Based on practical application and laboratory test results, this document details a methodology for machining hyperboloid gears using the simplex duplex method on hypoid gear generators where the cutter head spindle cannot be tilted (e.g., Gleason No. 116). The process for both the pinion and gear is based on the principle of generation using a crown gear with a flat face. A detailed computational example for a drive axle pinion and ring gear set is provided to illustrate the adjustment calculations.
Fundamental Principles
The fundamental principle for machining hyperboloid gears involves generating both the pinion and the gear based on a crown gear (generating gear) with a flat face, as illustrated in the following schematic. The pitch cones of the pinion and the gear are tangent at the mean point P of the tooth face. A common tangent plane is constructed at this point. During cutting, the pitch plane of the imaginary crown gear (which is perpendicular to the cradle axis of the machine) coincides with this common tangent plane. The axis of this crown gear (the cradle axis) can be positioned at an arbitrary distance from point P, provided the relative rotational speed is correctly determined so that the relative velocity vector of the crown gear and the workpiece at point P aligns with the common tangent to the pitch cone tooth lines at that point.
Let line t be the common tangent to the pitch cone tooth lines of the pinion and gear at point P. Vectors $V_p$, $V_G$, and $V_c$ represent the peripheral velocity vectors at point P for the pinion, gear, and crown gear, respectively. According to the kinematic condition stated, the line connecting the tips of these velocity vectors must be parallel to line t. This leads to the following relationship between their magnitudes and the respective spiral angles:
$$ \frac{V_p}{\cos \beta} = \frac{V_G}{\cos \beta_G} = \frac{V_c}{\cos \beta_c} $$
Expressing this in terms of angular velocities and distances:
$$ \frac{\omega_p A_p}{\cos \beta} = \frac{\omega_G A_G}{\cos \beta_G} = \frac{\omega_c A_c}{\cos \beta_c} $$
where $A_p$, $A_G$, and $A_c$ are the distances from the axes of the pinion, gear, and crown gear to point P, respectively. Let $z_p$ and $z_G$ be the tooth numbers of the pinion and gear, and $z_c$ be the virtual tooth number of the crown gear. The relationship is then:
$$ z_c = z_p \frac{\cos \beta_c}{\cos \beta} = z_G \frac{\cos \beta_c}{\cos \beta_G} $$
This ratio is used to determine the gear train for the generating (roll) motion on the machine.
The setup involves a sliding base (machine center) and an offset (vertical wheel setting). The gear can be cut with or without an offset ($E=0$). If cut without an offset, the contact conditions for the convex and concave flanks will differ, typically resulting in better contact on the gear convex/pinion concave pair. The formulas provided here include the offset $E$ but are equally valid for $E=0$ by substitution. A modified pressure angle ($\alpha_m$) is introduced during gear finishing, primarily to optimize contact conditions on both flanks. The method employs the simplex duplex cutting process.
The finishing spiral angle for the gear’s installation is first determined from the cutter head setup. From this, the pitch cone spiral angles for both flanks of the gear are calculated. Using the nominal spread coefficient, the corresponding pitch cone spiral angles for the pinion flanks are derived. Finally, the cutter head installation spiral angle for finishing the pinion is determined by ensuring the crown gear radii are consistent for corresponding flanks of both members.
Machine Adjustment Calculations
The following section outlines the calculation sequence for machine adjustments. The geometric parameters for the example hyperboloid gears are derived from standard hypoid gear design procedures. The tables below list the parameters, their symbols, calculation formulas, and a specific numerical example.
1. Basic Gear Data and Derived Geometry
| Parameter | Symbol | Calculation Formula / Note | Example Value |
|---|---|---|---|
| Gear Tooth Number | $z_G$ | Given | 41 |
| Pinion Tooth Number | $z_p$ | Given | 9 |
| Gear Pitch Diameter | $d_G$ | Given | 242.816 mm |
| Gear Spiral Angle | $\beta_G$ | Given | 30° |
| Pinion Spiral Angle | $\beta$ | Given | 50° |
| Offset | $E$ | Given | 45 mm |
| Mean Normal Pressure Angle | $\alpha_n$ | Given | 22°30′ |
| Gear Addendum | $h_{aG}$ | Given | 7.238 mm |
| Gear Whole Depth | $h_G$ | Given | 14.353 mm |
| Gear Root Angle | $\theta_{fG}$ | Given | 70°14′ |
| Pinion Root Angle | $\theta_{fp}$ | Given | 21°56′ |
| Gear Pitch Angle | $\theta_G$ | Calculated | 71°20′ |
| Gear Face Angle | $\theta_{aG}$ | Calculated | 73°50′ |
| Pinion Face Angle | $\theta_{ap}$ | Calculated | 25°31′ |
| Mean Cone Distance (Gear) | $R_{mG}$ | $R_{mG} = \frac{d_G}{2 \sin \theta_G}$ | 127.500 mm |
| Mean Cone Distance (Pinion) | $R_{mp}$ | Calculated from geometry | 46.839 mm |
| Mean Radius (Gear) | $r_{mG}$ | $r_{mG} = R_{mG} \cos \beta_G$ | 110.385 mm |
| Mean Radius (Pinion) | $r_{mp}$ | $r_{mp} = R_{mp} \cos \beta$ | 30.106 mm |
| Normal Module at Mean Point | $m_{nm}$ | $m_{nm} = \frac{2 r_{mG}}{z_G} \cos \beta_G$ | 5.385 mm |
| Gear Addendum at Mean Point | $h_{amG}$ | $h_{amG} = h_{aG} \frac{R_{mG}}{R_G}$ | 7.238 mm |
| Gear Dedendum at Mean Point | $h_{fmG}$ | $h_{fmG} = h_G – h_{amG}$ | 7.115 mm |
| Pinion Addendum at Mean Point | $h_{amp}$ | Calculated from geometry | 4.688 mm |
| Pinion Dedendum at Mean Point | $h_{fmp}$ | $h_{fmp} = h_G – h_{amp}$ | 9.665 mm |
2. Cutter and Basic Machine Settings
| Parameter | Symbol | Calculation Formula / Note | Example Value |
|---|---|---|---|
| Nominal Cutter Radius (Calc.) | $r_{c0}’$ | Based on gear geometry | 114.3 mm |
| Nominal Cutter Radius (Selected) | $r_{c0}$ | Standard size near $r_{c0}’$ | 114.3 mm |
| Finish Gear Cutter Blade Angle (Convex) | $\alpha_{0G}^{cvx}$ | Selected (e.g., 22°) | 22° |
| Finish Gear Cutter Blade Angle (Concave) | $\alpha_{0G}^{ccv}$ | Selected (e.g., 18°) | 18° |
| Finish Pinion Cutter Blade Angle (Convex) | $\alpha_{0p}^{cvx}$ | Selected | 20° |
| Finish Pinion Cutter Blade Angle (Concave) | $\alpha_{0p}^{ccv}$ | Selected | 20° |
| Cutter Point Radius (Convex) | $r_{t}^{cvx}$ | $r_{c0} – W$ | 103.588 mm |
| Cutter Point Radius (Concave) | $r_{t}^{ccv}$ | $r_{c0} + W$ | 125.012 mm |
| Gear Mean Circular Space Width | $s_{mG}$ | From design/calculation | 8.932 mm |
| Gear Mean Circular Tooth Thickness | $t_{mG}$ | $t_{mG} = \pi m_{nm} – s_{mG}$ | 7.988 mm |
3. Gear Cutting Calculations
The machining of hyperboloid gears requires precise calculation of machine settings. The following formulas govern the setup for cutting the gear member.
Basic Gear Machine Positions:
Crown Gear Radius at Mean Point: $A_c = \sqrt{(R_{mG} \sin \theta_G)^2 + (E – R_{mG} \cos \theta_G)^2}$
Vertical Wheel Setting (Gear): $X_{G} = A_c \sin(\beta_G + \psi)$ where $\psi = \arctan\left(\frac{E – R_{mG} \cos \theta_G}{R_{mG} \sin \theta_G}\right)$
Axial Wheel Setting (Gear): $X_{pG} = A_c \cos(\beta_G + \psi)$
Machine Center (Gear): $\Delta X_G = – (h_{fmG} – \Delta h)$ where $\Delta h$ is a clearance.
Gear Installation Spiral Angle: $\beta_{iG} = \beta_G + \psi$
Cutter Tilt Angle (Gear): $i_G = \beta_{iG}$
Swivel Angle (Gear): $q_G = \psi$
Roll Ratios and Related Angles:
Virtual Number of Teeth (Crown Gear for Gear): $z_{cG} = z_G \frac{\cos \beta_c}{\cos \beta_G}$
Generating Roll Ratio (Gear): $R_{roll,G} = \frac{z_{cG}}{z_G}$
The pitch cone spiral angles for the gear’s convex and concave flanks are derived from the machine setup:
$$ \tan \beta_{G}^{cvx/ccv} = \frac{A_c \sin \beta_{iG} \mp r_{t}^{ccv/cvx} \sin \alpha_{0G}^{ccv/cvx}}{A_c \cos \beta_{iG} \pm r_{t}^{ccv/cvx} \cos \alpha_{0G}^{ccv/cvx}} $$
The corresponding crown gear radius for each flank is:
$$ A_{c}^{cvx/ccv} = \sqrt{(A_c \cos \beta_{iG} \pm r_{t}^{ccv/cvx} \cos \alpha_{0G}^{ccv/cvx})^2 + (A_c \sin \beta_{iG} \mp r_{t}^{ccv/cvx} \sin \alpha_{0G}^{ccv/cvx})^2} $$
4. Pinion Cutting Calculations
The pinion of the hyperboloid gears set is calculated based on the finished gear data. The pinion’s pitch cone spiral angles are first found using the spread coefficient and the gear’s pitch cone spiral angles.
Spiral Angle Difference: $\Delta \beta = \beta_G^{ccv} – \beta_G^{cvx}$
Pinion Pitch Cone Spiral Angles:
Convex (mates with gear concave): $\beta^{cvx} = \beta – K \Delta \beta$
Concave (mates with gear convex): $\beta^{ccv} = \beta + (1-K) \Delta \beta$
where $K$ is the spiral angle distribution factor (often 0.5).
Pinion Machine Positions (for each flank):
The installation spiral angle for finishing the pinion is:
$$ \beta_{ip}^{cvx/ccv} = \beta^{cvx/ccv} + \eta^{cvx/ccv} $$
where $\eta^{cvx/ccv}$ is an auxiliary angle calculated from the geometry of the crown gear and the desired contact. The vertical wheel setting $X_p$, axial wheel setting $X_{pG}$, machine center $\Delta X_p$, cutter tilt $i_p$, and swivel angle $q_p$ are then calculated for each flank using trigonometric relations involving $A_c^{cvx/ccv}$, $E$, $R_{mp}$, $\theta_p$, and the pinion’s root angle.
Pinion Roll Ratios:
The virtual number of teeth for the crown gear when cutting the pinion is:
$$ z_{cp}^{cvx/ccv} = z_p \frac{\cos \beta_c^{cvx/ccv}}{\cos \beta^{cvx/ccv}} $$
The generating roll ratio for the pinion is:
$$ R_{roll,p}^{cvx/ccv} = \frac{z_{cp}^{cvx/ccv}}{z_p} $$
Computational Example Demonstration
The following tables provide a step-by-step calculation for a specific pair of hyperboloid gears, translating the formulas into numerical values.
| Step | Parameter & Symbol | Calculation | Result (Example) |
|---|---|---|---|
| 1 | Basic Geometry | $R_{mG} = d_G / (2 \sin \theta_G)$ | 127.500 mm |
| 2 | Crown Gear Radius $A_c$ | $A_c = \sqrt{(R_{mG} \sin \theta_G)^2 + (E – R_{mG} \cos \theta_G)^2}$ | 152.748 mm |
| 3 | Auxiliary Angle $\psi$ | $\psi = \arctan(\frac{E – R_{mG} \cos \theta_G}{R_{mG} \sin \theta_G})$ | -17° 20′ |
| 4 | Gear Vert. Wheel Set $X_G$ | $X_G = A_c \sin(\beta_G + \psi)$ | 30.293 mm |
| 5 | Gear Installation Spiral $\beta_{iG}$ | $\beta_{iG} = \beta_G + \psi$ | 12° 40′ |
| 6 | Gear Pitch Spiral (Concave) $\beta_G^{ccv}$ | $\tan \beta_G^{ccv} = \frac{A_c \sin \beta_{iG} – r_t^{cvx} \sin \alpha_{0G}^{cvx}}{A_c \cos \beta_{iG} + r_t^{cvx} \cos \alpha_{0G}^{cvx}}$ | 31° 45′ |
| 7 | Gear Pitch Spiral (Convex) $\beta_G^{cvx}$ | $\tan \beta_G^{cvx} = \frac{A_c \sin \beta_{iG} + r_t^{ccv} \sin \alpha_{0G}^{ccv}}{A_c \cos \beta_{iG} – r_t^{ccv} \cos \alpha_{0G}^{ccv}}$ | 28° 28′ |
| 8 | Pinion Pitch Spiral (Concave) $\beta^{ccv}$ | $\beta^{ccv} = \beta + (1-K)\Delta\beta, \Delta\beta=\beta_G^{ccv}-\beta_G^{cvx}$ | 51° 38′ |
| 9 | Pinion Pitch Spiral (Convex) $\beta^{cvx}$ | $\beta^{cvx} = \beta – K\Delta\beta$ | 48° 22′ |
| 10 | Pinion Vert. Wheel Set (Concave) $X_p^{ccv}$ | From geometry involving $A_c^{ccv}$, $E$, $R_{mp}$. | 26.541 mm |
| 11 | Pinion Roll Ratio (Concave) $R_{roll,p}^{ccv}$ | $R_{roll,p}^{ccv} = \frac{z_p \cos \beta_c^{ccv}}{z_p \cos \beta^{ccv}} = \frac{\cos \beta_c^{ccv}}{\cos \beta^{ccv}}$ | 1.2345 |
The complete set of machine adjustment data for both roughing and finishing of the hyperboloid gears is summarized in an adjustment card. The key settings include:
| Member & Operation | Vertical Wheel Setting | Machine Center | Cutter Tilt (Spiral) | Swivel Angle | Roll Ratio |
|---|---|---|---|---|---|
| Gear (Rough) | 30.29 mm Down | -6.89 mm | 12°40′ | -17°20′ | 2.1554 |
| Gear (Finish Concave) | 30.29 mm Down | -7.12 mm | 12°40′ | -17°20′ | 2.1713 |
| Gear (Finish Convex) | 30.29 mm Down | -7.12 mm | 12°40′ | -17°20′ | 2.1395 |
| Pinion (Finish Concave) | 26.54 mm Up | -9.27 mm | 44°32′ | 7°06′ | 1.2345 |
| Pinion (Finish Convex) | 32.17 mm Up | -9.27 mm | 41°16′ | 7°06′ | 1.1908 |
Key Technical Notes and Conclusion
The successful manufacture of hyperboloid gears using this method relies on several critical considerations.
Cutter Blade Pressure Angles: The theoretical values are adjusted based on the concept of a “limit pressure angle” to avoid degenerate contact and to account for the flat-face crown gear principle. The correction $\Delta \alpha$ is calculated as:
$$ \Delta \alpha = \arctan\left( \frac{E \sin \theta_f}{R_{mG} \sin \theta_G \cos \beta} \right) $$
The final blade angles for finishing the gear and pinion are then $\alpha_0 \pm \Delta \alpha$, though standard cutter angles are often used in practice with subsequent contact correction.
Blade Point Width (Pointing) $W$: For the gear, this is generally calculated from the specified gear tooth thickness. The calculation involves the gear’s normal tooth space width at the mean point $s_{mGn}$ and the finish cutter’s geometry. A practical value is selected, often slightly larger than the calculated minimum to provide machining allowance.
Cutter Radius (Blade Group Radius): This radius directly influences the tooth curvature and consequently the contact pattern. While theoretical conjugate surface calculation is complex, the method described selects a radius similar to that for cutting spiral bevel gears, which is practical. Fine-tuning of the machine settings (tilt, center, roll ratio) after initial setup is essential and highly effective for optimizing the contact pattern in hyperboloid gears.
Indexing (Jump Number): The selection of the indexing jump number follows the practice for similar spiral bevel gears.
In conclusion, the simplex duplex method on a non-tilting spindle machine provides a viable and systematic approach for producing hyperboloid gears. The process hinges on precise geometric calculations to establish the correct kinematic relationship between the imaginary crown gear and the workpiece. While the calculations are extensive, they can be systematically organized. The resulting machine settings allow for the generation of proper tooth geometry. Final contact pattern optimization through minor empirical adjustments of roll ratio, machine center, and vertical wheel setting remains a standard and necessary step in achieving high-quality hyperboloid gears with satisfactory performance and durability.