In my work on aerospace transmission systems, I focused on the difficult problem of designing a high-power, small transmission ratio spur face gear pair while respecting a tight space envelope. Face gears are attractive because they can transmit motion and torque between intersecting or offset axes, they offer good interchangeability, they permit compact layouts, and they are naturally suited to split-torque architectures. However, when the transmission ratio becomes small and the available volume is restricted, the usual design logic for face gears must be reconsidered. I therefore combined geometric synthesis, tooth flank modification, tooth contact analysis, loaded tooth contact analysis, and finite element based stress evaluation to produce a face gear pair that can operate under high torque without severe eccentric load, edge contact, or stress concentration at the tooth tip. The present article describes the complete reasoning, the parameter choices, the equations I used, the numerical experiments I performed, and the influence of installation errors on transmission error and contact path.
Design envelope and initial constraints. I began with the space limits imposed by the surrounding engine structure. The face gear had to fit within a minimum inner diameter of 187 mm, a maximum outer diameter of 250 mm, and a maximum face width of 31 mm. The mating cylindrical pinion had a maximum outer diameter of 167 mm. The transmitted torque at the face gear was \(T_1 = 1030.57\ \mathrm{N\cdot m}\). To account for dynamic and application effects, I adopted an application factor \(K_A = 1.1\) and a dynamic factor \(K_v = 1.1\). The torque used in the loaded simulations was therefore
$$ T = K_v K_A T_1 = 1.1 \times 1.1 \times 1030.57 = 1246\ \mathrm{N\cdot m}. $$
For a face gear pair, I define the transmission ratio as the ratio of the pinion tooth number to the face gear tooth number,
$$ i = \frac{z_2}{z_1}, $$
where \(z_2\) is the number of teeth on the cylindrical pinion and \(z_1\) is the number of teeth on the face gear. In my application, the required range was \(i = 0.695\) to \(0.927\), and the nominal pressure angle range was \(\alpha = 25^\circ\) to \(27.5^\circ\). These values are unusual for face gears because most face gear applications favor larger ratios. A small ratio means that the face gear has relatively few teeth compared with the pinion, and this changes the location of the contact path, the risk of undercut near the inner radius, and the risk of tip sharpening near the outer radius.
Because the pinion outer diameter is capped at 167 mm, the maximum allowable pinion addendum circle strongly constrains the module and tooth number. With an addendum coefficient \(h_a^* = 1\), the relation is
$$ d_{a2} = m \left( z_2 + 2 h_a^* \right) = 167\ \mathrm{mm}. $$
I also assumed a clearance coefficient \(c^* = 0.25\). To explore feasible combinations, I tabulated the pinion tooth number and the allowable face gear tooth number for several modules. The resulting screening table is given below.
| Module \(m\) / mm | Pinion teeth \(z_2\) | Face gear teeth \(z_1\) range |
|---|---|---|
| 3.0 | 53 | 58 to 76 |
| 3.5 | 45 | 49 to 64 |
| 3.9 | 40 | 44 to 57 |
| 4.2 | 37 | 40 to 53 |
| 4.5 | 35 | 38 to 50 |
I selected a module of \(3.9\ \mathrm{mm}\) as a promising compromise because it permits a reasonably large face gear tooth count while keeping the pinion outer diameter within the allowed 167 mm. With \(z_2 = 40\), the pinion outer diameter is
$$ d_{a2} = 3.9 \left( 40 + 2 \right) = 163.8\ \mathrm{mm}, $$
which remains inside the maximum. I then examined a face gear with \(z_1 = 51\), giving a nominal transmission ratio
$$ i = \frac{40}{51} = 0.784, $$
which lies within the required range. The corresponding preliminary face gear parameters are summarized in the following table.
| Parameter | Value |
|---|---|
| Module \(m\) | 3.9 mm |
| Pinion tooth number \(z_2\) | 40 |
| Face gear tooth number \(z_1\) | 51 |
| Minimum inner radius from undercut limit \(R_{\min}\) | 95.34 mm |
| Maximum outer radius from tip sharpening limit \(R_{\max}\) | 110.28 mm |
| Mid-radius of face width | 110.17 mm |
| Face gear pitch radius | 99.45 mm |
| Face width | 29.66 mm |
| Pressure angle range along face width | 19.02° to 43.86° |
The pressure angle on a face gear varies with radius. I approximated it by
$$ \cos \alpha = \frac{m_n z_n \cos \alpha_n}{d}, $$
where \(m_n\) is the normal module, \(z_n\) is the virtual tooth number, \(\alpha_n\) is the nominal pressure angle, and \(d\) is the local diameter on the face gear. This equation shows that the pressure angle increases toward the outer radius and decreases toward the inner radius. In a small-ratio face gear pair, the inner radius can be quite close to the pitch radius, while the outer radius can be far from it. As a result, the contact path tends to concentrate near the inner portion of the face gear unless the tooth flank is deliberately modified.
Geometric feasibility and the undercut and sharpening limits. For a face gear generated by a shaper or a grinding worm, the inner radius must be larger than the undercut limit, and the outer radius must be smaller than the tip sharpening limit. I used the standard conditions as inequality constraints. The undercut condition can be written in the form
$$ R_{\min} \ge R_{\mathrm{uc}} \left( z_1, z_s, \alpha, m \right), $$
and the tip sharpening condition as
$$ R_{\max} \le R_{\mathrm{tip}} \left( z_1, z_s, \alpha, m \right), $$
where \(z_s\) is the shaper tooth number. In my first trial, I set \(z_s = 40\), equal to the pinion tooth number, and examined a face gear with \(z_1 = 51\). The preliminary calculation used the parameters in the table below.
| Parameter | Value |
|---|---|
| Module | 3.9 mm |
| Pinion tooth number | 39 |
| Shaper tooth number | 40 |
| Face gear tooth number | 51 |
| Pressure angle | 25° |
| Inner radius | 97 mm |
| Outer radius | 125 mm |
Although this trial was geometrically admissible, the TCA results showed that most contact points were located close to the inner radius of the face gear, near the pitch region. In addition, the outer portion of the tooth showed a tendency toward edge contact. When I transferred the geometry to a finite element model and applied the loaded torque, the contact stress distribution was clearly biased toward the inner side. This is the classic eccentric load problem for a small-ratio face gear pair. The mid-radius of the face width differed from the pitch radius by more than 10 mm in my initial layout, which is large enough to shift the contact pattern away from the tooth center and to reduce the effective load-carrying area.
I also considered the common practice of tooth number difference modification, in which the shaper tooth number is chosen larger than the pinion tooth number by one to three teeth. In that approach, the face gear tooth flank receives a crowned modification whose center is near the pitch circle. The modification then extends toward both the inner and outer ends. However, in my case the distance from the inner end to the pitch circle is small, while the distance from the pitch circle to the outer end is much larger. The resulting modification amount at the outer end is therefore much larger than at the inner end. The contact impression is pushed toward the inner end, which is precisely the behavior I needed to avoid. I concluded that tooth number difference modification alone was not suitable for this small-ratio, space-limited face gear pair.
Modification strategy for the spur face gear pair. Because the face gear itself is difficult to modify after grinding or shaping, I chose to modify the cylindrical pinion. I set the shaper tooth number equal to the pinion tooth number, \(z_s = z_2\), and applied both profile modification and lead modification to the pinion. The modification curve was parabolic in both directions, and the resulting contact was designed to be a point contact that spreads into an elliptical area under load. The profile modification curve can be written as
$$ \Delta h_p (v) = a_1 \left( \frac{v}{h_1} \right)^2 \quad \mathrm{for} \quad 0 \le v \le h_1, $$
and the lead modification curve as
$$ \Delta h_l (w) = c_1 \left( \frac{w}{b_1} \right)^2 \quad \mathrm{for} \quad 0 \le w \le b_1, $$
where \(v\) is the profile coordinate, \(w\) is the lead coordinate, \(a_1\) and \(c_1\) are the maximum modification amounts, and \(h_1\) and \(b_1\) define the active modification lengths. I also introduced transition zones \(h_2\) and \(b_2\) to avoid abrupt changes at the boundaries. The purpose of the profile modification was to rotate the contact trace and prevent edge contact at the tooth tip. The purpose of the lead modification was to concentrate the contact trace near the middle of the face width and to reduce the eccentric load toward the inner radius. After several trial calculations, I selected the initial modification parameters listed below.
| Modification parameter | Value |
|---|---|
| \(a_1\) | 3 μm |
| \(a_2\) | 10 μm |
| \(h_1\) | 3.9 mm |
| \(h_2\) | 4.875 mm |
| \(b_1\) | 19 mm |
| \(b_2\) | 9 mm |
| \(c_1\) | 20 μm |
| \(c_2\) | 3 μm |
I also chamfered the tooth tips to reduce stress concentration. The face gear received a tip chamfer with a radius of 0.3 mm, and the cylindrical pinion received a tip chamfer with a radius of 0.5 mm. These small geometric details are important in a high-power face gear pair because the tooth tip can otherwise become a stress raiser during edge contact. The design parameters for the final face gear pair are given in the next table.
| Parameter | Value |
|---|---|
| Module | 3.9 mm |
| Pinion tooth number \(z_2\) | 40 |
| Shaper tooth number \(z_s\) | 40 |
| Face gear tooth number \(z_1\) | 51 |
| Inner radius \(R_1\) | 97.5 mm |
| Outer radius \(R_2\) | 124.5 mm |
| Pressure angle \(\alpha\) | 25° |
| Outer chamfer length \(L\) | 16.3 mm |
| Outer chamfer height \(h\) | 3.12 mm |
Tooth contact analysis of the modified face gear pair. I used a tooth contact analysis procedure based on the condition that the position vectors and normal vectors of the two tooth surfaces coincide at the instantaneous contact point. In the fixed coordinate system of the gear housing, the contact condition can be written as
$$ \mathbf{r}_1^{(f)} \left( u_1, v_1, \phi_1 \right) = \mathbf{r}_2^{(f)} \left( u_2, v_2, \phi_2 \right), $$
and
$$ \mathbf{n}_1^{(f)} \left( u_1, v_1, \phi_1 \right) = \mathbf{n}_2^{(f)} \left( u_2, v_2, \phi_2 \right), $$
where \(\mathbf{r}\) denotes a position vector, \(\mathbf{n}\) denotes a unit normal vector, \(u\) and \(v\) are surface parameters, and \(\phi_1\) and \(\phi_2\) are the rotation angles of the pinion and face gear. The transmission error is defined as
$$ \Delta \phi_2 = \phi_2 – \frac{z_2}{z_1} \phi_1. $$
For the initial modified design, the TCA calculation produced a transmission error curve with an amplitude on the order of 38 μrad. The contact trace was approximately a single upward-opening parabolic curve that crossed the face width. This was already a significant improvement over the unmodified design because the contact path no longer concentrated exclusively near the inner radius. However, the transmission error curve did not alternate in the ideal manner between adjacent tooth pairs. A non-alternating transmission error can cause an impact at meshing entry and can make the tooth tip carry a larger share of the load. I therefore treated the initial modification as a starting point rather than a final solution.
The loaded tooth contact analysis was performed with a finite element model containing five teeth in contact. The middle tooth was selected as the object of detailed stress evaluation. I applied the torque \(T = 1246\ \mathrm{N\cdot m}\) and extracted the contact pressure and root bending stress. The maximum contact stress and maximum root bending stress for the two members are listed below.
| Gear member | Maximum contact stress / MPa | Maximum root bending stress / MPa |
|---|---|---|
| Cylindrical pinion | 970 | 371 |
| Face gear | 975 | 212 |
The contact area occupied approximately 86.5 percent of the working tooth surface. The contact region was located near the middle of the tooth flank rather than at the inner edge. The contact ratio was estimated from the finite element results to be about 1.83. This value is acceptable for a spur face gear pair under high load, although it is lower than what one might obtain with a larger face width or a larger number of teeth. The stress levels were below the allowable values for the selected material, so the design satisfied the strength requirement. The modification therefore achieved two important goals simultaneously: it reduced the eccentric load toward the inner radius, and it avoided the severe tooth tip stress concentration that would otherwise occur at the outer edge.
Influence of installation errors on transmission error. A face gear pair in an aerospace transmission is never perfectly aligned. Misalignment can arise from manufacturing tolerances, bearing clearances, thermal distortion, and elastic deformation of the housing. I therefore introduced three independent installation errors: an axial offset error \(\Delta q\), a shaft angle error \(\Delta \gamma\), and an intersection error \(\Delta E\). The axial offset shifts the pinion along its own axis. The shaft angle error changes the angle between the pinion axis and the face gear axis. The intersection error moves the pinion axis relative to the face gear axis in the plane perpendicular to the face gear axis. The error cases I analyzed are given in the following table.
| Case | Axial offset \(\Delta q\) / mm | Shaft angle error \(\Delta \gamma\) / ° | Intersection error \(\Delta E\) / mm |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 0.025 | 0 | 0 |
| 3 | -0.05 | 0 | 0 |
| 4 | 0 | -0.005 | 0 |
| 5 | 0 | 0.015 | 0 |
| 6 | 0 | 0 | -0.03 |
| 7 | 0 | 0 | 0.05 |
| 8 | 0.025 | 0.005 | -0.03 |
| 9 | 0.025 | -0.005 | 0.03 |
In the TCA model with installation errors, the coordinate transformation between the pinion frame and the face gear frame includes the error parameters. For small errors, the additional displacement can be approximated by
$$ \Delta \mathbf{r} \approx \Delta q \mathbf{e}_q + \Delta \gamma \mathbf{e}_\gamma + \Delta E \mathbf{e}_E, $$
where \(\mathbf{e}_q\), \(\mathbf{e}_\gamma\), and \(\mathbf{e}_E\) are the sensitivity vectors of the contact point with respect to each error. The transmission error then becomes a function of the rotation angle and the error vector,
$$ \Delta \phi_2 = \Delta \phi_2 \left( \phi_1, \Delta q, \Delta \gamma, \Delta E \right). $$
The TCA results showed that the face gear tooth surface enters mesh near the outer radius and the tooth tip, then moves toward the inner radius as the mesh progresses. In the edge contact region, the transmission error was relatively large. In the middle and inner portions of the tooth flank, the transmission error was smaller. When the axial offset error was negative, the transmission error amplitude became much smaller than in the standard installation. A positive intersection error also improved the transmission error. This means that a small, deliberate negative axial offset or a small positive intersection error can be beneficial for this particular small-ratio face gear pair, provided that the contact path does not shift too far toward the tooth edge.
The sensitivity of the contact path to the installation errors was different for each error component. The shaft angle error \(\Delta \gamma\) was the most influential. The axial offset \(\Delta q\) was the second most influential. The intersection error \(\Delta E\) was the least influential. When the shaft angle error or the axial offset error became positive and increased, the contact trace moved along the face width toward the outer radius of the face gear. When either of these errors became negative and decreased, the contact trace moved toward the inner radius. In contrast, when the intersection error became positive and increased, the contact trace moved toward the inner radius, and when it became negative, the contact trace moved toward the outer radius. These trends are important because they allow a designer to compensate for one error by using another error of opposite effect.
The combined installation error can either amplify or cancel the individual effects. If all individual error components shift the contact path in the same direction, the combined error produces a larger shift. If the components shift the contact path in opposite directions, the combined error produces a smaller shift. This superposition and cancellation behavior is one of the main reasons why installation error analysis must be performed after the nominal tooth flank modification has been selected. In my study, the transmission error amplitude and curve shape under combined errors remained close to the standard installation case, which indicates that the point contact design is relatively robust with respect to the small errors that are typical of an aerospace transmission.
Optimization of the tooth flank modification. The initial modification parameters produced a contact trace that covered most of the tooth flank, but the transmission error curve was discontinuous in several error cases. I judged that this discontinuity could lead to noise, impact, and a reduction in effective contact ratio. I therefore modified the lead modification lengths while keeping the profile modification amounts and the transition zones unchanged. The initial lead modification lengths were \(b_1 = 19\ \mathrm{mm}\) and \(b_2 = 9\ \mathrm{mm}\). I changed them to \(b_1 = 12.4\ \mathrm{mm}\) and \(b_2 = 15.6\ \mathrm{mm}\). The optimized modification parameters are compared in the table below.
| Modification parameter | Initial value | Optimized value |
|---|---|---|
| \(a_1\) / μm | 3 | 3 |
| \(a_2\) / μm | 10 | 10 |
| \(h_1\) / mm | 3.9 | 3.9 |
| \(h_2\) / mm | 4.875 | 4.875 |
| \(b_1\) / mm | 19 | 12.4 |
| \(b_2\) / mm | 9 | 15.6 |
| \(c_1\) / μm | 20 | 20 |
| \(c_2\) / μm | 3 | 3 |
With the optimized lead modification, the transmission error curve became continuous and the alternation between adjacent tooth pairs improved. The contact trace remained spread across the working tooth surface, and the eccentric load toward the inner radius remained controlled. The optimized design also maintained the contact stress and root bending stress within acceptable limits. This shows that the lead modification length is a powerful parameter for tuning the transmission error of a small-ratio spur face gear pair. A shorter primary lead modification length and a longer transition zone can smooth the transition between teeth and reduce the discontinuity in the transmission error.
Design equations and performance measures. Throughout the design, I used several equations to evaluate the face gear pair. The nominal transmission ratio is
$$ i = \frac{z_2}{z_1}. $$
The pinion pitch diameter is
$$ d_2 = m z_2. $$
The face gear pitch radius at the pitch cone can be approximated by
$$ R_1 = \frac{m z_1}{2}. $$
The face width is
$$ B = R_2 – R_1, $$
where \(R_1\) is the inner radius and \(R_2\) is the outer radius of the face gear. The pressure angle at any radius \(r\) on the face gear is
$$ \alpha (r) = \arccos \left( \frac{m_n z_n \cos \alpha_n}{2 r} \right). $$
The contact ratio for the spur face gear pair can be estimated from the finite element solution by
$$ \varepsilon_\alpha = \frac{N_{\mathrm{in}} – N_{\mathrm{out}}}{\Delta N_{\mathrm{tooth}}}, $$
where \(N_{\mathrm{in}}\) and \(N_{\mathrm{out}}\) are the frame numbers at which a given tooth enters and leaves contact, and \(\Delta N_{\mathrm{tooth}}\) is the frame difference between adjacent teeth at the same contact position. In my analysis, this gave \(\varepsilon_\alpha \approx 1.83\). The transmission error is
$$ \Delta \phi_2 = \phi_2 – \frac{z_2}{z_1} \phi_1. $$
The contact stress and root bending stress were compared with allowable values. For the pinion, the maximum contact stress was 970 MPa and the maximum root bending stress was 371 MPa. For the face gear, the maximum contact stress was 975 MPa and the maximum root bending stress was 212 MPa. These values were below the allowable limits for the chosen material and heat treatment. The contact area fraction was
$$ A_f = \frac{A_{\mathrm{contact}}}{A_{\mathrm{working}}} \approx 0.865, $$
which indicates that the load was distributed over a large portion of the working tooth surface. The combination of a high contact area fraction and a moderate contact ratio is desirable for a high-power face gear pair because it reduces the peak pressure and improves the fatigue life.
Interpretation of the installation error results. The installation error study produced several practical conclusions. First, the shaft angle error is the most critical alignment parameter for this face gear pair. A small change in the shaft angle can move the contact trace across a significant portion of the face width. Second, the axial offset error is also important, but its effect is somewhat smaller. Third, the intersection error has the least influence on the contact path, although it still affects the transmission error. Fourth, a negative axial offset and a positive intersection error can reduce the transmission error amplitude in the nominal operating range. Fifth, combined errors can add or cancel, so the worst-case alignment condition is not always the sum of the worst individual errors. A proper tolerance stack-up analysis should therefore be based on the combined error state rather than on individual error limits alone.
I also observed that the contact trace moves toward the outer radius when the shaft angle error or the axial offset error is positive. This is a dangerous direction because the outer radius is where the tooth tip is thinner and where edge contact is more likely. Conversely, a negative shaft angle error or a negative axial offset error moves the contact trace toward the inner radius, where undercut and root interference are more likely. The design must therefore keep the contact trace within a safe band that avoids both the outer edge and the inner root. The point contact produced by the pinion modification helps to keep the contact trace inside this band, but the allowable installation error range must still be specified. Based on my TCA results, I recommend that the shaft angle error be kept within about \(\pm 0.005^\circ\) for this class of face gear pair, and that the axial offset error be kept within about \(\pm 0.025\ \mathrm{mm}\). The intersection error can be allowed a somewhat larger range, on the order of \(\pm 0.03\ \mathrm{mm}\), provided that the combined error does not push the contact trace to the tooth edge.
Manufacturing and assembly considerations. The design of a high-power small-ratio face gear pair is not complete without considering manufacturing and assembly. The face gear is usually generated by a shaper or a grinding worm. The shaper tooth number and the pressure angle determine the cutting geometry and the resulting tooth surface. In my design, I set the shaper tooth number equal to the pinion tooth number to avoid the unfavorable modification distribution that occurs when a tooth number difference is used. The pinion modification was then introduced by profile and lead grinding or by a controlled hard finishing process. The parabolic modification curves require accurate control of the grinding wheel position and the machine tool axes. The tooth tip chamfers were small but important; they reduced the risk of edge contact and stress concentration. The assembly should include a means of adjusting the axial position of the pinion, because the axial offset error has a significant influence on the contact path. A shim pack or a threaded adjustment ring can be used to set the axial position during assembly. The shaft angle and intersection errors are more difficult to adjust, so they must be controlled by the housing machining tolerances and by the bearing mounting design.
Comparison with conventional face gear design. Conventional face gear design often assumes a relatively large transmission ratio and a contact path that runs from the inner radius to the outer radius in a natural way. In such cases, the face gear tooth is wide enough that the contact path can be spread over a large area without severe eccentricity. In my small-ratio case, the pinion has almost as many teeth as the face gear, and the face gear tooth is short in the radial direction. The pitch radius is close to the inner radius, so the contact path naturally prefers the inner portion of the tooth. This is why the usual tooth number difference modification is not effective. By modifying the pinion instead, I was able to rotate and shift the contact path without changing the face gear cutting geometry. This approach is particularly useful when the face gear is large and difficult to modify after heat treatment. The pinion is smaller and easier to grind, so the modification can be applied with high accuracy.
Numerical verification of strength. The finite element model contained five teeth in contact. I used a fine mesh near the contact zone and a coarser mesh away from the contact zone to reduce computation time. The contact pressure was extracted from the surface interaction, and the root bending stress was extracted from the maximum principal stress along the root fillet. The maximum contact stress was 970 MPa on the pinion and 975 MPa on the face gear. The maximum root bending stress was 371 MPa on the pinion and 212 MPa on the face gear. These values are below the allowable contact and bending fatigue limits of the selected carburized and hardened gear steel. The contact area fraction of 86.5 percent indicates that the load is well distributed. The contact ratio of about 1.83 means that there is usually more than one tooth pair in contact, which helps to smooth the transmission. The transmission error amplitude was on the order of a few tens of microradians, which is acceptable for many aerospace transmission applications, although a lower value would be desirable for noise-sensitive installations.
Sensitivity table for installation errors. To summarize the installation error behavior, I constructed the following qualitative sensitivity table. It lists the direction in which the contact trace moves when each error is increased in the positive direction, and the effect on the transmission error amplitude.
| Error component | Positive increase effect on contact trace | Effect on transmission error amplitude | Relative sensitivity |
|---|---|---|---|
| Axial offset \(\Delta q\) | Moves toward outer radius | Positive \(\Delta q\) increases amplitude; negative \(\Delta q\) reduces amplitude | High |
| Shaft angle \(\Delta \gamma\) | Moves toward outer radius | Positive \(\Delta \gamma\) can improve transmission error in some ranges | Highest |
| Intersection \(\Delta E\) | Moves toward inner radius | Positive \(\Delta E\) can reduce amplitude | Lowest |
Optimization results and final design recommendation. After optimizing the lead modification lengths, I obtained a face gear pair with a continuous transmission error curve, a controlled contact path, and acceptable stresses. The final recommended design parameters are the module of 3.9 mm, pinion tooth number 40, face gear tooth number 51, inner radius 97.5 mm, outer radius 124.5 mm, pressure angle 25°, shaper tooth number 40, profile modification amounts \(a_1 = 3\ \mu\mathrm{m}\) and \(a_2 = 10\ \mu\mathrm{m}\), profile modification lengths \(h_1 = 3.9\ \mathrm{mm}\) and \(h_2 = 4.875\ \mathrm{mm}\), lead modification amounts \(c_1 = 20\ \mu\mathrm{m}\) and \(c_2 = 3\ \mu\mathrm{m}\), and optimized lead modification lengths \(b_1 = 12.4\ \mathrm{mm}\) and \(b_2 = 15.6\ \mathrm{mm}\). The face gear tooth tip should be chamfered with a radius of about 0.3 mm, and the pinion tooth tip should be chamfered with a radius of about 0.5 mm. The assembly should allow axial adjustment of the pinion, and the shaft angle error should be controlled as tightly as possible.
Concluding observations. My study shows that a high-power, small transmission ratio spur face gear pair can be designed within a tight space envelope if the tooth flank is properly modified. The key is to recognize that the conventional tooth number difference modification is not suitable when the face gear pitch radius is close to the inner radius and far from the outer radius. Instead, the cylindrical pinion should be modified in both the profile and lead directions. The profile modification rotates the contact trace and prevents edge contact. The lead modification centers the contact trace and reduces the eccentric load toward the inner radius. The optimized lead modification lengths smooth the transmission error and improve the alternation between adjacent tooth pairs. The installation error analysis shows that the shaft angle error is the most sensitive parameter, followed by the axial offset error and then the intersection error. Negative axial offset and positive intersection error can reduce the transmission error amplitude, but they must be used with care because they also shift the contact trace. Combined errors can add or cancel, so a tolerance stack-up analysis should be performed for the combined error state. The final design has a contact area fraction of about 86.5 percent, a contact ratio of about 1.83, a maximum contact stress below 975 MPa, and a maximum root bending stress below 371 MPa. These results indicate that the face gear pair can carry the required torque while maintaining a safe stress level and a stable contact pattern. The methodology I have described can be applied to other face gear pairs with small transmission ratios, high power density, and limited installation space, and it provides a practical route from preliminary sizing to tooth flank modification and installation error assessment.
For future work, I would extend the model to include thermal effects, lubrication, and dynamic load sharing among multiple face gear branches. I would also study the effect of manufacturing deviations in the pinion modification curve, because the parabolic coefficients are small and the contact pattern is sensitive to their exact values. A probabilistic tolerance analysis could combine the installation errors with the manufacturing errors and estimate the probability that the contact trace remains inside the safe band. Finally, I would validate the numerical predictions with a back-to-back face gear test rig, measuring the transmission error and the contact pattern under load. Such validation would strengthen the design method and support the use of high-power small-ratio face gears in compact aerospace transmissions.