Abstract:
In order to promote the application of face gears in various spatial layout transmission occasions and expand their application advantages in the transmission field, this paper derives the tooth surface equation of helical offset non-orthogonal face gears and completes the gear shaping simulation for this type of gear using VERICUT software. Based on the gear meshing principle, the tooth surface equation of helical offset non-orthogonal face gears is derived to calculate the coordinates of the tooth surface points. Then, the machine tool model, the tool, and the machining blank are constructed in 3D software and imported into VERICUT to write the corresponding numerical control machining program and complete the gear shaping machining simulation. Coordinate points are extracted from the machining model to analyze the tooth surface deviation from the calculated coordinate points. The results show that there is no overcut on the tooth surface, with a maximum residual of 74 μm and a minimum residual of 3.3 μm, verifying the correctness of NC gear shaping for helical offset non-orthogonal face gears.
1. Introduction
Face gear transmission is a transmission mechanism composed of a cylindrical gear and a bevel gear, invented by the Fellow Corporation. Nowadays, face gear transmission has been successfully applied in robot joint reducers, helicopter power distribution reducers, differentials in automotive transmission systems, and rear axle drive reducers. Compared with bevel gears, face gear transmission mechanisms have many advantages: lighter weight, smaller size, no axial positioning requirements, high carrying capacity, easy replacement, and low noise. Therefore, in recent years, many companies and researchers have invested considerable effort into the research of face gears.
The processing and manufacturing difficulties of face gears limit their application in the transmission field. Most existing research focuses on straight-tooth orthogonal face gears, neglecting the study of more complex and general offset non-orthogonal face gears. To further enrich the transmission forms of face gears, meet the requirements of diversified structural layouts in space-limited transmission systems, and expand the application advantages of face gears in the transmission field, this paper designs a face gear incorporating three parameters: helical teeth, offset, and non-orthogonality. The tooth surface equation of the helical offset non-orthogonal face gear is derived using the spiral gear shaper cutter tooth surface equation, and MATLAB is used for numerical modeling of the tooth surface of the helical offset non-orthogonal face gear, achieving tooth surface visualization and virtual simulation machining of the face gear. Through numerical methods, a comparative analysis of the tooth surfaces of the simulated face gear and the theoretical face gear is conducted, and the tooth surface deviations at corresponding points are calculated, laying a theoretical foundation for subsequent research on helical offset non-orthogonal gears.
Table 1: Parameters of Gear Shaper Cutter and Face Gear
| Parameters | Values | Parameters | Values |
|---|---|---|---|
| Number of teeth on gear shaper cutter (N_s) | 28 | Normal module (m) / mm | 6.35 |
| Number of teeth on face gear (N_2) | 160 | Normal pressure angle (α_n) / ° | 25 |
| Helix angle (β) / ° | 10 | Inner radius of face gear / mm | 561 |
| Offset distance (E) / mm | -50 | Outer radius of face gear / mm | 641 |
| Misalignment angle (γ_m) / ° | 110 | – | – |
2. Face Gear Machining Principle and Tooth Surface Equation
2.1 Face Gear Machining Principle
The gear shaping process of helical offset non-orthogonal face gears simulates the generating motion between a helical gear and a face gear. The machining principle is shown in Figure 1. Unlike the gear shaping of straight-tooth face gears, during the gear shaping of helical offset non-orthogonal face gears, the axis of the spiral gear shaper cutter is neither parallel nor intersecting with the axis of the face gear, with a misalignment angle (γ_m). When machining the face gear, according to the formation principle of the helical line of the helical gear, besides cutting motion, circumferential feed motion, radial feed motion, indexing generating motion, and tool withdrawal motion, the gear shaper machine also requires an additional rotation of the spiral gear shaper cutter to form the helical tooth profile. That is, when the spiral gear shaper cutter feeds axially by one helical lead (L_s), the spiral gear shaper cutter just completes one additional rotation around its axis.
Figure 1: Gear Shaping Machining Principle
(Insert image of gear shaping machining principle)
2.2 Derivation of Face Gear Tooth Surface Equation
2.2.1 Spiral Gear Shaper Cutter Tooth Surface Equation
The tool used for machining helical offset non-orthogonal face gears is an involute spiral gear shaper cutter, with its end face involute tooth profile and coordinate system shown in Figure 2. In coordinate system S_s, the position vector (R_s) of the spiral gear shaper cutter tooth surface can be expressed as Equation (1):
R_s(θ_s, λ_s) = ±R_bs sin(D – θ_s cos(D)) – R_bs cos(D – θ_s sin(D)) + (L_s λ_s / 2π)
D = (θ_cs + θ_s ± λ_s)
Where: R_ps = m_t N_s / 2 is the pitch circle radius on the end face; R_bs = R_ps cos(α_t) is the base circle radius on the end face; θ_cs = 2π/N_s – (tan(α_t) – α_t) represents the central angle corresponding to half the tooth space width on the base circle; θ_s represents the angular displacement of the spiral gear shaper cutter.
2.2.2 Face Gear Machining Coordinate System
The gear shaping of helical offset non-orthogonal face gears adopts the coordinate system shown in Figure 3. Where φ_2 = φ_s N_s / N_2, φ_2 and φ_s are the rotational angles of the face gear and the spiral gear shaper cutter, respectively; N_2 and N_s are the number of teeth on the face gear and the spiral gear shaper cutter, respectively; coordinate systems S_2 and S_s are fixed with the face gear and the spiral gear shaper cutter, respectively; auxiliary coordinate systems S_g, S_m, S_n, and S_p are all fixed with the machine frame; L represents the distance from the origin O_2 of the face gear coordinate system to the origin O_s of the spiral gear shaper cutter coordinate system; E represents the offset distance of the spiral gear shaper cutter; γ_m is the misalignment angle between the axis of the spiral gear shaper cutter and the axis of the face gear.
Figure 3: Gear Shaping Machining Coordinate System
(Insert image of gear shaping machining coordinate system)
2.2.3 Face Gear Tooth Surface Equation
The tooth surface of the helical offset non-orthogonal face gear is formed by the envelope of the spiral gear shaper cutter tooth surface. In Figure 3, S_s rotates clockwise by φ_s around the z_s axis to obtain S_g, S_g translates negatively along the z_g axis by L to obtain S_m, S_m translates positively along the x_m axis by E to obtain S_n, S_n rotates clockwise by γ_m around the x_n axis to obtain S_p, and S_p rotates counterclockwise by φ_2 around the z_p axis to obtain S_2.
3. MATLAB Simulation of Face Gear Tooth Surface
Through the theoretical derivation of the tooth surface equation of the helical offset non-orthogonal face gear, it can be known that in coordinate system S_w, the tooth surface equation and normal vector (n_w) of the face gear are Equations (9) and (10), respectively:
R_w(θ_s, λ_s, φ_s) = M_w2 R_2(θ_s, λ_s, φ_s)
f_w(θ_s, λ_s, φ_s) = n_w(∂R_w/∂φ_s) = 0
n_w(θ_s, λ_s, φ_s) = M_w2 n_2(θ_s, λ_s, φ_s)
Figure 4: Numerical Simulation Coordinate System
(Insert image of numerical simulation coordinate system)
4. Face Gear Shaping Machining Simulation
The virtual simulation process is basically the same as the actual machining process. By adjusting the parameters of the NC machine tool, the motion relationship between the spiral gear shaper cutter and the workpiece is obtained, realizing the control of the relationship between the motion axes of the spiral gear shaper cutter and the workpiece. The collaborative motion of the various components of the NC machine tool can machine the helical offset non-orthogonal face gear.
4.1 Construction of Gear Shaping Machine Tool Model