## 5 Verification Test
### Toth Contact Pattern Test  
When the loading torque T = 500 Nm, 1500 Nm, and 4500 Nm is applied separately, the real images of the contact pattern of the teeth surface are compared with the FEM simulation result as shown in Figure 16.  
As shown in Figure 16 above, we can find that the test and simulation results are basically consistent, and directional angle and contact trace change trends are similar, which preliminarily verifies the reliability of previously generated finite element models. The theoretical contact ratio of this spiral level gear pair is 2.26, which is calculated by Equation (45) according to gear design and manufacture parameters [26]. The theoretical contact ratio is the maximum contact ratio of gear pair can be achieved with the increase of load under the condition of no edge contact. Compared the real images of Figure 16a with Figure 16b above, we can find that, when the load T increases from 500 Nm to 1500 Nm, the contact pattern becomes larger but still within the tooth surface boundary, so the actual contact ratio of condition (a) and (b) are smaller than theoretical contact ratio, and the value are 1.53 and 1.74, which are calculated by Equations (46) and (47) [27; 28]. When load T is 4500 Nm, compare with 500 Nm and 1500 Nm, the contact pattern extend to the large end of the boundary and occur edge contact, so the actual contact ratio of condition (c) exceeds theoretical contact ratio, the result is 2.31.  
\[\varepsilon_{\gamma}=\sqrt{\varepsilon_{\alpha}{}^{2}+\varepsilon_{\beta}{}^{2}} \tag{45}\]  
\(\varepsilon_{\gamma}\) is theoretical contact ratio, \(\varepsilon_{\alpha}\) is transverse contact ratio, and \(\varepsilon_{\beta}\) is overlap contact ratio.  
\[\varepsilon_{\gamma}{}^{\prime}=\frac{\varepsilon_{\alpha}{}^{\prime 2}}{ \sqrt{\varepsilon_{\alpha}{}^{\prime 2}-I_{1}{}^{2}}} \tag{46}\]  
\[I_{1}=\sqrt[3]{\frac{C_{P}T_{1}}{C_{PD}T_{1D}}}\frac{\varepsilon_{\alpha} \varepsilon_{\beta}}{\varepsilon_{\gamma}} \tag{47}\]  
Figure 16: Results comparison of test and FEM of contact pattern: (**a**) Load T = 500 Nm, (**b**) Load T = 1500 Nm, and (**c**) Load T = 4500 Nm.  
\(\varepsilon_{y^{\prime}}\) is actual contact ratio, \(\varepsilon_{a^{\prime}}\) is revised transverse contact ratio, \(C_{P}\) is load distribution ratio, \(C_{PD}\) is load distribution ratio under full load, \(T_{1}\) is actual load and \(T_{1D}\) is full load.  
In summary, the actual contact ratio increases with the load T increases, which is beneficial to increase the meshing stiffness and improve the smoothness of transmission. When actual contact ratio is too large, the gear pair will occur severe edge contact, vibration, shock and noise. On the one hand, we need design theoretical contact ratio properly according to gear parameters and actual operating conditions, avoiding actual contact ratio greater than theoretical contact ratio. On the other hand, when we need loading large load, the actual installation position should make the contact zone of the tooth surface close to the small end when the light load is applied, so as to fully exert the loading capacity of the tooth surface, and avoiding edge contact under heavy load as much as possible.