# Transformation of cutting motion on the tooth surface of involute spiral bevel gear

It can be seen that the two rotary movements of the blade – rotation and revolution can be replaced by a linear motion and a rotary motion, as shown in Figure 1. The angular velocity of rotating motion (defined as ω 2) Equal to the angular velocity of the blade revolution, both ω。 Next, the moving speed of the blade endpoint a along the OA direction is solved.

For the convenience of calculation and analysis, take the time when the endpoint a of the arc blade coincides with the revolution center o as the initial position. As shown in Figure 1, the arc center of the initial position of the blade is o01. At this time, the connecting line oo01 is perpendicular to the junction line of the cutting area and the adjustment area. According to the cutting principle of spiral bevel gear tooth surface in Chapter 2, the two linkage movements of the blade are: in time t, the blade rotates at an angular speed around point o ω Rotate counterclockwise from position A1B1 to a2b2; At the same time, it rotates at an angular velocity around its geometric center O02 ω 0 rotates from position a2b2 to a3b3. Then, according to the rotation property of the figure: the angle formed by the connecting line between any pair of corresponding points and the rotation center is the rotation angle. It can be seen that ∠ a2o02a3= ω 0t。 According to isosceles Δ According to the edge angle relationship of a2o02a3, the moving distance of the end point a of the circular arc blade along the direction of in time t can be obtained:

Where: R – radius of circular arc blade.

Due to the rotation speed of the blade ω 0=2 ω， And the rotation speed of the gear blank around the end point a of the blade ω 2= ω， By substituting this relation into the formula and simplifying it, it can be obtained that the moving distance of the end point a of the blade after time t is:

By deriving the time t from the moving distance expression of the blade endpoint a in the formula, the moving speed V expression of the blade endpoint a in the direction of of of can be obtained:

Based on the above analysis, the three linkage movements of the tooth surface of the cutting spiral bevel gear can be equivalent converted from the linkage of three rotary movements to the linkage of one movement and two rotary movements: the movement V of the end point a of the circular arc blade along the boundary of the cutting area and the adjustment area, and the rotary movement of the blade around the end point a ω 2. And the rotation movement of the gear blank around its own rotation axis ω 1, as shown in Figure 2. Moreover, the speed of the above three must meet the following relationship:

According to the analysis in this section, only one linear movement V and two rotations are required（ ω 1 and ω 2) Through linkage, the tooth surface of spiral bevel gear with complete spherical involute tooth profile can also be cut. This makes it easier to realize and control the gear cutting movement, and the linkage between the gear cutting movement and the machine tool structure can be simplified to a certain extent compared with the previous three rotary movements.

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