# The Simulation Analysis of Cutting Characteristics and Chip Formation Mechanism in Gear Skiving
## 3 Analysis of Cutting Angles and Sliding Speed in Gear Skiving  
The working angles and cutting speed of the skiving tool are crucial parameters affecting the cutting state andhave a significant impact on tool wear. To study the variation laws of the working angles and cutting speed at different points along the cutting edges of the skiving tool during the skiving process, a calculation model for the working angles and cutting speed at a point \(O\) on the cutting edge was established. Figure 4 shows the schematic diagram for the calculation of cutting angles and cutting speed in gear skiving. Figure 3: Cutting pattern of gear skiving. Based on the algorithm from reference [10], the normal vector \(\mathbf{n}\) of the cutting plane can be expressed as the cross product of the cutting edge vector \(\mathbf{t_{c}}\) and the sliding velocity \(\mathbf{v}\) of the tool relative to the workpiece:  
\[\mathbf{n}=\mathbf{t_{c}}\times\mathbf{v} \tag{2}\]  
The unit vector \(\mathbf{v_{n}}\) of normal cutting velocity at a point on the cutting edge can be expressed as:  
\[\mathbf{v}_{n}=\frac{\mathbf{n}\times\mathbf{t_{c}}}{|\mathbf{n}\times \mathbf{t_{c}}|} \tag{3}\]  
In the orthogonal plane \(P_{0}\), the normal working rake angle \(\gamma_{n}\) at a point on the cutting edge can be expressed as the angle between the normal vector \(n\) and rake face; it can also be expressed as the angle between the rake face normal vector \(\mathbf{n_{r}}\) and the unit vector of normal cutting velocity \(\mathbf{v_{n}}\):  
\[\gamma_{n}=\arccos\frac{\mathbf{n_{r}}\cdot\mathbf{v_{n}}}{|\mathbf{n_{r}}|} \tag{4}\]  
The normal vector of the flank face \(\mathbf{n_{f}}\) at a point on the cutting edge can be expressed as the cross product of the cutting edge vector \(\mathbf{t_{c}}\) and the helix vector of the flank face \(\mathbf{t_{r}}\):  
\[\mathbf{n_{f}}=\mathbf{t_{c}}\times\mathbf{t_{r}} \tag{5}\]  
In the orthogonal plane \(P_{0}\), the normal working clearance angle at a point on the cutting edge \(\mathbf{\alpha_{n}}\) can be expressed as the angle between the normal vector of the flank face \(\mathbf{n_{f}}\) and the normal vector of the cutting plane \(\mathbf{n}\):  
\[\alpha_{n}=\arccos\frac{\mathbf{n_{f}}\cdot\mathbf{n}}{\left|\mathbf{n_{f}} \right|\cdot|\mathbf{n}|} \tag{6}\]  
In the base plane \(P_{s}\), the tool inclination angle at point \(O\) on the cutting edge \(\mathbf{\lambda_{s}}\) can be expressed as the angle between the cutting velocity \(\mathbf{v}\) and the normal cutting velocity \(\mathbf{v_{n}}\):  
\[\lambda_{s}=\arccos\frac{\mathbf{v}\cdot\mathbf{v_{n}}}{|\mathbf{v}|\cdot| \mathbf{v_{n}}|} \tag{7}\]  
The tangential cutting velocity can be expressed as:  
\[\mathbf{v_{f}}=\mathbf{v}\cdot\text{cos}(\lambda_{s}) \tag{8}\]  
Figure 4: Schematic diagram of cutting angles and cutting speed in gear skiving. The intersection of the cutting edge and top edge is defined as the tool tip. If \(\mathbf{v_{r}}\) leans to the tool tip, \(\mathbf{v_{r}}\) is negative; otherwise, it is positive. Based on the parameters in Table 1, the working rake angle of the skiving tool during the skiving process was calculated. The results are shown in Figure 4(a). The horizontal axis represents the tooth thickness half-angle at a point on the cutting edge, the vertical axis represents the tool rotation angle, and the \(z\)-axis represents the working rake angle of the tool. Figure 5: Working rake angle, clearance angle, and cutting thickness in gear skiving. The cutting edge of the tool starts cutting into the workpiece from the tooth meshing line until it exits the tooth space. During the process, the working rake angle decreases from \(-1.2^{\circ}\) to \(-47^{\circ}\). The corner portion connecting the top edge and the recessing edge has a high negative rake angle when exiting the cut. It is shown in deep red on the contour plot of the working rake angle. Generally, tools with negative rake angles are more prone to wear.