When to use: Enter tooth counts (or sprocket/pulley sizes) for one or more gear stages to find the overall gear ratio, output speed, output torque, and mechanical advantage — accounting for mesh efficiency losses per stage. Add stages to model a compound gear train (e.g. a two-stage speed reducer between a motor and a robot joint).
This calculator computes the output speed, output torque, and mechanical advantage of a gear train from tooth counts (or sprocket/pulley sizes) at each stage, including per-stage mesh efficiency losses. It supports simple single-stage reductions and multi-stage compound gear trains, the configuration most mechatronic systems actually use to get large speed reductions in a compact package.
For a single gear mesh, the gear ratio is the driven gear's tooth count divided by the driver's: ratio = N_driven / N_driver. Speed is inversely proportional to tooth count — a larger driven gear turns slower: ω_out = ω_in / ratio. Torque is directly proportional to ratio, minus mesh losses: T_out = T_in × ratio × η, where η is mesh efficiency. Power is conserved (minus losses): P_out = P_in × η, since power is torque × angular velocity and the ratio's effect on torque exactly offsets its inverse effect on speed.
For a compound gear train — multiple stages in series, such as a motor pinion driving an intermediate shaft that drives a final output gear — the overall ratio is the product of each stage's ratio, and overall efficiency is the product of each stage's efficiency. A two-stage 4:1 + 4:1 train gives a 16:1 overall reduction, not 8:1 — ratios multiply, they don't add. This is why compound trains reach high reductions (50:1, 100:1+) in far less space than a single stage would need.
Spur gears run at roughly 98–99% efficiency per mesh — the dominant loss is sliding friction at the tooth contact. Helical gears are similar or slightly better (98.5%+) due to smoother tooth engagement, at the cost of introducing axial thrust load. Bevel gears (used for right-angle drives) run around 97%. Worm gear sets are the outlier: efficiency ranges from roughly 90% for low-reduction, high-lead-angle worms down to 50% or less for high-reduction, self-locking worm sets — the trade-off is that a self-locking worm gear (typically below ~10:1 lead angle) can hold a load without back-driving, which spur and helical trains generally cannot do without a separate brake. Chain and timing-belt drives run around 96–97%, close to gears, but are used when the two shafts are far apart rather than meshed directly.
A gear train is almost always the missing link between an off-the-shelf DC or stepper motor's native torque/speed curve and what a mechatronic system actually needs at the output shaft — a robot joint typically needs high torque at low speed, while small motors are efficient at high speed and low torque. Use this calculator with the motor's rated torque and speed as the input-shaft values to find the torque and speed actually delivered at the load after your gearbox. Cross-check the result against the load's required torque (see the Actuator Sizing Calculator) with margin for the gearbox's own efficiency losses and any additional service factor for shock loading or duty cycle.
Enter the input torque and speed — typically the motor's rated (or stall/continuous, depending on your design point) values. Add one gear stage per mesh in the train and enter the driver and driven tooth counts (or, for a chain/belt stage, sprocket or pulley tooth/groove counts) along with the gear type, which sets a typical mesh efficiency. Add additional stages for compound trains — the calculator multiplies stage ratios and efficiencies automatically. Read the output torque, output speed, overall ratio, and overall efficiency in the results panel.
Gear ratios multiply across stages in a compound train, not add. If stage one reduces speed 4:1 and stage two reduces the already-slower intermediate shaft another 4:1, the overall reduction from input to output is 4 × 4 = 16:1. This is why even two modest gear stages can produce a large overall reduction in a small package — it's a common source of error to add ratios instead of multiplying them.
A worm gear mesh works by sliding contact along the worm's helical thread rather than the rolling contact of spur/helical gear teeth, which generates significantly more friction. The efficiency depends heavily on the worm's lead angle: low lead angles (high reduction ratios, often 20:1 to 100:1 in a single stage) can drop efficiency to 50% or lower, but the same friction that causes the loss also makes the mesh self-locking — the output shaft can't back-drive the worm, which is valuable in lifting or holding applications where you want the load to stay put with the motor off.
Start from the joint's required torque and speed (from the load — arm length, payload mass, desired angular velocity). Pick a motor with a torque/speed curve that intersects a good efficiency point, then work out the ratio needed: ratio ≈ (required output torque) / (motor torque at that operating point × gearbox efficiency). Enter that ratio here (as one or more real gear stages) to check the resulting output speed matches what the joint actually needs — if it doesn't, adjust the motor choice or add/remove a stage rather than using a single unrealistic single-stage ratio.
Gear ratio (N_driven/N_driver) describes the geometric relationship between speeds. Mechanical advantage is the practical torque multiplication you actually get at the output, which equals the gear ratio multiplied by mesh efficiency (ratio × η) — always slightly less than the raw ratio because of friction losses. For a compound train, use the overall mechanical advantage (product of all stages' ratio × efficiency) to size the output shaft, coupling, and any downstream components that need to handle the full output torque.
Tooth count ratio and diameter ratio give the same gear ratio for gears that mesh correctly, because two meshing gears must share the same module (metric) or diametral pitch (imperial) — that's what makes their teeth compatible. So ratio = N_driven/N_driver = D_driven/D_driver. Tooth counts are usually easier to get from a datasheet or by counting teeth directly, which is why this calculator uses them.
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