# Nonlinear Analysis and Reliability Analysis of Multilink Mechanism Considering Mixed Clearance
## 3 Establishment of Rigid Body Dynamics Model for Mechanisms with Clearances
### Establishment of a Rigid Body Dynamics Model with Mixed Clearances  
Rigid body dynamics analysis diagram of the eight-bar mechanism. In Figure 7, global coordinate system origin is set to point \(O\), while origins of local coordinate systems are typically set at centroid of each component. Global coordinates of for each component are indicated as:  
\[\mathbf{q}_{i}=(x_{i}\,y_{i}\,\theta_{i})^{\text{T}}(i=1,2,3,4,5,6,7) \tag{35}\]  
where \(x_{i}\) and \(y_{i}\) represent coordinates of the centroid of component \(i\) in global coordinate system, respectively, while \(\theta_{i}\) denotes angle of component \(i\) relative to \(x\)-axis in the global coordinate system. The velocities and accelerations of the components of each global coordinate system are expressed as follows, respectively:  
\[\dot{\mathbf{q}}_{i}=\Big{(}\dot{x}_{i}\,\dot{y}_{i}\,\dot{\theta}_{i}\Big{)}^{ \text{T}}(i=1,2,3,4,5,6,7) \tag{36}\]  
\[\ddot{\mathbf{q}}_{i}=\Big{(}\ddot{x}_{i}\,\ddot{y}_{i}\,\ddot{\theta}_{i}\Big{)}^ {\text{T}}(i=1,2,3,4,5,6,7) \tag{37}\]  
Generalized coordinates of an eight-bar mechanism system are represented as:  
\[\mathbf{q}=(q_{1}\,q_{2}\,q_{3}\,q_{4}\,q_{5}\,q_{6}\,q_{7})^{\text{T}} \tag{38}\]  
In the equation, there are a total of 21 generalized coordinates. The generalized velocities of system is \(\dot{\mathbf{q}}\) and generalized accelerations of system is \(\ddot{\mathbf{q}}\). When considering mixed clearances (composite revolute clearance A and prismatic clearance F), crank 1 and connecting rod 2 are joined by a rotating pair A\({}_{1}\), crank 1 and connecting rod 5 are joined by a rotating pair A\({}_{2}\), forming the composite revolute joint A. Slider and guide rail constitute prismatic joint F. Whenever a clearance exists, the constraint equations of the system will be reduced by two, meaning that the displacement constraints in both horizontal and vertical directions of the ideal planar kinematic pairs are released. At this point, the mechanism has a total of seven revolute joint constraints. Compared to constraint equations of ideal mechanism, the constraint equations of eight-bar mechanism with mixed clearances (composite revolute clearance A and prismatic clearance F) are reduced by six. The initial driving angles have been obtained through the inverse kinematics solution during the simulation process, and the system has a total of 15 constraint equations. Since crank 1 is directly driven by a motor, the prismatic joint clearance is directly related to the slider. Therefore, investigating the clearances at the composite revolute joint A and the prismatic joint F can effectively reflect the influence of mixed clearances on the dynamics of the mechanism. Figure 8 illustrates structure of an eight-bar mechanism with mixed clearances (composite rotation clearance A and prismatic clearance F). Each feature frame can be connected to feature frames on adjacent parts, and there are some mathematical constraints between adjacent frames. To ensure the position and orientation of a part with respect to its neighbors, it is necessary to impose the corresponding mechanical constraints between parts as well as assembly features [48]. The  
Figure 7: Schematic diagram of coordinates for a planar eight-bar mechanism. displacement constraint equation of mechanism when mechanism has a mixed clearances can be expressed as:  
\[\mathbf{\Phi}(\boldsymbol{q})=\left[