## 2 Cutting Engagement of Worm  
### Generating Surface Equation of Grinding Wheel  
Worm-cutting engagement is demonstrated via a right-handed worm in this section. To ensure meshing quality, both helical surfaces of the worm are ground using conical grinding wheels with specific taper angles [27; 28]. The working surfaces of the conical grinding wheel are designated as \(\Sigma_{\text{S}}^{(S)}\). The worm helicoids can be mathematically represented as \(\Sigma_{\text{1}}^{(S)}\). The subscript S (where S \(\in\) {1, 2}) identifies the two flanks of the worm. For S = 1, the corresponding flank engages with the convex surface of the face worm gear, hereafter termed the i-flank. For S = 2, it characterizes the opposing flank, designated as the e-flank, as depicted in Figure 2.  
Figure 1: A global flowchart of derivation strategy. To characterize the cutting engagement, a right-handed, fixed reference frame \(\sigma_{\text{01}}\Big{\{}O_{\text{1}};\overset{\rightarrow}{i}_{\text{01}}, \overset{\rightarrow}{j}_{\text{01}},\overset{\rightarrow}{k}_{\text{01}} \Big{\}}\) and a rotating reference frame \(\sigma_{\text{1}}\Big{\{}O_{\text{1}};\overset{\rightarrow}{i}_{\text{1}}, \overset{\rightarrow}{j}_{\text{1}},\overset{\rightarrow}{k}_{\text{1}} \Big{\}}\) are established on the worm. Two additional coordinate systems, \(\sigma_{\text{od}}\Big{\{}O_{\text{d}};\overset{\rightarrow}{i}_{\text{od}}, \overset{\rightarrow}{j}_{\text{od}},\overset{\rightarrow}{k}_{\text{od}} \Big{\}}\) and \(\sigma_{\text{d}}\Big{\{}O_{\text{d}};\overset{\rightarrow}{i}_{\text{d}}, \overset{\rightarrow}{j}_{\text{d}},\overset{\rightarrow}{k}_{\text{d}} \Big{\}}\), are established on the grinding wheel to describe its orientation after two sequential deflections for proper installation, as shown in Figure 3a. The angle \(\varepsilon_{s}\) between \(\overset{\rightarrow}{k}_{\text{od}}\) and the \(O_{\text{1}}-\overset{\rightarrow}{j}_{\text{01}}\overset{\rightarrow}{k}_{ \text{01}}\) plane is designated as the first tilt angle. The symbol \(\gamma\) represents the secondary tilt angle, which is the worm lead angle, as shown in Figure 3b. A local coordinate system, \(\sigma_{\text{a}}\Big{\{}O_{\text{a}};\overset{\rightarrow}{i}_{\text{a}}, \overset{\rightarrow}{j}_{\text{a}},\overset{\rightarrow}{k}_{\text{a}} \Big{\}}\), is defined on the conical grinding wheel to characterize its geometric profile. The parameter \(\delta_{\text{g}}^{(S)}\) denotes the semi-cone angle of the grinding wheel, while \(r_{\text{g}}^{(S)}\) indicates the radius at its larger end. Figure 4 presents the complete configuration of these coordinate systems, as shown in Figure 4.  
Figure 3: Setting of coordinate system for cutting engagement. (**a**) Coordinate systems for worm grinding process; (**b**) first tilt angle \(\gamma_{\text{1}}\) and second tilt angle \(\varepsilon_{\text{3}}^{(S)}\) of the grinding wheel. Figure 2: Schematic of cutting engagement of worm and meshing of face worm gear drive. Using the geometrical relationship in Figure 4, the equation of \(\Sigma_{\text{g}}^{(S)}\) can be expressed in \(\sigma_{\text{a}}\) as:  
\[\left(\overset{\rightarrow}{\boldsymbol{r}}_{\text{a}}\right)_{\text{a}}=u\sin \delta_{\text{g}}\cos\theta\overset{\rightarrow}{\boldsymbol{i}}_{\text{a}}+u \sin\delta_{\text{g}}\sin\theta\overset{\rightarrow}{\boldsymbol{j}}_{\text{a }}^{(I)}+u\cos\delta_{\text{g}}\overset{\rightarrow}{\boldsymbol{k}}_{\text{a}}, \tag{1}\]  
where \(u\) and \(\theta\) are the two variables of \(\Sigma_{\text{g}}^{(S)}\), whose normal vector \(\left(\overset{\rightarrow}{\boldsymbol{n}}\right)_{\text{a}}=\cos\delta_{ \text{g}}\cos\theta\overset{\rightarrow}{\boldsymbol{i}}_{\text{a}}+\cos \delta_{\text{g}}\sin\theta\overset{\rightarrow}{\boldsymbol{j}}_{\text{a}}- \sin\delta_{\text{g}}\overset{\rightarrow}{\boldsymbol{k}}_{\text{a}}\) and its direction points from the entity to space. Therefore, the fundamental quantities of \(\Sigma_{\text{g}}^{(S)}\) are \(F=0\) and \(M=0\)[29].