## 2 Calculation of the Surface of the Drum-Shaped Grinding Tool
### Two-Degree-of-Freedom Meshing Equation  
To obtain the correct tooth surface of the internal helical gear ground by the drum-shaped grinding tool, two independent motions are necessarily required, which are the generation motion, where the drum-shaped grinding tool and the internal helical gear rotate around their respective axes according to the transmission ratio, and the feed motion, where the drum-shaped grinding tool moves along the axial direction of the internal helical gear. The tooth height profile and width profile of the internal helical gear are obtained through the generating motion and the axial feed motion, respectively. The whole process of generation grinding of the internal helical gear with the drum-shaped grinding tool is completed by the combination of these two motions. Figure 3 shows the model for grinding the internal helical gear using the drum-shaped grinding tool. Figure 3: Internal helical gear grinding using the drum-shaped grinding tool. Figure 2: Discretization of the tooth surface of internal helical gear. Interference between the grinding tool and the internal helical gear due to the presence of shaft angle can be avoided by using the drum-shaped grinding tool. The diameter of the drum-shaped grinding tool gradually increases from both the upper and lower transverse faces towards the middle. Therefore, the transverse face profiles of the drum-shaped grinding tool are different and need to be calculated separately. The coordinate system between the internal helical gear and the drum-shaped grinding tool can be established, as illustrated in Figure 4.  
As shown in Figure 4, \(S_{w}\)(\(O_{w}\)-\(x_{w}y_{w}z_{w}\)) represents the coordinate system at the transverse face of the drum-shaped grinding tool, and it can be moved along the axial direction, and \(S_{g}\)(\(O_{s}\)-\(x_{g}y_{g}z_{g}\)) represents the fixed coordinate system. \(S_{1}\)(\(O_{1}\)-\(x_{1}y_{1}z_{1}\)) and \(S_{2}\)(\(O_{2}\)-\(x_{2}y_{2}z_{2}\)) rotate with the drum-shaped grinding tool and the internal helical gear, respectively. At the initial, the coordinate system \(S_{1}\) coincides with \(S_{w}\), and \(S_{2}\) coincides with \(S_{g}\), \(b^{\prime}\) is the tooth width at the transverse face of the drum-shaped tool, and \(a\) is the center distance from the drum-shaped grinding tool to the internal helical gear. \(z_{1}\) and \(z_{2}\) coincide with the rotational axes of the drum-shaped grinding tool and the internal helical gear, respectively, and the angle between the two axes is \(\Sigma\). The drum-shaped grinding tool rotates around the \(z_{1}\) axis with angular velocity \(\omega_{1}\), while the internal helical gear rotates around the \(z_{2}\) axis with angular velocity \(\omega_{2}\) and moves along the \(z_{2}\) axis with velocity \(v_{02}\) (in actual processing, this motion is performed by the drum-shaped grinding tool). \(\Delta\) means the horizontal distance from \(S_{1}\) to \(S_{2}\), which can be obtained as:  
\[\Delta=b^{\prime}\sin\sum \tag{5}\]  
\(\lambda\) denotes the rotation angle of the internal helical gear at different transverse faces meshing positions, which can be acquired as follows:  
\[\lambda=\arctan\frac{\Delta}{a} \tag{6}\]  
The number of degree-of-freedom in the spatial meshing motion refers to how many motion parameters are independent. There are three parameters, \(\omega_{1}\), \(\omega_{2}\), and \(v_{02}\), in the process of generation grinding of the internal helical gear by the drum-shaped grinding tool. Due to the existence of a certain transmission relationship between the drum-shaped grinding tool and the internal helical gear, when \(\omega_{1}\) is determined, \(\omega_{2}\) is also determined (or when \(\omega_{2}\) is determined, \(\omega_{1}\) is also determined). Therefore, two motion parameters are independent of each other and are called two-degree-of-freedom meshing. Figure 4: Coordinate system applied for continuous generation grinding of internal helical gear.