# Nonlinear Analysis and Reliability Analysis of Multilink Mechanism Considering Mixed Clearance
## 2 Creation of Kinematic Pair Clearance Model
### Creation of Kinematic Model for Revolute Joint Clearance  
Because of the existence of clearances of the revolution joints, based on the geometrical relations between the bearings with the shaft, the motion state between the shaft and the bearing can be classified into the state of freedom, the state of continuous contact, and the state of collision, as illustrated in Figure 1. _XOY_ denotes global coordinate system, while _xoy_ denotes local coordinate system on the component. Components 1 and 2, as well as components 1 and 5, are bearings and shafts connected via rotary joints, respectively. \(O_{1}\) denotes the bearing center and \(O_{2}\) represents the shaft center. The radius of the bearing is represented by \(R_{1}\) and the radius of the shaft is represented by \(R_{2}\). In event of a collision, the contact point on the bearing hole is labeled as \(Q_{1}\), while the contact point on shaft is labeled as \(Q_{2}\).  
The coordinates of the centers \(O_{1}\) and \(O_{2}\) of the rotating bearing and shaft in the revolute joint can be represented as:  
\[r_{k}^{O}=r_{k}+A_{k}r_{k}^{O} \tag{1}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Joint Type** & **Normal Impact Force Model** & **Tangential Friction Force Model** \\ \hline Revolute joint & L-N model & Modified Coulomb friction force model \\ Prismatic joint & Hertz Model and L-N Model & Modified Coulomb friction force model \\ \hline \hline \end{tabular}
\end{table}
Table 1: Assumptions about the clearance model.  
Figure 1: Kinematic states of revolute clearance: (**a**) freedom; (**b**) continual contact; (**c**) collision.  
where the coordinates of member \(k\) in the global and local coordinate systems are \(r_{k}\) and \(r_{k}^{o}\), respectively, and \(A_{k}\) is a rotational matrix, which is expressed as:  
\[A_{k}=\begin{pmatrix}\cos\theta_{k}&-\sin\theta_{k}\\ \sin\theta_{k}&\cos\theta_{k}\end{pmatrix} \tag{2}\]  
Eccentricity between the bearing and shaft is denoted as:  
\[\mathbf{e}=\mathbf{r}_{2}^{O}-\mathbf{r}_{1}^{O} \tag{3}\]  
The unit vectors of the excenter vectors for bearings and shafts are as follows:  
\[\mathbf{n}_{R}=\frac{\mathbf{e}}{e} \tag{4}\]  
where \(e=\sqrt{e^{\intercal}e}\) denotes the magnitude of the excenter vector.  
If there is a collision between the hole in bearing and shaft, relative collision depth is expressed as:  
\[\delta_{R}=e-c_{R} \tag{5}\]  
where \(c_{R}\) represents the clearance size of the revolute joint, which is the dimensional error between the bearing and the shaft, and can be expressed as:  
\[c_{R}=R_{1}-R_{2} \tag{6}\]  
Therefore, the criterion for determining collisions among the components within a revolute joint with clearance is:  
\[\begin{cases}\delta_{R}<0&\text{Dissociative state}\\ \delta_{R}=0&\text{Contact state}\\ \delta_{R}>0&\text{Impact state}\end{cases}\]  
The precise moment of collision between \(t_{n}\) and \(t_{n+1}\) is determined by the following equation:  
\[\delta_{R}(t_{n})\delta_{R}(t_{n+1})<0 \tag{7}\]  
In a system of fixed coordinates, the location of the collision point vector can be represented as:  
\[\mathbf{r}_{j}^{O}=\mathbf{r}_{j}^{O}+R_{j}\mathbf{n}_{R}\;(j=1,2) \tag{8}\]  
The velocity vector at point of collision when bearing and shaft collide with each other is:  
\[\dot{\mathbf{r}}_{j}^{O}=\dot{\mathbf{r}}_{j}^{O}+R_{j}\dot{\mathbf{n}}_{R}\;(j=1,2) \tag{9}\]  
where \(\dot{\mathbf{n}}_{R}\) represents the first derivative of eccentric vector with unit vector \(\dot{\mathbf{n}}_{R}=\frac{\dot{e}e-\dot{e}e}{e^{2}}\).  
The collision point on bearing has velocity components in the normal and tangent directions with respect to the axes, and these are described as:  
\[\left\{\begin{array}{l}v_{n}=\left(\dot{\mathbf{r}}_{2}^{O}-\dot{\mathbf{r}}_{1}^{O }\right)^{\intercal}\mathbf{n}_{R}\\ v_{t}=\left(\dot{\mathbf{r}}_{2}^{O}-\dot{\mathbf{r}}_{1}^{O}\right)^{\intercal}\mathbf{t }_{R}\end{array}\right. \tag{10}\]  
where rotate vector \(\mathbf{n}_{R}\) ninety degrees counterclockwise to get tangential vector \(t_{R}\).