*3.3. TSCS Analysis of Gear with LCM*

The purpose of the LCM is to ensure that when the gears are at the maximum inclination, the teeth engaged with each other do not have an edge point contact (tangent but not intersecting), so as to reduce edge contact.

It should be admitted that there is a certain load concentration phenomenon on the tooth surface with LCM, so the LCM should meet the most basic requirements as follows: first, when the gear tooth has the maximum inclination, the gear tooth shall not have the edge embedding phenomenon; second, when the gear tooth does not incline or has a certain inclination, the gear tooth shall have the minimum

*Metals* **2020**, *10*, 1370

load concentration, that is, the lead crowning radius should be the largest (the LCM quantity is the smallest).

According to the ISO standard [4], the LCM mainly considers the ME of the original gear, and the formula is as follows:

$$C = 0.5 \times F\_{\text{\ $xx\$ }} \tag{8}$$

where *<sup>F</sup>*β*xcv* is the original equivalent misalignment for the determination of the crowning height. According to the range of ME set in this paper, the LCM quantities are taken as 3.5 μm, 7 μm and 10.5 μm, respectively, for the FEM contact analysis.

Figure 20 shows the LCM of the tooth surface, where *a* is the tooth width, *l* = 0.5 × *a*, *C* is the LCM quantity, and the lead crowning radius is:

$$R = \frac{l^2}{2 \times \mathbb{C}}\tag{9}$$

Adjust the tooth surface nodes to input the LCM to the FEM model. Then, the calculation formula of the modification quantity of node *i* is:

$$C\_i = C \times \left(\frac{l\_i}{l}\right)^2\tag{10}$$

where *li* is the distance from the node numbered *i* to the center line of the tooth width.

**Figure 20.** The lead crowning modifications (LCM) of the tooth surface.

In this paper, selecting 3.5 μm, 7 μm and 10.5 μm for the LCM quantities, respectively, and conducting the gear FEM contact analysis under the applied torque of 10 N·m, the analysis results are shown in Figure 21.

In Figure 21, from Figure 21a to Figure 21c are shown the TSCS nephograms of *C* = 3.5 μm, *C* = 7 μm and *C* = 10.5 μm. It can be seen that the TSCS with LCM presents stress the concentration phenomenon in the center of the tooth surface, and with the increase in LCM quantity, 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 gears with different LCM quantities, and the results of the maximum TSCS are compared as shown in Table 8 and Figure 22. It can be seen that the influence of LCM on the TSCS is very large. At the same time, it is found that with the increase in applied load, the ratio of the maximum TSCS of each gear with different ME to the ideal gear gradually decreases.

**Figure 21.** The TSCS nephograms of *C* = 3.5 μm, *C* = 7 μm and *C* = 10.5 μm.

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**Table 8.** The results of the maximum TSCS under different torques. Unit: MPa.


**Figure 22.** The results of the maximum TSCS under different torques.

The object of the above FEM contact analysis results is a pair of engaged teeth with only LCM and no ME. However, as we all know, in the actual gear engagemen<sup>t</sup> process, this must be accompanied by ME and TPD, and the lead crowning machining is also an essential process of gear machining. This is

because the LCM can effectively reduce the stress concentration at the edge point caused by the ME and the TPD; therefore, there must be an interaction between these factors, which needs further study.

### **4. Analysis of the Influence of TPD, ME and LCM on TSCS by TM**

According to the content of the third chapter, it can be found that when the TPD, ME and LCM exist separately, the influence on the TSCS is relatively large. However, when the above three factors exist in the gear at the same time, due to the interaction between the factors, the influence degree on the contact stress is not a simple superposition.

This chapter mainly studies the interaction between these three factors, and the influence degree of each factor on the TSCS when the above three factors exist at the same time. As a powerful tool, the Taguchi method can effectively study the above problems, and its steps are shown in Figure 23.

**Figure 23.** The steps of the Taguchi method (TM).
