Abstract
The equal-distance spiral bevel gear, as a novel type of bevel gear, features a normal equal-distance spiral tooth surface, making it suitable for mass production through the metal powder injection molding (MIM) process. Parametric equations for the spherical involute and equal-distance conical spiral curves were derived based on coordinate transformation theory. A tooth surface mathematical model was established using the principle of tooth surface formation. The mathematical model was programmed in MATLAB to calculate the coordinates of discrete points on the tooth surface, and the precise modeling of the equal-distance spiral bevel gear was accomplished through reverse engineering in UG. ANSYS was utilized to simulate the meshing contact of the equal-distance spiral bevel gear, revealing its transmission performance. Finally, the trial production of the equal-distance spiral bevel gear was completed using the MIM process. The results indicate that the combination of the tooth surface mathematical model and the inverse modeling of discrete points ensures the accuracy of the 3D model, and the MIM process is feasible for mass production of equal-distance spiral bevel gear.
1. Introduction
Spiral bevel gear is primarily used to transmit motion and power between two intersecting or staggered shafts. They offer advantages such as high contact ratio, strong load-bearing capacity, smooth transmission, and low noise, making them widely used in mechanical transmission applications . The tooth surface of a spiral bevel gear is a spatial curved surface, and tooth surface contact is a local point contact, which complicates the 3D modeling and makes it difficult to ensure modeling accuracy.
Research on spiral bevel gear has primarily focused on circular arc spiral bevel gear, prolonged cycloid spiral bevel gear, and logarithmic spiral bevel gear. Circular arc spiral bevel gear and prolonged cycloid spiral bevel gear have tooth traces that are circular arcs and cycloids, respectively, neither of which theoretically belongs to a conical spiral line. The tooth trace of a logarithmic spiral bevel gear adopts a logarithmic spiral line, which is a conical spiral line. In mold forming of small gears (small cone angle gears) in spiral bevel gear transmissions, rotational demolding is challenging, necessitating machining on 5-axis CNC centers or specialized machines, which increases production costs and manufacturing cycles. Additionally, for the production of small modulus spiral bevel gear, a method involving double-sided milling for small gears and mold forming for large gears has been proposed in China (note: this mold forming is only an approximate tooth profile suitable for transmissions with low requirements). However, only large gears have achieved mass production, while small gears still require costly machining .
Metal powder injection molding (MIM) is a novel metal component manufacturing technology that introduces plastic injection molding technology into the field of powder metallurgy. It is suitable for producing small, complex 3D shapes and near-net-shape components with special performance requirements .
This study designs a pair of equal-distance spiral bevel gear, both of which are suitable for the MIM process, particularly the small cone angle gear, and achieves mass production through the MIM process for the first time. The tooth profile of the equal-distance spiral bevel gear adopts a spherical involute, and the tooth trace adopts an equal-distance conical spiral line. The tooth surface mathematical model is derived using coordinate transformation, and MATLAB is used to calculate the coordinates of discrete points on the tooth surface, which are then imported into UG for 3D modeling of the gear . This ensures the model accuracy of the equal-distance spiral bevel gear. Furthermore, the meshing transmission performance of the equal-distance spiral bevel gear is studied through ANSYS. Finally, an appropriate MIM process is selected to trial-produce small cone angle equal-distance spiral bevel gear, verifying the accuracy of the model and the feasibility of the MIM process for producing small cone angle spiral bevel gear, realizing mass production of spiral bevel gear.
2. Design Parameters Selection for Equal-Distance Spiral Bevel Gear
Based on actual usage and gear characteristics, and with reference to the relevant parameters of Gleason spiral bevel gear and relevant literature , a pair of small modulus equal-distance spiral bevel gear was designed. The relevant geometric parameters are shown in Table 1.
Table 1. Design parameters of the equal-distance spiral bevel gear
| Parameter | Small gear | Large gear |
|---|---|---|
| Rotation direction | Left | Right |
| Number of teeth, N | 16 | 23 |
| Shaft intersection angle, Σ | 90° | 90° |
| Pitch cone angle, δ | 34.82° | 55.176° |
| Module, m | 0.905 mm | 0.905 mm |
| Midpoint helix angle, β | 30° | 30° |
| Tooth width, b | 5 mm | 5 mm |
| Pressure angle, α | 20° | 20° |
3. Mathematical Modeling of the Tooth Surface of the Equal-Distance Spiral Bevel Gear
3.1 Spherical Involute
The equal-distance spiral bevel gear employs a spherical involute as the tooth profile. Its formation process is illustrated in Figure 1, where t is the rotation angle of the circular plane Q rolling purely on the base cone K, and ψ is the angle between the straight line OB and the instantaneous rotation axis OC. The circular plane Q is tangent to the base cone K and rolls purely around it. During movement, a point B on the circular plane Q forms a spatial curve, which is the spherical involute .
The circular plane Q is tangent to the base cone K (cone angle δb) at the straight line OC. A right-handed coordinate system (O-XYZ) is established with the cone axis OO’ as the Z-axis, fixed to the cone. Another right-handed coordinate system (O-X’Y’Z’) is established with the instantaneous rotation axis OC as the Z’-axis, fixed to the circular plane Q. The base cone K remains stationary, while the circular plane Q rolls purely, making the coordinate system (O-X’Y’Z’) the moving coordinate system.
The coordinate transformation matrix (M) for transforming the coordinate system (O-X’Y’Z’) into the coordinate system (O-XYZ) is shown in Equation (1).
M = \begin{pmatrix} cos t & -\sin t \cos \delta_b & \sin t \sin \delta_b \\ sin t & \cos t \cos \delta_b & -\cos t \sin \delta_b \\ 0 & \sin \delta_b & \cos \delta_b end{pmatrix}
Where t is the rotation angle of the circular plane Q, and δb is the cone angle of the base cone K.
Assuming the radius of the circular plane Q is l, which is also the cone distance at the starting point of the spherical involute on the base cone K, the coordinates (X’, Y’, Z’) of point B when the circular plane Q rolls to any position in the moving coordinate system (O-X’Y’Z’) are shown in Equation (2).
beginpmatrixX′Y′Z′endpmatrix=beginpmatrixlsinψ0lcosψendpmatrix
Where ψ is the angle between OB and the instantaneous rotation axis OC (variable), and ψ = t sin δb.
The coordinates of point B are transformed into the coordinate system (O-XYZ) using the coordinate transformation matrix M, as shown in Equation (3). Where (X, Y, Z) are the coordinates of any point on the spherical involute, and the parametric equation of the spherical involute can be expressed as Equation (4).
beginpmatrixXYZendpmatrix=M×beginpmatrixX′Y′Z′endpmatrix
beginpmatrixXYZendpmatrix=beginpmatrixl(sinδbcosψcost+sinψsint)l(sinδbcosψsint−sinψcost)lcosδbcosψendpmatrix
Where l is the cone distance at the starting point of the spherical involute.
3.2 Equal-Distance Conical Spiral Line
3.2.1 Equation of the Equal-Distance Conical Spiral Line
The equal motion of an Archimedean spiral moving uniformly along a horizontal plane and a uniform motion along a direction perpendicular to this horizontal plane combine to form an equal-distance conical spiral line (referred to as an equal-distance spiral line) . This study employs it as the tooth trace of the gear teeth. As shown in Figure 2, a right-handed coordinate system (o-xyz) is established, where the cone angle of the reference cone is δ, and the equal pitch is p.
The rotation angle of the Archimedean spiral is φ, which is the angle of rotation of the equal-distance spiral line around the z-axis. The Archimedean spiral starts from the origin of the coordinates, with a polar radius of 0 when φ = 0. The equation of the equal-distance spiral line is established as shown in Equations (5) and (6).
begin{aligned} x &= k \varphi \cos \varphi \\ y &= k \varphi \sin \varphi \\ z &= k \varphi \cot \delta end{aligned}
psinδ=2πk
Combining Equations (5) and (6), the equation of the equal-distance spiral line can be obtained as shown in Equation (7).
begin{aligned} x &= \frac{p}{2\pi} \varphi \sin \delta \cos \varphi \\ y &= \frac{p}{2\pi} \varphi \sin \delta \sin \varphi \\ z &= \frac{p}{2\pi} \varphi \cos \delta end{aligned}
3.2.2 Solution for the Equal Pitch
Since the equal pitch p is unknown, it needs to be solved based on the design parameters of the equal-distance spiral bevel gear in Table 1. Assuming the midpoint of the tooth trace is n, then the helix angle at point n is βn = 30°. The vector at point n is shown in Equation (8).
vecon=(x,y,z)
The tangent (vector) of the equal-distance spiral line at point n can be obtained using Equation (7), as shown in Equation (9).
vecτ=(x˙,y˙,z˙)
begin{aligned} dot{x} &= \frac{p}{2\pi} \sin \delta (\cos \varphi – \varphi \sin \varphi) \\ dot{y} &= \frac{p}{2\pi} \sin \delta (\sin \varphi + \varphi \cos \varphi) \\ dot{z} &= \frac{p}{2\pi} \cos \delta end{aligned}
Given βn = 30°, Equation (10) can be derived. At point n, Equations (11) and (12) can be obtained.
angle(on,τ)=arccos(∣∣on∣∣×∣∣τ∣∣on×τ)=30∘
fracp2πφcosδ=2cosδR1+R2
begin{aligned} R_1 &= \frac{mN}{2 \sin \delta} \\ R_2 &= R_1 – b end{aligned}
By combining Equations (7) to (12) and using MATLAB for programming calculations, the equal pitches p1 = 63.26 mm and p2 = 90.93 mm for the small and large gears, respectively, can be obtained.
3.3 Establishment of the Tooth Surface Mathematical Model
Since the tooth surface intercept at any point on the equal-distance spiral line is a spherical involute, multiple spherical involutes distributed along the equal-distance spiral line can be used to represent the gear tooth surface, as shown in Figure 3. A right-handed coordinate system (o-x’y’z’) is established with the cone vertex as the origin o and the cone axis as the z’-axis, fixed to the cone. Another moving coordinate system (o-x1y1z1) initially coincides with the coordinate system (o-x’y’z’), and the spherical involute with a starting cone distance of rb is fixed to it. The moving coordinate system (o-x1y1z1) is rotated around the z1-axis by an angle φ (the spherical involute rotates together around the z1-axis by φ). Therefore, only the parametric equation of any one spherical involute distributed along the equal-distance spiral line needs to be found, which is the tooth surface intercept equation at any point on the equal-distance spiral line, to obtain the tooth surface equation.
By adjusting the starting cone distance l of the spherical involute or rotating the spherical involute around the z1-axis by an angle φ along the equal-distance spiral line, multiple different tooth profiles can be determined. From the large end to the small end of the gear teeth, the spherical involute at any point on the spiral line of the tooth surface is determined not only by the constantly changing starting cone distance l of the spherical involute but also by the constantly changing rotation angle φ of the equal-distance spiral line. The distance rb from any point on the equal-distance spiral line to the origin o is rb=x2+y2+z2, and combining with Equation (5), it can be obtained that rb=2πpφ. Taking rb as the starting cone distance of the spherical involute, i.e., l = rb. At this time, the coordinate transformation matrix (M’) for transforming the coordinate system (o-x1y1z1) into the coordinate system (o-x’y’z’) is shown in Equation (13).
M’ = \begin{pmatrix} cos \varphi & -\sin \varphi & 0 \\ sin \varphi & \cos \varphi & 0 \\ 0 & 0 & 1 end{pmatrix}
In the moving coordinate system (o-x1y1z1), the spherical involute is fixed to it, and its parametric equation is shown in Equation (14).
beginpmatrixx1y1z1endpmatrix=beginpmatrixfracp2πφ(sinδbcosψcost+sintsinψ)fracp2πφ(sinδbsintcosψ−costsinψ)fracp2πφcosδbcosψendpmatrix
Using the coordinate transformation matrix, the parametric equation of the spherical involute at any point on the equal-distance spiral line after rotating by an angle φ around the z1-axis can be obtained, which is the tooth surface equation (Equation (15)) of the equal-distance spiral bevel gear.
beginpmatrixx′y′z′endpmatrix=M′×beginpmatrixx1y1z1endpmatrix
4. Precise Modeling of the Equal-Distance Spiral Bevel Gear
4.1 Solving Discrete Points on the Tooth Surface Using MATLAB
Since the tooth surface of the equal-distance spiral bevel gear is a spatial spiral curved surface, it is difficult to ensure accuracy when modeling with 3D software UG. Therefore, this study uses the established tooth surface equation to program in MATLAB, sets appropriate step sizes for calculation, and solves for the coordinates of discrete points on the tooth surface. The smaller the step size in the program, the more precise the tooth surface modeling. The calculated tooth surface is shown in Figure 4. The discrete point coordinate data is saved in a txt format recognizable by UG using the dlmwrite function.
4.2 Accurate 3D Modeling of the Gear
The discrete point coordinate data saved in txt format is then imported into the 3D software UG. In UG, the discrete points are utilized to fit a smooth curved surface, resulting in a group of tooth surfaces, as shown in Figure 6. Subsequently, the top cone surface, root cone surface, front cone surface, and back cone surface of the gear teeth are constructed. By employing commands such as trimming and stitching, a single tooth of the equal-distance spiral bevel gear is obtained, as depicted in Figure 7. Once the modeling of a single tooth is completed, the fillet transition at the tooth root can be optimized with variable radius rounding. After that, the single tooth is circumferentially arrayed to produce the precise model of the equal-distance spiral bevel gear, as shown in Figure 8.
This method, which combines the mathematical model of the tooth surface with the inverse modeling of discrete points, ensures the accuracy of the 3D model. By leveraging the precision of MATLAB calculations and the modeling capabilities of UG, the complex spatial spiral curved surface of the equal-distance spiral bevel gear is accurately represented, laying a solid foundation for subsequent analyses and manufacturing processes.