## 3 Meshing Stiffness and Vibration Response for Cracked Gears in Different Gear Shape Parameters  
### Meshing Stiffness and Vibration Response for Cracked Gears in Different Pressure Angles  
For calculating the meshing stiffness, most work has been based on fixed gear shape parameters. However, gear shape parameters for an actual manufacture process need to change. Different pressure angles (14.5\({}^{\circ}\), 20\({}^{\circ}\), and 25\({}^{\circ}\)) were taken into consideration for the meshing stiffness and vibration response investigation in this part. The installation of the meshing gear needs high precision. However, in the actual machining process, the axis of the meshing gear is usually unparalleled due to the manufacturing error and the limited skill of installation. Then, a non-uniform meshing force will be operating on the gear profile, causing crack propagation along the tooth width with non-uniform depths. In this paper, the non-uniform depth of the crack was also introduced for the investigation of meshing stiffness and vibration response. The expression of the crack depth is shown below. \[q_{w}=q\sqrt{\frac{w-w_{c}}{w}},\ wc\in[0-w] \tag{21}\]  
where  
\(q\) is the initial crack depth and \(w\) is the width of the gear. Since changing the different pressure angles could not affect the length of the tooth, the depths of the crack in different pressure angles were set the same (0.3 mm, 0.6 mm, 0.9 mm, 1.2 mm, 1.5 mm and 1.8 mm). The crack direction was set as 63.3\({}^{\circ}\) with the central line of the tooth, as shown in Figure 6[33]. The meshing stiffness in different pressure angles of different crack depths is shown in Figures 7-9. The meshing stiffness tended to increase with the increase of pressure angle, but tended to decrease with the angle rotating in each signal and double tooth interval. Figure 6: Gear tooth microcrack model. Figure 8: Meshing stiffness of the 20\({}^{\circ}\) pressure angle. Figure 7: Meshing stiffness of the 14.5\({}^{\circ}\) pressure angle. Figure 9: Meshing stiffness of the 25\({}^{\circ}\) pressure angle. Then, the vibration signal could be obtained by the dynamic model presented in Section 2.2, as shown in Figures 10-12. To more clearly express the effect of a crack on the vibration response, the method in which the healthy vibration signals are to be removed from the original signal was used in this paper for the vibration tendency investigation. The residual signals in the different crack depths with different pressure angles are shown below. From the residual, we can see that the residual signal under the pressure angle 20\({}^{\circ}\) had the lowest amplitude. Figure 11: Vibration response for the cracked gear under the 20\({}^{\circ}\) pressure angle. Figure 10: Vibration response for the cracked gear under the 14.5\({}^{\circ}\) pressure angle. To see the tendency clearly, the dealing result of Root Mean Square (RMS), the kurtosis and the crest factor (CF) are shown in Figures 13-15. The calculated method of the fault detection indicators is shown from Equations (22)-(24). The kurtosis had an unstable tendency in a small crack, and the CF not only had an unstable tendency in a small crack but also had an uncertain tendency in a large crack. Moreover, the value of CF turned out to be the largest before the 1.2-mm crack under the pressure angle 20\({}^{\circ}\) when compared with the value of CF under the other two pressure angels. It does not meet with the conclusion that the residual signal under the pressure angle 20\({}^{\circ}\) has the lowest amplitude. However, the RMS had a monotonically increasing tendency with an increase in crack depth. Then, we chose RMS as the best fault detection indicator of the three for the investigation of vibration response. From the RMS result, we see that with an increasing tendency of the pressure, the vibration response of the gearbox changed erratically.