Abstract
As a new type of bevel gear, the equal-distance spiral bevel gear is suitable for mass production by metal powder injection molding (MIM) due to the characteristic of the normal equal-distance of its spiral tooth surface. According to the coordinate transformation theory, the parametric equations of spherical involute and equal-distance conical spiral curves were derived. The mathematical model of the tooth surface was established by the formation principle of the tooth surface. The mathematical model of the tooth surface was programmed in MATLAB to calculate the coordinates of discrete points on the tooth surface, and the accurate modeling of the equal-distance spiral bevel gear was completed by reverse engineering in UG. The meshing contact of the equal-distance spiral bevel gear was simulated to obtain its transmission performance in ANSYS. Finally, the trial production of the equal-distance spiral bevel gear was completed based on the MIM process. The results show that the mathematical model of the tooth surface combined with the inverse modeling of discrete points can ensure the accuracy of the 3D model, and the MIM process can be used to produce the equal-distance spiral bevel gears for mass production.
1. Introduction
Spiral bevel gears are mainly used to transmit motion and power between two intersecting or staggered shafts. They have the advantages of large contact ratio, strong load-bearing capacity, smooth transmission, and low noise, making them widely used in mechanical transmission fields. The tooth surface of a spiral bevel gear is a spatial curved surface, and the tooth surface contact is local point contact, making the 3D model relatively complex, and it is difficult to ensure the accuracy of gear modeling. Research on spiral bevel gears has primarily focused on circular arc tooth bevel gears, prolate epicycloid bevel gears, and logarithmic spiral bevel gears. Among them, the tooth trace lines of circular arc tooth bevel gears and prolate epicycloid bevel gears are circular arcs and cycloids, respectively, which theoretically do not belong to conical helices. The tooth trace line of logarithmic spiral bevel gears adopts a logarithmic spiral, which is a conical helix. When using molds to form the small gear (small cone angle gear) in spiral bevel gear transmission, it is difficult to achieve rotational demolding, necessitating machining production on a 5-axis CNC machine center or a dedicated machine tool, which increases production costs and manufacturing cycles. In addition, for the production and processing of small-module spiral bevel gears, domestic methods currently propose using double-sided hobbing for the small gear and mold forming for the large gear (note: this mold forming is only an approximate tooth shape suitable for transmission types with low requirements). However, only the large gear has achieved mass production, and the small gear still requires costly machining production.
Metal powder injection molding (MIM) is a new metal component preparation technology that introduces plastic injection molding technology into the field of powder metallurgy to form a novel component processing technology. It is suitable for producing small, complex 3D shapes, and near-net-shape components with special performance requirements.
This paper studies and designs a pair of equal-distance spiral bevel gears, both of which are suitable for the metal powder injection molding process, especially the small cone angle gear, and for the first time, realizes mass production through the metal powder injection molding process. The tooth profile of the equal-distance spiral bevel gear adopts a spherical involute, and the tooth trace line adopts an equal-distance conical spiral. The tooth surface mathematical model is derived using coordinate transformation, and the coordinates of discrete points on the tooth surface are calculated and imported into UG to complete the 3D modeling of the gear, ensuring the accuracy of the model of the equal-distance spiral bevel gear. On this basis, the meshing transmission performance of the equal-distance spiral bevel gear is studied through ANSYS. Finally, an appropriate metal powder injection molding process is selected to trial-produce the small cone angle equal-distance spiral bevel gear, verifying the accuracy of the model and the feasibility of using the metal powder injection molding process for small cone angle spiral bevel gears, realizing the mass production of spiral bevel gears.
2. Selection of Design Parameters for Equal-Distance Spiral Bevel Gear
According to actual usage and gear characteristics, while referring to relevant parameters of Gleason spiral bevel gears and consulting relevant literature, a pair of small-module equal-distance spiral bevel gears was designed, with the relevant geometric parameters shown in Table 1.
Table 1. Design Parameters of the Equal-Distance Spiral Bevel Gear
| Name | Direction | Number of Teeth, N | Shaft Intersection Angle, Σ / (°) | Pitch Cone Angle, δ / (°) | Module, m / mm | Midpoint Helix Angle, β / (°) | Tooth Width, b / mm | Pressure Angle, α / (°) |
|---|---|---|---|---|---|---|---|---|
| Small Gear | Left | 16 | 90 | 34.82 | 0.905 | 30 | 5 | 20 |
| Large Gear | Right | 23 | 90 | 55.176 | 0.905 | 30 | 5 | 20 |
3. Mathematical Modeling of Tooth Surface of Equal-Distance Spiral Bevel Gear
3.1 Spherical Involute
The equal-distance spiral bevel gear adopts a spherical involute as the tooth profile. Its formation process is shown 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, which is 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, which is fixed to the circular plane Q. While keeping the base cone K stationary, the circular plane Q rolls purely, and at this time, the coordinate system (O-X’Y’Z’) is 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} sin t \cos \delta_b & \cos t \cos \delta_b & -\sin \delta_b \\ sin t \sin \delta_b & \cos t \sin \delta_b & \cos \delta_b \\ 0 & 0 & 1 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 that the radius of the circular plane Q is l, according to the formation principle of the spherical involute in Figure 1, l is also the cone distance at the starting point of the spherical involute on the base cone K. Then, in the moving coordinate system (O-X’Y’Z’), the coordinates (X’, Y’, Z’) of point B when the circular plane Q rolls to any position are shown in Equation (2).
begin{pmatrix} X’ \\ Y’ \\ Z’ end{pmatrix} = \begin{pmatrix} l \sin \psi \\ 0 \\ l \cos \psi end{pmatrix}
Where: ψ is the angle between OB and the instantaneous rotation axis OC (variable), and ψ = t sin δb.
Using the coordinate transformation matrix M, the coordinates of point B are transformed into the coordinate system (O-XYZ), as shown in Equation (3). Where (X, Y, Z) are the coordinates of any point, i.e., any point on the spherical involute, and the parametric equation of the spherical involute can be expressed as Equation (4).
begin{pmatrix} X \\ Y \\ Z end{pmatrix} = M \times \begin{pmatrix} X’ \\ Y’ \\ Z’ end{pmatrix}
begin{pmatrix} X \\ Y \\ Z end{pmatrix} = \begin{pmatrix} l (\sin \delta_b \cos \psi \cos t + \sin \psi \sin t) \\ l (\sin \delta_b \cos \psi \sin t – \sin \psi \cos t) \\ l \cos \delta_b \cos \psi end{pmatrix}
Where: l is the cone distance at the starting point of the spherical involute.
3.2 Equal-Distance Conical Spiral Curve
3.2.1 Equation of Equal-Distance Conical Spiral Curve
The motion can be decomposed into two uniform motions: one is uniform motion along an Archimedean spiral on a horizontal plane, and the other is uniform motion along a direction perpendicular to this horizontal plane. The combined motion route of these two motions is an equal-distance conical spiral curve (abbreviated as equal-distance spiral curve) [21-22], which is used as the tooth trace line of the gear teeth in this study. As shown in Figure 2, a Cartesian coordinate system (o-xyz) is established, where the cone angle of the reference cone is δ, and the equal-distance lead is p.
The rotation angle of the Archimedean spiral is φ, which is the angle of rotation of the equal-distance spiral curve around the z-axis. The Archimedean spiral starts from the coordinate origin, with a polar radius of 0 when φ = 0. The equation of the equal-distance spiral curve is established as shown in Equations (5) and (6).
begincasesx=kφcosφy=kφsinφz=kφcotδendcases
psinδ=2πk
Combining Equations (5) and (6), the equation of the equal-distance spiral curve is obtained as shown in Equation (7).
begincasesx=2πpφsinδcosφy=2πpφsinδsinφz=2πpφcosδendcases
3.2.2 Solution of Equal-Distance Lead
Since the equal-distance lead p is unknown, it needs to be solved based on the design parameters of the equal-distance spiral bevel gear in Table 1. Assuming that the midpoint of the tooth trace line of the gear teeth is n, then the helix angle at point n is βn = 30°, and the vector at point n is shown in Equation (8).
vecon=(x,y,z)
Through Equation (7), the tangent (vector) of the equal-distance spiral curve at point n can be obtained, as shown in Equation (9).
vecτ=(x˙,y˙,z˙)
begincasesdotx=2πpsinδ(cosφ−φsinφ)doty=2πpsinδ(sinφ+φcosφ)dotz=2πpcosδendcases
From 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
begincasesR1=2sinδmNR2=R1−bendcases
By combining Equations (7) to (12) and using MATLAB programming to calculate, the equal-distance leads p for the small and large gears can be obtained, which are p1 = 63.26 mm and p2 = 90.93 mm, respectively.
3.3 Establishment of Tooth Surface Mathematical Model
Since the intercept of the tooth surface at any point on the equal-distance spiral curve is a spherical involute, multiple spherical involutes distributed along the equal-distance spiral curve 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 apex as the origin o and the cone axis as the z’-axis, which is fixed to the cone. Another moving coordinate system (o-x1y1z1) is established, which initially coincides with the coordinate system (o-x’y’z’) and is fixed to the spherical involute with a starting cone distance of rb. 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 an angle φ). Therefore, only the parametric equation of any spherical involute distributed along the equal-distance spiral curve needs to be solved, i.e., the intercept equation of the tooth surface at any point on the equal-distance spiral curve, 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 curve, multiple different tooth profile lines can be determined. From the large end to the small end of the gear teeth, the spherical involute at any point on the spiral curve of the tooth surface is determined not only by the continuously changing starting cone distance l of the spherical involute but also by the continuously changing rotation angle φ of the equal-distance spiral curve. The distance from any point on the equal-distance spiral curve to the origin o is rb = (x2 + y2 + z2)1/2, and rb can be obtained from Equation (5) as rb = pφ/2π. 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).
begin{pmatrix} x_1 \\ y_1 \\ z_1 end{pmatrix} = \begin{pmatrix} frac{p}{2\pi} \varphi (\sin \delta_b \cos \psi \cos t + \sin t \sin \psi) \\ frac{p}{2\pi} \varphi (\sin \delta_b \sin t \cos \psi – \cos t \sin \psi) \\ frac{p}{2\pi} \varphi \cos \delta_b \cos \psi end{pmatrix}
Using the coordinate transformation matrix, the parametric equation of the spherical involute at any point on the equal-distance spiral curve 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.
begin{pmatrix} x’ \\ y’ \\ z’ end{pmatrix} = M’ \times \begin{pmatrix} x_1 \\ y_1 \\ z_1 end{pmatrix}
4. Accurate Modeling of Equal-Distance Spiral Bevel Gear
4.1 MATLAB Solution for Discrete Points on Tooth Surface
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 paper uses the established tooth surface equation to program in MATLAB, sets appropriate step sizes for calculation, and solves to obtain the coordinates of discrete points on the tooth surface. The smaller the step size in the program, the more accurate the tooth surface modeling. The operation program obtains the tooth surface shown in Figure 4. The discrete point coordinate data is saved in a txt format recognizable by UG using the dlmwrite function.
4.2 3D Modeling of Gear
The discrete point coordinate data saved in txt format is imported into UG, as shown in Figure 5. The fitting surface tool is used to fit the discrete points into a smooth surface to obtain a set of tooth surfaces, as shown in Figure 6. Next, the tip cone surface, root cone surface, front cone surface, and back cone surface of the gear teeth are established. Through commands such as trimming and stitching, a single tooth of the equal-distance spiral bevel gear is obtained, as shown in Figure 7. After completing the modeling of a single tooth of the equal-distance spiral bevel gear, a variable radius fillet optimization is performed on the tooth root transition section. Subsequently, a circular array is performed on the single tooth to obtain the precise model of the equal-distance spiral bevel gear, as shown in Figure 8.
By following these steps, the 3D modeling of the equal-distance spiral bevel gear is achieved with high precision. The discrete point data generated from the mathematical model ensures that the tooth surfaces are accurately represented, and the use of advanced CAD software like UG facilitates the creation of a smooth and realistic gear model. This precise modeling is crucial for subsequent analysis and manufacturing processes, ensuring that the designed gear meets all performance requirements.