In this guide
Every robot wheel, conveyor, mill, and vehicle drivetrain sits between a motor that spins too fast with too little torque and a load that needs to turn slower with far more torque. Gears are how you trade one for the other — and the mathematics is simpler than most students fear: it is mostly one ratio, applied carefully. This guide builds gear-ratio calculation from the ground up — terminology, the core formulas, compound trains, and module selection — with worked examples sized for real student projects.
The vocabulary: teeth, module, pitch diameter
You cannot calculate with gears until the terms are exact:
- Teeth (z): the count of teeth on a gear. The only number that matters for ratio.
- Module (m): the size of the teeth, in millimetres. Definition: m = pitch diameter / teeth. Two gears mesh only if their modules (and pressure angle, almost always 20° in student stock) match.
- Pitch diameter (D): the effective rolling diameter: D = m × z. A 20-tooth gear of module 1.5 has D = 30 mm.
- Pressure angle: the tooth profile angle — 20° is the modern standard. Do not mix 14.5° and 20° gears; they will not mesh properly.
- Driver / driven: the driver is the input gear (on the motor), the driven is the output gear (on the load).
The practical consequence of module: it sets tooth size and therefore strength. Module 0.5–1 gears are for instruments and small robots; module 1–2 for desktop mechanisms and robot drivetrains; module 2.5+ for vehicle and mill work. When buying stock gears, match module first, teeth second.
The one ratio
For two meshing gears:
Gear ratio i = teeth on driven / teeth on driver = z_driven / z_driver = speed of driver / speed of driven
A 12-tooth pinion driving a 60-tooth gear: i = 60/12 = 5. The output spins 5× slower than the motor. And — conservation of power, minus losses — it delivers roughly 5× the torque:
T_driven = T_driver × i × η
where η is the mesh efficiency (~0.95–0.98 per spur-gear mesh; use 0.95 for honest student math). Power is conserved: P = T × ω (torque times angular velocity in rad/s) is the same on both sides except for the efficiency loss. You never get something for nothing — halving the speed doubles the torque, minus a few percent to friction and noise.
Direction note: two external gears reverse direction; three in a row (with an idler) restore it. An idler gear between driver and driven changes direction without changing the ratio — its tooth count cancels out of the math.
Compound trains: multiplying ratios
One gear pair rarely gives enough reduction — a 5:1 single stage needs a driven gear five times the driver's diameter, which gets enormous fast. Instead, stack stages: the driven gear of stage 1 shares a shaft with the driver pinion of stage 2. Ratios multiply:
i_total = i_1 × i_2 × i_3 × …
Efficiency multiplies too (and therefore drops): η_total = η_1 × η_2 × … — three 0.95 stages give 0.857 overall. Keep stage counts low; each mesh costs you ~5% and adds backlash.
Worked example 1: robot drive wheel
A DC gearmotor spins at 300 rpm with 0.35 N·m (3.5 kg·cm) torque. You want a 100 mm wheel turning at 60 rpm for a sensible robot speed.
Required ratio: i = 300 / 60 = 5. Options: single stage 12T → 60T (i = 5 exactly). Output torque: 0.35 × 5 × 0.95 ≈ 1.66 N·m. Check the speed: wheel circumference = π × 0.1 m ≈ 0.314 m; at 60 rpm (1 rev/s) the robot moves ≈ 0.31 m/s — a brisk walking pace, right for a demo robot. One stage, done. But suppose the motor were 1500 rpm: i = 25 needed. Single stage would be 12T → 300T — absurd. Compound: 12T → 60T (i=5), then 12T → 60T again (i=5): i_total = 25, η = 0.95² ≈ 0.90, output torque = 0.35 × 25 × 0.90 ≈ 7.9 N·m at 60 rpm. Two compact stages beat one giant gear.
Worked example 2: pedal-powered grain mill
Human pedalling delivers roughly 75–100 W at a comfortable 60–70 rpm cadence. A pedal-powered grain mill needs its grinding plates at ~200–250 rpm. That is a speed-increasing drive: i = 70/225 ≈ 0.31, or expressed as a reduction ratio, 1:3.2. A chain drive (chainring to sprocket, same math as gears: teeth ratio) handles this naturally — and notice the torque consequence: the mill shaft gets only ~31% of the pedal torque (minus efficiency), which is why mill builders use a flywheel to smooth the grinding pulses. The same ratio math governs the coconut dehusking machine model, where a motor's speed is reduced to the slow, high-torque rotation the dehusking spikes need.
Worked example 3: sizing against a motor datasheet
Motor: 12 V DC, rated 5000 rpm, rated torque 0.06 N·m, stall torque 0.3 N·m. Load: needs 0.9 N·m at ~250 rpm. Required i = 5000/250 = 20. Check torque: 0.06 × 20 × 0.9 (two stages ≈ 0.9) ≈ 1.08 N·m — covers the 0.9 N·m need with a small margin; good. Now the critical check beginners skip: can the motor actually deliver its rated torque continuously at that operating point? Rated torque is the continuous thermal rating — running at stall torque (0.3 N·m here) for more than seconds overheats the motor. Size the ratio so the motor works near its rated torque, not its stall torque, and keep 20–30% margin for friction you did not model.
Module selection and centre distance
Once the ratio is known, pick the module for strength and the tooth counts for ratio, then compute the shaft spacing:
Centre distance C = m × (z_1 + z_2) / 2
Example: module 1.5, 12T pinion + 60T gear: C = 1.5 × 72 / 2 = 54 mm. That is the exact shaft spacing your bracket or gearbox plate must hold — get it wrong by even 0.5 mm and the mesh is either binding or sloppy. Design the mounting holes from this number, not the other way round.
Module selection guideline for student builds:
| Module | Tooth size | Typical use |
|---|---|---|
| 0.5–0.8 | Fine | Small robots, instruments, servo gearboxes |
| 1.0–1.5 | Medium | Desktop mechanisms, robot drivetrains, model gearboxes |
| 2.0–3.0 | Coarse | Mills, dehuskers, vehicle drivetrains |
Bigger module = stronger teeth but a bulkier, heavier gearbox for the same ratio. When in doubt between two modules, take the larger — stripped plastic or aluminium gear teeth are a depressingly common demo-day failure, and the weight penalty is small.
Gear types: which one for your project
| Type | Character | Use it for |
|---|---|---|
| Spur | Simple, cheap, noisy at speed; shafts parallel | Model gearboxes, robot drives — the default |
| Helical | Quieter, stronger (gradual engagement); produces axial thrust | Quieter drivetrains; needs thrust bearings |
| Bevel | Turns the corner (usually 90°); shafts intersect | Right-angle drives, differential-like layouts |
| Worm | Huge single-stage reduction (20:1–100:1); usually self-locking | Winches, lifts — holding without power; note efficiency is poor (50–80%) |
| Rack and pinion | Rotation → linear motion | CNC axes, linear actuators, steering |
Two properties worth memorising: worm drives self-lock — the load cannot back-drive the motor, which is why they are used in lifts and adjustable mounts (but never where you need efficiency or back-drivability). And planetary (epicyclic) sets pack high reduction into a coaxial package — the gearboxes on cordless drills are planetary; lovely, but harder to build from stock parts than a spur train.
Backlash, and why precision mechanisms care
Backlash is the small free play between meshing teeth — the output shaft can wiggle slightly with the input held. For a grain mill or a conveyor it is irrelevant. For a robot arm joint or a CNC axis doing bidirectional positioning, backlash is positioning error: every direction reversal loses a fraction of a degree. Mitigations: split (anti-backlash) gears with spring-loaded halves, preloaded bearings, or — the honest student answer — put the encoder on the output shaft (closed loop) rather than counting motor steps. A cutaway gearbox trainer is the ideal teaching rig here: you can see and feel backlash, staging, and meshing in a way no diagram conveys.
Efficiency and heat: the losses you must budget
No transmission is free. Honest per-stage efficiencies for student-grade hardware:
| Stage | Efficiency | Notes |
|---|---|---|
| Spur mesh, greased | 0.95–0.98 | Use 0.95 in calculations |
| Helical mesh | 0.94–0.97 | Slightly below spur |
| Chain drive | 0.95–0.98 | When clean, aligned, and tensioned |
| V-belt | 0.90–0.95 | Slip and flex losses |
| Timing belt | 0.96–0.98 | Near-chain efficiency, no slip |
| Worm drive | 0.50–0.80 | The price of huge reduction + self-locking |
Multiply stage efficiencies for the train total, and remember the lost power becomes heat in the gearbox. A 100 W drive through two 0.9-efficient stages dumps ~19 W as heat — fine in open air, a problem in a sealed plastic box. Ventilate enclosed gearboxes on anything above a few tens of watts.
Build checklist: from calculation to working gearbox
- Ratio computed from required output speed and available motor speed.
- Torque checked at the motor's rated (continuous) torque with 20–30% margin, efficiency included.
- Stage count minimised: prefer fewer, larger stages over many small ones (each mesh costs ~5% and adds backlash).
- Module chosen for tooth strength; meshing gears share module and pressure angle.
- Centre distance C = m(z_1+z_2)/2 computed and used to position shafts — not eyeballed.
- Shafts supported on both sides of each gear (cantilevered pinions flex and skip under load); bearings, not bare holes in acrylic.
- Backlash acceptable for the application; anti-backlash or closed-loop sensing where it is not.
- Lubrication: light grease on metal gears; dry or silicone on plastic gears (petroleum grease attacks some plastics).
- Guard over the mesh — exposed gears catch fingers, sleeves, and wires. Non-negotiable for any demo with an audience.
Gears are also the natural companion to motor selection: work through the servo vs stepper companion guide (same series) first so the motor's torque–speed character is known, then use this guide's math to bridge the gap to the load. The cutaway gearbox trainer project makes an excellent physical companion to this guide — build or study one and every formula above becomes something you can point at.
Planetary gears: big reduction in a small package
A planetary (epicyclic) set — sun gear in the centre, planet gears around it, ring gear outside — gives 3:1 to 10:1 in a single coaxial stage, which is why cordless drills and robot wheel hubs use them. The ratio depends on which member is fixed; the common student case (ring fixed, sun input, carrier output) gives i = 1 + z_ring/z_sun. A 12T sun with a 48T ring: i = 1 + 48/12 = 5. Coaxial input and output make them ideal where space is tight, but they are harder to build from stock spur gears than a simple two-stage train — buy a ready-made planetary gearbox (commonly sold for NEMA 17/23 motors in 5:1, 10:1, 20:1) rather than machining one.
Worm gears: self-locking, quantified
A worm drive's self-locking is not magic — it is friction winning over the thread's lead angle. If the lead angle is smaller than the friction angle (roughly: lead angle < ~5–6° for steel-on-bronze with grease), the gear cannot back-drive the worm. That is why a worm-driven lift stays put with the power off. The price: sliding contact means efficiency of 50–80% (worse at high ratios), so the lost power becomes heat — size the housing and lubrication for it, and never use a worm where you need back-drivability (a robot joint that must go limp on power loss, for instance).
Gear materials: printed, aluminium, steel, brass
| Material | Strength | Wear life | Cost/effort | Use for |
|---|---|---|---|---|
| 3D-printed (PLA/PETG) | Low–medium | Fair (keep loads light, grease with silicone) | Cheapest, fastest to iterate | Prototypes, light robot drivetrains, demos |
| 3D-printed (nylon/PA) | Medium | Good (self-lubricating-ish) | Needs dry storage, hotter nozzle | Longer-life printed gears |
| Aluminium stock gears | Medium–high | Good | Moderate | Model gearboxes like the trainer rig |
| Steel/brass stock gears | High | Excellent | Higher cost, need proper shafts/bearings | Mills, dehuskers, anything with real torque |
The honest progression: prototype the geometry in printed plastic (cheap iteration on ratio and centre distance), then replace the loaded stages with metal stock gears for the final build. The cutaway gearbox trainer demonstrates exactly this with real metal gears you can inspect.
Measuring backlash in a built gearbox
Lock the input shaft, then rock the output shaft back and forth and measure the free play with a dial indicator — or, with no instruments, mark the output pulley, rock it, and measure the arc with a protractor. A few tenths of a degree per stage is normal for stock gears; several degrees means worn teeth, loose shaft fits, or bearings with play. If your mechanism hunts or oscillates around its setpoint, measure backlash before blaming the control loop — no PID tuning fixes mechanical slop.
Lubrication, concretely
- Metal gears: light lithium or synthetic gear grease, a thin film — not packed full. Excess grease flings off and collects dust into grinding paste.
- Plastic gears: silicone grease or dry PTFE spray. Petroleum-based greases attack PLA, ABS, and polycarbonate — check compatibility before applying.
- Chains (covered in the companion guide): oil, regularly — but keep oil off nearby belts and plastic gears.
- Re-lubrication interval: student machines run intermittently, so "every few months of use, or when it gets noisy" is an honest schedule. Noise is the wear alarm — a gearbox that suddenly gets louder is telling you something changed.
Laying out a gearbox plate: a worked example
Take the Worked Example 1 compound train: 12T → 60T, then 12T → 60T, module 1.5. Three shafts: motor shaft (12T pinion), intermediate shaft (60T + 12T fixed together), output shaft (60T).
- Stage 1 centre distance: C1 = 1.5 × (12 + 60)/2 = 54 mm.
- Stage 2 centre distance: C2 = 54 mm (same pair).
- Lay the three shafts in a triangle or a line with 54 mm spacing — a triangle keeps the box compact; a line is easier to drill accurately.
- Shafts: 8 mm steel rod is plenty for this torque class. Support each shaft in bearings on both plates (front and back) — 608 skate bearings (8×22×7 mm) are cheap, everywhere, and perfectly adequate. Never run a loaded shaft in a bare drilled hole; the hole wallows out and the centre distance drifts.
- Plates: 6–8 mm acrylic or aluminium, drilled on a drill press. Drill the bearing seats with the plates clamped together so both sides align — misaligned plates are the number-one cause of binding in student gearboxes.
- Assemble dry first: every shaft should spin freely by hand with the gears meshed. If any stage binds, find the tight spot (usually a mis-drilled hole or a gear not seated square on its shaft) before adding grease — grease masks binding that will eat the motor's torque margin.
This plate-layout discipline — computed centre distances, bearings both sides, clamped-together drilling — is what separates a gearbox that runs for years from one that eats itself in a week. The same approach scales up to the coconut dehusking machine and down to printed robot gearboxes.
Buying stock gears: a checklist
- Module and pressure angle match across every meshing pair (20° standard).
- Bore matches your shaft (or buy with a smaller bore and ream — never drill a gear bore oversize by hand and hope).
- Hub with set screw, or a keyway — press-fit alone walks off under reversing loads.
- Face width ≥ module × 8–10 for loaded stages (narrower gears strip faster).
- Count spares: one extra pinion of each size. Pinions wear first, and a stripped pinion the night before a demo with no spare is a familiar tragedy.
Gear ratios in vehicles: reading a gearbox like an engineer
A cutaway gearbox trainer usually demonstrates a 3- or 4-speed sliding-mesh or constant-mesh box. The ratios tell the whole story of what each gear is for:
- 1st gear (~3.5:1): maximum torque multiplication for starting from rest and climbing. Engine torque × 3.5 × final-drive ratio — this is why a small engine can move a loaded vehicle.
- 2nd–3rd (~2.2:1, ~1.4:1): progressively less multiplication as speed builds.
- 4th/top (~1:1, direct drive): no reduction — maximum efficiency for cruising, since every gear mesh avoided is ~5% loss avoided.
The final drive (differential) adds its own reduction (~3–4:1), multiplying everything. Total ratio in 1st: 3.5 × 3.7 ≈ 13:1 — thirteen times the engine's torque at the wheels, at one-thirteenth the engine's rpm. When you look at a trainer's gear pairs, count the teeth on each pair and compute these numbers yourself — it turns a display model into a readable engineering document, and it is excellent viva material because it shows you understand why the ratios are staged the way they are.
Ratio planning worksheet
For any new design, fill this in before buying a single gear:
- Motor speed (rpm) and rated torque (N·m): ___
- Required output speed (rpm): ___ → required ratio i = ___
- Stages: ___ (keep ≤ 3 if possible)
- Per-stage ratios: ___ × ___ = i_total ___
- Output torque = motor torque × i_total × 0.95^stages = ___ (must exceed load torque × 1.3)
- Module: ___ ; tooth counts: ___ ; centre distances: ___
- Shaft diameters and bearing choice: ___
- Lubrication and guarding plan: ___
Eight lines, ten minutes, and it catches every common error — wrong ratio direction, torque checked at stall instead of rated, forgotten efficiency, centre distance eyeballed — before money is spent.
One ratio to remember
If this guide leaves you with a single number, make it this: torque multiplies by the ratio, speed divides by it, power is conserved minus ~5% per mesh. Every gearbox decision — compound staging, module choice, motor matching — is an application of that sentence. Compute it on paper before you cut metal, and the gearbox becomes the most predictable part of your build instead of the most mysterious.