In the field of high-performance mechanical transmissions, spiral bevel gears are critical components used in aerospace, automotive engineering, and robotics due to their high angular accuracy, large single-stage transmission ratios, flexible shaft angle designs, and compact structures. However, traditional spiral bevel gears, which are generated based on the virtual shaper principle using an involute cylindrical gear as the cutter, typically exhibit line contact. This leads to unavoidable edge contact and high sensitivity to assembly errors, compromising meshing stability and longevity. To address these issues, we propose a computer-aided design (CAD) method for three-dimensional (3D) topological modification of the tooth flanks in spiral bevel gears, achieving point contact and eliminating edge contact. Our approach leverages advanced foundry technology, where the modified tooth surface geometry is mapped onto the mold cavity for precision casting, enabling the mass production of gears with预设的啮合性能. This integration of CAD and foundry technology ensures high manufacturing accuracy and performance reliability.
The design process begins with the establishment of the baseline tooth surface for the spiral bevel gear using the virtual shaper principle. The gear is generated by a virtual helical cylindrical gear cutter, and the mathematical model involves coordinate transformations and meshing equations. The position vector of the cutter tooth surface, denoted as Σs, is transformed to the gear tooth surface Σ2 through a series of coordinate systems. The meshing equation ensures the conjugate relationship between the surfaces. The transmission ratio is defined as:
$$m_{2s} = \frac{\phi_s}{\phi_2} = \frac{\omega_s}{\omega_2} = \frac{N_2}{N_s}$$
where φs and φ2 are the rotation angles of the cutter and gear, respectively, ωs and ω2 are their angular velocities, and Ns and N2 are the number of teeth. The tooth surface equations for the left-hand helical cylindrical gear cutter are given as follows for the two flank sides I and II:
$$r_{I,II} = \begin{bmatrix} \mp [r_{b1} \sin(\theta_{01} + \theta_1) – u_1 \sin(\lambda_{b1}) \cos(\theta_{01} + \theta_1)] \\ – [r_{b1} \sin(\theta_{01} + \theta_1) – u_1 \cos(\lambda_{b1}) \sin(\theta_{01} + \theta_1)] \\ \mp [u_1 \sin(\lambda_{b1}) – p_1 \theta_1] \end{bmatrix}$$
and the corresponding unit normal vectors are:
$$n_{I,II} = \begin{bmatrix} \mp \sin(\lambda_{b1}) \cos(\theta_{01} + \theta_1) \\ – \sin(\lambda_{b1}) \sin(\theta_{01} + \theta_1) \\ \mp \cos(\lambda_{b1}) \end{bmatrix}$$
Here, θ1 and u1 are the Gaussian parameters, rb1 is the base circle radius, λb1 is the base cylinder lead angle, p1 is the spiral parameter, and θ01 is the half-angle of the tooth space on the base cylinder. The upper signs apply to flank I, and the lower signs to flank II. By substituting subscript “1” with “s”, we obtain the equations for the virtual cutter. The gear tooth surface Σ2 is derived through coordinate transformation matrices and the meshing condition:
$$r_{2}^{I,II}(u_s, \theta_s, \phi_s) = M_{2s}(\phi_s) r_s^{I,II}(u_s, \theta_s)$$
$$f(u_s, \theta_s, \phi_s) = n_s^{I,II} \cdot v_{s}^{I,II} = 0$$
where M2s(φs) is the transformation matrix, and vs(s2) is the relative velocity between the cutter and gear surfaces. This baseline design results in line contact, which is prone to edge contact and sensitivity to misalignments.
To achieve point contact and improve meshing stability, we introduce a topological modification method involving two steps: modification along the contact path and modification along the contact line. The first step involves presetting a transmission error function along a desired contact path to implement profile modification. The transmission error Δφ2 is defined as the difference between the actual rotation angle of the output gear and its theoretical value based on the input gear rotation:
$$\Delta\phi_2 = [\phi_2 – \phi_2^{(0)}] – \frac{N_s}{N_2} (\phi_s – \phi_s^{(0)})$$
where φs(0) and φ2(0) are the initial rotation angles. By treating the virtual cutting process as actual meshing, Δφ2 is expressed as a function of φs, and a fourth-order polynomial is used for the preset transmission error function:
$$\Delta\phi_2(\phi_s) = b_0 + b_1 \phi_s + b_2 \phi_s^2 + b_3 \phi_s^3 + b_4 \phi_s^4$$
The coefficients bi (i = 0 to 4) are determined based on the design of the contact path on the projection plane. The contact path is defined by points A, B, M, C, D, with distances ltA, ltB, ltC, ltD from the initial contact point M, and an angle η between the contact path and the tooth root direction. By applying this modified roll function during virtual generation, a new tooth surface Σ2n is obtained, which includes profile modification along the contact path. The modification amount δi at discrete grid points is calculated as the normal direction difference between the fully conjugate surface Σ2m and Σ2n:
$$\delta_i(\phi_{sm}, \theta_{sm}, \phi_{sn}, \theta_{sn}) = n_2^{(m,i)}(\phi_{sm}, \theta_{sm}) \cdot [r_2^{(m,i)}(\phi_{sm}, \theta_{sm}) – r_2^{(n,i)}(\phi_{sn}, \theta_{sn})]$$
for i = 1, 2, …, p, where p is the number of grid points. This step eliminates edge contact but retains line contact, so further modification is needed.
The second step involves computer-aided design of modification along the contact line direction to achieve point contact. For each contact line HQ, which intersects the contact path AD at point H, the tangent direction T is considered. A parabolic modification function is applied along the direction L parallel to T, with the center C of the instantaneous contact ellipse and semi-major axis a. The modification amount ξ at any point Q on the contact line is given by:
$$\xi = \frac{\zeta}{a^2} l^2$$
where ζ = 0.0068 mm is the contact deformation amount, and l is the parameter along the contact ellipse major axis. The modification amounts ξi at all grid points form a modification surface, which is superimposed onto the profile-modified surface Σ2n to obtain the bidirectional modified surface Σ2c:
$$r_2^{(c,i)}(\phi_{sc}, \theta_{sc}) = r_2^{(n,i)}(\phi_{sn}, \theta_{sn}) + n_2^{(m,i)}(\phi_{sm}, \theta_{sm}) \xi_i$$
This results in a point contact double-convex tooth surface, which enhances meshing stability and reduces sensitivity to assembly errors. The entire process is tailored for foundry technology, where the 3D topological modification is replicated in the mold cavity through CNC machining, enabling precision casting of high-performance gears.
To validate our method, we developed a CAD software system and performed tooth contact analysis (TCA) for two cases with different modification parameters. The main transmission parameters for the spiral bevel gear pair are summarized in the table below:
| Design Parameter | Value |
|---|---|
| Number of teeth on spiral cylindrical gear, N1 | 7 |
| Number of teeth on spiral bevel gear, N2 | 36 |
| Normal pressure angle, αn (°) | 20 |
| Shaft angle, γ (°) | 90 |
| Helix angle, β (°) | 52 (left-hand) |
| Center distance, E (mm) | 7 |
| Normal module, mn (mm) | 0.65 |
| Radial shift coefficient, xn | 0.277 |
| Inner radius, R1 (mm) | 11.7 |
| Outer radius, R2 (mm) | 14.5 |
The modification design parameters for the two cases are as follows:
| Parameter | Case 1 (Convex Side) | Case 1 (Concave Side) | Case 2 (Convex Side) | Case 2 (Concave Side) |
|---|---|---|---|---|
| Δφ2A (arcmin) | 12 | 12 | 12 | 24 |
| Δφ2B (arcmin) | 4 | 4.5 | 4 | 9 |
| Δφ2M (arcmin) | 0 | 0.8 | 0 | 0.8 |
| Δφ2C (arcmin) | 3.5 | 4 | 3.5 | 9 |
| Δφ2D (arcmin) | 12 | 12 | 12 | 24 |
| 2a (mm) | 1.14 | 0.96 | 1.14 | 0.96 |
| η (°) | 51 | -41 | 63 | -41 |
TCA results show that for Case 1, the contact path on the convex side has an angle η1 = 51°, and the contact ellipse major axis is 2a = 1.14 mm, matching the design. The transmission error for one meshing cycle is 5.0 arcsec, and the contact ratio is ε1r = 1.5. For the concave side, the transmission error is 7.4 arcsec, and the contact ratio is ε1c = 1.68. In Case 2, the convex side contact path angle is η2 = 63°, with a transmission error of 5.52 arcsec and contact ratio ε2r = 1.4, while the concave side has a transmission error of 12.6 arcsec and contact ratio ε2c = 1.6. The discrete contact spots form the desired contact paths, confirming point contact without edge contact. The increase in η amplifies the modification amount, and higher preset transmission error amplitudes Δφ2 increase the profile modification along the contact path, but not along the contact line.
Based on these parameters, we manufactured prototype spiral bevel gears using Cu-Zn alloy material and CNC machining. The gears were applied in fishing reel transmission mechanisms, resulting in smooth operation and elimination of jamming issues observed in previous designs. This demonstrates the practicality of our CAD-based topological modification method for foundry technology, enabling the production of high-performance gears through precision casting.
In conclusion, our computer-aided design approach for topological tooth flanks in spiral bevel gears effectively achieves point contact by combining modifications along the contact path and contact line. The preset transmission error function and parabolic modification function ensure controlled contact patterns and reduced sensitivity to assembly errors. The integration with foundry technology allows for the replication of these modifications in mold cavities, facilitating mass production of gears with enhanced meshing stability. Future work could focus on optimizing modification parameters for specific applications and extending the method to other gear types, further leveraging the capabilities of foundry technology in advanced manufacturing.