# Cutting principle of concave tooth surface based on generation line of spiral bevel gear tooth surface

Taking the concave tooth surface of right-hand spiral bevel gear as an example, the generation principle of concave tooth surface cutting is analyzed and studied. From the analysis, it can be seen that the generation process of the concave tooth surface of the right-hand spiral bevel gear is: the tooth surface generation line pure rolls on the base cone with the tangent plane of the base cone – (q) plane, winds from the big end to the small end, and unfolds from the base cone to the face cone, so as to generate the concave tooth surface of the spiral bevel gear. As we all know, in the process of substantial tooth cutting, the blade cannot feed from the base cone of the tooth blank according to the generation direction of the tooth surface, and then cut out in the direction of the face cone. However, if the inverse motion of the generated motion of the tooth surface is taken as the cutting motion of the tooth surface, that is, the generation line of the tooth surface is taken as the cutting edge, and the cutting generation of the tooth surface is carried out from the face cone to the root cone and from the small end to the large end of the tooth blank, the cutting process of the whole tooth surface is just opposite to the generation process of the tooth surface. This cutting method of tooth surface is completely feasible, and the tooth surface of spiral bevel gear without principle error can be cut according to this method. Based on the concept of generating line cutting on the tooth surface, this paper breaks through the Gleason cutting system and creatively puts forward a new machining method of spiral bevel gear.

In order to more intuitively understand the two key areas in the process of tooth surface cutting: adjustment area ψ And cutting area angle μ， Use CATIA software to establish a three-dimensional model in which the (q) plane is tangent to the base cone and intersects with the root cone and face cone, as shown in Figure 1. From the figure, it is intuitively found that the (q) plane is tangent to the straight line OC with the base cone, and intersects with the root cone and the face cone on the straight lines og and OE respectively, so as to form the adjustment area between the base cone and the root cone and the cutting area between the root cone and the face cone.

In order to facilitate the analysis of the relationship between various motions required for tooth surface cutting generation based on the spherical involute tooth surface generation principle, the engineering drawing function of CATIA is used to project the front view of the three-dimensional view of Fig. 1 with the (q) plane as the reference plane, as shown in Fig. 2.

The coordinate system is established on the basis of Figure 2. Take point o as the origin of the rectangular coordinate system. The horizontal right direction is the positive direction of X axis and the vertical upward direction is the positive direction of Y axis. And establish the cutting area and adjustment area required in the actual cutting process, as shown in Figure 3. The shaded part of area dhjf and area gkie is the angle, and the size is μ The trapezoidal jkgf area has two symmetrical angles, and the size is ψ The adjustment area is also a non machining area.

According to the analysis of the tooth surface generation principle and the tooth surface generation line cutting principle in this section, the tooth surface of spiral bevel gear without principle error can be machined by taking the tooth surface generation line – arc segment AB as the cutting edge and the inverse motion of the tooth surface generation motion as the cutting motion. As shown in Figure 3, the arc-shaped blade – arc segment AB is located on the (q) plane, the blade radius is r, and the tool position, that is, the distance between the blade center and the (q) plane center, is Q. When the occurrence line ab of the tooth surface rotates at an angular speed around point o with the (q) plane ω Rotate the gear blank to ω 1, and ω And ω 1. Maintain relationship: ω/ω 1 = sin δ B, the generating line of the tooth surface will roll on the base cone, so as to cut the concave tooth surface of the spiral bevel gear along the direction of the surface cone to the base cone and the small end to the large end. In order to completely cut the tooth surface of spiral bevel gear, the motion track of tooth surface generation line AB must cover the whole cutting area, that is, the tooth surface generation line AB must rotate to A1B1 position in the figure with plane (q). However, it is obvious from Figure 3 that when the tooth surface line moves from position AB to position A1B1, while cutting the complete spiral bevel gear tooth surface, overcutting the shaded part of the grid in the adjustment area is absolutely not allowed. Therefore, a motion must be added in the process of tooth surface cutting, Prevent overcutting in the adjustment area in the process of cutting the tooth surface.

Because the cutting edge is a circular arc-shaped cutting edge, and the circular arc has the characteristics of rotating and overlapping around its own geometric rotation center, after comprehensive analysis, a tooth surface can be added in the cutting process of the tooth surface to generate a rotary retraction movement around its own rotation center. It can not only prevent over cutting, but also coincide with the arc line after rotating around the geometric center of the arc blade, which does not change the shape of the tooth surface of the generated spiral bevel gear. Next, calculate the rotation speed of the blade around its own geometric center, and define the speed as the rotation speed of the blade, which is: ω 0

In order to facilitate the motion analysis, the speed that should be carried out synchronously in the actual machining is ω The blade rotates around point O and the speed is ω The blade rotation of 0 is divided into two steps, which can greatly simplify the calculation of motion, and does not violate the motion relationship of the blade, but the above two steps can be carried out synchronously in the actual tooth surface cutting process.

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