## 3 Tooth Probability Strength Fitting Based on Gear Fatigue Test
### Tooth Probability Strength Fitting  
Tooth root stress peak is used as an evaluation index for stress grade, and the bending fatigue performance of tooth root is tested by the group method under four stress grades. The selected stress levels and the number of test points under various stress levels are 649 MPa (17 points), 618 MPa (22 points), 586 MPa (29 points), and 555 MPa (38 points). During the test, if any tooth on the gear sample fails first, the testing machine will automatically stop, and the direct data obtained from this is gear life rather than tooth life. It represents the ability of the individual gear to maintain excellent transmission function under current stress level, so gear life is also \"first broken tooth\" life. From the perspective of probability, the more teeth on a gear, the more potential for failure links. Therefore, under the same stress and revolution conditions, the failure risk of the gear will increase with the increase in the number of teeth. In the process of this test, a complete rotation of the sample is recorded as one gear life. In this conventional counting mode, the statistical characteristics of the number of teeth lead to the difference in the probability life between the gear and tooth. In order to obtain direct strength input variables for the reliability prediction model, a probabilistic statistical transformation method is proposed to fit tooth P\(-\)S\(-\)N curves based on gear life data. The probability life relationship between the gear and tooth is established based on the concept of minimum order statistics. The fracture of any tooth on a gear will cause  
Figure 6: Gear bending fatigue test: (**a**) test platform; (**b**) vibration monitoring; (**c**) revolution counter; (**d**) loading device; (**e**) acceleration sensors; (**f**) gear sample; (**g**) gear installation; (**h**) root cracks; (**i**) fracture morphology. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Items** & **Parameters** & **Items** & **Parameters** \\ \hline Module (mm) & 5 & ISO quality grade & 5 \\ Number of teeth & 25 & Material brand & 1 Cr1N39T1 \\ Pressure angle (\({}^{\circ}\)) & 20 & Carburized depth (mm) & 0.8 \(\pm\) 0.13 \\ Helix angle (\({}^{\circ}\)) & 0 & Tooth surface hardness & 59–63 HRC \\ Face width (mm) & 32 & Tooth core hardness & 35–48 HRC \\ Root fillet radius (mm) & 2.7 & Precision machining & Grinding \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter list of test gear. the gear to lose excellent transmission capacity. For this reason, it can be considered that the life of a gear depends on the minimum life of its teeth. Suppose \(X_{1},\ X_{2},\ \ldots,\ X_{n}\) is a set of samples from a parent \(X\), then \(X_{\text{min}}=\text{min}(X_{1},\ X_{2},\ \ldots,\ X_{n})\) is the minimum order statistics of the parent. This probability model will be applied to the life transformation calculation under the failure mode of tooth root bending fatigue, then gear probability life is equal to the minimum order statistics of tooth probability life. Assuming that the cumulative distribution function of random variable \(X\) is \(F(x)\) and its probability density function is \(f(x)\), then the probability density function of the minimum order statistics of \(X\) can be expressed as  
\[g_{\text{min}}(x)=z\cdot[1-F(x)]^{z-1}\cdot f(x) \tag{8}\]  
where \(z\) indicates the number of gear teeth. If the two-parameter Weibull distribution is adopted to express tooth probability life, then the cumulative distribution function can be expressed as  
\[F(x)=1-\exp\left[-(x/\theta)^{\beta}\right] \tag{9}\]  
and the probability density function is  
\[f(x)=\left(\beta\cdot x^{\beta-1}/\theta^{\beta}\right)\exp\left[-(x/\theta)^ {\beta}\right] \tag{10}\]  
where \(\beta\) and \(\theta\) is respectively the shape parameter and scale parameter of the tooth life distribution.