2.3.2. Determination of Mesh Transition Boundary Position

As shown in Figure 6, take the length of *b* as the basic unit of the refined area size, and set the MPC boundary of the contact area as *w* × *h*. Firstly, set *w* × *h* as 8*b* × 8*b* to refine the mesh, and the FEM model of a pair of ideal teeth is shown in Figure 7.

**Figure 7.** Finite element method (FEM) model.

Apply a torque of 25 Nm to the FEM model and conduct FEM solution. The Von Mises stress nephogram is shown in Figure 8. It can be seen from Figure 8 that the stress gradient of the gear at the engagemen<sup>t</sup> position is very large, but the stress decreases sharply at a certain distance from the maximum stress position, and at a position less than 4*b* below the tooth surface, the stress gradient is very gentle. Therefore, when the size of the refined area is *w* × *h* = 8*b* × 8*b*, the influence of the MPC on the solution accuracy of the model can be ignored. It can be seen from Figure 9 that the maximum TSCS solved by the FEM model is 1676.23 MPa, and the relative error is only 1.36% when compared with the 1653.75 MPa calculated by the Hertz contact theory.

**Figure 8.** The Von Mises stress nephogram of solution result.

**Figure 9.** The maximum tooth surface contact stress (TSCS) solved by the FEM model.

When the size of the refined area is *w* × *h* = 8*b* × 8*b*, the solution accuracy meets the requirements, so the size of the refined area is further reduced. Figure 10 shows the Von Mises stress nephograms corresponding to the refined areas of different sizes, and Table 3 is the corresponding maximum TSCS. It can be seen from Figure 10 and Table 3 that when *w* × *h* = 4*b* × 4*b* and *w* × *h* = 4*b* × 3*b*, the stress gradients at the MPC connection position are relatively gentle, and the errors between the maximum TSCS and that calculated by the Hertz contact theory are within 2.5%, the solution accuracy of the model still meets the requirements. With a further reduction in the size of the refined area, when *w* × *h* = 4*b* × 2*b* and *w* × *h* = 3*b* × 4*b*, the stress gradients at the MPC connection position are relatively large, and the errors between the maximum TSCS and that calculated by the Hertz contact theory are also relatively large.

(**c**) ݓ ൈ ݄ ൌ Ͷܾ ൈ

ʹܾ (**d**) ݓ ൈ ݄ ൌ ͵ܾ ൈ Ͷܾ

**Figure 10.** The Von Mises stress nephogram of solution result.


In addition, the solution time shows that the larger the size of the refined area, the longer the solution time, especially when the solution time of *w* × *h* = 8*b* × 8*b* is more than 10 times that of *w* × *h* = 4*b* × 3*b*, so if the full tooth mesh is refined, the calculation time will be much longer.

In conclusion, for the TSCS analysis of the ideal gear researched in this paper, the reasonable size of the refined area should be at least *w* × *h* = 4*b* × 3*b*, but the TSCS values of the gears with TPD, ME or LCM are much larger than that of the ideal gear. Therefore, according to the Hertz contact theory, assuming that the maximum TSCS of the gear with influence factors is 1.5 times that of the ideal gear, the contact half-width should also be increased by 1.5 times, as should the half-width of the contact zone *b*. Therefore, when analyzing the TSCS of the gears with influence factors, the reasonable size of the refined area should be at least *w* × *h* = 6*b* × 4.5*b* = 0.60 mm × 0.45 mm.

In addition, referring to the calculation method of TSCS in ISO6336 [3,4], the accuracy of FEM can be further verified. According to Equation (6) and Table 4, the TSCS of the gear is 1657.28 MPa, and the TSCS error between the *w* × *h* = 4*b* × 3*b* and Equation (6) is only 2.19%.

$$
\sigma\_{\rm H} = Z\_{\rm D} Z\_{\rm H} Z\_{\rm E} Z\_{\rm E} \sqrt{\frac{F\_t}{d\_1 b} \frac{u+1}{u}} \sqrt{\mathcal{K}\_A \mathcal{K}\_{\uparrow} \mathcal{K}\_V \mathcal{K}\_{H\beta} \mathcal{K}\_{Ha}} \tag{6}
$$


### **Table 4.** Meanings of the symbols used in Equation (6).

### **3. TSCS Analysis of Gear with TPD, ME or LCM**

## *3.1. TSCS Analysis of Gear with TPD*

All actual manufactured gears have TPD. According to the definition of TPD in the ISO 1328-1:2013 standard [20], TPD refers to the amount of actual tooth profile deviating from the designed tooth profile, which is calculated in the end plane and perpendicular to the involute tooth profile. According to the gear precision grade, gear module and pitch circle diameter, the range of the total TPD value under the precision grade is specified in the ISO standard [20].

Figure 11 shows the node distribution form of the tooth surface refined area. Each node on the tooth surface of the area is numbered, then the corresponding TPD is input to each node, and these TPD values come from the measured data of the gear factory. Figure 12 shows the tooth surface state after inputting the measured TPD of a grade 4 precision gear, and the unevenness of the tooth surface can be vaguely seen from the figure. We enlarged the tooth surface (shadow part) of the refined area in Figure 12, and show the enlarged effect in Figure 13. In Figure 13, the X axis direction is the direction from tooth root to tooth top, the Y axis direction is the direction of the tooth width, and the Z axis represents the TPD value. After enlarging the refined area, it can be clearly seen that the tooth surface is uneven.

**Figure 11.** The node distribution form of the tooth surface's refined area.

**Figure 12.** The tooth surface state after inputting the measured tooth profile deviations (TPD).

**Figure 13.** The enlarged effect of the tooth surface refined area with TPD.

This paper also proposes a method to obtain the TPD of other precision grade gears when the TPD value of the grade 4 precision gear is known. As shown in Table 5, according to the range of the total TPD value of each precision grade gear, it is considered that:

$$f\_{\rm ni} : f\_{4i} = F\_{\rm n} : F\_4 \tag{7}$$

where *F*n is the upper limit value of the total TPD value of the grade *n* precision gear, and *fni* is the actual TPD value of each node of the grade *n* precision gear (*i* is the node number), so the TPDs of the refined areas of other precision grade gears can be obtained by Equation (7).


**Table 5.** The total tooth profile deviations (TPD) value of each grade precision gear.

In this paper, the FEM contact analyses for grade 2, 4 and 6 precision gears with only TPD are carried out, respectively. Figure 14 is the result of the FEM solution when the torque is 10 Nm, and, from Figure 14a to Figure 14d, shows the TSCS nephograms of ideal gear, grade 2 precision gear, grade 4 precision gear and grade 6 precision gear. It can be seen that the TSCS with TPD presents irregular distribution on the tooth surface, and with the reduction of gear precision, the gradient of the local TSCS increases gradually, and the maximum TSCS also increases accordingly.

In this paper, the FEM contact analysis is conducted under different torques for different precision grades of gears, and the results of the maximum TSCS are compared as shown in Table 6 and Figure 15. It can be seen that the influence of TPD on the TSCS is very large, and the difference between the maximum TSCS corresponding to the different precision grades is large. At the same time, it is found that with the increase in applied load, the ratio of the maximum TSCS of each precision grade gear to the ideal gear gradually decreases.

**Table 6.** The results of the maximum TSCS under different torques. Unit: MPa.


**Figure 15.** The results of the maximum TSCS under different torques.
