## 4 The Effect of Error on the Meshing Force in Star Gear Train
### Relative Displacement of Meshing Line Caused by Error  
In order to analyze the difference in the error on the meshing pairs, the coordinate system is established as shown in Figure 5. Taking the center \(O_{\text{s}}\) of the sun wheel under standard installation as the coordinate origin, the connecting line between the center of the star wheel and the sun gear is the \(x\)-axis, and the \(y\)-axis meets the Cartesian coordinate system to establish the static coordinate system. Then, the position angle of the five star wheels can be expressed as:  
\[\delta=\frac{2\pi(i-1)}{5} \tag{10}\]  
The coordinates of the actual center of the star wheels can be expressed as:  
\[\left\{\begin{array}{l}x_{i}=a\cos\delta\\ y_{i}=a\sin\delta\\ \end{array}\right. \tag{11}\]  
Here, \(a\) is the standard center distance.  
When the installation error of the sun gear appears in the actual installation, the error causes the relative displacement of the meshing pair on each branch. In order to study the influence of the installation error on the five branches, the actual installation coordinate system is established as \(x^{\prime}\) O\({}_{s}\)'\(y^{\prime}\). The connection line between the center of the star wheel and the sun gear on the first branch in the actual installation is the \(x^{\prime}\) axis, as shown in Figure 6. Here, the assembly error of the sun gear is defined as the center deviation distance \(A_{\mathrm{s}}\) and the error position angle \(\gamma_{\mathrm{s}}\). Therefore, the coordinates of the actual center of the sun gear are:  
\[\left\{\begin{array}{l}x_{si}=A_{\mathrm{s}}\cos\gamma_{\mathrm{s}}\\ y_{si}=A_{\mathrm{s}}\sin\gamma_{\mathrm{s}}\end{array}\right. \tag{12}\]  
At this time, the actual center distance between the sun wheel and the star wheels becomes:  
\[{a_{i}}^{\prime}=\sqrt{\left(x_{i}-x_{si}\right)^{2}+\left(y_{i}-y_{si}\right) ^{2}} \tag{13}\]  
The corresponding actual meshing angle can be derived from the following formula:  
\[{\alpha_{i}}^{\prime}=\arccos\frac{a\cdot\cos\alpha}{{a_{i}}^{\prime}} \tag{14}\]  
The distance between the center of star wheel and the center of the first star wheel can be calculated as:  
\[l_{i}=\sqrt{\left(x_{i}-a\right)^{2}+{y_{i}}^{2}} \tag{15}\]  
Therefore, the position angle of each star wheel under the actual installation coordinate system can be expressed as:  
\[\varphi_{i}=\arccos\frac{a_{1}^{2}+a_{i}^{2}-l_{i}^{2}}{2a_{1}a_{i}} \tag{16}\]  
Now, the position angle \(\beta_{ii}\) of the meshing line between the sun gear and the star wheel can be expressed as:  
\[\beta_{ii}=\frac{\pi}{2}-{\alpha_{i}}^{\prime}+\varphi_{i} \tag{17}\]  
Figure 5: Diagram of gear meshing position relationship.  
Then, the equivalent displacement on the meshing line caused by the installation error of the sun gear can be expressed as:  
\[\left\{\begin{array}{l}\overline{A}_{\mathrm{s}x}=A_{\mathrm{s}x}\cdot\cos\beta _{\mathrm{i}i}\\ \overline{A}_{\mathrm{s}y}=A_{\mathrm{s}y}\cdot\sin\beta_{\mathrm{i}i}\end{array}\right. \tag{18}\]  
After superposition of \(\overline{A}_{\mathrm{s}x}\) and \(\overline{A}_{\mathrm{s}y}\), the equivalent displacement of the installation error on the meshing line can be expressed as:  
\[e_{\mathrm{s}pi}=A_{\mathrm{s}}\cdot\cos\gamma_{\mathrm{s}}\cdot\cos\beta_{ \mathrm{i}i}+A_{\mathrm{s}}\cdot\sin\gamma_{\mathrm{s}}\cdot\sin\beta_{\mathrm{ i}i} \tag{19}\]  
Similarly, when the inner ring gear has an installation error, the relative displacement of the installation error on the meshing line can also be expressed as:  
\[e_{\mathrm{p}ir}=A_{\mathrm{r}}\cdot\cos\gamma_{\mathrm{r}}\cdot\cos\beta^{ \prime}_{\mathrm{i}i}+A_{\mathrm{r}}\cdot\sin\gamma_{\mathrm{r}}\cdot\sin\beta ^{\prime}_{\mathrm{i}i} \tag{20}\]  
Here, \(\overline{A}_{\mathrm{r}}\) is the installation error of the ring gear, \(\gamma_{\mathrm{r}}\) is the position angle of the installation error of the ring gear, and \(\beta^{\prime}_{\mathrm{i}i}\) is the position angle of the meshing line between the star gear and the internal gear.