A servo actuator in an aerospace vehicle must convert motor speed into high torque within a fixed installation envelope. The reducer considered here uses a spiral bevel gear stage followed by an NGW-type 2K-H planetary stage. The input shaft is driven by the servo motor, and the planet carrier is the output. The spiral bevel stage transfers motion through a right angle at the input speed, while the planetary stage splits torque among four planets to achieve high torque density, fast response, and high transmission accuracy. Because the external dimensions are fixed by the aircraft installation, the design problem is not simply to enlarge the gears; it is to verify the highly loaded teeth, planet carrier, and housing without introducing excessive mass or axial compliance.
Spiral bevel stage and planetary architecture
The reducer is a compound train. Gears 1 and 2 form the spiral bevel pair; gears 3, 4, and 5 form the planetary stage, with gear 3 as the sun gear, four gear-4 planets, and gear 5 as the ring gear. The spiral bevel pair has a 21-tooth pinion and a 45-tooth gear, module 1 mm, and a 35° helix angle in opposite hands. The planetary stage uses module 0.6 mm spur teeth: 24 teeth on the sun, 32 on each planet, and 88 on the ring. The gears are made of 20CrMnTi, quenched and tempered, then carburized and quenched to obtain high surface hardness and tooth-root strength.
Table 1. Gear parameters of the compound reducer.
| Gear | Element | Teeth | Module (mm) | Helix angle (°) |
|---|---|---|---|---|
| 1 | Spiral bevel pinion | 21 | 1 | 35, left hand |
| 2 | Spiral bevel gear | 45 | 1 | 35, right hand |
| 3 | Sun gear | 24 | 0.6 | 0 |
| 4 | Planet gear, four units | 32 | 0.6 | 0 |
| 5 | Ring gear | 88 | 0.6 | 0 |
Load cases and classical bending checks
Two operating conditions govern the strength assessment: rated operation with an output torque \(T_H=50\ \mathrm{N\cdot m}\), and an emergency-stop condition with \(T_H=150\ \mathrm{N\cdot m}\). For the planetary stage, the sun-gear torque is obtained from the output torque, the planetary ratio \(i_{3H}=4.67\), and a transmission efficiency \(\eta=0.9\). The tangential force on the sun gear is divided among four planets:
$$T_3=\frac{T_H}{i_{3H}\eta},\qquad F_{t21}=\frac{2T_3}{4d_3}$$
where \(d_3=mz_3=0.6\times24=14.4\ \mathrm{mm}\). The resulting loads are summarized in Table 2.
Table 2. Loads on the sun gear in the planetary stage.
| Condition | \(T_H\) (N·m) | \(T_3\) (N·m) | \(F_{t21}\) (N) |
|---|---|---|---|
| Rated | 50 | 11.9 | 413.19 |
| Emergency stop | 150 | 35.7 | 1,239.57 |
The duty is short, so tooth-surface pitting is not treated as the governing failure mode; the hand check focuses on tooth-root bending. Because the four planets share the tangential load, the sun gear 3 is the critical member in the planetary stage. Using standard root-bending factors, the rated bending stress is 170 MPa and the emergency-stop stress is 510 MPa. The corresponding bending limit is 2,029.46 MPa, giving safety factors of 11.94 and 3.98. With a minimum required safety factor of 2, the sun gear satisfies the bending-strength requirement.
For the spiral bevel pair, the rated root bending stress is 105.37 MPa. With a limiting safety factor \(S_{F\lim}=2\), the allowable working stress is 399.5 MPa, so the rated condition is acceptable. Under emergency stop, the calculated root bending stress rises to 316.12 MPa and remains below the allowable basis. The bevel stage therefore also satisfies the root-bending check.
Finite-element contact model of the planetary mesh
A transient dynamic contact model of the meshing planet pair was built in ANSYS Workbench. Chamfers and light shaft sections were removed to reduce model size without changing the load path. The tooth flank of gear 3 was defined as the contact surface and the tooth flank of gear 4 as the target surface. The contact was frictional, with a friction coefficient of 0.05, and the mesh was refined in the contact region to resolve the tooth-root stress gradient.
The input speed is 6,000 r/min. The bevel stage reduces this to the sun-gear speed:
$$\omega_3=\omega_2=\omega_1\frac{z_1}{z_2}=6000\frac{21}{45}=2800\ \mathrm{r/min}$$
The torque applied to the planet in the rated case follows the tooth-number ratio:
$$T_4=T_3\frac{z_4}{z_3}=11.9\frac{32}{24}=15.9\ \mathrm{N\cdot m}$$
For the emergency-stop case, the applied planet torque is 47.7 N·m. The maximum von Mises equivalent stress at the root of gear 3 is 313.56 MPa under rated load and 675.18 MPa under emergency stop. The gear material has a yield strength of 850 MPa, so both cases retain a positive margin against yielding.
Planet carrier and housing verification
The planet carrier transmits torque from four planet pins. Deformation of the carrier can offset the planet axes and produce uneven load distribution across the face width, so the carrier was checked as a three-dimensional static problem. Each pin carries one quarter of the output torque divided by the center distance \(a_{34}\):
$$F_H=\frac{T_H}{4a_{34}}$$
With \(a_{34}=(d_3+d_4)/2=16.8\ \mathrm{mm}\), the pin load is 744.05 N in rated operation and 2,232.14 N in emergency stop. The maximum equivalent stress in the carrier is 84.3 MPa under rated load and 252.9 MPa under emergency stop, both below the carrier material yield strength of 785 MPa.
The housing was checked under a 20 kN axial load. Its maximum equivalent stress is 332.03 MPa, below the 455 MPa yield strength of the 7075 aluminum alloy. The maximum axial deformation at the output shaft is 0.00927 mm, giving an axial stiffness of
$$K=\frac{20\,000\ \mathrm{N}}{9.27\times10^{-6}\ \mathrm{m}}=2.16\times10^{9}\ \mathrm{N/m}$$
This exceeds the required \(1\times10^{8}\ \mathrm{N/m}\), so the housing deformation does not impair normal transmission.
Table 3. Finite-element equivalent-stress checks.
| Component | Load case | Maximum equivalent stress (MPa) | Yield strength (MPa) | Result |
|---|---|---|---|---|
| Sun gear 3 tooth root | Rated | 313.56 | 850 | Pass |
| Sun gear 3 tooth root | Emergency stop | 675.18 | 850 | Pass |
| Planet carrier | Rated | 84.3 | 785 | Pass |
| Planet carrier | Emergency stop | 252.9 | 785 | Pass |
| Housing | 20 kN axial load | 332.03 | 455 | Pass |
Interpretation of stress measures and design implications
The classical tooth-root formula treats the tooth as a cantilever beam and returns a nominal bending stress. The finite-element result is a von Mises equivalent stress obtained from the three principal stresses through the distortion-energy criterion:
$$\sqrt{\frac{1}{2}\left[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2\right]}\le\sigma_s$$
These two quantities are not directly interchangeable. The finite-element stress includes the local tooth geometry, contact conditions, and multi-axial stress state, whereas the hand calculation uses a simplified beam model and empirical factors. The finite-element equivalent stress was 184.45% of the beam-theory bending stress in the rated case and 132.39% in the emergency-stop case. The condition-dependent ratio should not be read as a simple calculation error; it reflects the different stress definitions and the sensitivity of local stress to contact and mesh modeling. For acceptance, the relevant comparison is between each calculated stress and its corresponding allowable or yield value.
The engineering value of the finite-element check is therefore complementary rather than substitutive. The hand calculation provides a transparent first sizing screen for the spiral bevel and planetary teeth, while the contact model reveals the local stress field at the tooth root and the static analyses capture the load paths in the carrier and housing. In a fixed envelope, this combination helps avoid both under-design and unnecessary oversizing when handbook factors span a wide range.
Design outcome and scope
The spiral bevel stage and the planetary stage both satisfy the root-bending requirements. The sun gear 3 has safety factors of 11.94 under rated load and 3.98 under emergency stop, above the required minimum of 2. The bevel stage remains below its allowable bending working stress in both cases. The finite-element checks give maximum equivalent stresses below the respective yield strengths for the sun gear, planet carrier, and housing. The housing axial stiffness of \(2.16\times10^{9}\ \mathrm{N/m}\) exceeds the \(1\times10^{8}\ \mathrm{N/m}\) requirement.
These results establish static and peak-load strength margins for the key parts. They do not by themselves constitute a fatigue-life validation, a tooth-contact durability assessment, or a transmission-error analysis over the complete duty cycle. The omission of pitting in the hand check is consistent with the short-duty assumption, but long-term or repeated actuation should be assessed with a dedicated contact-fatigue and load-spectrum analysis.