Gear Ratios and Torque

Part of Gear Making

Calculating gear ratios, speed changes, and torque multiplication for practical machinery design.

Why This Matters

The gear ratio is the fundamental variable that determines what a gear drive does. It sets the output speed relative to the input speed, and by conservation of energy, the inverse relationship between output torque and input torque. Getting the ratio right is the difference between a machine that works efficiently and one that either runs too fast to control or too slowly to be useful.

In practical rebuilding, almost every machine needs a specific speed relationship between its power source and its working element. A waterwheel turning at 15 rpm must drive a millstone at 120 rpm — a 1:8 speed increase. A hand crank operating at 60 rpm must drive a grinding wheel at 1,800 rpm — a 1:30 increase. A windmill turning at 40 rpm must power a water pump requiring 120 rpm — a 1:3 increase. Designing the gear train to achieve these ratios accurately and efficiently is the first step in building functional machinery.

The relationship between gear ratio, torque, and power is exact and fundamental. Understanding it prevents the common mistake of designing a gear train for the right ratio but then mounting inadequate bearings and shafts to handle the changed torque level.

The Gear Ratio Equation

For a gear pair, the gear ratio i is:

i = N₂/N₁ = ω₁/ω₂ = d₂/d₁

where:

• N₁, N₂ = number of teeth on gear 1 (driving) and gear 2 (driven)
• ω₁, ω₂ = rotational speed of gear 1 and gear 2
• d₁, d₂ = pitch circle diameters

All three expressions are equivalent. You can calculate the ratio from tooth counts, speeds, or diameters.

Speed calculation: ω₂ = ω₁ / i = ω₁ × (N₁/N₂)

If gear 1 has 12 teeth and runs at 60 rpm, and gear 2 has 48 teeth: i = 48/12 = 4; ω₂ = 60/4 = 15 rpm.

Ratio terminology:

• i > 1: speed reduction (gear 2 slower than gear 1) — the driven gear is larger
• i < 1: speed increase (gear 2 faster than gear 1) — the driven gear is smaller
• i = 1: same speed, used only to reverse rotation direction or transfer drive between parallel shafts

Torque and Gear Ratio

In an ideal (frictionless) gear pair, power is conserved:

P = T₁ × ω₁ = T₂ × ω₂

Since ω₂ = ω₁/i, we have:

T₂ = T₁ × i

The output torque equals the input torque multiplied by the gear ratio. This is the fundamental law of mechanical advantage. A 4:1 gear reduction multiplies torque by 4 while dividing speed by 4.

Example: A waterwheel produces 500 N·m torque at 15 rpm. After a 4:1 gear increase to drive a millstone at 60 rpm:

• Output torque = 500 / 4 = 125 N·m
• Output speed = 15 × 4 = 60 rpm
• Power (ideal): 500 × 15 = 125 × 60 = 7,500 N·m/rpm (same)

In practice, friction reduces output power by 2–5% per gear stage, but the relationship remains approximately true.

Choosing Tooth Counts for a Target Ratio

Given a target gear ratio, choose tooth counts that:

1. Give the exact ratio (or as close as needed)
2. Use practical tooth counts (minimum ~10–12 teeth on the pinion)
3. Are achievable to cut with available tools

For simple ratios (2:1, 3:1, 4:1): direct tooth count selection is easy. 12 and 24 teeth give 2:1. 12 and 36 give 3:1. 10 and 40 give 4:1.

For decimal ratios (1.5:1, 2.5:1): use multiples. 1.5:1 → 20 and 30 teeth. 2.5:1 → 12 and 30, or 16 and 40.

For irrational ratios: approximate. A ratio of π (needed for certain dividing calculations) can be approximated as 355/113 or simpler fractions: 22/7 ≈ 3.143 is close to π. The gear pair of 14 and 44 teeth gives 3.143:1 within 0.05%.

For very high ratios in a single stage: impractical beyond about 6:1–8:1. Use compound trains (multiple stages in series) as described in the compound trains article.

Gear Ratio and Contact Ratio

The number of tooth pairs in contact simultaneously (the contact ratio) affects smoothness of operation. Contact ratio greater than 1.0 means at least one pair of teeth is always in contact. Contact ratio between 1.0 and 2.0 means occasionally two pairs share the load. Higher contact ratio → smoother, quieter, stronger.

Contact ratio is a function of tooth geometry, not gear ratio per se. However, using a large number of teeth (fine module, small individual teeth) increases contact ratio compared to coarse teeth at the same pitch diameter. For low-speed, heavily loaded gears, contact ratio matters less. For higher-speed gears, aim for contact ratio of at least 1.3–1.5.

Practical Ratio Selection for Common Machines

Machine	Typical source speed	Required output speed	Approximate ratio
Waterwheel to millstone	10–20 rpm	120–150 rpm	1:8 to 1:15 (increase)
Windmill to grain mill	30–50 rpm	120–150 rpm	1:3 to 1:5 (increase)
Treadle lathe (foot to spindle)	40–80 rpm	300–600 rpm	1:5 to 1:10 (increase)
Hand winch	60 rpm	(very slow, heavy load)	10:1 to 50:1 (reduction)
Clock minute to hour hand	1 rpm	1/60 rpm	60:1 (reduction)
Clock escape wheel to minute	(complex)	(complex)	multi-stage

Verify the selected ratio against actual operating conditions. A millstone that is theoretically designed to run at 120 rpm but is driven by a waterwheel turning at only 10 rpm under partial water supply will actually run at 10 rpm × ratio — check the minimum expected power source speed, not just the design speed.