Compound Trains
Part of Gear Making
Designing and building multi-stage gear trains to achieve large speed ratios or precise mechanical advantages.
Why This Matters
A single pair of gears can achieve a speed ratio up to about 6:1 or sometimes 8:1 before the size disparity becomes impractical — a very small pinion meshing with a very large wheel. To achieve ratios of 20:1, 100:1, or more, a compound gear train uses multiple stages in series, where the output of one stage drives the input of the next.
Compound gear trains are essential for clocks (which need ratios of hundreds to thousands of hours’ worth of rotation to seconds of rotation), for mill drives (where a waterwheel turning at 10–20 rpm must drive a millstone at 120–150 rpm — a ratio requiring at least two stages), for lifting mechanisms (where large mechanical advantages are needed), and for any machinery requiring fine speed control.
Understanding compound trains allows you to design the gear arrangement before cutting a single tooth — choosing the number of stages, the teeth counts for each gear, and the physical arrangement to fit within the available space. This planning prevents building a gear train only to find that the final speed ratio is wrong or that the stages physically interfere with each other.
The Compound Train Speed Ratio
In a simple gear train (each shaft carries only one gear), the overall ratio is the product of all individual ratios:
i_total = (N₂/N₁) × (N₄/N₃) × (N₆/N₅) …
where N₁, N₃, N₅ are the driving gears on each shaft and N₂, N₄, N₆ are the driven gears.
In a compound train, pairs of gears are mounted on the same intermediate shaft so they rotate together. The first-stage driven gear and the second-stage driving gear are on the same shaft. This compounding multiplies the ratios:
If Stage 1 has ratio i₁ = N₂/N₁ and Stage 2 has ratio i₂ = N₄/N₃ (where N₃ and N₂ are on the same shaft), the overall ratio is i_total = i₁ × i₂.
Example: To achieve 40:1 reduction in two stages:
- Stage 1: 8-tooth pinion driving 32-tooth gear = 4:1
- Stage 2: 10-tooth pinion driving 40-tooth gear = 4:1
- Total: 4 × 4 = 16:1 — not enough
- Try: Stage 1: 8-tooth driving 40-tooth = 5:1; Stage 2: 8-tooth driving 40-tooth = 5:1 → 25:1
- Or: Stage 1: 8-tooth driving 48-tooth = 6:1; Stage 2: 8-tooth driving 48-tooth = 6:1 → 36:1
- For exactly 40:1: Stage 1: 5:1; Stage 2: 8:1 → 8-tooth pinion with 40-tooth wheel and 6-tooth pinion with 48-tooth wheel
Designing for Specific Ratios
The design problem is: given a target ratio, find tooth counts for each stage that:
- Multiply to give (close to) the target ratio
- Use practical tooth counts (minimum ~8–10 teeth for strength)
- Fit within the space constraints
Factoring the ratio: Factor the target ratio into the product of smaller ratios, each achievable in one stage. For prime number target ratios, this may be impossible exactly — use a close approximation. For a clock requiring 3,600:1 (ratio from hour hand to second hand), factor as: 60 × 60 = 60:1 × 60:1, achievable as 3 stages of 60:1 (but 60:1 in one stage is impractical) or reconfigured as 5 stages of approximately 5:1 each (5⁵ = 3,125, not quite; adjust one stage: 5×5×5×5×6 = 3,750 — still close enough for a draft clock).
Practical tooth count ranges: Minimum teeth: 8–12 (fewer causes interference and weak teeth). Maximum teeth: 80–120 for a wheel in a compact train (larger is fine if space allows). Convenient intermediate shaft pinions: 10–20 teeth. Large wheels: 48–96 teeth (giving 4:1 to 8:1 with a 12-tooth pinion).
Physical Arrangement of Compound Trains
The shafts must be positioned so that:
- Each gear pair meshes at the correct center distance
- Intermediate shafts are accessible for bearing support
- Gears do not physically interfere with each other
- The train can be assembled and disassembled for maintenance
Linear arrangement: Shafts are arranged in a row. Simple to build but becomes long with many stages.
Back-gearing: A common arrangement for machine tools where a high-speed normal drive and a slower reduced drive share the same spindle. The back gears fold the train back on itself to save space.
Epicyclic (planetary) train: A highly compact arrangement where one or more planet gears orbit around a central sun gear, inside a ring gear. Very high ratios in small space. More complex to make but extremely space-efficient. Used in clocks, winches, and compact multipliers.
Torque and Power Through the Train
As the gear train reduces speed, torque increases proportionally (ignoring friction). In an ideal gear train:
Power_out = Power_in (no losses) Speed_out = Speed_in / i_total Torque_out = Torque_in × i_total
In a real gear train, each stage has friction losses of approximately 1–3% for well-made, lubricated gears. A 4-stage train might have total efficiency of 0.97⁴ ≈ 0.88 — 12% power loss to friction. For heavily loaded trains with marginal lubrication, losses can reach 3–5% per stage.
The output shaft and its bearings must be designed to carry the full output torque — which may be many times larger than the input torque. Bearing sizes and shaft diameters must increase toward the output end of the train.
Assembly Order and Backlash Management
Assemble compound trains from the output (slow) end toward the input (fast) end. This way, you can check each stage’s backlash and mesh condition before adding the next. The slowest gear pair is usually the largest and sets the physical layout; work inward from there.
Check overall backlash by holding the output shaft fixed and rocking the input shaft. The apparent input shaft movement (amplified by the total ratio) shows the sum of all individual stage backlashes. This is normal — each stage contributes a small amount.
Total system backlash should not cause unacceptable position error for the application. In a clock, backlash must be nearly zero (hence the importance of maintaining spring or weight tension to take up backlash). In a mill drive, some backlash is acceptable and even desirable as it prevents shock loads from breaking teeth.