Tooth Profiles
Part of Gear Making
The geometry of gear tooth shapes — why the involute profile dominates modern gearing and how to lay out teeth without specialized tools.
Why This Matters
Gear teeth must have exactly the right shape to transmit power smoothly. If two meshing gears have poorly shaped teeth, several problems occur: the velocity ratio isn’t constant (causing jerky, vibrating motion), the contact between teeth generates excessive sliding friction (causing rapid wear and heat), and the tooth root stresses are concentrated in ways that cause early fracture.
The tooth profile determines the fundamental quality of the gearing. A gear with the correct involute profile will run quietly, wear slowly, and tolerate small errors in center distance. A gear with the wrong profile — even if it looks close — will perform poorly and wear out quickly.
For post-collapse gear making, you need to understand: what profile is needed, why it works, and how to approximate it well enough with the tools available. Perfect involute geometry requires a gear-cutting machine or carefully made form cutters. But approximations using circular arcs work surprisingly well for low-speed applications, and they can be laid out with compass and straightedge.
The Involute Profile
The involute is a curve traced by the end of a string unwinding from a cylinder (the base circle). This sounds arbitrary, but it has a crucial property: two involute gears in mesh have a constant velocity ratio regardless of small errors in center distance. You can move the gears slightly apart and they still transmit motion at the correct ratio (with only slightly increased backlash). This tolerance for center distance variation is why involutes became universal in industrial gearing.
The base circle diameter for a standard gear: Base Circle = Pitch Diameter × cos(pressure angle). For the standard 20° pressure angle: Base Circle = PD × cos(20°) = PD × 0.940.
Pressure angle: The angle of the force between meshing teeth relative to a tangent at the pitch circle. The standard is 20° (older designs use 14.5°). Higher pressure angle teeth are stronger but produce more bearing loads. The pressure angle determines the shape of the tooth flank.
Every involute tooth has:
- Flank: The active working surface below the pitch circle, following the involute curve
- Face: The active working surface above the pitch circle, also involute
- Root: The fillet at the base of the tooth — must have adequate radius to avoid stress concentration
- Tip: The top land, usually a small flat or slight radius
Contact between meshing teeth occurs along the line of action, a straight line tangent to both base circles, inclined at the pressure angle. All tooth contact, from approach to recess, takes place on this line. A single pair of gear teeth are in contact for only part of each revolution; the ratio of the contact arc to the tooth pitch is the contact ratio. Contact ratio should be above 1.2 for smooth operation; 1.4-1.6 is typical.
Laying Out an Involute Tooth Template
Without a gear-cutting machine, make a template to guide filing or carving:
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Draw the pitch circle at radius = PD/2
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Draw the base circle at radius = PD/2 × cos(20°) = PD/2 × 0.940
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Mark tooth spacing: Circumference of pitch circle = π × PD. Tooth pitch = π × PD ÷ N (number of teeth). Mark teeth at equal intervals around the pitch circle.
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Construct the involute: From a point on the pitch circle, draw the pressure angle line (20° from radial). The involute at the pitch circle is tangent to this line. The approximate involute can be drawn as a circular arc with its center on the base circle, radius equal to the perpendicular distance from the base circle to the tooth flank point.
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Simpler approximation (Willis method): The tooth flank can be approximated by a circle whose center lies on the pitch circle at a distance of 0.54 × pitch from the tooth center, with a radius of about 0.29 × pitch for a standard 14.5° pressure angle system. This gives adequate results for wooden gears and low-speed metal gears.
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Trace the template: Cut the profile from thin sheet metal or stiff cardboard. Use this to mark the tooth spaces on the gear blank, then cut by filing or chiseling.
Circular Arc Teeth
For wooden millwork and slow-speed applications, circular arc teeth are perfectly adequate and much easier to lay out and cut:
The tooth flank is a circular arc, and the mating tooth face is the matching circular arc (called an epicycloid, but approximated by a simple circle). For the classic pin-wheel mills:
- Teeth on the large wooden gear (spur wheel) are simply cylindrical pegs (pins) of hardwood set into holes in the rim
- Mating gear (lantern pinion) has cylindrical staves parallel to the shaft axis — the pins engage the staves
- No tooth profile calculation needed — the geometry works automatically for any size pins
This lantern pinion design was used in mills for over 1,000 years and remains valid today. The wooden pins on the spur wheel wear and can be individually replaced; the cylindrical bars of the lantern pinion wear evenly. The contact is rolling with some sliding — acceptable friction for greased wooden gears.
Profile Shift and Non-Standard Gearing
When cutting gears with fewer than 14 teeth, standard tooth proportions cause undercutting — the tool cuts into the base of the tooth, weakening it. Profile shift corrects this by displacing the cutting tool outward relative to the blank.
Positive profile shift (used on small pinions): The tool starts cutting at a larger radius, leaving more material at the root. The teeth become thicker at the base, with a more pointed tip and a working depth slightly less than standard.
In practice: if making a pinion with 10-13 teeth, add a positive shift of 0.3-0.5 × Module. This requires the mating gear to receive a corresponding negative shift to maintain proper mesh. Pre-collapse gear cutting software (or gear design tables) handles these calculations; in a workshop, using 14+ teeth on pinions avoids the issue entirely.
Inspecting Tooth Contact Pattern
After gears are cut and assembled, inspect actual tooth contact by applying marking compound (prussian blue, or a thin layer of grease mixed with lampblack) to one gear’s teeth and running the mesh under light load.
Good contact: The marking compound transfers to the mating tooth in a broad band centered on the pitch line, covering 60-80% of the tooth face width.
High contact (marking near tooth tip): Center distance too small, or teeth too thick. Gears binding at engagement.
Low contact (marking near root): Center distance too large, or teeth too thin. Contact only near the bottom of the tooth — load concentrated on weakest part.
End contact (marking at only one end): Shafts not parallel. Correct alignment before running under load.
Interrupted pattern: Rough tooth surface, or a periodic error in tooth spacing. Run gears lightly loaded for a break-in period to smooth surfaces; correct pitch errors in future gear sets.