## 4 Results and Discussions
### Effects of Single Tooth Surface Crack Parameter on Mesh Stiffness and Vibration Characteristics  
#### 4.1.1 Effects of a Single Crack Parameter on Mesh Stiffness  
As displayed in Figure 5, taking the surface crack parameters as \(L_{1}\) = 1, 2, 3 mm, \(L_{2}\) = 40 mm, \(D\) = 3 mm, and \(x_{1}\) = 2 mm, the influence of the crack length (\(L_{1}\)) on the meshing stiffness of single-tooth pair and double-tooth pairs is studied. It can be obtained that the total meshing stiffness of the cracked gear is smaller than that of the healthy gear at the beginning of the meshing process. The stiffness drop zone is enlarged with the increase in the crack length, while the decrease in the magnitude of the meshing stiffness under different crack lengths remains the same. The maximum ratios of the stiffness reduction of single and double tooth pairs are 8.2% and 7.0%, respectively.  
As displayed in Figure 6, the influence of the crack width (\(L_{2}\)) on the meshing stiffness of single-tooth pair and double-tooth pair is investigated when the surface crack parameters are set as \(L_{1}\) = 2 mm, \(L_{2}\) = 15, 30, 45 mm, \(D\) = 3 mm, and \(x_{1}\) = 2 mm. It can be observed from the calculation results that the reduction in the magnitude of the meshing stiffness gradually goes up with the increase in the crack width, but the mesh period where the meshing stiffness decreases remain unchanged. The maximum ratios of the stiffness reduction of single and double tooth pairs are 13.0% and 8.7%, respectively.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Pinion** & **Gear** \\ \hline Teeth number & 23 & 39 \\ Module (mm) & 3 & 3 \\ Teeth width (mm) & 50 & 50 \\ Pressure angle (\({}^{\circ}\)) & 20 & 20 \\ Poisson’s ratio & 0.3 & 0.3 \\ Addendum coefficient & 1 & 1 \\ Dedendum coefficient & 0.25 & 0.25 \\ Hub radius (mm) & 25 & 25 \\ Young’s modulus \(E\) (MPa) & 2.06 \(\times\) 10\({}^{5}\) & 2.06 \(\times\) 10\({}^{5}\) \\ Mass (kg) & 1.32 & 3.16 \\ Mass moment of inertia (kg\(\cdot\)m\({}^{2}\)) & 9.8 \(\times\) 10\({}^{-4}\) & 68 \(\times\) 10\({}^{-4}\) \\ Bearing radial stiffness (N/m) & \(K_{yx}\) = \(K_{yy}\) = 5.8 \(\times\) 10\({}^{8}\) & \(K_{gx}\) = \(K_{gy}\) = 5.8 \(\times\) 10\({}^{8}\) \\ Bearing radial damping (N\(\cdot\)s/m) & \(C_{yx}\) = \(C_{yy}\) = 5 \(\times\) 10\({}^{3}\) & \(C_{gx}\) = \(C_{gy}\) = 5 \(\times\) 10\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Spur gear system parameters.  
Figure 5: Effects of surface crack length on meshing stiffness: (**a**) single-tooth pair, (**b**) double-tooth pairs.  
The effects of the crack depth (\(D\)) on the meshing stiffness of the single-tooth pair and double-tooth pair are analyzed when setting the surface crack parameters as \(L_{1}\) = 2 mm, \(L_{2}\) = 40 mm, \(D\) = 1, 2, 3 mm and \(x_{1}\) = 2 mm. We can find in Figure 7 that the reduction in the magnitude of the meshing stiffness is gradually enlarged when the crack depth increases, while the areas where the mesh stiffness decreases remain the same. The maximum ratios of the stiffness reduction of single and double tooth pairs are 10.8% and 7.2%, respectively.