# Optimization Design of RV Reducer Crankshaft Bearing
## 2 Performance of crankshaft Bearings of Rotate Vector (RV) Reducers
### Force Analysis of Crankshaft Bearings  
A Cartesian coordinate system, \(xoy\), is employed to analyze the load of the crankshaft bearings. The origin is located at the center of the distribution circle of the pin gears. The direction of the \(y\) axis coincides with the eccentric direction of the crankshaft. The cycloidal gear bears the forces from the pin gear and crankshaft bearings. The component forces of pin gears on cycloid gear along the \(x\)-axis and \(y\)-axis are \(\sum P_{ix}\) and \(\sum P_{iy}\), respectively. The forces acting on the cycloidal gear by the crankshaft bearing are decomposed into, \(F_{j1}\), \(F_{j2}\), and \(F_{j3}\). The torque caused by the force \(F_{j1}\) is balanced with the torque caused by the force \(\sum P_{ix}\). The force \(F_{j2}\) is balanced with force \(\sum P_{ix}\) and \(F_{j3}\) is balanced with \(\sum P_{iy}\). This can be obtained from the force balance on the cycloidal gear:  
\[\left\{\begin{array}{l}F_{j1}=\frac{1}{n}\sum P_{ix}\frac{e_{xg}}{r_{o}}\\ F_{j2}=\frac{1}{n}\sum P_{ix}\\ F_{j3}=\frac{N_{y}}{n}\sum P_{iy}\end{array}\right. \tag{1}\]  
where \(\sum P_{iy}=K_{y}\sum P_{ix}\) and \(K_{y}=\frac{2}{n}\left[\frac{1}{k}+\frac{k^{2}-1}{2k^{2}}\ln\frac{1+k}{1-k}\right]\), \(k\) is the short width coefficient and \(k=eZ_{b}/R_{z}\), \(z_{b}\) is the number of pin teeth, \(e\) is the eccentric distance of the crankshaft, \(z_{g}\) is the number of cycloidal gear teeth, and \(R_{z}\) and is the radius of distribution circle of the pin gear.  
It is assumed that the output torque of the reducer is \(T_{v}\). For the torque equilibrium of the cycloidal gear,  
\[\sum p_{ix}=\frac{T_{v}}{2 ez_{g}} \tag{2}\]  
Figure 2: Schematic diagram of crankshaft bearing structure: (**a**) Cross-section of the RV reducer with crankshaft bearing; (**b**) Structure of the crankshaft bearing.  
In Figure 3b, \(o_{1}\) is the center of the cycloidal gear. \(o_{c}\) is the center of the crankshaft bearing. The coordinate system \(x_{1}o_{c}y_{1}\) is rigidly connected with the crankshaft bearing. The angle between the \(y_{1}\) and \(y\) axes is \(\theta\). It is assumed that the force between the crankshaft bearing and the cycloidal gear is \(F\). \(F\) can be decomposed into the normal force \(F_{r}\) and tangential force \(F_{t}\). \(F_{r}\) is in the direction of the axis \(x_{1}\). \(F_{t}\) is in the direction of the axis \(y_{1}\). The forces acting on the cycloidal gear by the crankshaft bearing are decomposed into \(F_{j1}\), \(F_{j2}\), and \(F_{j3}\). According to the balance between the acting and reaction forces,  
\[\left\{\begin{array}{c}F_{t}=F_{j1}+F_{j2}\cos\theta-F_{j3}\sin\theta\\ F_{r}=F_{j2}\sin\theta+F_{j3}\cos\theta\end{array}\right. \tag{3}\]  
Therefore, the resultant force acting on the crank bearing is  
\[F=\sqrt{F_{t}^{2}+F_{r}^{2}}=\frac{Tv}{2 ez_{g}r_{o}}\sqrt{\left(ez_{g}\right) ^{2}+{r_{o}}^{2}+{k_{y}}^{2}{r_{o}}^{2}+2 ez_{g}r_{o}\cos\theta-2k_{y}ez_{g}r_ {o}\sin\theta} \tag{4}\]  
From the above equation, the load of the crankshaft bearing changes periodically as the crank shaft rotates. The load change period was \(2\pi\). According to [PERSON] et al. [19], the equivalent load of the crankshaft bearing can be calculated by  
\[F_{m}=\left(\frac{1}{2\pi}\int_{0}^{2\pi}F^{4}d_{\theta}\right)^{1/4} \tag{5}\]  
The equivalent load of the bearing obtained through integration is  
\[F_{m}=\frac{T_{p}}{2}\left(\frac{\left(ez_{g}\right)^{4}+4(1+k_{y}^{2})\left( ez_{g}\right)^{2}r_{o}^{2}+\left(1+k_{y}^{2}\right)^{2}r_{o}^{4}}{n^{4} \left(ez_{g}\right)^{4}r_{o}^{4}}\right)^{1/4} \tag{6}\]