BIZOMAT MAKEOS · Gears
Planetary and Epicyclic Gearing Ratios for the 24 Handbook Arrangements
For whichever of the handbook’s 24 planetary, epicyclic and differential gear arrangements matches mine, how many turns does the follower make for one turn of the driver, and what speed does the follower run at?
Speed ratios of planetary or epicyclic gearing by the formula the handbook prints with each of its 24 arrangements (Figs. 1 to 24): A the driving gear, B the driven gear or follower, C the fixed gear, x and y the planet gears, z the secondary driving gear and S the turns of the secondary driver per turn of the initial driver (negative when they turn in opposite directions), giving F, the turns of the follower per turn of the driver, or D, the turns of the driver per turn of the follower, with F = 1 ÷ D, and the follower speed for your driver speed.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Arrangement (handbook figure) | choice | The figure in the handbook’s table whose gears, arm and fixed member match your train. |
| Driving gear A (teeth or pitch diameter) | Size of the driving gear, as teeth or pitch diameter. | |
| Driven gear or follower B | Size of the driven gear or follower. | |
| Fixed gear C | Size of the fixed gear. | |
| Planet gear x | Size of the planet gear marked x in the figure. | |
| Planet gear y | Size of the planet gear marked y in the figure. | |
| Secondary driving gear z | Size of the secondary driving gear when the follower is driven from two members. | |
| Turns of the secondary driver per turn of the initial driver S | Turns of the secondary driver for one turn of the initial driver; negative if they turn opposite ways. | |
| Angle E of the inclined planet (Fig. 4) | ° | Angle of the inclined planet axis in Fig. 4. |
| Speed of the driver | rpm | Speed of the driving member. |
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: Arrangement (handbook figure)
- What it is
- The figure in the handbook planetary gearing table that matches your gear train.
- Units and format
- Choose one of 24 options from the list: Fig. 1 (F = 1 + C/B); Fig. 2 (F = 1 − C/B); Fig. 3 (F = C/B); Fig. 4 (F = cos E + C/B); Fig. 5 (F = 1 + xC/yB); Fig. 6 (F = 1 + yC/xB); Fig. 7 (D = 1 + xC/yA); Fig. 8 (D = 1 + yC/xA); Fig. 9 (F = 1 + C/B); Fig. 10 (D = 1 + C/A); Fig. 11 (D = 1 + C/A); Fig. 12 (F = 1 + C/B); Fig. 13 (F = 1 − Cx/yB); Fig. 14, with idler (D = (1 + C/A) ÷ (1 − Cx/yB)); Fig. 15 (D = (1 + C/A) ÷ (1 − Cy/xB)); Fig. 16 (D = 1 − Cx/yA); Fig. 17 (F = 1 − Cx/yB); Fig. 18 (D = 1 − Cx/yA); Fig. 19 (F = 1 + z(1 − S)/B); Fig. 20 (D = (A + z) ÷ (A + Sz)); Fig. 21 (D = (A + z) ÷ (A + Sz)); Fig. 22 (F = 1 + z(1 − S)/B); Fig. 23 (D = 1 + C/A); Fig. 24, bevel (D = (1 + C/A) ÷ (1 − Cy/xB)).
- Example
- The form opens with “Fig. 10 (D = 1 + C/A)” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Chooses the formula used for the ratio.
- Get it right
- Match which member drives, which follows and which gear is held still before choosing.
What this means: Driving gear A (teeth or pitch diameter)
- What it is
- Size of the driving gear, as a tooth count or a pitch diameter.
- Units and format
- Enter a plain number with no unit. The form accepts 0.000001 to 100000. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the ratio through the formula of the chosen figure.
- Get it right
- Use the same kind of size, teeth or pitch diameter, for every gear.
What this means: Driven gear or follower B
- What it is
- Size of the driven gear that is the follower of the train.
- Units and format
- Enter a plain number with no unit. The form accepts 0.000001 to 100000. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the ratio through the formula of the chosen figure.
- Get it right
- Use the same kind of size, teeth or pitch diameter, for every gear.
What this means: Fixed gear C
- What it is
- Size of the gear that is held still in the arrangement.
- Units and format
- Enter a plain number with no unit. The form accepts 0.000001 to 100000. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 80 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the ratio through the formula of the chosen figure.
- Get it right
- Use the same kind of size, teeth or pitch diameter, for every gear.
What this means: Planet gear x
- What it is
- Size of the planet gear marked x in the handbook figure.
- Units and format
- Enter a plain number with no unit. The form accepts 0.000001 to 100000. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the ratio through the formula of the chosen figure.
- Get it right
- Only used in the compound planet figures.
What this means: Planet gear y
- What it is
- Size of the planet gear marked y in the handbook figure.
- Units and format
- Enter a plain number with no unit. The form accepts 0.000001 to 100000. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the ratio through the formula of the chosen figure.
- Get it right
- Only used in the compound planet figures.
What this means: Secondary driving gear z
- What it is
- Size of the secondary driving gear when the follower has two drivers.
- Units and format
- Enter a plain number with no unit. The form accepts 0.000001 to 100000. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 60 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the ratio through the formula of the chosen figure.
- Get it right
- Only used in Figs. 19 to 22.
What this means: Turns of the secondary driver per turn of the i…
- What it is
- Turns of the secondary driver for each turn of the initial driver.
- Units and format
- Enter a plain number with no unit. The form accepts -1000 to 1000. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.5 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the ratio of the differential arrangements.
- Get it right
- Enter it as negative when the two drivers turn in opposite directions.
What this means: Angle E of the inclined planet (Fig. 4)
- What it is
- Angle between the inclined planet axis and the main axis in Fig. 4.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0° to 180°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 90° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Adds cos E to the ratio of Fig. 4.
- Get it right
- Only used in Fig. 4; 90° makes cos E zero.
What this means: Speed of the driver
- What it is
- Rotational speed of the member that drives the train.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0 rpm to 1000000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1000 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The follower speed is this times F.
- Get it right
- Use the motor or input shaft speed.
Method and formulas
- Choose the figure whose arrangement matches yours; its formula gives F or D.
- Simple trains: F = 1 ± C/B, C/B or cos E + C/B (Figs. 1–4, 9, 12); D = 1 + C/A (Figs. 10, 11, 23).
- Compound planets: F = 1 + xC/yB or 1 + yC/xB (Figs. 5, 6), D = 1 + xC/yA or 1 + yC/xA (Figs. 7, 8), F = 1 − Cx/yB (Figs. 13, 17), D = 1 − Cx/yA (Figs. 16, 18), D = (1 + C/A) ÷ (1 − Cx/yB) or (1 + C/A) ÷ (1 − Cy/xB) (Figs. 14, 15, 24).
- Secondary driver: F = 1 + z(1 − S)/B (Figs. 19, 22); D = (A + z) ÷ (A + S z) (Figs. 20, 21).
- F = 1 ÷ D; follower speed = F × driver speed; a negative value turns the follower the opposite way to the driver.
- Example: Fig. 10 gives D = 5, so the follower turns at 200 rpm.
Limitations
- The member called the driver, the follower and the fixed gear must match the figure; the same gears give different ratios with a different member held.
- In Figs. 1 to 4, F is the rotation of the planet follower about its own axis.
- Gear sizes may be teeth or pitch diameters, but use the same kind for all of them.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the ratio of a planetary or epicyclic gear train from its arrangement
- Check the speed of the follower of a compound planetary train
- Work out a differential with a secondary driver
Who it is for
- Machine designers
- Gearbox designers
- Maintenance engineers
What you get
- Follower turns per driver turn F
- Driver turns per follower turn D
- Follower speed for the driver speed
- Direction of the follower
Interpreting the result
- A ratio near zero means the follower barely moves, typical of trains with nearly equal gear sizes, which give large reductions.
- A negative ratio means the follower turns the opposite way to the driver.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2212–2214. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Useful alongside · BIZOMAT original book
Ratios, speeds and gear trains
What it covers: Calculate gear ratio, output speed and torque for simple and compound trains, and use tooth numbers to spread wear with a hunting tooth.
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Open the Planetary and Epicyclic Gearing Ratios for the 24 Handbook Arrangements calculator