In the realm of power transmission for intersecting or, more critically, offset axes, hypoid bevel gears stand as a pinnacle of engineering design. Their ability to transmit high torque smoothly and quietly, particularly in automotive driveline applications such as rear axles, makes them indispensable. Among the various manufacturing systems, the Oerlikon method for generating cycloidal teeth (Cyclo-Palloid or Spiro®) offers distinct advantages in production efficiency and performance. This article delves into the comprehensive mathematical modeling of the tooth surfaces generated by this method, encompassing both the active working flanks and the crucial tooth root fillets. Furthermore, it introduces a modified and more pragmatic approach for Tooth Contact Analysis (TCA), specifically for hypoid bevel gears, that bypasses the traditional complexities associated with surface curvature derivation. The validity of the proposed methodologies is substantiated through comparative analysis with established commercial software and physical rolling tests.
The Oerlikon continuous indexing, face hobbing process is renowned for its efficiency. Unlike the “five-cut” method for circular arc teeth, it requires only two machines to produce a gear pair, significantly reducing capital investment and cycle time. The process simultaneously generates both sides of a tooth space using a cutter head equipped with multiple tool groups, each containing an inside and an outside blade. This discussion focuses on establishing a precise digital twin of the physical gear, a prerequisite for accurate performance simulation.
Comprehensive Tooth Surface Generation Model
The mathematical modeling of hypoid bevel gears manufactured via the Oerlikon method requires a systematic derivation through a series of coordinate transformations, tracing the path from the cutting tool edge to the final gear blank. A complete model must account for the precise geometry of the cutter, including the blade edge and tip radii, which are essential for profile modifications and defining the stress-critical root region.
Cutter Geometry and Coordinate Systems
The fundamental cutting element is the blade, characterized by a circular cutting edge (for profile crowning) and a tip fillet. The geometry is defined in a local blade coordinate system \( S_l(x_l, y_l, z_l) \). For a point \( P \) on the circular cutting edge with radius \( R_{BH} \), the position vector is given by:
$$
\mathbf{r}_l(u) = \begin{bmatrix}
2R_{BH}\sin(u/(2R_{BH})) \cos(\alpha_0 – u/(2R_{BH})) \\
0 \\
2R_{BH}\sin(u/(2R_{BH})) \sin(\alpha_0 – u/(2R_{BH}))
\end{bmatrix}
$$
where \( u \) is the profile parameter and \( \alpha_0 \) is the tool pressure angle. For a point on the tip fillet with radius \( r_e \), the vector is:
$$
\mathbf{r}_l(\theta) = \begin{bmatrix} X_M \\ 0 \\ Z_M \end{bmatrix} + r_e \begin{bmatrix} \cos \alpha_1 – \cos(\theta + \alpha_1) \\ 0 \\ \sin(\theta + \alpha_1) – \sin \alpha_1 \end{bmatrix}
$$
Here, \( \theta \) is the fillet parameter, and \( (X_M, 0, Z_M) \) are the coordinates of the transition point \( M \) between the cutting edge and the fillet. This blade is then mounted on the rotating cutter head. The transformation from the blade system \( S_l \) to the cutter head system \( S_t(x_t, y_t, z_t) \) involves rotations accounting for the blade’s initial setting angle \( \beta_i \) and the tooth direction angle \( \delta_0 \). The general transformation is a sequence: \( \mathbf{M}_{tl} = \mathbf{M}_{tp}\mathbf{M}_{pn}\mathbf{M}_{nm}\mathbf{M}_{ml} \). Thus, the tool edge in the cutter head system is:
$$
\mathbf{r}_t(u) = \mathbf{M}_{tl} \mathbf{r}_l(u) \quad \text{or} \quad \mathbf{r}_t(\theta) = \mathbf{M}_{tl} \mathbf{r}_l(\theta)
\tag{1}
$$
The key coordinate transformations involved in the generation of hypoid bevel gears are summarized below.
| Transformation | Description | Key Parameters |
|---|---|---|
| \( S_l \to S_t \) | Blade to Cutter Head | Blade setting angle \( \beta_i \), Direction angle \( \delta_0 \) |
| \( S_t \to S_d \) | Cutter Head to Generating Gear (Cradle) | Cutter rotation \( \beta \), Cradle angle \( \phi_{c1} \), Machine settings (S, j, i) |
| \( S_d \to S_1 \) | Generating Gear to Workpiece (Gear Blank) | Cradle roll angle \( \phi_{c2} \), Workpiece rotation \( \phi_1 \), Machine offsets (Em, ΔX, ΔB, γm) |
From Cutter Head to the Generating Gear and Final Workpiece
The cutter head is mounted on a simulated generating gear (or cradle). The transformation to the generating gear coordinate system \( S_d \) incorporates the cradle rotation \( \phi_{c1} \) and the cutter rotation \( \beta \), which are kinematically linked by the ratio of the number of cutter groups \( z_0 \) to the number of teeth on the generating gear \( z_p \): \( \phi_{c1} = (z_0 / z_p) \beta \). Therefore, the surface on the generating gear is:
$$
\mathbf{r}_d(u, \beta) = \mathbf{M}_{dt}(\beta) \mathbf{r}_t(u)
\tag{2}
$$
Finally, the generating gear envelopes the tooth surface onto the actual gear blank. The relationship between the cradle roll angle \( \phi_{c2} \) and the gear blank rotation \( \phi_1 \) is given by the ratio of their tooth numbers: \( \phi_{c2} = (z / z_p) \phi_1 \), where \( z \) is the number of teeth on the gear being cut. The tooth surface of the gear (e.g., the pinion) in its own coordinate system \( S_1 \) is therefore:
$$
\mathbf{r}_1(u, \beta, \phi_1) = \mathbf{M}_{1d}(\phi_1) \mathbf{r}_d(u, \beta)
\tag{3}
$$
For the root fillet surface generated by the tool tip, the equation is analogous: \( \mathbf{r}_1(\theta, \beta, \phi_1) = \mathbf{M}_{1d}(\phi_1) \mathbf{M}_{dt}(\beta) \mathbf{M}_{tl} \mathbf{r}_l(\theta) \).
The Equation of Meshing and Surface Normals
For a generated gear, not every point on the cutter path becomes part of the final tooth surface. Only points satisfying the equation of meshing are retained. This equation states that the relative velocity between the generating tool and the workpiece must be orthogonal to the common surface normal at the point of contact:
$$
f_1(u, \beta, \phi_1) = \mathbf{n}_1 \cdot \mathbf{v}_1^{(l1)} = \mathbf{n}_1 \cdot \left( \dot{\phi}_1 \frac{\partial \mathbf{r}_1}{\partial \phi_1} \right) = 0
\tag{4}
$$
The surface unit normal vector \( \mathbf{n}_1 \) and a tangent vector \( \mathbf{t}_1 \) are derived from the partial derivatives of the surface equation:
$$
\mathbf{n}_1(u, \beta, \phi_1) = \frac{\partial \mathbf{r}_1 / \partial u \times \partial \mathbf{r}_1 / \partial \beta}{||\partial \mathbf{r}_1 / \partial u \times \partial \mathbf{r}_1 / \partial \beta||}, \quad \mathbf{t}_1(u, \beta, \phi_1) = \frac{\partial \mathbf{r}_1 / \partial \beta}{||\partial \mathbf{r}_1 / \partial \beta||} \times \mathbf{n}_1
\tag{5}
$$
The complete pinion tooth surface \( \Sigma_1 \) is defined by Equations (3) and (4). The mating gear tooth surface \( \Sigma_2 \) is derived following a similar procedure. In the Spirac® method, the gear is often formate-cut (non-generated), while the pinion is generated, requiring a specific mating process. In the Spiroflex® method, both members are generated.
Modified Tooth Contact Analysis (TCA) for Hypoid Bevel Gears
Tooth Contact Analysis is the computational simulation of the meshing of two gear teeth. The primary goals are to determine the transmission error, the path of contact, and the size and orientation of the instantaneous contact ellipse under unloaded or lightly loaded conditions. Traditional TCA for complex surfaces like those of hypoid bevel gears often relies on second-order approximations at a reference point, requiring the computation of principal curvatures and relative curvature, which is algebraically complex and less accurate away from the reference point.
Mathematical Foundation of Mesh Condition
The core of TCA is solving for the conditions where two surfaces are in continuous point contact. Consider the pinion surface \( \Sigma_1 \) and the gear surface \( \Sigma_2 \). Their position vectors and unit normals are transformed into a fixed global coordinate system \( S_s \), incorporating their respective rotations \( \varphi_1 \) and \( \varphi_2 \):
$$
\begin{aligned}
\mathbf{r}_s^{(1)} &= \mathbf{M}_{s1}(\varphi_1) \mathbf{r}_1(u, \beta, \phi_1), \quad & \mathbf{n}_s^{(1)} &= \mathbf{L}_{s1}(\varphi_1) \mathbf{n}_1(u, \beta, \phi_1) \\
\mathbf{r}_s^{(2)} &= \mathbf{M}_{s2}(\varphi_2) \mathbf{r}_2(\tilde{u}, \tilde{\beta}, \tilde{\phi}_2), \quad & \mathbf{n}_s^{(2)} &= \mathbf{L}_{s2}(\varphi_2) \mathbf{n}_2(\tilde{u}, \tilde{\beta}, \tilde{\phi}_2)
\end{aligned}
\tag{6}
$$
The conditions for contact are the coincidence of position vectors and the alignment of unit normals (considering they are already unit vectors):
$$
\mathbf{r}_s^{(1)}(u, \beta, \phi_1, \varphi_1) = \mathbf{r}_s^{(2)}(\tilde{u}, \tilde{\beta}, \tilde{\phi}_2, \varphi_2), \quad \mathbf{n}_s^{(1)}(u, \beta, \phi_1, \varphi_1) = \mathbf{n}_s^{(2)}(\tilde{u}, \tilde{\beta}, \tilde{\phi}_2, \varphi_2)
\tag{7}
$$
Since \( \mathbf{n}_s^{(1)} \) and \( \mathbf{n}_s^{(2)} \) are unit vectors, the second vector equation provides only two independent scalar equations. A more robust formulation is achieved by projecting the normal alignment condition onto two orthogonal vectors within the tangent plane of one gear, say \( \Sigma_2 \). Using the tangent vector \( \mathbf{t}_s^{(2)} \) and its perpendicular \( \mathbf{n}_s^{(2)} \times \mathbf{t}_s^{(2)} \), the system becomes:
$$
\begin{cases}
\mathbf{r}_s^{(1)}(u, \beta, \phi_1, \varphi_1) – \mathbf{r}_s^{(2)}(\tilde{u}, \tilde{\beta}, \tilde{\phi}_2, \varphi_2) = 0 \\
(\mathbf{n}_s^{(2)} \times \mathbf{t}_s^{(2)}) \cdot \mathbf{n}_s^{(1)} = 0 \\
\mathbf{t}_s^{(2)} \cdot \mathbf{n}_s^{(1)} = 0
\end{cases}
\tag{8}
$$
This system of five independent scalar equations (3 from position, 2 from normal) is combined with the two equations of meshing (4) for the pinion and the gear (if generated). For a Spirac® gear pair (gear formate-cut, pinion generated), we have 6 equations with 7 unknowns: \( u, \beta, \phi_1, \tilde{u}, \tilde{\beta}, \tilde{\phi}_2, \varphi_2 \), with \( \varphi_1 \) as the input. Solving this system for a sequence of \( \varphi_1 \) values yields the contact path and the corresponding output rotation \( \varphi_2 \). The transmission error (TE) is then calculated as:
$$
\Delta \varphi_2 (\varphi_1) = [\varphi_2(\varphi_1) – \varphi_{20}] – \frac{N_1}{N_2} [\varphi_1 – \varphi_{10}]
\tag{9}
$$
where \( N_1, N_2 \) are the numbers of teeth, and \( \varphi_{10}, \varphi_{20} \) are the initial contact angles at the design reference point.
A Novel Method for Instantaneous Contact Ellipse Determination
The traditional method for calculating the contact ellipse involves computing the principal curvatures and directions of both surfaces at the contact point, deriving the relative curvature, and then finding the axes of the ellipse. This process is mathematically intensive and is based on a second-order approximation of the surfaces.
The proposed modified method operates directly on the exact surface equations without curvature calculation or second-order approximation. The procedure is as follows:
Step 1: At a solved contact point \( C \) with common unit normal \( \mathbf{u}_3 \), consider a plane \( Q \) that contains the normal line (i.e., its normal is perpendicular to \( \mathbf{u}_3 \)). The intersection of plane \( Q \) with the two tooth surfaces \( \Sigma_1 \) and \( \Sigma_2 \) yields two space curves. We define a separation function along the line within \( Q \) that is perpendicular to \( \mathbf{u}_3 \). By iteratively searching along this line from point \( C \), we find the two points, \( c_1 \) on \( \Sigma_1 \) and \( c_2 \) on \( \Sigma_2 \), where the normal distance (separation) between the two surfaces equals a predefined small value \( \delta \). This value \( \delta \) typically represents the thickness of marking compound used in physical roll testing (e.g., 0.0064 mm). The total distance \( c = c_1 + c_2 \) defines two boundary points of the contact ellipse along the direction associated with plane \( Q \).
Step 2: Rotate the plane \( Q \) around the common normal axis \( \mathbf{u}_3 \) with a small angular step (e.g., 1 degree). For each angular position, repeat Step 1. After rotating through 180 degrees, a complete set of boundary points is obtained, defining the full instantaneous contact ellipse. The major axis length is the maximum value of \( c \) found, and its direction is given by the angular orientation of plane \( Q \) at that maximum.
This method is computationally straightforward, uses the exact geometry, and provides a more realistic representation of the contact area, especially for highly conjugate or localized contact patterns common in optimized hypoid bevel gears.
Numerical Example and Experimental Correlation
To validate the developed model and TCA method, a high-speed automotive axle hypoid bevel gear pair designed with the Spirac® method was analyzed. The basic geometry, cutter, and machine setup parameters are condensed in the following tables.
| Parameter | Gear (Right-Hand, Convex) | Pinion (Left-Hand, Concave) |
|---|---|---|
| Shaft Angle | 90° | |
| Offset | 22 mm | |
| Number of Teeth | 39 | 9 |
| Normal Module at Ref. Point | 3.251 mm | |
| Face Width | 28 mm | 31.53 mm |
| Spiral Angle at Ref. Point | 49.997° | 34.046° |
| Setting | Gear Side | Pinion Side |
|---|---|---|
| Cutter Radius (Convex/Concave) | 88.024 mm / 87.578 mm | 87.697 mm / 87.951 mm |
| Blade Pressure Angle (Concave/Convex) | -23.61° / 17.516° | -23.007° / 18.059° |
| Cutter Blade Edge Radius | ~444 mm | ~98,400 mm / ~99,300 mm |
| Machine Root Angle | 72.026° | -0.531° |
| Sliding Base Setting (ΔX) | 0.0024 mm | 0.903 mm |
Using the described methodology, the complete tooth surfaces for both the gear and pinion were generated, including the root fillets. Three-dimensional solid models were constructed from these surfaces. The subsequent TCA was performed using the modified algorithm.
The predicted contact pattern (instantaneous contact ellipses aggregated over a small rotation) and transmission error from our model were compared against results from the commercial Klingelnberg KIMoS5 design software. The comparison showed excellent agreement. The contact patterns were similar in location, shape, and orientation. The calculated unloaded transmission errors were within 2% of the software’s predictions.
Furthermore, physical gears were manufactured on a Gleason Phoenix 600HC hypoid generator and subjected to a rolling test on a gear roll tester. The actual contact pattern obtained from the test using marking compound was photographed. The experimental contact pattern on both the convex and concave sides of the gear correlated very well with the TCA-predicted pattern, confirming the accuracy of the tooth surface model and the effectiveness of the contact analysis. This successful validation for hypoid bevel gears demonstrates that the model is reliable for predicting meshing behavior under no-load conditions.
Conclusion
This work presents a rigorous and complete mathematical framework for modeling Oerlikon-type cycloidal hypoid bevel gears. By explicitly incorporating the cutting blade edge and tip geometry, the model generates a full tooth surface, encompassing the active flank and the root transition surface, which is vital for subsequent strength and durability analysis (e.g., Loaded Tooth Contact Analysis and finite element stress analysis). The proposed modified TCA method represents a significant simplification and enhancement over classical approaches. By eliminating the need for complex principal curvature calculations and operating directly on the exact surface equations, it provides a more direct and accurate means of determining the instantaneous contact ellipse for hypoid bevel gears. The method’s validity is firmly established through favorable comparisons with industry-standard software outputs and physical rolling test results. This integrated approach to modeling and simulation forms a robust digital foundation for the design, analysis, and performance optimization of high-quality hypoid bevel gear drives.