and \(P_{r}\) are the lowest point of the right end circle of the cylinder inside the bearing and the lowest end circle of the right end circle of the journal in Figure 4, respectively. Obviously, point \(P_{r}\) is the closest point of the journal to the bearing in the radial direction. The auxiliary plane \(\tau_{2}\) along point \(Q\) is perpendicular to the right end face of the inner cylinder of the bearing. The auxiliary plane \(\tau_{1}\) along point \(P_{r}\) is parallel to plane \(\tau_{2}\), thus the distance from plane \(\tau_{1}\) to plane \(\tau_{2}\) is the distance from point \(P_{r}\) to plane \(\tau_{2}\), which is called the minimum distance between the bearing and the journal in the radial direction, denoted as \(\delta_{r}\), as shown in Equation (4):  
\[\delta_{r}=R_{i}-|\mathbf{d_{i}}|\sin\theta-R_{j}\cos\theta \tag{4}\]  
\(m\) is the projection of the mode of vector \(\mathbf{d_{1}}\) on the axis \(A_{i}B_{i}\). \(\mathrm{n}\) is the projection of journal radius \(A_{j}P_{a}\) on the axis \(A_{i}B_{i}\) of the bearing. \(P_{a}\) is the most right point of the journal and the closest point to the bearing in the axial direction. The distance between \(P_{a}\) and the inner surface of the bearing is expressed by \(\delta_{a}\), which is called the minimum distance between the bearing and the journal in the axial direction.