# Tooth surface detection formula of hourglass worm gear

Let the circumferential matching angle between the measuring coordinate system and the theoretical coordinate system be φ P. Then the coordinate value of the measured point in the theoretical coordinate system is:

According to the coordinate value of the measured point in the theoretical coordinate system, the torus helix of the point is calculated

The geometric relationship between the measured points and the theoretical points is shown in Figure 1. In the figure, PIJ is the measured point, qij is the normal theoretical point corresponding to the theoretical torus helix, SIJ is the equal section theoretical point corresponding to the theoretical torus helix, and the length of pijqij is the tooth surface deviation Δ fij。

Since the measured point PIJ and the point SIJ on the torus helix are in the same section, there are two points

Where: (z2c) PIJ — coordinate value of PIJ point in Z direction;

(z2) SIJ — the coordinate value of SIJ point in Z direction.

The mathematical model of the transmission pair is derived in detail. Here, only the helix equation of the left and right tooth surfaces of the enveloping hourglass worm is given

Where: RGL2 — position vector of left tooth surface point of enveloping hourglass worm;

Rgr2 — position vector of right tooth surface point of enveloping hourglass worm;

RB — base circle radius of involute gear with variable tooth thickness;

u， θ——— The tooth surface parameters of involute gear with variable tooth thickness;

δ——— Position angle;

φ 1 — rotation angle of involute gear with variable tooth thickness;

φ 2 — rotation angle of enveloping hourglass worm;

β L — pitch circle helix angle of left tooth surface of involute gear with variable tooth thickness;

β R — helix angle of indexing circle on right tooth surface of involute gear with variable tooth thickness;

Φ L — meshing function of left tooth surface of worm pair;

Φ R — meshing function of right tooth surface of worm pair; 