## 3 Curvature Interference Limit Function of Meshing
\begin{array}{l}\left(\stackrel{{\rightarrow}}{{ \omega}}_{12}\right)_{01}=\left(\stackrel{{\rightarrow}}{{ \omega}}_{1}\right)_{01}-\left(\stackrel{{\rightarrow}}{{ \omega}}_{2}\right)_{01}=-\frac{1}{i_{12}}\stackrel{{ \rightarrow}}{{j}}_{01}+\stackrel{{\rightarrow}}{{k}}_{01}\\ \left(\stackrel{{\rightarrow}}{{V}}_{12}\right)_{01}=\left( \stackrel{{\rightarrow}}{{\omega}}_{12}\right)_{01}\times \left(\stackrel{{\rightarrow}}{{r}}_{1}\right)_{01}-\left( \stackrel{{\rightarrow}}{{\omega}}_{2}\right)_{01}\times \left(\stackrel{{\rightarrow}}{{O}}_{2}\right)_{01}=V_{x} \stackrel{{\rightarrow}}{{i}}_{01}+V_{y}\stackrel{{ \rightarrow}}{{j}}_{01}+V_{z}\stackrel{{\rightarrow}}{{k}}_{01} \end{array} ,\right. \tag{12}\]  
where \(V_{x}=-y_{01}^{*}-\frac{1}{i_{12}}\left(z_{01}-p\varphi+z_{\rm A}+\frac{L_{W}} {2}\right)\), \(V_{y}=x_{01}^{*}\), and \(V_{z}=\frac{x_{01}^{*}+a}{i_{12}}\). From Equations (11) and (12), the meshing function \(\phi\) can be obtained as:  
\[\phi=\left(\stackrel{{\rightarrow*}}{{n}}\right)_{01}\cdot \left(\stackrel{{\rightarrow}}{{V}}_{12}\right)_{01}=\frac{1}{i_{12 }}[A\sin(\varphi_{1}-\varphi)+B\cos(\varphi_{1}-\varphi)+C], \tag{13}\]  
where the components can be obtained after calculation as \(A=n_{y}\Big{(}z_{01}-p\varphi+z_{\rm A}+\frac{L_{W}}{2}\Big{)}-n_{z}(y_{01}+a_ {\rm d})\), \(B=-n_{x}\Big{(}z_{01}-p\varphi+z_{\rm A}+\frac{L_{W}}{2}\Big{)}+n_{z}x_{01}\), and \(C=n_{z}(a-i_{12}p)\). Using the coordinate transformation \(\sigma_{01}\rightarrow\sigma_{2}\), the equation for \(\Sigma_{2}^{(S)}\) can be determined in \(\sigma_{2}\) from Equations (11) and (13) as:  
\[\left\{ \begin{array}{l}\left(\stackrel{{\rightarrow}}{{r}}_{2} \right)_{2}=R\left[\stackrel{{\rightarrow}}{{k}}_{2},\varphi_{2} \right]\bigg{\{}R\left[\stackrel{{\rightarrow}}{{i}}_{02},- \frac{\pi}{2}\right]\left(\stackrel{{\rightarrow}}{{r}}_{1}^{*} \right)_{01}+\left(\stackrel{{\rightarrow}}{{O}}_{2}\stackrel{{ \rightarrow}}{{O}}_{1}\right)_{02}\bigg{\}}=x_{2}\stackrel{{ \rightarrow}}{{i}}_{2}+y_{2}\stackrel{{\rightarrow}}{{j}}_{2}-y_{01}^{* }\stackrel{{\rightarrow}}{{k}}_{2}\\ \phi_{\rm d}(u,\theta)=0\\ \phi(u,\theta,\varphi,\varphi_{1})=0\end{array} ,\right. \tag{14}\]  
whose components are \(x_{2}=\left(x_{01}^{*}+a\right)\cos\varphi_{2}-\left(z_{01}-p\varphi+z_{\rm A}+ \frac{L_{W}}{2}\right)\sin\varphi_{2}\), \(y_{2}=\left(x_{01}^{*}+a\right)\sin\varphi_{2}+\left(z_{01}-p\varphi+z_{\rm A}+ \frac{L_{W}}{2}\right)\cos\varphi_{2}\), and \(z_{2}=-y_{01}^{*}\). When \(S=1\), Equation (14) denotes the convex flank of the face worm gear, while for \(S=2\), it represents the concave flank. Through the computation of the partial derivative of \(\phi\) with \(\varphi_{1}\), the meshing limit function can be determined as:  
\[\frac{\partial\phi}{\partial\varphi_{1}}=\frac{1}{i_{12}}[A\cos(\varphi_{1}- \varphi)-B\sin(\varphi_{1}-\varphi)], \tag{15}\]  
A moving orthogonal frame \(\sigma_{01}^{*}\left\{P_{i}\overset{\rightarrow}{\boldsymbol{\alpha}}_{ \xi},\overset{\rightarrow}{\boldsymbol{\alpha}}_{\eta},\overset{\rightarrow }{\boldsymbol{n}}^{*}\right\}\) is built at an arbitrary contact point of \(\Sigma_{1}^{(S)}\) during meshing. \(\overset{\rightarrow}{\boldsymbol{\alpha}}_{\xi}\) and \(\overset{\rightarrow}{\boldsymbol{\alpha}}_{\eta}\) are arbitrarily two orthogonal directions on \(\Sigma_{1}^{(S)}\). According to Equation (9), \(\overset{\rightarrow}{\boldsymbol{\alpha}}_{\xi}\) and \(\overset{\rightarrow}{\boldsymbol{\alpha}}_{\eta}\) can be obtained as:  
\[