## 3 Establishment of the Installation Error Model for the Drum-Shaped Grinding Tool
### Deviation Calculation of Internal Helical Gear Tooth Surface  
According to the surface equation of the internal helical gear in the absence of drum-shaped grinding tool installation errors, and by discrete tooth surface for calculating the coordinates of tooth surface points, the radial vector \(\boldsymbol{r}_{t(i,j)}\) (\(i=1\)-\(5\), \(j=1\)-\(9\)) and the normal \(\boldsymbol{n}_{t(i,j)}\) of the theoretical tooth surface grid points can be obtained. The same method can be used to obtain the radial vector \(\boldsymbol{r}_{e(i,j)}\) of the error tooth surface grind points of the internal helical gear in the presence of drum-shaped grinding tool installation errors. Tooth surface normal deviation refers to the distance by which the tooth surface deviates from the theoretical value in the normal direction which can be determined as follows:  
\[K_{(i,j)}^{h}=(\boldsymbol{r}_{t(i,j)}-\boldsymbol{r}_{e(i,j)})\cdot \boldsymbol{n}_{t(i,j)} \tag{16}\]  
The degree of deviation between the error tooth surface and the theoretical tooth surface of the internal helical gear is measured by calculating the average normal deviation value \(K\) of the tooth surface, which can be expressed as:  
\[K=\sum\limits_{i=1,j=1}^{i=5,j=9}\left|K_{(i,j)}^{h}\right| \tag{17}\]  
Analysis of the Impact of Drum-Shaped Grinding Tool Installation Errors on Internal Helical Gear Tooth Surface Deviation  
The impact of drum-shaped grinding tool installation errors on the distribution of the tooth surface deviation of the internal helical gear is analyzed, based on the parameters of the drum-shaped grinding tool and the internal helical gear provided in Table 1, and the installation errors of the drum-shaped grinding tool detailed in Table 2.  
Taking the left tooth surface as an example, the distribution of deviation was researched. As shown in Figure 8, the distribution of normal deviation of the tooth surface of the internal helical gear involving error \(A\) is presented. From a numerical point of view, it can be seen in Figure 8 that the maximum normal deviation of the tooth surface caused by error \(A\) occurs at the tooth dependum on the upper transverse face and the tooth addendum on the lower transverse face. At the tooth addendum, the normal deviation decreases from the upper transverse face to the lower transverse face and then increases, while at the tooth dedendum, it gradually decreases. The normal deviation varies significantly along both the tooth height and width directions. As shown in Figure 9, the distribution of normal deviation of the tooth surface of the internal helical gear involving error \(B\) is presented. From Figure 9, it can be observed that the maximum normal deviation of the tooth surface caused by error \(B\) occurs at the tooth addendum on both the upper and lower transverse faces. At both the addendum and the tooth dedendum, the normal deviation decreases from the upper transverse face to the lower transverse face and then increases. Thus, normal deviation varies significantly along both the tooth height and width directions. As shown in Figure 10, the distribution of normal deviation of the tooth surface of the internal helical gear involving error \(C\) is presented. \begin{table}
\begin{tabular}{c c c c c} \hline
**Group. No** & **Eccentricity** & **Eccentric Angle** & **Perpendicularity** & **Perpendicularity** \\  & \(e\) **(mm)** & \(a\) **(°)** & _\(\mathbf{\varphi_{x}}\) **(°)** & _\(\mathbf{\varphi_{y}}\) **(°)** \\ \hline \(A\) & \(0.5\) & \(30\) & \(0\) & \(0\) \\ \(B\) & \(0.5\) & \(120\) & \(0\) & \(0\) \\ \(C\) & \(0\) & \(0\) & \(-0.5\) & \(0\) \\ \(D\) & \(0\) & \(0\) & \(0\) & \(-0.5\) \\ \hline \end{tabular}
\end{table} 
Table 2: Drum-shaped grinding tool installation errors. Figure 8: Deviation distribution of internal helical gear tooth surface involving error \(A\). Figure 9: Deviation distribution of internal helical gear tooth surface involving error \(B\).