# Tooth surface construction of parallel shaft pure rolling gear

The different choice of sign in the tooth profile section curve equation will affect the convexity and convexity of tooth surface; According to the center position and arc radius of the profile curve of pinion and ring gear ρ 1、 ρ Three kinds of contact surfaces can be constructed: convex concave contact, convex convex contact and convex flat contact. The positive sign of pinion profile section curve equation is selected as an example.

(1) Convex concave contact. When equation (1) takes positive sign and equation (2) takes positive sign, the circle centers OS and OS2 of profile section arc of pinion and ring gear are located on the same side. At this time, the tooth surface of the ring gear is concave, and the contact type is convex concave contact, as shown in Figure 1.

(2) Convex convex contact. When equation (1) takes a positive sign and equation (2) takes a negative sign, the circle centers OS and OS2 of the tooth profile arc of the pinion and the ring gear are on the opposite side. At this time, the tooth surface of the ring gear is convex, and the contact type is convex, as shown in Figure 2.

(3) Convex flat contact. When formula (1) takes a positive sign, formula (2) takes a negative sign and ρ When 2 →∞, the arc AB evolves into a straight line AB, and the tooth surface of the ring gear becomes a tangent plane. At this time, the contact type is convex flat contact, as shown in Figure 3.

For the three kinds of contact surfaces, the contact stress is different due to the different relative curvature radius of the contact points. According to Hertz contact principle, the calculation formula of contact stress at meshing point is as follows:

Where Qmax is the maximum contact stress; Q is the external force; E1 and E2 are the elastic modulus of pinion and ring gear respectively; ν 1、 ν 2 is Poisson’s ratio; Σρ Is the relative curvature radius; “+” The symbol indicates the contact form of convex tooth face to convex tooth face; “-” The symbol indicates the contact form of convex tooth face to concave tooth face.

It can be seen from equation (3): under the same load condition, the contact stress of tooth surface is inversely proportional to the relative curvature radius. The tooth surface with convex concave contact has the largest relative curvature radius and the smallest contact stress. Suppose E1 = E2 = 2 × 105Mpa、 ν 1= ν 2=0.3、Q=20N、 ρ 1=1mm、 ρ 2 = 4mm, the maximum contact stress of different contact surfaces can be obtained by taking the relevant parameters into equation (13). The contact stress of convex concave contact is 28.95% lower than that of convex convex contact and 19.17% lower than that of convex flat contact. For the line gear whose tooth surface is point contact, it can effectively improve the tooth surface contact strength. Therefore, the convex concave contact form of internal meshing line gear pair has the most research value, and is also the main object of this paper. 