A crown wheel presents teeth on the face of a disk to a spur pinion on a perpendicular shaft. The pinion meshes at a selected pitch radius of the face wheel. This calculator shows that historical face-wheel arrangement, with both real shaft directions and the gear ratio tied to the entered tooth counts.
Crown Wheel and Spur Gear Interactive Calculator
A spur pinion engages teeth on the face of a crown wheel. Change the tooth counts to update the actual drawing and ratio, then compare motion, torque and power.
Equation Used
- Rigid perpendicular shafts and matched pitch at the chosen crown radius.
- Simplified teeth; flank generation and interference are not evaluated.
- Uniform input rotation; playback may be slowed.
- User-entered efficiency applied to steady input power.
Simplified face-gear tooth illustration, not manufactured flank geometry or a load rating. Shaft rotation magnitudes follow the actual tooth counts.
Face teeth turn the drive through a right angle
The blue spur pinion turns about the horizontal shaft. Its teeth engage the raised face teeth of the gold crown wheel, whose shaft is perpendicular to the disk. The crown has a flat face; the drawing is not a pair of conical bevel gears.
The pinion’s pitch-cylinder bottom touches the crown pitch plane at its pitch circle. At that point, both pitch velocities are equal. More crown teeth give slower crown motion and a larger ideal torque ratio. The animation draws every entered tooth and uses the same tooth ratio for the moving shafts.
The historical plate is a construction reference. The illustration uses a narrow face band and simplified tooth bodies. Manufacturing a mating pair requires generated flank geometry and interference checks.
Compare tooth counts and right-angle motion
Use the crown and pinion tooth controls to compare ratios while watching both shafts. This page represents the light-duty crown-wheel-and-spur arrangement described in the historical references. It does not assign it an automotive differential load rating or substitute it for a spiral-bevel or hypoid final drive.
Zero input holds the mechanism still. Playback is slowed on one common time scale so the output always follows the selected ratio. The two small end-on views make the separate shaft rotations easier to follow.
Tooth ratio, revolutions and power
Let Zc be crown teeth and Zp pinion teeth. The reduction ratio is i=Zc/Zp. With the pinion as input, crown revolutions and speed are Nout=Nin/i and nout=nin/i. These are magnitudes about different shaft axes.
For a lossless pair, Tout/Tin=i. With an entered efficiency η, Tout=Tin×i×η. Input power is Pin=Tin×2πnin/60, and power loss is Pin(1−η). Efficiency is a user-supplied operating estimate, not a value predicted from tooth count.
The pitch radii satisfy Rc/rp=Zc/Zp. Equal pitch velocity gives ωcRc=ωprp. The illustrated size changes only to make the pair fit the window; this calculator does not specify a production module or tooth profile.
Example: 60 crown teeth and 20 pinion teeth
The ratio is 3:1. Three pinion revolutions give one crown revolution. With 300 input rpm, crown speed is 100 rpm.
For 0.2 N·m input torque and an assumed 85% efficiency, output torque is 0.51 N·m. Input power is about 6.28 W and loss about 0.94 W. At zero rpm, mechanical power is zero; the static torque ratio still follows the selected idealized model.
Ratio calculation is not a tooth-strength rating
Face width, generated flank shape, contact ratio, undercut, bearing alignment and material determine whether a physical pair will mesh and carry its load. The simplified visible teeth are not manufacturing geometry. The calculator verifies the pitch kinematics and arithmetic, not those design checks.
The illustration exposes the gear mesh and shafts. Bearing housings and guards are omitted for clarity. It assumes rigid shafts and steady rotation, with no elastic deflection, backlash or starting transient.
Crown-wheel questions
Why are the shafts perpendicular?
The spur pinion engages the disk’s face teeth, changing the rotation axis by 90°.
Does changing the tooth count affect the drawing?
Yes. Tooth count, relative pitch size and the shaft speed ratio all update together.
Is this an automotive crown-wheel design?
No. It represents the historical face wheel meshing with a spur pinion. Automotive final drives commonly use different bevel or hypoid geometry.
Does greater reduction create more power?
No. Ideal torque rises as speed falls. An entered efficiency below 100% reduces output power.
Construction references
Brown, movement 26, identifies the crown wheel meshing with a spur gear. Hiscox, Mechanical Movements, Powers and Devices, printed page 42, figure 130 (PDF page 52), illustrates the same light-duty arrangement.
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