*4.2. Analysis of FEM Simulation Results*

According to Table 9, 27 FEM simulation models with di fferent factor level combinations are established and solved.

Figure 24 is the TSCS nephogram of the first FEM model in Table 10, and compared with the TSCS nephogram with only one single influence factor, it has the characteristics of irregularity of TPD, an edge stress concentration of ME and a central stress concentration of LCM. Table 11 shows the results of 27 FEM simulation tests.

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**Figure 24.** The TSCS nephogram of the first FEM model.



Minitab (17.1.0, Minitab, LLC, State College, PA, USA) was used to analyze the test results. The Minitab response tables for signal to noise (SN) ratios and means are shown in Table 12. From the table, whether it is the SN ratios response table or the mean response table, it can be seen that the TPD has the greatest influence on the TSCS, followed by the LCM; the <*b* and <*a* are relatively much smaller.

**Table 12.** The Minitab response tables for signal to noise (SN) ratios and means.


The main e ffects plots for SN ratios and means are shown in Figure 25. From the Figure 25a,b, within the set value range, it can be seen that the influence degree of TPD on the TSCS is much greater than that of LCM, and the influence degree of LCM is much greater than that of ME. The influence degree of the two kind of ME is very small, and the di fference is not big. In addition, it can be seen that when the precision grade changes from 4 to 6, it has a grea<sup>t</sup> influence on the TSCS; when changing from 2 to 4, the influence degree is significantly reduced. Similarly, for the LCM, when the LCM quantity changes from 3.5 μm to 7 μm, the influence degree is also slightly larger.

### *4.3. The Interaction between TPD, ME and LCM*

The interaction between di fferent factors can be obtained through Minitab. Figure 26 shows the interaction plot for means between the TPD and other factors. The nonparallel lines in the interaction plot indicate the interaction degree between the TPD and other factors. If the three lines are completely parallel, it means that there is no interaction between them; on the contrary, the more nonparallel the three lines, the greater the interaction degree between them.

(**a**) The main effects plots for signal to noise (SN) ratios.

**Figure 25.** The main effects plots for signal to noise (SN) ratios and means.

(**a**) The interaction plot for means between TPD and <ܽ.

(**b**) The interaction plot for means between TPD and <ܾ. 

(**c**) The interaction plot for means between TPD and LCM. 

**Figure 26.** The interaction plot for means between the TPD and other factors.

As can be seen from Figure 26a,b, there is basically no interaction between the TPD and the ME, and from Figure 26c, there is a certain interaction between the TPD and the LCM, but the interaction degree is not significant.

Figure 27a,b show the interaction plots for means between the LCM and the ME, and Figure 27c shows the interaction plot for means between <*a* and <*b*. From Figure 27a, it can be seen that there is an obvious interaction between the LCM and the <*a*; in particular, when the <*a* changes from 0.4◦ to 0.6◦, the interaction degree is more significant. From Figure 27b, it can be seen that there is also an obvious interaction between the LCM and the <*b*. Therefore, there is an obvious interaction between the LCM and the overall ME. From Figure 27c, it can be seen that the interaction degree between meshing errors <*a* and <*b* is very significant, even though the influence degree of these two kinds of ME on the TSCS is very small.

(**a**) The interaction plot for means between <ܽ and LCM. 

(**b**) The interaction plot for means between <ܾ and LCM. 

**Figure 27.** The interaction plot for means.

It can be known from the third chapter that when the <*a* and <*b* exist alone, they have a grea<sup>t</sup> influence on the TSCS. However, when all the influence factors exist, because of the obvious interaction between them, the influence degree of the <*a* and <*b* on the TSCS becomes very small.

In addition, in order to further verify the importance of LCM for the engaged gears with ME, two FEM simulation models are established. The two models have the same precision grade of TPD (grade 2), <*a* (0.2◦) and <*b* (−0.04◦), but the first model does not have the LCM, and the second model has an LCM of 3.5 um. The TSCS analysis of the two models is carried out, and Figure 28a is the stress nephogram of the first model, while Figure 28b is the stress nephogram of the second model. Through comparison, it can be found that the LCM can effectively alleviate the edge stress concentration caused by ME, and the maximum TSCS of the model with LCM is significantly smaller than that without LCM. This also demonstrates that there is an obvious interaction between LCM and ME.

(**b**) The model with LCM. 

**Figure 28.** The TSCS nephogram.

### *4.4. Determining the Optimal Combination of Influence Factor Levels*

The maximum TSCS of each combination of factor levels can also be predicted by the Minitab. As shown in Figure 29, the approximate maximum TSCS corresponding to the combination of different factor levels can be obtained by inputting the level of each factor.


**Figure 29.** The Taguchi results are predicted by Minitab.

It can be seen from Figure 25 that the TPD grade is 2 and the LCM quantity is 3.5 μm in the optimal combination of factor levels. Then, the prediction results of partial combinations are shown in Table 13. It can be seen from Table 13 that when the TPD grade is 2 and the LCM quantity is 3.5 μm, the ME has little effect on the maximum TSCS, and the maximum TSCS range is 1605.57–1622.67 MPa. Therefore, it is sufficient to control the <*a* and <*b* of ME within the specified range. However, in conclusion, it is determined that the optimal combination of factor levels is grade 2 for the TPD, −0.12◦ for the <sup>&</sup>lt;*b*, 0.2◦ for the <*a*, and 3.5 μm for the LCM quantity.

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**Table 13.** The prediction results of different combinations.

### *4.5. Comparative Analysis of Di*ff*erent Combinations of Factor Levels*

According to the original process requirements of precision gear manufacturing in a gear factory, the TPD, <*a*, <*b* and LCM are required to be controlled at grade 4, 0.4◦, −0.08◦ and 3.5 μm, respectively, and the maximum TSCS of the original combination of factor levels is 2062.00 MPa, which is 28.43% larger than the 1605.57 MPa of the optimal combination.

In order to better show the influence of the TSCS on the gear fatigue life, according to the contact fatigue SN curve (Figure 30) and the corresponding curve equation (Equation (11)) provided by the gear factory, the gear fatigue life corresponding to different TSCS can be obtained.

$$\text{lg}\sigma = -0.0908 \text{lgN} + 3.8767 \tag{11}$$

According to Equation (11), when the σ = 1605.57 MPa, the corresponding fatigue life *N* = 2.46 × 107, while when the maximum TSCS is 2062 MPa, the corresponding fatigue life *N* = 1.5 × 106. It can be found that the gear contact fatigue life of the optimal combination of factor levels is much longer than that of the original combination.

In addition, when the minimum fatigue life of the gear is required to be 1 × 107, the maximum acceptable TSCS is 1743 MPa, as calculated by Equation (11). Therefore, only when the TPD is grade 2 and the LCM is 3.5 μm can the fatigue life meet the requirements. It also shows that the contact fatigue performance of the optimal combination of factor levels is better than that of the original combination.

**Figure 30.** The contact fatigue SN curve.
