2.3.1. Hertz Contact Theory

In order to determine the optimal position of the mesh transition boundary, it is necessary to establish di fferent-sized refined areas to verify the accuracy of the model. The engagemen<sup>t</sup> of involute cylindrical gears is similar to the contact between two cylinders, so the accuracy of the FEM solution can be verified by Hertz contact theory [19]. Figure 5 is a Hertz contact model, where *F* is the normal force applied, σ*Hmax* is the maximum contact stress on the contact surface, *b* is the half width of the

contact zone, *a* is the contact length, and ρ1 and ρ2 are the curvature radius of the two cylinders. σ*Hmax* and *b* can be calculated by Equations (1) and (2).

$$
\sigma\_{Hmax} = \sqrt{\frac{F}{\pi \times a} \times \frac{\left(\frac{1}{\rho\_1} + \frac{1}{\rho\_2}\right)}{\left(\frac{1-\mu\_1^2}{E\_1} + \frac{1-\mu\_2^2}{E\_2}\right)}}\tag{1}
$$

$$b = \sqrt{\frac{4 \times F}{\pi \times a} \times \frac{\left(\frac{1 - \mu\_1^2}{E\_1} + \frac{1 - \mu\_2^2}{E\_2}\right)}{\left(\frac{1}{\rho\_1} + \frac{1}{\rho\_2}\right)}}\tag{2}$$

**Figure 5.** Hertz contact model.

In Equations (1) and (2), μ1 and μ2 are the Poisson's ratio of the materials of cylinder 1 and cylinder 2, and *E*1 and *E*2 are the elastic modulus of the materials of cylinder 1 and cylinder 2. Treat the cylindrical contact as the engaged contact of two gear teeth; *a* is equivalent to the tooth width, and *F* is equivalent to the normal force of the tooth surface contact. In Equations (3)–(5), *Ft* is the circumferential force produced by torque, α is the pressure angle, and *d*1 and *d*2 are the diameters of the pitch circles.

$$F = \frac{F\_t}{\cos \alpha} = \frac{2 \times T}{d\_1 \times \cos \alpha} \tag{3}$$

$$\rho\_1 = \frac{d\_1 \times \sin \alpha}{2} \tag{4}$$

$$
\rho\_2 = \frac{d\_2 \times \sin \alpha}{2} \tag{5}
$$

Themaximum torque*T* applied to the engaged gearis defined as 25N·m. Combining Equations (1)–(5), the TSCS of the engaged gear is calculated to be 1653.75 MPa, and the half width of the contact zone is 0.10 mm. The TSCS has basically reached the contact fatigue limit of carburized alloy steel gears with high material quality and heat treatment quality. Therefore, in this paper, the maximum torque applied to the gears will not exceed 25 N·m. If the size of the refined area determined under this torque can meet the solution accuracy, the solution accuracy can also be guaranteed when the torque is less than 25 N·m.
