Planetary gear design (Willis equation)
Design a simple minus-planetary gear set: the tooth counts of sun and ring gear plus the number of planets, together with the chosen operating mode (which member is held stationary), yield the stationary ratio, the ratio, all three shaft speeds and optionally the output torque via the Willis equation - including a check of the planet assembly and neighbour conditions.
Calculation
24
Design rating is an approximation: tooth root/pitting load capacity and profile shift (ISO 6336) as well as the bearings of carrier and planet pins are not included here.
Stationary ratio and tooth counts
- Stationary ratio i0
- -3
- Planet tooth count z_P
- 24
- Ratio i (operating mode)
- 4
Ratio and output
- Output speed n_out
- 375 1/min
- Output rotation direction
- same as input
Shaft speeds
- Sun speed n_sun
- 1,500 1/min
- Carrier speed n_carrier
- 375 1/min
- Ring gear speed n_ring
- 0 1/min
Sketch: planetary gear set cross-section (sun, planets, ring gear, carrier, schematic)
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Formulas and fundamentals
Layout and Stationary Ratio
A simple planetary gear set consists of the central sun gear (external teeth), p evenly spaced planet gears (external teeth, meshing with both sun and ring gear), the ring gear (internal teeth), and the carrier, which holds the planet axles. The stationary ratio is the sun/ring-gear ratio with the carrier held fixed (fixed-carrier gear train):
The minus sign follows from the ring gear's internal teeth: with the carrier fixed, sun and ring gear rotate in opposite directions. The planet tooth count follows from the centre-distance condition as z_P = (zH - zS)/2 - the difference zH - zS must therefore be even, otherwise no whole-number planet tooth count exists.
Willis Equation
The kinematic coupling of the three shafts (sun, ring gear, carrier) is described by the Willis equation:
Holding one of the three members stationary (speed 0) gives the ratio of that operating mode directly from the Willis equation, and hence all three shaft speeds from a given input speed.
The Three Operating Modes
Ring gear fixed (input at the sun, output at the carrier) - the most common configuration, a large positive reduction:
Sun fixed (input at the ring gear, output at the carrier) - a smaller ratio than with the ring gear fixed:
Carrier fixed (input at the sun, output at the ring gear, fixed-carrier gear train) - the ratio equals the stationary ratio directly, with a reversal of rotation direction:
The output speed follows from the ratio as n_out = n_in/i, and, given an input torque, the output torque follows via the per-stage efficiency eta (guideline value 0.97):
Assembly and Neighbour Conditions
For the p planets to be fitted evenly and without backlash between sun and ring gear, the assembly condition must hold:
In addition, adjacent planets must not overlap at their tip circles (simplified neighbour, i.e. clearance, condition, including tip clearance):
If the assembly condition is violated, the chosen tooth-count set cannot be assembled evenly with this number of planets (choose a different planet count or tooth counts). If the neighbour condition is violated, the planets collide at their tip circles - more planets reduce the space needed per planet, and more teeth on sun or ring gear also help.
Scope and Multi-Stage Trains
This calculator covers only the Willis kinematics. Tooth root and pitting load capacity as well as profile shift of the gearing must be verified with the spur gear calculator per ISO 6336, and the bearings of carrier and planet pins separately. For multi-stage planetary trains, the ratios of the individual stages multiply to the overall ratio:
Worked example
Reference example: A minus-planetary gear set with sun zS = 24, ring gear zH = 72 and p = 3 planets. The planet tooth count is zP = (72-24)/2 = 24, the stationary ratio i0 = -72/24 = -3. The assembly condition (zS+zH)/p = 96/3 = 32 is a whole number and satisfied, the neighbour condition (24+24)·sin(60°) = 41.6 > 24+2 = 26 is also satisfied - both indicators are green.
With the ring gear fixed (input at the sun, output at the carrier) and an input speed of n_in = 1500 rpm, the ratio is i = 1 + zH/zS = 1 + 72/24 = 4, giving an output speed n_out = n_in/i = 1500/4 = 375 rpm at the carrier. At an input torque of M_in = 10 Nm and eta = 0.97, the output torque follows as M_out = 10·4·0.97 = 38.8 Nm.
For comparison of the operating modes with the same tooth counts: with the sun fixed (input at the ring gear, output at the carrier), the ratio is markedly smaller at i = 1 + zS/zH = 1 + 24/72 = 1.33. With the carrier fixed (fixed-carrier gear train, input at the sun, output at the ring gear), i = i0 = -3 - the minus sign indicates the reversal of rotation direction between sun and ring gear.
Frequently asked questions
How do I calculate the ratio of a planetary gear set?
First determine the stationary ratio i0 = -zH/zS from the tooth counts of sun and ring gear. The Willis equation then gives the actual ratio depending on which member is held stationary: i = 1 - i0 with the ring gear fixed, i = 1 - 1/i0 with the sun fixed, or i = i0 with the carrier fixed (fixed-carrier gear train). The output speed then follows from n_out = n_in/i.
What is the stationary ratio of a planetary gear set?
The stationary ratio i0 is the ratio between sun and ring gear with the carrier notionally held fixed (fixed-carrier gear train) - i0 = -zH/zS. It is negative because sun and ring gear rotate in opposite directions with the carrier fixed (external versus internal teeth). All ratios of the three practically used operating modes follow from the stationary ratio via the Willis equation.
Why are three planets usually used?
Three (or more) evenly spaced planets split the transmitted load across several parallel tooth meshes (load sharing) - for the same size, the gear set transmits a multiple of the torque of a simple spur gear pair, and the symmetric load distribution leaves the sun shaft nearly free of radial force. More planets further increase power density, but are limited by the neighbour condition (tip-circle clearance) and the assembly condition.
What is the assembly condition for planetary gear sets?
For p planets to be fitted at equal angular spacing between sun and ring gear, (zS+zH)/p must be a whole number. If it is not, the chosen planet count does not match the tooth counts - either adjust the tooth counts of sun or ring gear, or choose a different planet count. In addition, zH - zS must be even, so that a whole-number planet tooth count zP = (zH-zS)/2 exists at all.
Planetary gear versus spur gear - what is the difference?
A spur gear pair transmits the load through a single tooth mesh and becomes correspondingly large and heavy at high torques. A planetary gear set splits the load across p parallel planets (load sharing), making it markedly more compact and lighter for the same torque, runs coaxially (input and output shafts aligned), and achieves a larger ratio in a single stage. In return, the gearing geometry (internal ring gear, planet bearings in the carrier) is more complex and costlier to manufacture.
How do I calculate multi-stage planetary gear trains?
Each stage is first calculated individually like a simple planetary gear set using the Willis equation. The overall ratio of a multi-stage train is the product of the individual ratios: i_total = i1·i2·…·in. The output of one stage is the input of the next; the overall torque follows analogously from the product of the individual ratios and the product of the efficiencies of all stages.
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