Vocabulary and geometry that every variant in this folder uses. Type-specific notes link here instead of restating the definitions.

• What a gear actually does
• Involute tooth
• Circles and tooth height
• Module and pitch
• Pressure angle
• Helix and spiral angles
• Contact ratio
• Backlash
• Ratio, torque, and speed
• Efficiency and losses
• Materials and lubrication
• 3D-printed gears

• A pair of meshing gears enforces a constant angular-velocity ratio (ideally) by rolling/sliding along conjugate tooth profiles
• They also set torque (inverse of speed), rotation sense, and sometimes axis direction
• Friction devices (belts, traction drives) can slip; gears are positive engagement — they skip only if a tooth fails or jumps

Related: torque in the structure notes.

Most power gears use an involute profile: the path of a point on a taut string unwinding from a base circle.

Why it won:

• Line of action is a straight line; the common normal at contact always passes through the pitch point
• Velocity ratio stays constant even if center distance is a little off (unlike cycloidal clock teeth)
• One cutter (or one 3D-print profile) can mesh with any other involute of the same module and pressure angle

Cycloidal profiles (clocks, some pumps) are not the same thing as a cycloidal drive, which uses a lobed disc and pins.

For a spur (the picture every other type is a deformation of):

Undercut: too few teeth (roughly $N < 18$ at $20^\circ$ full-depth involute) makes the cutter nibble the root. Profile shift / addendum modification is the usual fix.

Metric module $m$ (mm):

$$ m = \frac{d}{N} $$

• $d$ — pitch diameter (mm)
• $N$ — number of teeth
• Two gears mesh only if they share $m$ (and pressure angle, and usually helix)

Circular pitch $p = \pi m$ (arc length from tooth to tooth on the pitch circle).

Imperial diametral pitch $P_d = N / d$ (teeth per inch of pitch diameter). Conversion: $m = 25.4 / P_d$.

• Angle between the line of action and the common tangent to the pitch circles
• Modern default: $20^\circ$; older: $14.5^\circ$; heavy / stub teeth: $25^\circ$
• Larger pressure angle → stubbier, stronger teeth, higher separating force on bearings
• Mating gears must share pressure angle

• Helix angle $\beta$: tooth slant on a cylindrical gear (helical, herringbone)
• Spiral angle: analogous slant on a conical gear (spiral bevel, hypoid)
• Axial pitch and transverse module show up once $\beta \neq 0$
• Normal module $m_n = m_t \cos\beta$ — the module you would measure in the plane perpendicular to the tooth

• Average number of tooth pairs in contact
• Spur: typically ~1.2–1.6; below 1.0 the drive would theoretically separate between teeth
• Helical / spiral / planetary (multiple planets) raise effective contact and smoothness
• High contact ratio is why helical gears are quieter than spur at the same speed

• Circumferential play between mating teeth (measured on the pitch circle)
• Needed: oil film, thermal growth, machining/print tolerance
• Harmful: positioning error, impact at reversal (robots feel this as joint “slop”)
• Reduced by: tighter center distance, split pinions, preload, harmonic / cycloidal kinematics, or anti-backlash springs

For a printed pair, design intentional backlash (often $0.2$–$0.5,\mathrm{mm}$ at the pitch circle, scaled to module) rather than hoping slicer tolerance lands on zero.

For an external pair (both gears toothed on the outside):

$$ i = \frac{\omega_\text{in}}{\omega_\text{out}} = \frac{N_\text{out}}{N_\text{in}} = \frac{T_\text{out}}{T_\text{in} \cdot \eta} $$

• External mesh reverses rotation; internal mesh preserves it
• $i > 1$ is reduction (speed down, torque up)
• $\eta$ is stage efficiency — do not ignore it on worm, harmonic, or cycloidal stages
• Idler gears change direction (or span a gap) without changing $|i|$

Power (ideal): $P = T\omega$. A reduction that multiplies torque by $i$ divides speed by $i$; power is only lost to friction, not to the ratio itself.

Main loss mechanisms:

• Sliding at the tooth flank (highest near the tip/root; zero at the pitch point for involute)
• Bearing / thrust loads (helical, worm, hypoid)
• Windage and churning in oil
• Hysteresis in a flexing element (harmonic flexspline)

Involute spur/helical pairs are in the high 90s % because sliding is modest. Worm and strain-wave stages live much lower because sliding (or elastic deformation) is the point of the geometry.

Lubrication is a design input, not an afterthought: hypoid and worm require EP (extreme-pressure) or dedicated oils because of high sliding.

Constraints that dominate Eragon-style parts:

• Module floor: very small $m$ (≲ $1,\mathrm{mm}$) disappears into nozzle width and slicer error. Prefer $m \ge 1.5$–$2,\mathrm{mm}$ on FDM
• Pressure angle: $20^\circ$ is fine; $25^\circ$ can be more forgiving of undercut and weak roots
• Orientation: print spurs with the axis vertical (teeth as walls) for strength along the tooth; helical teeth introduce overhangs
• Backlash: oversize holes / undersize teeth in CAD; do not rely on “exact” involute from a screenshot
• Wear: printed gears are for low duty or as a mockup; load-bearing joints usually move to purchased metal planetary / harmonic units
• Fillet and root: add explicit fillets; FDM stress-concentrates at sharp roots

Involute generators (FreeCAD Gear workbench, involute.py scripts, or a purchased profile) beat hand-drawn trapezoids once you care about smoothness.