## 4 Numerical Examples  
For the high-precision gear pair, in order to increase the actual coincidence degree after loading, the design of transmission error is considered firstly to reduce the vibration, and then the topological modification tooth surface is employed to change the shape, size, and position of the contact mark. Take a pair of high precision hypoid gears as an example. See Table 1 for basic parameters and Table 2 for theoretical cutting parameters (the rated torque of gear is 600 N\(\cdot\)m). Figure 4: Geometrical transmission error and loaded transmission error curve. The TCA and ease-off modified deviation are shown in Figure 5 for the theoretical tooth, the conjugate tooth, and the best ease-off tooth. The tooth contact path presents a large angle diagonal contact (see Figure 5a,b). The contacted path for the conjugate tooth surface is along the pitch circle with little transmission error (see Figure 5c). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Pinion** & **Gear** \\ \hline Tooth number & 8 & 41 \\ Mean helix angle/(deg) & 48.93 & 30.63 \\ Addendum height/(mm) & 5.77 & 1.05 \\ Dedendum height/(mm) & 1.16 & 5.73 \\ Pitch cone angle/(deg) & 12.53 & 76.82 \\ Face cone angle/(deg) & 17.45 & 77.73 \\ Root cone angle(deg) & 11.67 & 71.68 \\ Outer cone distance/(mm) & 97.19 & 84.72 \\ Face width/(mm) & 28 & 24 \\ Axial setting & 13.84 & \(-\)1.43 \\ Generation type & Generated & Formate \\ Offset distance/(mm) & & 23 \\ Shaft angle/(deg) & & 90 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric parameters of the hypoid gear pair. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Pinion (Concave Surface)** & **Gear (Convex Surface)** \\ \hline Tilt angle/(deg) & 16.7 & 0 \\ Swivel angle/(deg) & 346.7 & 0 \\ Cutter radius/(deg) & 80.5 & 75.4 \\ Cutter angle/(deg) & 20 & 22.5 \\ Radial setting/(mm) & 73.95 & 74.04 \\ Basic cangle/(deg) & 83.6 & \(-\)66.3 \\ Vertical/(mm) & 19.6 & 0 \\ Axial/(mm) & 2.4 & 0.6 \\ Machine center to back/(mm) & 2.7 & 0 \\ Machine root angle/(deg) & \(-\)2 & 70.9 \\ Roll ratio & 5.064418 & — \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Machining parameters for theoretical tooth surfaces. Figure 5: _Cont._  
The ease-off modification of the theoretical tooth is shown in Figure 5d. The best ease-off modification of the optimized tooth is shown in Figure 5e0, and is composed of contact gaps from the contact line of the tooth (see Figure 5e1) and gaps of the tooth from the unload transmission error (see Figure 5e2). Conclusions can be drawn from Figure 5 that ease-off modification are consistent with the contact area for the theoretical tooth, the conjugate tooth, and the best ease-off tooth, respectively. The larger the amplitude of transmission error, the larger the tooth surface deviation. There are 22 contact points on the tooth surface, and a meshing period is divided into eight equal parts, and then the design of coincidence degree \(\epsilon_{\mathbf{t}}\) is close to \((22-1)/8=2.6\). The ease-off modification is the largest for the theoretical tooth, which can reduce the sensitivity of assembly error, but leads to a decrease in the effective degree of contact \(\epsilon_{\mathbf{c}}\). The ease-off modification refers to the zeros for the conjugate tooth, which contributes to the biggest coincidence degree, but leads to greater sensitivity to assembly error. The best ease-off modification can reduce the sensitivity of the assembly error, with an effective coincidence degree greater than the theoretical tooth. Tooth load distribution based on LTCA is shown in Figure 6. The load distribution of the theoretical tooth is shown in Figure 6a, and is concentrated into the central. The load distribution of the conjugate tooth is shown in Figure 6b, and has obvious contact with the edge of tooth, while the load value is basically the same in the middle, top, and root of the tooth.