Envelope-Limited Architecture
A steering-gear actuator sets the flight path and attitude of a spacecraft, and its speed-reduction stage sits between a high-speed servo motor and a control surface that demands torque at low speed. Because the actuator is packaged inside the vehicle, the outer dimensions of the reducer are fixed before the gear design begins; the engineering problem is therefore not to size a transmission freely but to extract the highest torque capacity, stiffness and accuracy from a volume that cannot grow.
Two transmission principles meet that constraint particularly well in combination. A spiral bevel stage transfers drive through 90° with a high contact ratio and quiet running, while an NGW-type 2K-H negative planetary train (sun gear as input, ring gear fixed, carrier as output) divides the torque across several planet meshes and keeps the radial package small. In the reducer examined here the bevel pair has a ratio of 45/21 ≈ 2.14 and the planetary stage 1 + 88/24 ≈ 4.67, giving an overall reduction of about 10:1; with a 6000 r/min motor the sun gear turns at 2800 r/min and the carrier at roughly 600 r/min. Input shaft and motor are coupled directly, and the output shaft is the planet carrier itself, which avoids an additional bearing-supported output member.
Gear Set and Material Selection
The bevel pair carries the higher speed and the planetary stage the higher torque, so both are made from 20CrMnTi, a case-hardening gear steel chosen for core toughness combined with a hard, wear-resistant carburized surface after quenching and tempering. The five gears of the train are defined as follows.
| Gear | Teeth | Module (mm) | Spiral angle / hand | Function |
|---|---|---|---|---|
| 1 | 21 | 1 | 35° left | Bevel pinion, motor side |
| 2 | 45 | 1 | 35° right | Bevel wheel |
| 3 | 24 | 0.6 | 0 (straight) | Sun gear, four planet meshes |
| 4 | 32 | 0.6 | 0 (straight) | Planet, four units |
| 5 | 88 | 0.6 | 0 (straight) | Internal ring gear, fixed |
The duty cycle is short and intermittent, so tooth-surface pitting is not treated as the governing failure mode; the verification concentrates on root bending. Within the planetary stage the load is shared by four planets, which lowers the tangential force on each mesh and makes the sun gear the critical member of that stage.
Analytical Root Bending Checks
The conventional check follows the standard root-stress formulation. The nominal stress at the root is built from the tangential load, face width b and module m, modified by form and stress-concentration factors and by the load-sharing factor:
$$\sigma_{F0}=\frac{F_t}{b\,m}\,Y_{Fa}Y_{Sa}Y_{\varepsilon}$$
$$\sigma_F=\sigma_{F0}K_AK_VK_{F\beta}K_{F\alpha}$$
$$\sigma_{FG}=\sigma_{Flim}Y_{ST}Y_{NT}Y_{\delta relT}Y_{RrelT}Y_X \qquad S_F=\frac{\sigma_{FG}}{\sigma_F}$$
For the sun gear the torque is obtained from the carrier torque through the planetary ratio and efficiency, i3H = 4.67 and η = 0.9, and the tangential force at one mesh divides the sun torque among the four planets, with d3 = 14.4 mm:
$$T_3=\frac{T_H}{i_{3H}\,\eta}, \qquad F_{t21}=\frac{2T_3}{4\,d_3}$$
| Quantity | Rated operation | Emergency stop |
|---|---|---|
| Output torque TH (N·m) | 50 | 150 |
| Sun torque T3 (N·m) | 11.9 | 35.7 |
| Tangential load Ft (N) | 413.19 | 1239.57 |
| Root bending stress σF (MPa) | 170 | 510 |
| Bending limit stress σFG (MPa) | 2029.46 | 2029.46 |
| Safety factor SF | 11.94 | 3.98 |
Against a minimum acceptable factor of 2, the sun gear passes in both load cases; the emergency stop, at three times rated torque, still retains a factor of nearly 4 on this nominal basis. The limit stress of about 2029 MPa is not the yield strength of the steel — it is the fatigue bending strength raised by life, notch-relative, reliability and size factors, and it is valid only for the specified duty and reliability target.
The bevel pair is checked with a comparable formulation in which the tangential load follows from the torque and pitch diameter, Ft = 2000T1/d1 = 2000T2/d2, and the bending factors include the bevel geometry and load-distribution terms. With a minimum required factor of 2, the permissible working stress is σ′Flim = 399.5 MPa. The computed root stress is 105.37 MPa at rated load and 316.12 MPa at the emergency stop, both below the permissible value, so the bevel pair also satisfies the root bending requirement.
Contact Finite Element Model of the Planet Mesh
The analytical check gives a single nominal number per tooth and says nothing about the stress gradient at the fillet, the真实 contact patch, or the compliance of the surrounding structure. A transient dynamic contact analysis of the planetary meshes addresses these points. Chamfers and plain shaft sections are removed to keep the model tractable, the sun gear tooth flank is defined as the contact surface and the planet flank as the target, and the contact is frictional with a coefficient of 0.05. The mesh is refined in the contact and fillet regions, since element size there controls the peak stress more strongly than anywhere else in the model.
Load steps reproduce the kinematics and the torque path: the sun gear rotates at 2800 r/min (the bevel stage transforms the 6000 r/min motor speed by 21/45), and the resisting torque applied to the planet is scaled from the sun torque by the tooth ratio:
$$T_4=T_3\frac{z_4}{z_3}=11.9\times\frac{32}{24}=15.9\ \text{N·m}\quad(\text{rated}),\qquad 47.7\ \text{N·m}\quad(\text{emergency stop})$$
Interpreting the Stress Results
The peak von Mises equivalent stress at the sun gear root is 313.56 MPa at rated load and 675.18 MPa at the emergency stop. Against a yield strength of 850 MPa for the case-hardened gear steel, these correspond to margins of about 2.7 and 1.26 respectively, so the elastic response remains admissible in both cases.
| Load case | Nominal beam-theory stress (MPa) | FE von Mises stress (MPa) | Ratio |
|---|---|---|---|
| Rated | 170 | 313.56 | 1.84 |
| Emergency stop | 510 | 675.18 | 1.32 |
The two numbers are not the same quantity and should not be read as a validation of one against the other. The conventional formula treats the tooth as a cantilever beam, applies a nominal bending stress at a simplified section, and folds local effects into empirical factors. The finite element result is a local equivalent stress derived from the three principal stresses through the distortion-energy criterion:
$$\sqrt{\tfrac{1}{2}\left[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2\right]}\le\sigma_s$$
It therefore includes the stress concentration at the root fillet, the real contact pressure distribution and the compliance of the tooth and rim, none of which appear explicitly in the beam model. Beyond the difference in definition, the ratio itself changes between load cases because the contact path, the number of simultaneously loaded tooth pairs and the friction contribution do not scale linearly with torque, whereas the nominal formula does. A ratio near 1.8 at rated load and 1.3 at the emergency stop is thus expected behaviour rather than an inconsistency; what the comparison shows is that the handbook method is a conservative sizing tool whose absolute values are not transferable to a local stress criterion. Where a validated model and solution settings are available, finite element results track measured and analytical reference solutions more closely, which is the practical argument for using them at the final verification stage rather than for preliminary sizing alone.
Carrier and Housing Verification
The carrier deforms under load, and any tilt of the planet pins redistributes load along the face width, so the carrier is analysed as a complete solid including the four pins. Each pin carries FH = TH/(4a34), with the centre distance a34 = 16.8 mm set by the 0.6 mm module and the 24/32 tooth counts, giving 744.05 N at rated load and 2232.14 N at the emergency stop.
| Component | Load case | Max equivalent stress (MPa) | Yield strength (MPa) | Margin |
|---|---|---|---|---|
| Sun gear 3 root | Rated | 313.56 | 850 | 2.71 |
| Sun gear 3 root | Emergency stop | 675.18 | 850 | 1.26 |
| Carrier with pins | Rated | 84.3 | 785 | 9.3 |
| Carrier with pins | Emergency stop | 252.9 | 785 | 3.10 |
| Housing | 20 kN axial load | 332.03 | 455 (7075 aluminium) | 1.37 |
The housing is a 7075 aluminium casting and is checked for two separate requirements. Under a 20 kN axial load its peak equivalent stress of 332.03 MPa stays below the 455 MPa yield strength. Axial compliance matters equally, because deflection of the housing displaces the output shaft and degrades transmission accuracy: the computed maximum axial deformation is 0.00927 mm, which corresponds to an axial stiffness of 2.16 × 109 N/m against a design requirement of 1 × 108 N/m, a margin of more than twenty times.
What the Two Methods Contribute
The case illustrates a division of labour rather than a competition. Hand calculations with standard factors are fast, transparent and conservative, and they are the right tool for establishing tooth counts, module and face width under a fixed envelope; their weakness is that the range of recommended factor values leaves the designer with a wide band of permissible dimensions and no information about local stress or structural stiffness. Finite element analysis closes that gap: it returns the fillet stress, the contact pressure distribution, the carrier deflection and the housing stiffness, and it allows a part to be judged against yield and stiffness limits directly rather than through empirical knock-down factors.
The limits of the present verification should be stated as clearly as its results. The contact model is linear elastic, uses a single friction coefficient, and omits manufacturing misalignment, thermal distortion, lubricant film behaviour and the effect of surface treatment on local strength; the comparison between nominal and equivalent stress is a comparison of criteria, not an experimental validation. The thinnest margin in the assembly, 1.26 on the sun gear root during the emergency stop, is a reminder that the emergency case governs the sizing of the planetary stage even though the rated case dominates the fatigue assessment. Within those qualifications, the analytical checks and the finite element results agree that the bevel stage, the planetary stage, the carrier and the housing all meet their strength and stiffness targets, and the combination of the two methods gives a more reliable basis for fixing dimensions in an envelope that cannot be enlarged.