A compound gear train transmits rotation through successive gear pairs. Here, the first driven gear and second driving gear share an intermediate shaft, so the two stage ratios multiply. Change the four tooth counts and input speed to compare the ideal output speed and rotation direction.
Gear Train Interactive Calculator
Calculate ideal speed ratios for a two-stage compound external gear train in which the stage-one driven gear and stage-two driver are rigidly fixed to the same intermediate shaft.
Equation Used
- N2 and N3 are rigidly fixed to one intermediate shaft and therefore have equal angular speed.
- Both stages use external gears with compatible geometry within their own mesh.
- Ideal rigid teeth with no slip or backlash.
- No module, center-distance, interference, load, efficiency, strength, lubrication or life model.
Ideal no-slip speed kinematics only. Tooth counts define ratios, but do not by themselves establish compatible module, center distance, tooth form, interference clearance, face width, load capacity, efficiency, backlash, lubrication or life.
How the two stages connect
Input gear N1 drives N2. Gear N2 is rigidly fixed to N3 on the intermediate shaft, so N3 shares N2's angular speed and drives output gear N4. Each external mesh reverses direction; after two reversals the output rotates in the same direction as the input.
The animation separates the two meshes into axial layers so both remain visible. Within each mesh, pitch radius is shown proportional to tooth count.
Ratio exploration
The calculator is useful for ideal speed-ratio exploration before detailed design. A buildable gearbox additionally requires compatible modules and pressure angles, center distances, interference and contact checks, materials, face widths, shafts, bearings, housing, lubrication, load spectrum and thermal capacity.
Ideal compound ratio
For the first external mesh, speed reduction magnitude is N2/N1. For the second it is N4/N3. Because N2 and N3 share a shaft, the overall magnitude is their product: i=(N2/N1)(N4/N3). Ideal output speed is input speed divided by i.
KHK's technical references relate pitch dimensions to module and tooth count. This calculator does not select a module, so its diagram is a ratio schematic rather than a manufacturing drawing.
Example: two 4:1 stages
With N1 = 16, N2 = 64, N3 = 18 and N4 = 72, the first stage ratio is 64/16 = 4 and the second is 72/18 = 4. Overall ratio is 16:1. An input speed of 1440 rpm therefore gives an ideal output speed of 90 rpm.
The intermediate shaft turns at 360 rpm opposite to the input. The second mesh reverses direction again, leaving the output turning in the input direction.
The controls can also describe a speed increase. For N1 = 40, N2 = 20, N3 = 50 and N4 = 20, the ratios are 0.5 and 0.4, giving 0.2 overall. At 1440 rpm input, ideal output is 7200 rpm. This arithmetic does not establish that the selected gears or bearings can operate at that speed.
Scope and limitations
The output is speed only. Torque cannot be claimed without input torque and efficiency, and capacity cannot be claimed without detailed tooth, shaft, bearing and housing design. Extreme tooth-count combinations may be impossible or impractical with a chosen module, center distance or tooth form.
Frequently asked questions
Do N2 and N3 mesh with each other?
No. They are fixed to one intermediate shaft and rotate together. The two rows separate them in the drawing so both gear pairs are visible.
Why does the output turn in the same direction as the input?
Each external gear pair reverses rotation. Two successive reversals restore the input direction.
Can the result be a speed increase?
Yes. An overall ratio below 1 gives output speed greater than input speed. The formula handles both reduction and increase.
Does this calculate torque or a gearbox rating?
No. It calculates ideal speed ratios only. Torque, losses, tooth loading and component strength require further inputs and analysis.
Are the drawn teeth manufacturing profiles?
No. The marks illustrate rotation. Exact tooth profiles, module, clearances and interference are not generated.
Gear-manufacturer technical references
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