## 2 Theoretical Framework for Numerical Analysis of Oil Injection Lubrication  
### 2.1. Gear Modeling Theory  
Variable tooth thickness gears, if designed based on the instant axis surface, cannot ensure the requirement of the center distance between the main and driven gears by selecting the pitch surface and, therefore, must rely on the pitch cone for design. The working pitch cone of the variable tooth thickness gear is shown in Figure 1, where the point \(M\) lies on the pitch cone line, the working cone angle is \(\gamma_{\omega j}\), and \(n\) is the unit normal vector. In Figure 2, \(S_{f}(x_{f},y_{f},z_{f})\) represents the fixed coordinate system, \(S^{\prime}_{1}(x^{\prime}_{1},y^{\prime}_{1},z^{\prime}_{1})\) is the coordinate system of the driving gear, and \(S^{\prime}_{2}(x^{\prime}_{2},y^{\prime}_{2},z^{\prime}_{2})\) is the coordinate system of the driven gear. Even though the meshing between the variable tooth thickness gear and the forming rack is linear contact, its actual meshing transmission is point contact, with the point \(P\) being the contact point. \(E\) is the shortest distance between cones; \(d_{1}\) is the shortest distance along the axis direction of the cone top; and \(\tau_{1}\), \(\tau_{2}\), \(\gamma_{\omega 1}\), \(\gamma_{\omega 2}\), \(r_{\rho\omega 1}\), and \(r_{\rho\omega 2}\) are the unit vectors, working pitch cone angles, and working pitch circle radii on gear 1 and gear 2, respectively.  
The equation of the pitch cone, \(R_{j}(j=1,2)\), can be expressed in the \(S\) coordinate system as  
\[R_{j}=\begin{bmatrix}x_{j}\\ y_{j}\\ z_{j}\end{bmatrix}=\begin{bmatrix}u_{j}\cos\theta_{j}\sin\gamma_{\omega j}\\ u_{j}\sin\theta_{j}\sin\gamma_{\omega j}\\ u_{j}\cos\gamma_{\omega j}\end{bmatrix} \tag{1}\]  
where the coordinates of point \(M\) are \((x_{j},y_{j},z_{j})\), \(u_{j}\) is the distance from point \(M\) along the cone line to the cone apex, and \(\theta_{j}\) is the angle of extension of point \(M\) in the plane \(x-y\). Then, the unit normal vector, \(n_{j}\), on the pitch cone line can be expressed as follows:  
\[n_{j}=\frac{N_{j}}{\left|N_{j}\right|}=\begin{bmatrix}\cos\theta_{j}\cos\gamma _{\omega j}\\ \sin\theta_{j}\cos\gamma_{\omega j}\\ -\sin\gamma_{\omega j}\end{bmatrix} \tag{2}\]  
Based on the transformation relationships between coordinate systems, the contact equation of the contact points can be derived as follows:  
\[\begin{cases}R_{fw}^{1}=R_{fw}^{2}\\ n_{fw}^{1}=-n_{fw}^{2}\end{cases} \tag{3}\]  
Figure 1: Working pitch cone of variable tooth thickness gear.  
Figure 2: Fixed coordinate system.  
where \(R_{fw}^{j}\) is the position vector of the contact point \(P\) in gear \(j\), and \(n_{fw}^{j}\) is the unit normal vector of the contact point \(P\) in gear \(j\).  
Given that \(r_{pw\imath}=\sin\gamma_{\omega j}u_{j}\), it follows that  
\[\left\{\begin{array}{l}r_{pw\imath 1}\cos\theta_{1}=r_{pw\imath 1}\cos\theta_{2} +E\\ r_{pw\imath 1}\sin\theta_{1}=-r_{pw\imath 2}\sin\theta_{2}\\ d_{1}=r_{pw\imath 1}\cot\gamma_{\omega 1}+r_{pw\imath 2}\cot\gamma_{\omega 2}\\ \sin\theta_{1}\cos\gamma_{\omega 1}=\sin\theta_{2}\cos\gamma_{\omega 2}\\ \cos\theta_{1}\cos\gamma_{\omega 1}=-\cos\theta_{2}\cos\gamma_{\omega 2}\\ \sin\gamma_{\omega 1}=\sin\gamma_{\omega 2}\end{array}\right. \tag{4}\]  
Additionally, \(\left|n_{fw\imath}^{1}\right|=\left|n_{fw\imath}^{2}\right|\), and the mutual relationship between the two pitch cone angles is obtained as follows:  
\[\gamma_{\omega 1}=\gamma_{\omega 2} \tag{5}\]  
Simplifying the above, the mathematical expressions for \(E\) and \(d_{1}\) are obtained as follows:  
\[d_{1}=(\gamma_{pw\imath 1}+\gamma_{pw\imath 2})\cot\gamma_{\omega 1} \tag{6}\]  
\[E=\gamma_{pw\imath 1}+\gamma_{pw\imath 2} \tag{7}\]