In modern mechanical transmission systems, straight bevel gears play a critical role in transmitting motion and power between intersecting shafts. Their applications span across aerospace, automotive, and heavy machinery industries due to their efficiency and reliability. However, the manufacturing of high-precision straight bevel gears remains a complex challenge, particularly when aiming for optimized tooth contact characteristics and minimal transmission errors. This study focuses on the development of a comprehensive mathematical model for straight bevel gears produced using concave cutters, along with a detailed tooth contact analysis (TCA) methodology. The primary goal is to establish a theoretical foundation for determining optimal machining parameters and cutter configurations, enabling the production of high-quality straight bevel gears on domestically developed CNC spiral bevel gear milling machines.
The concave cutter method, notably employed in Gleason’s Coniflex® system, introduces a unique approach to generating crowned tooth surfaces, which enhance the gear meshing performance by accommodating misalignments and reducing edge loading. Unlike traditional gear cutting methods such as planing or double-disc milling, the concave cutter technique involves a specialized tool with an inwardly angled cutting edge relative to the cutter’s face. This configuration results in a conical surface with an internal cone angle during cutter rotation, ultimately producing a crowned tooth profile on the straight bevel gear. The crown amount, denoted as ΔS, is a key parameter influencing the tooth’s longitudinal curvature and is derived from the relationship between the cutter’s structural parameters and the gear’s geometry. The fundamental equation for the crown amount is given by:
$$ \Delta S = \Delta H \left[ \tan (\alpha_f + \delta) – \tan \alpha_0 \right] $$
where αf represents the pressure angle of the gear being machined, δ is the concave angle of the main cutting edge, α0 is the tool pressure angle, and ΔH is the depth variation. Alternatively, the crown amount can be expressed in terms of the gear’s design parameters:
$$ \Delta S = \frac{b^2 \cos \alpha_f \tan \delta}{4 D_e} $$
Here, b is the face width of the gear, and De is the cutter diameter. The cutter diameter itself is determined based on the gear’s face width and pressure angle, ensuring proper root line formation during machining. The formula for the cutter diameter is:
$$ D_e = \frac{b^2 \cos \alpha_f}{4 \Delta H} $$
Additionally, the tool’s top width must be carefully selected to accommodate the gear’s root channel widths at both the toe and heel, ensuring effective material removal without interference. These parameters collectively define the cutter’s geometry and its interaction with the gear blank during the cutting process.
To model the tooth surface of the straight bevel gear, the concept of a virtual generating gear is employed. The cutting process is analogous to the meshing between this virtual generating gear and the actual gear being machined. The coordinate system for the virtual generating gear is established with its apex as the origin. In this coordinate system Sc (Oc-XcYcZc), the position vector rc and unit normal vector nc of the tooth surface are defined as functions of parameters u (along the tooth height) and v (along the tooth width):
$$ \mathbf{r_c}(u, v) = \begin{bmatrix} v \cos \lambda_0 \\ \pm (v \sin \lambda_0 + u \sin \alpha_0) \\ u \cos \alpha_0 \end{bmatrix} $$
$$ \mathbf{n_c}(u, v) = \frac{\mathbf{r_{cu}} \times \mathbf{r_{cv}}}{\left| \mathbf{r_{cu}} \times \mathbf{r_{cv}} \right|} $$
The angles λ0 and α0 are calculated from the basic gear parameters:
$$ \lambda_0 = \frac{s_e / 2 – h_a \tan \alpha_f}{R_e / \cos \theta_f} $$
$$ \alpha_0 = \arctan \left( \frac{\tan \alpha_f}{\cos \lambda_0} \right) $$
where se is the toe end tooth thickness, ha is the addendum, Re is the outer cone distance, and θf is the root angle. These equations ensure the accurate representation of the generating gear’s tooth surface, which is essential for subsequent coordinate transformations and machining simulations.
The cutting process involves multiple coordinate systems to describe the relative motions between the cutter, the virtual generating gear, and the workpiece. The cutter coordinate system St (Ot-XtYtZt) is defined with its origin at the intersection of the cutter’s axis and the tip plane. For any point m on the cutting edge, the position vector rt and unit normal vector nt are expressed as:
$$ \mathbf{r_t} = \begin{bmatrix} \sin \theta (r – s \cos \delta) \\ \cos \theta (r – s \cos \delta) \\ s \sin \delta \\ 1 \end{bmatrix} $$
$$ \mathbf{n_t} = \begin{bmatrix} -\sin \theta \sin \delta \\ \cos \theta \sin \delta \\ \cos \delta \end{bmatrix} $$
Here, r is the cutter radius, s is the distance along the cutting edge from the tip, and θ is the rotation angle of the cutting edge around the cutter axis. The derivatives with respect to s and θ are computed to derive the normal vector. During machining, the workpiece coordinate system Sg is related to the machine tool coordinate system Sm through a series of transformations. The key transformation matrices include:
- Rotation from St to S1: M1t
- Rotation from S1 to S2: M21
- Translation from S2 to Sc: Mc2
- Rotation from Sc to Sm: Mmc
- Rotation from Sm to S3: M3m
- Rotation from S3 to Sg: Mg3
The machine settings, such as the cutter location parameters (L, E, D), the tilt angle α, and the swivel angle λ, are determined by solving the system of equations that ensure the cutter’s cutting edge aligns with the reference point on the generating gear’s tooth surface. The gear ratio I12, which defines the relationship between the generating gear rotation φ1 and the workpiece rotation ψ1, is constant and given by:
$$ I_{12} = \frac{\cos \theta_f}{\sin \delta_d} $$
where δd is the pitch angle of the workpiece. The relationship between the rotations is:
$$ \psi_1 = I_{12} \phi_1 $$
The tooth surface of the machined straight bevel gear is obtained by transforming the cutter’s surface to the workpiece coordinate system:
$$ \mathbf{r_g} = \mathbf{M_{g3}} \mathbf{M_{3m}} \mathbf{M_{mc}} \mathbf{M_{c2}} \mathbf{M_{21}} \mathbf{M_{1t}} \mathbf{r_t} $$
$$ \mathbf{n_g} = \mathbf{L_{g3}} \mathbf{L_{3m}} \mathbf{L_{mc}} \mathbf{L_{c2}} \mathbf{L_{21}} \mathbf{L_{1t}} \mathbf{n_t} $$
The meshing condition during cutting is enforced through the equation of meshing, which states that the relative velocity between the generating gear and the workpiece must be orthogonal to the common normal vector at the point of contact:
$$ \mathbf{v_{12}} \cdot \mathbf{n_m} = 0 $$
Solving this equation yields the parameter s as a function of φ1 and θ, which defines the contact lines on the tooth surface. By varying φ1, the entire tooth surface is generated as the envelope of these contact lines.
For the tooth contact analysis, the mathematical models of both the pinion and gear tooth surfaces are utilized. The TCA is performed in a fixed coordinate system Sf, which coincides with the pinion coordinate system Sp. The position and normal vectors of the gear tooth surface are transformed into Sf using a series of rotations accounting for the shaft angle Σ and the rotational positions η1 and η2 of the pinion and gear, respectively. The contact conditions require that the position vectors and unit normal vectors of both surfaces coincide at the contact point:
$$ \mathbf{r_{fp}} = \mathbf{r_{fg}} $$
$$ \mathbf{n_{fp}} = \mathbf{n_{fg}} $$
The transmission error is defined as the deviation from the ideal linear relationship between the rotations of the pinion and gear:
$$ \Delta \eta_2 = (\eta_2 – \eta_{20}) – \frac{z_1}{z_2} (\eta_1 – \eta_{10}) $$
To assess the sensitivity of the gear pair to misalignments, contact patterns and transmission error curves are computed at three designated points along the tooth width: near the toe (25% of face width), the midpoint (50%), and near the heel (75%). The results demonstrate that the contact ellipse covers approximately half of the face width, and the transmission error curve exhibits a parabolic shape, indicating the effectiveness of the crowning introduced by the concave cutter.
The following table summarizes the basic parameters of the example straight bevel gear pair and the concave cutter used in the analysis:
| Parameter | Pinion | Gear |
|---|---|---|
| Number of Teeth | 35 | 35 |
| Module (mm) | 2.5 | 2.5 |
| Cutter Pressure Angle (°) | 20 | 20 |
| Face Width (mm) | 15 | 15 |
| Pitch Angle (°) | 45 | 45 |
| Root Angle (°) | 42.22 | 42.22 |
| Addendum (mm) | 3 | 3 |
| Cutter Radius (mm) | 72.78 | 72.78 |
| Concave Angle (°) | 2 | 2 |
The machine tool settings for cutting the gears are provided in the table below:
| Setting | Pinion | Gear |
|---|---|---|
| Gear Ratio | 1.412554 | 1.412554 |
| Workpiece Installation Angle (°) | 42.22 | 42.22 |
| L (mm) | 54.30 | 54.30 |
| E (mm) | 28.05 | 28.05 |
| D (mm) | 67.12 | 67.12 |
| Swivel Angle (°) | 1.612 | 1.612 |
The TCA results, obtained through a custom-developed program, show that the contact patterns are consistently located within the central region of the tooth surface, and the transmission error curves are smooth and parabolic, confirming the beneficial effects of the crowned tooth profile. The maximum transmission error is within acceptable limits, ensuring stable and quiet operation of the straight bevel gear pair.
Experimental validation was conducted by machining a pair of straight bevel gears on a H350C CNC gear milling machine using the specified concave cutter parameters. The machined gears were then assembled and tested on a gear rolling tester to inspect the contact patterns. The experimental contact patterns observed at the toe, midpoint, and heel locations closely match the theoretical predictions from the TCA. Furthermore, the tooth profile error was measured on a gear measurement center, revealing a maximum deviation of 11.9 μm, which falls within the permissible range for practical applications. The slight discrepancies are attributed to factors such as tool wear, machine tool inaccuracies, and workpiece material variations. These experimental results affirm the accuracy of the developed mathematical model and the TCA methodology.
In conclusion, this research presents a robust mathematical framework for modeling and analyzing straight bevel gears manufactured with concave cutters. The derived tooth surface equations, coupled with the comprehensive TCA technique, provide valuable insights into the meshing behavior and performance of straight bevel gear pairs. The methodology facilitates the determination of optimal machining and tool parameters, thereby enhancing the quality and reliability of straight bevel gears produced on advanced CNC milling machines. Future work may explore the integration of this model with real-time monitoring systems for adaptive control during the machining process, further improving the manufacturing precision of straight bevel gears.