Ratio Calculation

Part of Gear Making

How to calculate gear ratios to match power source speed to the requirements of driven machinery.

Why This Matters

Every mechanical power transmission problem reduces to a ratio question: you have a power source spinning at one speed, and you need machinery turning at a different speed or with different torque. A water wheel turns at 10 rpm; a millstone needs 120 rpm. A hand crank delivers 40 rpm; a drill needs 800 rpm. Getting these ratios wrong means either destroying machinery with excessive speed, or failing to generate enough torque to do useful work.

Gear ratio calculation is pure arithmetic — no calculus, no advanced engineering required. But it requires careful systematic thinking. A mistake in ratio calculation wastes weeks of fabrication effort building the wrong gearset, or worse, destroys expensive or hard-to-replace equipment. Understanding ratios also lets you design compound gear trains that achieve very high or very low final ratios using gears of manageable size.

This knowledge extends beyond gears to pulleys, sprockets, and any rotating transmission system. The same math applies whether you’re driving a grain mill, a forge bellows, a water pump, or an electric generator.

The Fundamental Ratio

The gear ratio between two meshing gears is simply the number of teeth on the driven gear divided by the number of teeth on the driving gear:

Ratio = Driven Teeth ÷ Driver Teeth

If the driving gear has 20 teeth and the driven gear has 60 teeth, the ratio is 60 ÷ 20 = 3:1. This means for every 3 turns of the driver, the driven gear makes 1 turn. Speed is reduced by a factor of 3; torque is multiplied by a factor of 3 (ignoring friction losses).

Alternatively, if the driver has 60 teeth and the driven has 20, the ratio is 20 ÷ 60 = 1:3. Speed is increased by 3; torque is reduced by 3. This is a speed-increasing or “step-up” gear pair.

The same formula applies to pulleys using diameter instead of tooth count:

Ratio = Driven Diameter ÷ Driver Diameter

A 4-inch pulley driving a 12-inch pulley gives a 3:1 reduction. A 12-inch driving a 4-inch gives 1:3 step-up.

Speed and Torque Calculations

Given the input speed and ratio, output speed is:

Output RPM = Input RPM ÷ Ratio

A water wheel turning at 15 rpm through a 10:1 speed-increasing gearset produces 150 rpm at the output shaft.

Torque follows the inverse relationship (conservation of energy, minus losses):

Output Torque = Input Torque × Ratio × Efficiency

Where efficiency is typically 0.95-0.98 for a single well-made gear pair (2-5% friction loss). For multiple stages, multiply the efficiencies together.

Power remains essentially constant through a gear train (minus friction losses). This is the key insight: you cannot get more power out than you put in. Gears trade speed for torque or torque for speed, but total power (torque × speed) stays the same.

If a water wheel delivers 500 ft-lbs of torque at 15 rpm, total power is 500 × 15 = 7,500 ft-lbs/min ≈ 0.23 horsepower. A 10:1 step-up gives 50 ft-lbs torque at 150 rpm — same power, different speed-torque combination.

Compound Gear Trains

When a single gear stage can’t achieve the required ratio (because it would require impractically large or small gears), you use compound stages in series.

Total Ratio = Stage 1 Ratio × Stage 2 Ratio × Stage 3 Ratio…

Example: you need 100:1 reduction. A single stage would require a 100-tooth gear driving a 1-tooth pinion — impossible. Instead, use three stages:

• Stage 1: 5:1 (50-tooth gear, 10-tooth pinion)
• Stage 2: 4:1 (40-tooth gear, 10-tooth pinion)
• Stage 3: 5:1 (50-tooth gear, 10-tooth pinion)

Total: 5 × 4 × 5 = 100:1

In a compound gear train, each intermediate shaft carries both a large driven gear (output from previous stage) and a small driving pinion (input to next stage) keyed together on the same shaft. The compound arrangement allows any ratio to be achieved with gears of reasonable, fabricable size.

Practical limits: avoid ratios greater than 6:1 or 7:1 per stage in wooden gear trains (structural limits). Metal gears can handle 8:1 to 10:1 per stage. For very high ratios, worm gears can achieve 20:1 to 60:1 in a single stage but have lower efficiency (50-90%).

Designing for Required Speed

Working backwards from requirements:

1. Identify required output speed (e.g., grain mill: 90-150 rpm; generator for 50Hz power: 1,500 or 3,000 rpm for 2-pole or 4-pole machines; pump: 300-600 rpm)

2. Measure available input speed (water wheel revolution count over 60 seconds with a counter, or timed by counting)

3. Calculate required total ratio: Total Ratio = Input RPM ÷ Required Output RPM

4. Factor into stages: Decompose the ratio into manageable per-stage ratios. Aim for 2-6 stages maximum.

5. Select tooth counts: For each stage, choose tooth counts that give the stage ratio, keeping tooth counts between 12 (minimum for strength) and 80-100 (practical fabrication limit for large wooden gears).

Example: Water wheel at 8 rpm, generator needs 1,500 rpm. Total ratio = 1,500 ÷ 8 = 187.5:1.

• Stage 1: 6:1 (60 teeth ÷ 10 teeth)
• Stage 2: 6.25:1 — round to 6:1 or adjust: try 5:1 + 5:1 + 7.5:1
• Revised: 5 × 5 × 7.5 = 187.5 — but 7.5:1 requires non-integer teeth ratio

Adjust: Stage 1: 5:1, Stage 2: 5:1, Stage 3: 8:1 = 200:1 → output = 8 × 200 = 1,600 rpm (slightly high, acceptable if generator handles it). Or use a final belt drive to fine-tune.

Checking Your Work

Before cutting a single gear tooth, verify:

1. Tooth count check: Stage ratio = driven teeth ÷ driver teeth. Confirm it equals target.

2. Speed check: Trace RPM through each shaft. Input → ÷ ratio 1 → ÷ ratio 2 → etc. = output RPM.

3. Center distance check: Two meshing gears must have their shaft centers separated by exactly (Pitch Diameter 1 + Pitch Diameter 2) ÷ 2. Pitch diameter = Number of teeth ÷ Diametral Pitch (or = Module × Number of teeth in metric). If you’re laying out a gear train, compute all center distances before drilling shaft holes.

4. Torque capacity check: Each gear must handle the torque at its shaft. Higher torque = larger and wider gears needed at low-speed stages.

Document all calculations in a simple table: shaft number, RPM, torque, gear pair, tooth counts. Review it before committing to fabrication. A single transcription error can invalidate days of work.