BIZOMAT MAKEOS · Gears
Planetary Gear Set Design (Ratio, Assembly, Forces and Bending)
Does this planetary gear set assemble with equally spaced planets, what ratio and torque does it give, and are the sun teeth within their bending allowable?
Design check of a simple planetary gear set: planet teeth, ratio, output speed and torque for the chosen fixed member (as the planetary gear tool), the equal-spacing assembly condition and the planet adjacency (tip clearance) check, pitch diameters and centre distance, sun torque and the tangential force per planet with a load-sharing factor, the meshing speed relative to the carrier, the planet pin load and the Lewis bending stress on the sun teeth.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Sun teeth | Teeth on the sun gear. | |
| Ring teeth | Teeth on the internal ring gear. | |
| Number of planets | Number of planet gears. | |
| Arrangement | choice | Which member drives, which is held and which is the output. |
| Input speed | rpm | Speed of the input member. |
| Input torque | N·m | Torque on the input member. |
| Efficiency | % | Efficiency of the stage. |
| Module | mm | Module of all three gears. |
| Face width | mm | Face width of the sun mesh. |
| Load-sharing factor | Factor for uneven load sharing between planets (1 for a floating sun and perfect sharing; higher for rigid mounting). | |
| Lewis form factor (sun) | Lewis form factor for the sun teeth. | |
| Tooth finish | choice | Manufacturing quality, which sets the velocity factor. |
| Allowable bending stress | MPa | Allowable bending stress of the sun material (your design value). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Sun teeth
- What it is
- The number of teeth on the sun gear, the central external gear the planets mesh around.
- Units and format
- Enter a plain number with no unit. The form accepts 9 to 300. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 24 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ring minus sun teeth must be a positive even number so the planets are standard gears.
What this means: Ring teeth
- What it is
- The number of teeth on the ring gear, the internal gear that surrounds the planets. The planet tooth count follows as (ring − sun) ÷ 2.
- Units and format
- Enter a plain number with no unit. The form accepts 30 to 1000. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 72 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Count the internal teeth of the ring; (sun + ring) ÷ planets must be whole for equal spacing.
What this means: Number of planets
- What it is
- How many planet gears share the load between the sun and the ring, mounted on the carrier at equal spacing.
- Units and format
- Enter a plain number with no unit. The form accepts 2 to 12. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The tangential force on each planet falls in proportion to the number of planets.
- Get it right
- The tool checks that this count can be equally spaced and that neighbouring planets clear each other.
What this means: Arrangement
- What it is
- Which member is driven, which is held still and which delivers the output. It decides the ratio, the direction of the output and the sun torque.
- Units and format
- Choose one of 3 options from the list: Sun in, ring fixed, carrier out (reduction); Ring in, sun fixed, carrier out; Sun in, carrier fixed, ring out (reverses).
- Example
- The form opens with “Sun in, ring fixed, carrier out (reduction)” selected as a worked example. Choose the option that matches your case.
What this means: Input speed
- What it is
- The speed of the member that drives the set (sun or ring, depending on the arrangement).
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0.000001 rpm to 100000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1500 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher speed raises the mesh speed and the velocity factor; tooth forces come from torque, not speed.
What this means: Input torque
- What it is
- The torque applied to the input member of the set. In the ring-input arrangement it is the torque on the ring; otherwise it is the torque on the sun.
- Units and format
- Enter a number in N·m: newton-metres. The form accepts 0.000001 N·m to 10000000 N·m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 N·m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sun torque, tooth forces, pin load and bending stress rise in proportion.
- Get it right
- Use the torque at this stage’s input, after any stage before it.
What this means: Efficiency
- What it is
- The share of input power the stage delivers at its output after mesh and bearing losses, from the gearbox maker or your estimate.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 50% to 100%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 97% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only the output torque uses it; the tooth forces are worked from the input torque.
What this means: Module
- What it is
- The tooth size shared by the sun, planets and ring: pitch diameter divided by the number of teeth.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.5 mm to 50 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger module enlarges the sun and lowers the tooth force and bending stress.
- Get it right
- All three gears must have the same module and pressure angle to mesh.
What this means: Face width
- What it is
- The axial width over which the sun teeth mesh with the planets. It carries the tooth load in the bending check.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sun bending stress falls in proportion to face width.
- Get it right
- Use the width actually in mesh, the narrower of sun and planet.
What this means: Load-sharing factor
- What it is
- A multiplier on the tooth force that allows for the planets not sharing the load equally, set by how the sun and planets are mounted and how accurately they are made.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 2. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.15 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Tooth force, pin load and bending stress rise in proportion.
What this means: Lewis form factor (sun)
- What it is
- A dimensionless factor for the bending strength of the sun’s tooth shape, read from a Lewis form factor table for its tooth count and pressure angle.
- Units and format
- Enter a plain number with no unit. The form accepts 0.1 to 0.6. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sun bending stress is inversely proportional to the form factor.
What this means: Tooth finish
- What it is
- How the teeth were made or finished, which sets the velocity factor that allows for dynamic loading from tooth errors at speed.
- Units and format
- Choose one of 4 options from the list: Cast teeth; Cut teeth; Hobbed or shaped teeth; Shaved or ground teeth.
- Example
- The form opens with “Hobbed or shaped teeth” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- A finer finish gives a smaller velocity factor and a lower bending stress.
- Get it right
- Choose the finish the gears will actually have.
What this means: Allowable bending stress
- What it is
- The bending stress you allow at the root of the sun teeth, from material data or the gear standard your design follows, with your margin included.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 2000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 150 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use a bending allowable, not a surface contact value; the tool checks bending only.
Method and formulas
- Planet teeth z_p = (z_r − z_s) ÷ 2; ratio 1 + z_r ÷ z_s (ring fixed), 1 + z_s ÷ z_r (sun fixed), −z_r ÷ z_s (carrier fixed) (as the planetary gear tool).
- Equal spacing needs (z_s + z_r) ÷ n_p to be a whole number; neighbouring planets clear if (z_s + z_p) sin(π ÷ n_p) > z_p + 2.
- Sun torque from the torque split T_s : T_r : T_c = 1 : z_r ÷ z_s : (1 + z_r ÷ z_s); tangential force per planet F_t = 2T_s ÷ (d_s n_p) × K_s; planet pin load ≈ 2F_t.
- Lewis σ = K_v F_t ÷ (b m Y) with the velocity factor from the pitch-line speed relative to the carrier.
- Example: 24/72 teeth with 3 planets give 4:1; each planet carries 798.61 N and the sun bending stress is 97.985 MPa.
Limitations
- Lewis method for a first sizing; full ratings use ISO 6336 with the planetary load factors.
- Standard (unshifted) gears with planets of equal size.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Lay out a planetary stage for a gearmotor or winch
- Check tooth counts for equal planet spacing
- Find the planet bearing load
Who it is for
- Design engineers
- Maintenance engineers
- Power transmission engineers
What you get
- Planet teeth, ratio, speeds and torque
- Assembly and adjacency checks
- Tooth force per planet, pin load and sun bending stress
Interpreting the result
- More planets cut the load per tooth but need better load sharing (a floating sun or flexible pins).
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2211, 2122–2160. Topic reference; no source tables or recommended settings are reproduced.
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Open the Planetary Gear Set Design (Ratio, Assembly, Forces and Bending) calculator