Abstract: A mathematical model for checking the generating interference during internal gear shaving is established using spatial meshing theory. The relationship between generating interference and the difference in the number of teeth between the internal gear and the shaving cutter, the shaft intersection angle, and the displacement coefficient of the internal gear is given. This model can not only be used for checking the generating interference during internal gear shaving, but also provides a method for checking the interference of interleaved shaft helical gear internal meshing transmission. Keywords: Spatial meshing theory, internal gear shaving, interference checking Introduction Internal gear shaving can be studied as a spatial helical gear meshing transmission. When internal helical gears mesh, three types of interference may occur. First, interference occurs between the transition line of the pinion tooth tip and the internal gear tooth root; second, interference occurs between the transition curve of the internal gear tooth tip and the pinion tooth root; third, meshing interference occurs when the pinion tooth collides with the internal gear tooth tip before disengaging. However, before internal gear shaving, the tooth root has already undergone a small amount of backing after pre-shaving and shaping, so the first type of interference generally does not occur. Furthermore, since the shaving cutter used for pinions has a retraction hole at the tooth root, a second type of interference will not occur. The only possible interference during internal gear shaving is a third type. This paper refers to this interference as generating interference and analyzes and discusses it. Unlike internal gear shaping, internal gear shaving is processed according to the principle of interleaved-axis helical gear transmission, and cannot be studied as a planar meshing transmission within a certain cross-section. The relative motion of the two gears is a helical motion along a rotating-sliding axis—therefore, the analytical method for generating interference in internal gear shaping cannot be applied; a new mathematical model must be established to study the generating interference problem during internal gear shaving. 1 Mathematical Model As shown in Figure 1, a coordinate system is established, where S₀ (o₀ - x₀y₀z₀) is the stationary coordinate system of the workpiece, with the o₀ axis coinciding with the workpiece axis; S₂ (o₂ - x₂y₂z₂) is the moving coordinate system fixed to the workpiece, with o₂z₂... The ub axis coincides with the o0z0 axis: S(0-xyz) is the stationary coordinate system of the shaving cutter, with the oz axis coinciding with the shaving cutter axis and the ox axis collinear with the o0x0 axis; S1(o1-x1y1z1) is the moving coordinate system fixed to the shaving cutter, with the 01z1 axis coinciding with the oz axis. The distance between the origins of the workpiece system and the tool system is α, and the angle between the workpiece axis and the tool axis is Σ. Let the shaving cutter rotate around the 01z1 axis with the motion coordinate system S[sub]1[/sub] and pass through the angle Φ[sub]1[/sub], at this time the internal gear rotates around the 02z2 axis with its motion coordinate system S[sub]2[/sub] and pass through the angle Φ[sub]2[/sub]: If the transmission ratio between the shaving cutter and the internal gear is i[sub]12[/sub], then z[sub]1[/sub] —— number of teeth of the shaving cutter z[sub]2[/sub] —— number of teeth of the internal gear From Figure 1, the transformation relationship between the motion coordinate system S[sub]2[/sub] of the internal gear and the motion coordinate system S[sub]1[/sub] of the shaving cutter can be obtained[3] where is the coordinate transformation matrix, which can be written as If it is required that the o[sub]1[/sub]- of the tool motion coordinate system S[sub]1[/sub] is 0[sub]1[/sub]- The transformation relationship between a point in the x1-x1-y1 plane and the workpiece motion coordinate system S2 can be obtained by substituting equations (3), (4), (5), (6), and (7) into equation (2), performing matrix multiplication, and rearranging. Equation (8) gives the transformation relationship between the coordinates in the o1-x1-y1 plane of the tool motion coordinate system S1 and the coordinates in the workpiece motion coordinate system S2. If a fixed point is given in the o1-x1-y1 plane of S2, equation (8) represents the motion trajectory curve of that point in S2. As shown in Figure 2, a shaving cutter end section is drawn through the origin O1 in the tool motion coordinate system S1. The coordinates of the tooth vertex Q in this section can be written as follows. Substituting equation (9) into equation (8) yields the trajectory curve equation of the shaving cutter tooth tip p in the workpiece motion coordinate system S2. Assuming the internal gear is a spur gear, as shown in Figure 3, in the workpiece motion coordinate system S2, a plane σ is drawn through the tooth vertex B of the internal gear and the o2z2 axis. The equation can be written as follows. When shaving, the trajectory curve of the shaving cutter tooth tip Q in the coordinate system S2 must intersect the plane σ, as shown in Figure 4. Let the intersection point be E. To find the coordinates of the intersection point, we can substitute equation (9) into equation (8), and then substitute equation (8) into equation (10). After rearranging, we can consider that Φ1 = i12Φ2. Equation (11) is a transcendental equation with parameter Φ2, which can be solved numerically. Then, we can substitute the obtained values ​​of Φ2 and Φ1 back into equation (8) to find the coordinates of the intersection point E (x2E, Y2E, z2E). Let L be the distance from point E to the o[sub]2[/sub]z[sub]2[/sub] axis. If the difference between L and the radius R[sub]φ2[/sub] of the internal gear tip circle is represented by ΔR, then the value of equation (12) can be used to determine whether the internal gear and the shaving cutter will generate interference during shaving. 2 Calculation of center distance a and shaft angle Σ Before performing the above interference check, it is necessary to calculate the shaft angle Σ and center distance a when the shaving cutter and the internal gear being processed mesh. For this purpose, a cross section is drawn through the ox axis. Then, the backlash-free meshing equation of the cross-shaft transmission is β[sub]2[/sub]=0 when the internal gear is a spur. Substituting equations (14), (15), and (16) into equation (13) and rearranging, we get equation (18) into equation (17). Equation (17) is a transcendental equation containing unknowns and can be solved by numerical methods. After solving, the pitch circle radius of the internal gear is the pitch circle radius of the shaving cutter. From equations (19) and (20), the center distance a is 3. Calculation example : The basic parameters of the internal gear and the shaving cutter used are as follows: Internal gear: module m=4, pitch circle pressure angle α2=20°, pitch circle helix angle β2=0, number of teeth z2=70, tooth tip height coefficient f=1, displacement coefficient ξ2=0.5, pitch circle shaving allowance δ=0; Shaving cutter: normal module mn=4, normal pressure angle αn=20°, pitch circle helix angle β1=5°, number of teeth z1=41. To investigate the influence of various parameters (such as tooth difference ΔZ, shaft angle Σ, and displacement coefficient ξ²) on the probability of generating interference, while keeping the other parameters constant, the number of teeth z² of the internal gear was changed, thus altering the tooth difference between the internal gear and the shaving cutter. A series of ΔZ values ​​were calculated and plotted as the curve shown in Figure 5. Similarly, keeping other parameters constant, only the displacement coefficient of the internal gear was changed, and a series of ΔZ values ​​were calculated and plotted as the curve shown in Figure 6. Figure 7 shows the relationship curve between the shaft angle Σ and ΔR. 4. Analysis and Discussion As can be seen from Figure 5, generating interference will occur when the tooth difference between the internal gear and the shaving cutter is less than or equal to 9. Furthermore, the calculation results above show that generating interference during internal gear shaving is not only related to the tooth difference between the internal gear and the shaving cutter, but also to the displacement coefficient of the internal gear and the shaft angle during shaving. Increasing the tooth difference between the internal gear and the shaving cutter can reduce the probability of generating interference. Similarly, increasing the displacement coefficient of the internal gear can reduce the possibility of generating interference. Furthermore, increasing the shaft angle during gear shaving can also reduce the possibility of generating interference. References: 1. Sichuan Provincial Machinery Industry Bureau. Fundamentals of Gear Tool Design (Volume 1). Beijing: Machinery Industry Press, 1982. 2. Yuan Zhejun et al. Gear Tool Design (Volume 1). Beijing: New Era Press, 1983. 3. Yao Nantong et al. Application of Mathematics in Tool Design. Beijing: Machinery Industry Press, 1988. 4. Wu Xutang. Gear Meshing Principle. Beijing: Machinery Industry Press, 1982.