Two compatible external spur gears transmit rotary motion at a tooth-count ratio and rotate in opposite directions. This calculator models one ideal lossless mesh; it does not establish tooth compatibility, load capacity or durability.
Transmission of Rotary Motion Interactive Calculator
Enter input speed, driver and driven tooth counts, and input torque. Results assume one lossless external gear mesh with compatible tooth geometry.
Equation Used
- Exactly two external spur gears mesh on fixed parallel axes.
- Both gears have compatible module or diametral pitch, pressure angle and tooth form.
- Tooth counts are positive integers over the supplied ranges.
- The mesh is rigid and lossless for the displayed speed and torque magnitudes.
- Backlash, interference, undercut, profile shift, contact ratio, efficiency, dynamics, lubrication, strength and safe design are omitted.
One External Spur-Gear Mesh
For compatible gears, tooth size fixes pitch diameter in proportion to tooth count. At the mesh, equal pitch-line velocity gives an angular-speed magnitude ratio inverse to tooth count. Two externally meshing gears rotate in opposite directions.
The orange contact marker in the animation is illustrative only; the model does not resolve individual tooth contact.
What This Calculator Does
The tool reports driven/driver tooth ratio, ideal output-speed magnitude, ideal output-torque magnitude and the signed output/input speed ratio. A negative sign indicates reversal across the external mesh.
It does not select gear module, pressure angle, face width, material, centre distance or lubrication.
Ideal External-Gear Relations
|Nout| = Nin/i
|Tout| = Tini
Nout/Nin = -Zin/Zout
The torque relation follows ideal lossless power balance and reports magnitude only.
Worked Ideal Example
With 12 driver teeth, 48 driven teeth, 1465 rpm input and 1 N*m input torque, i=4.00. The ideal output-speed magnitude is about 366 rpm, the ideal output-torque magnitude is 4.00 N*m and the signed speed ratio is -0.250, showing opposite rotation.
Model Boundary
- Mechanical arrangement: compatible external spur gears have pitch diameter proportional to tooth count and reverse rotation across one mesh.
- Calculated: ideal tooth ratio, speed magnitude, torque magnitude and direction sign.
- Not calculated: module or diametral pitch selection, pressure angle, involute geometry, centre distance, backlash, clearance, profile shift, undercut, interference, contact ratio, mesh efficiency, sliding, dynamic factor, noise, vibration, lubrication, temperature, tooth bending, pitting, scuffing, wear, face load, shaft or bearing load, fatigue, material, manufacturing tolerance, guarding or safe design.
Frequently Asked Questions
Why is the signed speed ratio negative?
A single external gear mesh reverses rotation.
Can the driven gear turn faster?
Yes. If it has fewer teeth than the driver, its speed magnitude is higher.
Does torque always increase when speed decreases?
Only ideally by the inverse speed ratio. Real output torque is lower because of losses.
Will any two tooth counts mesh?
No. Tooth size, pressure angle, tooth form and geometry must also be compatible.
Does the calculator check undercut or interference?
No.
Can the result rate a gearbox?
No. Load capacity and durability require a complete gear design.
References
- NASA, Spur-Gear System Efficiency at Part and Full Load: external spur-gear geometry, pitch-circle relationships, diametral pitch P=N/D, centre distance and backlash discussion.
- NASA/TM-2018-219929, Design and Test of a Two-Speed Transmission for Vertical Lift Applications: external-gear train examples and explicit opposite-direction rotation without reversing idlers.
- NASA, Flywheel Energy Storage System for an Electric Vehicle: fixed-ratio spur-gear reducer example with pinion/gear tooth counts, pitch diameters and centre distance.
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