In the automotive industry, curved-tooth hyperboloid gears have found widespread application due to their ability to transmit motion between non-intersecting and non-parallel shafts with high efficiency and compact design. These gears are manufactured under different tooth systems, such as the Gleason system with contracted spiral teeth, and the Oerlikon and Klingelnberg systems, which feature constant-height epicycloidal teeth. While these systems belong to the same class of transmissions, their geometric parameter calculations exhibit distinct characteristics. The geometric parameter calculation encompasses the determination of pitch cone geometry, selection of modification coefficients, undercut checking, tooth tip sharpening verification, and blank dimension computation. For large pitch cone angle constant-height epicycloidal bevel gears, secondary cutting verification is also necessary. Among these, the pitch cone geometric parameters form the foundation for all subsequent calculations. However, the formulas differ significantly across the three tooth systems, necessitating a critical evaluation. In this paper, we investigate the underlying principles of pitch cone geometric parameter calculation for curved-tooth hyperboloid gears, identify commonalities among the three systems, and propose a precise calculation method tailored to the characteristics of epicycloidal curves. The goal is to enhance the accuracy and consistency in designing hyperboloid gears, which are crucial for advanced drivetrain systems.
The accurate design of hyperboloid gears relies heavily on the precise determination of pitch cone parameters. These parameters include the pitch circle radii \(r_{m1}\) and \(r_{m2}\), pitch cone angles \(\delta_1\) and \(\delta_2\), and spiral angles \(\beta_{m1}\) and \(\beta_{m2}\) at the reference point \(M\). The relative position of the two hyperboloid gears is defined by the shaft angle \(\Sigma\) and offset distance \(E\), while the relative motion is governed by the gear ratio \(i_{12} = z_2 / z_1\), where \(z_1\) and \(z_2\) are the tooth numbers of the pinion and gear, respectively. The reference point \(M\) specifies the location where these geometric parameters are evaluated. From the conjugate condition of the two tooth surfaces at point \(M\), we derive three relational equations among the nine parameters. Additionally, if symmetric meshing on both tooth flanks is required, one constraint equation is imposed. Thus, five parameters must be given to solve for the remaining four using these four equations.
Typically, based on product structure and strength estimates, the following parameters are provided: \(\Sigma\), \(E\), \(z_1\), \(z_2\), \(d_{e2}\) (the outer pitch diameter of the gear), \(b_2\) (the gear face width), and the spiral angle \(\beta_{m2}\) at the reference point for the gear (for spiral bevel hyperboloid gears, \(\beta_{m1}\) for the pinion is given). If an initial value for the gear pitch cone angle \(\delta_2\) is assumed, then \(r_{m2} = (d_{e2} – b_2 \sin \delta_2) / 2\). At this stage, six parameters are known: \(\Sigma\), \(E\), \(i_{12}\), \(\beta_{m2}\) (or \(\beta_{m1}\)), \(r_{m2}\), and \(\delta_2\), which is one more than required. Therefore, an iterative solution using \(\delta_2\) must be employed.
To ensure conjugate tooth surfaces at the reference point \(M\), given \(\delta_2\), we perform an inner iteration to solve for \(\delta_1\), \(r_{m1}\), and \(\beta_{m1}\) using the three relational equations. Starting with an initial value \(\delta_1 = \delta_{10}\), we compute the following:
$$ \beta^\circ = \beta_{m1} – \beta_{m2} = \arccos(\tan\delta_1 \cdot \tan\delta_2 + \cos\Sigma \cos\delta_1 \cos\delta_2) $$
$$ \beta_{m1} = \beta^\circ + \beta_{m2} \quad \text{(or } \beta_{m2} = \beta_{m1} – \beta^\circ \text{)} $$
$$ r_{m1} = r_{m2} \cdot \frac{z_1 \cos \beta_{m2}}{z_2 \cdot \cos \beta_{m1}} $$
$$ \Delta E = E – \frac{\sin \beta^\circ}{\sin \Sigma} (r_{m1} \cos \delta_2 + r_{m2} \cos \delta_1) $$
If \(|\Delta E| \leq 10^{-4}\), the iteration terminates; otherwise, we update \(\delta_1 = \delta_{10} + \Delta \delta_1\) for the next iteration. The increment \(\Delta \delta_1\) is calculated as:
$$ \Delta \delta_1 = \frac{-\Delta E}{\frac{\sin \beta^\circ}{\sin \Sigma} \left( r_{m2} \sin \delta_1 – \frac{\Delta r_{m1}}{\Delta \delta_1} \cos \delta_2 \right) – \frac{E}{\tan \beta^\circ} \cdot \frac{\Delta \beta^\circ}{\Delta \delta_1}} $$
where
$$ \frac{\Delta \beta^\circ}{\Delta \delta_1} = -\frac{\pi}{180 \sin \beta^\circ \cos^2 \delta_1} \left( \tan \delta_2 + \frac{\cos \Sigma \sin \delta_1}{\cos \delta_2} \right) $$
$$ \frac{\Delta r_{m1}}{\Delta \delta_1} = r_{m1} \tan \beta_{m1} \cdot \frac{\Delta \beta^\circ}{\Delta \delta_1} $$
Note that the spiral directions of the two gears must be opposite, ensuring \(\beta_{m1} < 90^\circ\) (or \(\beta_{m2} < 90^\circ\)). To simplify the initial value selection and accelerate convergence, we introduce an auxiliary parameter, the gear deviation angle \(\varepsilon\). Starting with \(\varepsilon = 0\), we iterate using:
$$ \delta_1 = \arcsin(\cos \varepsilon \sin \Sigma \cos \delta_2 – \cos \Sigma \sin \delta_2) $$
$$ \beta^\circ = \arcsin(\sin \varepsilon \sin \Sigma / \cos \delta_1) $$
Then, compute \(\beta_{m1}\) (or \(\beta_{m2}\)), \(r_{m1}\), and \(\Delta E\) using equations (2) to (4). If \(|\Delta E| > 10^{-4}\), update \(\varepsilon\) as:
$$ \varepsilon = \arcsin\left( \frac{E}{r_{m2} + r_{m1} \frac{\cos \delta_2}{\cos \delta_1}} \right) $$
This method guarantees conjugate tooth surfaces at point \(M\) when the absolute values of the pressure angles are equal at the reference point. It is applicable to Klingelnberg and Oerlikon epicycloidal hyperboloid gears (typically given \(\beta_{m2}\)) and spiral bevel hyperboloid gears (typically given \(\beta_{m1}\)).
For symmetric meshing on both tooth flanks, we derive conditions based on the geometry of hyperboloid gears. The contact angle \(\theta_{21}\), induced normal curvature in the profile direction \(K_{21G}\), induced normal curvature in the tooth line direction \(K_{21v}\), and induced geodesic torsion \(\tau_{21v}\) at point \(M\) are given by:
$$ \tan \theta_{21} = \frac{K_{2v} – Q}{\tau_{2v} + S_a \cdot W \cdot E \sin \Sigma} $$
$$ K_{21G} = \frac{(\tau_{2v} + S_a \cdot W \cdot E \sin \Sigma)^2}{K_{2v} – Q} + W c_o \sin(\alpha_n – \alpha_o) $$
$$ K_{21v} = K_{21G} \tan^2 \theta_{21} $$
$$ \tau_{21v} = K_{21G} \tan \theta_{21} $$
where
$$ c_o = a_o^2 + b_o^2 $$
$$ W = \frac{\cos \beta_{m1} \cos \beta_{m2}}{r_{m1} \cdot r_{m2} \cdot \sin^2 \beta^\circ} $$
$$ a_o = r_{m1} \sin \beta_{m1} \sin \delta_2 – r_{m2} \sin \beta_{m2} \sin \delta_1 $$
$$ b_o = \cos \beta^\circ (r_{m1} \cos \delta_2 + r_{m2} \cos \delta_1) $$
$$ \alpha_o = \arctan(a_o / b_o) $$
Here, \(\alpha_o\) is the limit pressure angle, \(K_{2v}\) and \(\tau_{2v}\) are the normal curvature and geodesic torsion of the gear tooth surface \(\Sigma_2\) at point \(M\) along the tooth line direction. For left-hand drives (pinion left-hand, gear right-hand), \(S_a = 1\); for right-hand drives, \(S_a = -1\). The pressure angle \(\alpha_n\) is positive for the gear convex flank and pinion concave flank, and negative for the gear concave flank and pinion convex flank. After transformation, \(Q\) is expressed as:
$$ Q = K_{jv} \cdot \frac{\cos \alpha_n}{\cos \alpha_o} – W e_o \sin(\alpha_n – \alpha_o) \cos \alpha_o $$
where
$$ e_o = \sin \beta^\circ (r_{m1} \cos \delta_2 \tan \beta_{m2} + r_{m2} \cos \delta_1 \tan \beta_{m1}) $$
$$ K_{jv} = \cos \alpha_o \sin \beta^\circ \left( \frac{\sin \delta_1 \cos \beta_{m2}}{r_{m1}} – \frac{\sin \delta_2 \cos \beta_{m1}}{r_{m2}} \right) – e_o W \sin \alpha_o $$
\(K_{jv}\) is the limit normal curvature. Substituting into the earlier equations yields:
$$ \tan \theta_{21} = \frac{K_{2v} \cos \alpha_o – K_{jv} \cos \alpha_n + W e_o \sin(\alpha_n – \alpha_o)}{\cos \alpha_o (\tau_{2v} + S_a \cdot W \cdot E \sin \Sigma)} $$
$$ K_{21G} = \frac{\cos \alpha_o (\tau_{2v} + S_a \cdot W \cdot E \sin \Sigma)^2}{K_{2v} \cos \alpha_o – K_{jv} \cos \alpha_n + W (e_o + c_o) \sin(\alpha_n – \alpha_o)} $$
Symmetric meshing requires that for line contact, the contact angle, induced normal curvatures, and geodesic torsion on both flanks satisfy \(\theta_{21i} = -\theta_{21e}\), \(K_{21Gi} = -K_{21Ge}\), \(K_{21vi} = -K_{21ve}\), and \(\tau_{21vi} = \tau_{21ve}\). This leads to three conditions:
- \(\alpha_{ni} – \alpha_o = -(\alpha_{ne} – \alpha_o)\), giving the gear pressure angles:
$$ \alpha_{ni} = \alpha_\Delta / 2 + \alpha_o $$
$$ \alpha_{ne} = -\alpha_\Delta / 2 + \alpha_o $$
where \(\alpha_\Delta\) is the sum of the absolute pressure angles on the gear convex and concave flanks. Typically, \(\alpha_o < 0\), so \(\alpha_{ni} < |\alpha_{ne}|\). - \(K_{2vi} \cos \alpha_o – K_{jv} \cos \alpha_{ni} = K_{jv} \cos \alpha_{ne} – K_{2ve} \cos \alpha_o\), which simplifies to the average curvature \(K_o\):
$$ K_o = \frac{K_{2vi} + K_{2ve}}{2 \cos(\alpha_\Delta / 2)} = K_{jv} $$ - \(\tau_{2vi} = \tau_{2ve}\).
Thus, for symmetric meshing, the pressure angles and average curvature at the reference point are determined by the pitch cone geometric parameters, which then define the cutter blade angles and cutter radius. The geodesic torsion condition is satisfied if the tooth line direction is a principal direction, where \(\tau_{2vi} = \tau_{2ve} = 0\).
For epicycloidal hyperboloid gears, the curvature at the reference point is calculated using:
$$ K_o = \frac{1}{r_b} \left( 1 + \frac{E_b \sin \Delta}{r_b (1 + i_{jo})} \right) $$
where
$$ r_b = r_o \cos \delta_o – E_b \sin \Delta $$
$$ \delta_o = \arcsin\left( \frac{z_o \cdot m_n}{2 r_o} \right) $$
$$ m_n = \frac{2 r_{m2} \cos \beta_{m2}}{z_2} $$
$$ \Delta = \arcsin\left[ \frac{1}{E_{x2}} (r_o \cos \delta_o – R_{m2} \sin \beta_{m2}) \right] $$
$$ R_{m2} = r_{m2} \sin \delta_2, \quad E_b = \frac{i_{jo} E_{x2}}{1 + i_{jo}} $$
$$ E_{x2} = \sqrt{R_{m2}^2 + r_o^2 – 2 R_{m2} \cdot r_o \sin(\beta_{m2} – \delta_o)} $$
$$ i_{jo} = z_o / z_{p2}, \quad z_{p2} = z_2 / \sin \delta_2 $$
Given an initial \(\delta_2\) from outer iteration, we perform inner iteration to obtain pitch cone parameters, then compute \(K_{jv}\) and \(K_o\). Through outer iteration, we enforce \(K_o = K_{jv}\) to solve for \(\delta_2\), with tolerance \(\Delta \rho = |1/K_o – 1/K_{jv}| \leq 10^{-3}\).
Now, let us analyze the existing calculation methods for the three tooth systems. The Gleason system provides calculation cards for geometric parameters of spiral bevel hyperboloid gears, but only for shaft angle \(\Sigma = 90^\circ\). The formulas are accurate yet cumbersome. The cutter blade angles and cutter radius match those from our equations, ensuring symmetric meshing. The Klingelnberg system uses an inner iteration similar to ours, but the outer iteration relies on an approximate formula for the angle between the meshing line and gear pitch cone generatrix, given by \(\tan \theta^*_m = \sin \delta_2 \tan \varepsilon\). This aims to ensure symmetric meshing by matching tooth line curvature with the limit normal curvature, but the approximation introduces errors. The Oerlikon system employs complex formulas that are approximate for \(\Sigma \neq 90^\circ\) and does not fully address symmetric meshing. It requires satisfying \(\tan \phi^*_{12} = \sin \Sigma \tan q_2 / \sin \delta_2\), where \(q_2\) is derived from cutter geometry. However, this condition does not guarantee \(K_o = K_{jv}\), leading to significant asymmetry in meshing.
To illustrate, consider a design example for epicycloidal hyperboloid gears with parameters: \(\Sigma = 90^\circ\), \(E = 40 \text{ mm}\), \(z_1 = 12\), \(z_2 = 49\), \(d_{e2} = 400 \text{ mm}\), \(b_2 = 60 \text{ mm}\), \(\beta_{m2} = 30^\circ\) (giving \(m_n = 6.0654 \text{ mm}\)), \(z_o = 5\), \(r_o = 135 \text{ mm}\), and \(\alpha_\Delta / 2 = 20^\circ\). We compute geometric parameters using three methods: our proposed method, the Klingelnberg method, and the Oerlikon method. The results are summarized in the table below.
| No. | Parameter | Symbol | Our Method | Klingelnberg | Oerlikon |
|---|---|---|---|---|---|
| 1 | Gear pitch cone angle (°) | \(\delta_2\) | 71.3468 | 71.2613 | 70.3260 |
| 2 | Gear reference pitch radius (mm) | \(r_{m2}\) | 171.5758 | 171.5902 | 171.7513 |
| 3 | Pinion pitch cone angle (°) | \(\delta_1\) | 18.2124 | 18.2962 | 19.2130 |
| 4 | Pinion reference spiral angle (°) | \(\beta_{m1}\) | 42.9218 | 42.9213 | 42.9176 |
| 5 | Pinion reference pitch radius (mm) | \(r_{m1}\) | 49.6927 | 49.6965 | 49.7401 |
| 6 | Reference normal module (mm) | \(m_n\) | 6.0649 | 6.0654 | 6.0711 |
| 7 | Cutter orientation angle (°) | \(\delta_o\) | 6.4486 | 6.4492 | 6.4552 |
| 8 | Limit pressure angle (°) | \(\alpha_o\) | -1.7251 | -1.6806 | -1.1934 |
| 9 | Limit normal curvature (1/mm) | \(K_{jv}\) | \(7.8809 \times 10^{-3}\) | \(7.9562 \times 10^{-3}\) | \(8.7782 \times 10^{-3}\) |
| 10 | Meshing line angle (°) | \(\theta_m\) | 11.4338 | 11.4267 | 11.3504 |
| 11 | Gear tooth line curvature (1/mm) | \(K_o\) | \(7.8809 \times 10^{-3}\) | \(7.8802 \times 10^{-3}\) | \(7.8720 \times 10^{-3}\) |
| 12 | Curvature error (1/mm) | \(\Delta K\) | \(-2.2 \times 10^{-8}\) | \(7.6 \times 10^{-5}\) | \(9.1 \times 10^{-4}\) |
| 13 | Curvature radius error (mm) | \(\Delta \rho\) | \(3.6 \times 10^{-4}\) | \(-1.2120\) | \(-13.1149\) |
Our method yields \(\delta_2 = 71.3468^\circ\) with \(\Delta \rho = 3.6 \times 10^{-4} \text{ mm}\), indicating high precision. The Klingelnberg method gives \(\delta_2 = 71.2613^\circ\) and \(\Delta \rho = -1.212 \text{ mm}\), showing noticeable error due to approximations. The Oerlikon method results in \(\delta_2 = 70.3260^\circ\) and \(\Delta \rho = -13.1149 \text{ mm}\), which is substantial and may lead to asymmetric meshing. This is critical for hyperboloid gears manufactured via double-side cutting, as separate contact zone adjustments are not feasible.
In conclusion, the proposed method offers a precise and unified approach for calculating geometric parameters of epicycloidal hyperboloid gears. It ensures conjugate tooth surfaces and symmetric meshing by rigorously enforcing \(K_o = K_{jv}\) and proper pressure angle determination. The Klingelnberg method, while similar in intent, relies on approximate formulas that introduce errors. The Oerlikon method, particularly for non-90° shaft angles, lacks accuracy and does not guarantee symmetric meshing, which is detrimental for double-side cutting processes. Therefore, we recommend adopting our method to enhance the design and performance of hyperboloid gears in automotive and other high-precision applications. Future work could explore extensions to other gear types or dynamic loading conditions, further solidifying the role of hyperboloid gears in advanced mechanical systems.