# Tooth surface equation of parallel shaft pure rolling gear

Taking the points on the contact lines R1 and R2 as the coordinate origin and the base vectors of fixture coordinate systems s and SP as the X, y and Z axes, the tooth profile coordinate systems os-xsecyseczsec and os-xsec2ysec2zsec2 (abbreviated as δ 1、 δ 2)。 At the meshing point n, the profile section of the pinion and ring gear is shown in Fig. 1, ρ 1、 ρ 2 is the arc radius of tooth profile section of pinion and ring gear respectively; γ Is the contact angle formed by OSN and xsec axis, which determines the position of contact point n on the arc. When the contact points fall at a and B, there will be edge contact. In order to reduce the edge contact and improve the contact strength of gear, it is suggested that the contact point should be set near the midpoint of arc ab.

The tooth profile section curve of pinion and ring gear is in the coordinate system δ 1、 δ The parameter equations are: 1

Where: μ 1、 μ 2 is the variable parameter of the tooth profile section curve equation. Taking the inner side of the profile curve of the pinion as the positive direction, when the center OS2 of the profile curve of the ring gear is on the same side of OS, the formula takes the positive sign; On the contrary, a negative sign is used. Symbols will affect the concavity and convexity of the tooth surface, which will be discussed in detail.

As shown in Figure 3, the circular tooth surface is formed by the helical movement of the tooth profile section curve along the axial direction

Where: when I = 1 and 2, the formula represents the tooth surface of pinion and ring gear respectively. MSi- δ I is the tooth profile coordinate system δ The transformation matrix from I to Si. 