## 5 Numerical Studies
### Effect of Comprehensive Error on Transmission Error (TE)  
A spur gear pair with a comprehensive error consisting of a machining error, thermal deformation, and a pitch error is generated, and LTCA is conducted on plastic and steel. Figures 16-19 show the transmission errors of the spur gear pair as one of the samples. Samples of the spur gear pair have a machining error (\(\epsilon=0.5\)), thermal deformation (\(\eta=0.0125\)), and a pitch error \(e_{p}\). The pitch error of the driving gear is \(e_{p}=-0.001\) deg in Figures 16 and 17 and \(e_{p}=0.001\) deg in Figures 18 and 19. The driven gear has no pitch error.  
Transmission errors with a comprehensive error under unloaded and loaded conditions are calculated to be larger than the transmission errors of the three uncertainties. In the case of a steel gear under the unloaded condition, the transmission error with thermal deformation is the smallest among the three uncertainties. Although the transmission error caused by only thermal deformation is small, thermal deformation coupled with mechanical uncertainties causes a significant transmission error. When thermal deformation is coupled with other uncertainties, it has a more considerable impact on the transmission error than magnitude. From Figures 16 and 17, it is found that the effect of the interaction of thermal deformation is different according to the material properties. The effect of the interaction of each uncertainty is clearly shown in Figure 20. There is a difference between the comprehensive uncertainty (red line) and the linear sum of each uncertainty (blue line). Note that the effect of the interaction of each uncertainty leads to the difference between the geometric error of the comprehensive uncertainty and the linear sum of each uncertainty. In the case of a plastic gear under the unloaded condition, the tendency of the transmission error with the comprehensive error is similar to that of thermal deformation. From this fact, it is found that thermal deformation has the greatest effect on the transmission error among the three uncertainties. In the case of a plastic gear under the loaded condition, all transmission errors with uncertainty have similar tendencies owing to elastic deflection. The gear tooth, which has a pitch error, causes an error in the tooth contact pattern. When the gear tooth contact is late or early, the difference in the contact pattern appears as a transmission error. From comparison between Figures 16 and 17 (or Figures 18 and 19), it is found that the magnitude of pitch error affects the major tendency of transmission error, not a detailed waveform.  
Figure 15: Peak-to-peak transmission error of (**a**) steel and (**b**) plastic spur gear pairs with regard to change in \(\eta\). There is no machining error or pitch error (\(\epsilon=0,e_{p}=0\)).  
Figure 16: Transmission error of steel spur gear pair under **(a)** unloaded and **(b)** loaded conditions (Comprehensive uncertainty: \(\epsilon=0.5\), \(\eta=0.0125\), \(e_{p}=-0.001\) deg, \(ME\): \(\epsilon=0.5\), \(\eta=0\), \(e_{p}=0\) deg, \(TD\): \(\epsilon=0\), \(\eta=0.0125\), \(e_{p}=0\) deg, \(PE\): \(\epsilon=0\), \(\eta=0\), \(e_{p}=-0.001\) deg, \(ME+TD\): \(\epsilon=0.5\), \(\eta=0.0125\), \(e_{p}=0\) deg).