You need an overall ratio for a multi-stage compound train, a simple path with idlers, or the ratio that hits a target output RPM. This pad takes up to four compound stages with per-stage efficiency, or a simple driver/driven pair plus idlers, plus input speed and torque, then returns overall ratio, train value, cascade speeds, output torque, and rotation direction.
Defaults open on Compound train: two stages 20T/60T × 20T/60T, 1440 rpm, 10 N·m, 95% efficiency per stage. CALCULATE returns overall i = 9:1, output 160 rpm, output torque 81.225 N·m, and same-as-input direction (two meshes). Switch to Simple + idlers for a single mesh with direction extras, or Target RPM to reverse-solve required ratio and search nearby integer tooth pairs.
It sits under Mechanical Calculators next to the gear ratio calculator (pair / two-stage first pass) and motor gearbox calculator. Math stays in your browser.
Formula
- Compound stage ratio: i_k = Z_driven,k / Z_driver,k. Overall i = Π i_k (up to four stages).
- Train value = 1 / i = n_out / n_in (ideal, before losses).
- Output speed n_out = n_in / i. Cascade: after stage k, rpm_k = rpm_{k-1} / i_k.
- Output torque T_out = T_in × i × η_total, with η_total = Π η_k (enter each stage as percent).
- Simple train: i from first and last gears only. Idlers add meshes and flip direction; they do not change i.
- Target RPM: i_req = n_in / n_out. Equal stage split i_stage ≈ i_req^(1/N). Tooth search finds nearby integer pairs.
- Direction: odd mesh count → opposite to input; even → same. Mesh count = stages + extra idlers.
Reproduce the default two-stage compound path on CALCULATE:
| Quantity | Value |
|---|---|
| Stage 1 / Stage 2 | 20/60 × 20/60 |
| Stage ratios | 3 × 3 |
| Overall i | 9 : 1 |
| η per stage | 95% → η_total 90.25% |
| Input | 1440 rpm · 10 N·m |
| Output speed | 160 rpm |
| Output torque | 81.225 N·m |
How it works
Builds a multi-stage gear train: compound (up to 4 stages with per-stage efficiency), simple train with idlers for direction, or target RPM to required ratio with a clock-style tooth-pair search. Returns overall ratio, train value, cascade speeds, output torque, and rotation direction.
Pick Input mode: Compound train (default), Simple train + idlers, or Target RPM → required ratio. Each mode has its own live 3D sketch (compound same-shaft or cascade, simple path with idlers, or n_in → i_req → n_out flow). Play/Pause on the diagram. For compound, set 1–4 stages and enter driver/driven teeth plus efficiency per stage. CALCULATE fills Results. Editing a field or changing mode clears Results. RESET restores the 20/60 × 20/60 defaults.
Compound stages multiply. Package size stays sane.
In a compound train, a shaft carries the driven gear of one mesh and the driver of the next, so ratios multiply: (Z2/Z1) × (Z4/Z3). Two 3:1 stages give 9:1 overall without a single huge wheel.
This pad lets you stack one to four stages with a separate efficiency on each mesh. Cascade RPM after every stage is listed in Results so you can check intermediate shafts before you freeze a housing layout.
Default compound path vs a simple pair:
| Layout | Tooth path | Overall i |
|---|---|---|
| Compound (default) | 20/60 × 20/60 | 9.0 |
| Simple pair | 20 → 180 | 9.0 (one huge gear) |
| Three stage example | 20/40 × 20/60 × 25/50 | 12.0 |
Default CALCULATE path: i = 3 × 3 = 9, n_out = 1440/9 = 160 rpm, T_out = 10 × 9 × 0.95 × 0.95 = 81.225 N·m.
Two external meshes without extra idlers leave the output turning the same way as the input.
Idlers flip direction. They do not change the ratio.
In a simple train, only the gears that change speed enter the ratio product. An idler between driver and driven cancels from the magnitude of i and adds one mesh, so rotation flips again.
Use Simple train + idlers when the sketch is a single mesh with spacers for center distance or direction. Set Extra idlers on any mode when you add those meshes outside the counted stages.
20T → idler → 60T keeps i = 3. Two meshes → output same direction as input.
Odd mesh count without that idler would reverse the output.
Target RPM reverse-solves ratio. Tooth search suggests pairs.
Clock and instrument designers often hunt integer tooth counts for a hard ratio. This pad's Target RPM mode first computes i_req = n_in / n_out, splits equal stages as i_req^(1/N), then searches nearby integer pairs (pinions 12–40, wheels 18–120) for a close product.
Treat suggestions as a starting sketch, not a finished gearbox. Confirm module, center distance, and strength on the spur or helical pads before you cut metal.
1440 rpm in and 160 rpm out → i_req = 9. Two equal stages → about 3:1 each.
Paste a suggested pair into Compound mode to verify cascade RPM and torque with your efficiencies.
Efficiency stacks. Stress sizing is a different pad.
Ideal torque scales with i. Real meshes lose a few percent to friction and churning. Enter each compound stage efficiency (defaults 95%). Path efficiency is the product. A 9:1 train at 95% × 95% turns 10 N·m into 81.225 N·m, not 90.
This pad does not size tooth bending, contact stress, worm starts, or planetary tables. After you lock i and speeds, continue with spur/helical geometry or a catalog gearbox. Pinions under about 17 teeth on standard full-depth spur profiles can undercut — the pad warns when a stage driver is under 17.
Pair pad vs train pad
Use the gear ratio calculator for a quick pair by teeth or pitch diameter, or a fixed two-stage compound with one path efficiency. Use this gear train calculator when you need three or four stages, per-stage η, cascade speeds, idler-aware simple trains, or a target-RPM reverse solve with tooth suggestions.
Neither pad is a vehicle MPH tool. For motor power after you know load torque and RPM, open the motor gearbox calculator.
Worked example
Default Compound train: 20/60 × 20/60, 1440 rpm, 10 N·m, 95% per stage. Reproduce on CALCULATE.
- Leave Input mode on Compound train with 2 stages. Confirm 20/60 and 20/60, efficiencies 95, 1440 rpm, 10 N·m, idlers 0.
- CALCULATE. Stage ratios 3 and 3. Overall i = 9.
- Cascade: 1440 → 480 rpm after stage 1 → 160 rpm out.
- η_total = 0.95 × 0.95 = 0.9025. T_out = 10 × 9 × 0.9025 = 81.225 N·m.
- Direction same as input (two meshes). Optional: Target RPM 1440 → 160 also returns i_req = 9.
Result: Default: ratio 9.0:1, 160 rpm out, 81.225 N·m, same direction.
When to use
- Sizing a two- to four-stage compound reducer before housing layout
- Checking cascade RPM on intermediate shafts
- Estimating output torque with per-stage efficiency
- Simple trains where idlers set direction or span
- Finding required ratio from target output RPM and nearby tooth pairs
Limitations
- Spur/external mesh model. No planetary, worm, or bevel kinematics.
- Maximum four compound stages in the UI.
- Does not size tooth bending, contact stress, backlash, or center distance.
- Tooth-pair search is a bounded integer screen, not a full clock DFS with shaft constraints.
- Not a vehicle MPH tool (no axle ratio or tire height).
- Not a substitute for a signed gearbox selection on safety-critical drives.
FAQ
- How is this different from the gear ratio calculator?
- Gear ratio covers pair-by-teeth, pair-by-diameter, and a fixed two-stage compound with one efficiency. This pad adds up to four stages, per-stage η, cascade RPM, simple+idler mode, and target-RPM reverse solve with tooth suggestions.
- How do you calculate a multi-stage gear train ratio?
- Multiply stage ratios. Each stage is driven teeth ÷ driver teeth. Overall i = i1 × i2 × … . Output speed is n_in / i.
- Do idler gears change gear ratio?
- No. In a simple train, idlers change direction and spacing only. Only compound gears fixed on the same shaft multiply the ratio.
- What does train value mean?
- Train value is 1/i, equal to ideal n_out/n_in. Some textbooks quote train value instead of reduction ratio.
- How does efficiency affect torque here?
- T_out = T_in × i × η_total. On the default 9:1 with 95% per stage, η_total = 90.25% and 10 N·m becomes 81.225 N·m.
- What does Target RPM mode do?
- It sets i_req = n_in / n_out, shows an equal-stage split, and searches nearby integer tooth pairs you can paste into Compound mode.
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