The type of spiral bevel gear studied is spherical involute involute involute involute spiral bevel gear with the intersection angle of 90 °, and the vertices of base cone, root cone, pitch cone and face cone coincide at one point. Taking right-hand spiral bevel gear as an example, this paper expounds the generation principle of convex and concave tooth surfaces of bevel gear.

In order to visually analyze the generation principle of spiral bevel gear tooth surface, CATIA software is used to establish its spatial three-dimensional generation model, as shown in Figure 1. In the wireframe and curved surface design environment under CATIA mechanical design module, the base cone model and tangent (q) plane model of spiral bevel gear are established respectively, and the tooth surface generation line arc segment a ‘B’ is established on the (q) plane. The bus length of the base cone is equal to the radius of the (q) plane. Then add constraints in the assembly design module to make the base cone fit tangentially with the (q) plane, and the geometric center of the base cone top and the (q) plane coincide with the o ‘point. Then, in the DMU kinematics module under CATIA digital model, add the fixed constraint on the base cone model and the rolling curve joint constraint on the (q) plane model, and select the driving command in the rolling curve joint command, so that the (q) plane follows the direction in Fig. 1 on the base cone ω‘ Pure rolling motion in direction. Finally, the editing simulation function under the DMU kinematics module is used to generate the simulation animation and replay of the pure rolling of the (q) plane on the base cone, and the track function is used to track the spatial motion track of the tooth surface generating line – arc segment a ‘B’ with the pure rolling of the (q) plane on the base cone, The curved surface filling operation is carried out on the motion trajectory of arc segment a ‘B’, so as to generate the concave tooth surface model of right-hand spiral bevel gear. Figure 1 is the generation model of concave tooth surface of right-hand spiral bevel gear. It can be seen from this that the generation process of the concave tooth surface of the right-hand spiral bevel gear is as follows: the tooth surface generation line – the arc segment a’B ‘in the figure carries out pure rolling motion on the base cone with the tangent plane of the base cone – (q) plane, the tooth surface generation line gradually wraps on the base cone from the point B’ at the big end to the point a A’ at the small end, and develops from the base cone to the face cone, Thus, the concave tooth surface of the right spiral bevel gear is formed.

To ensure the pure rolling relationship between the base cone and the (q) plane, the following three schemes can be realized: ⑴ the base cone does not move, (q) the plane carries out pure rolling on the base cone (as described above); (2) the (q) plane is fixed, and the base cone is pure rolling on the (q) plane; ⑶ the base cone maintains a certain speed ratio relationship with the (q) plane and rotates around its own geometric rotation axis to realize the pure rolling relationship between them. Compared with the above three schemes, the common point is that they can realize the generation movement of tooth surface. The difference is: the first two schemes can intuitively show the generation process of tooth surface, but the motion of the required (q) plane or base cone is spatial motion, and the analysis and calculation are complex; The latter decomposes a spatial motion into two plane motions. Although the tooth surface generation process is not intuitive, the generated motion is relatively simple, and the tooth surface cutting generated motion based on this scheme is easy to realize.

According to scheme (3), the generation model of tooth surface of right-hand spiral bevel gear is established, as shown in Figure 2. When the base cone angle is δ The base cone of B revolves around its geometric rotation centerline o’o1 ‘ ω The (q) plane rotates at the speed of 1 around the vertical line of the (q) plane passing through the o ‘point of its geometric center ω When rotating at the speed of, the tooth surface occurrence line a’B ‘can also roll on the base cone, so as to generate the concave tooth surface of right-hand spiral bevel gear—— Σ 1。

In order to ensure the pure rolling relationship between the base cone and the (q) plane, the rotation speed of the base cone is calculated ω Rotation speed of planes 1 and (q) ω The relationship between them is deduced. In any time period T of pure rolling between the base cone and the (q) plane, the length turned by the outer diameter of the base cone bottom circle with radius R is:

In the same time period T, the arc length turned by the outer diameter of the (q) plane with radius q is:

As can be seen from Fig. 2, the (q) plane is tangent to the base cone and the bus o’c of the base cone, and the radius of the (q) plane is equal to the bus length of the base cone. Then, when the base cone and the (q) plane rotate around their own geometric axis in a pure rolling relationship, the arc length º L1 rotated by the outer diameter of the base cone bottom circle in any time period T is equal to the arc length º L2 rotated by the outer diameter of the (q) plane. Combined with the formula, the relationship can be obtained:

The formula can be obtained from the geometric relationship between the edges and corners of the right triangle o’o1’c in Figure 2:

Combined with the formula, in order to ensure the pure rolling relationship between the base cone and the (q) plane, ω 1 and ω The following relationship must be satisfied:

Where: δ B — base cone angle