Gear output speed calculator

Zero or negative gear ratio is rejected with a clear invalid_input error instead of inventing speed from an impossible mesh.

How gear-pair output speed is defined from n_in and GR

Rearranging that identity gives the single division this calculator performs: output shaft speed n_out equals input shaft speed n_in divided by GR. Velocity ratio is reported as the reciprocal of GR so you can write n_out equals n_in times velocity_ratio if that form matches your notes. This endpoint does not count teeth; if you only know tooth numbers, compute GR first or use a dedicated gear-ratio tool, then pass the ratio here. The response always echoes the rounded inputs, publishes output_speed, velocity_ratio, transmission class, formula, message, and steps, and never invents a second stage or a unit conversion. Keep n_in and n_out in the same unit—RPM with RPM, rad/s with rad/s—because the engine is unitless arithmetic by design.

Input contract, response shape, and error behaviour

Prefer sending numbers rather than pre-formatted strings when you control the client; when you must send strings, avoid unit labels such as RPM that would fail the finite-number pattern. Send a JSON body with two required fields: input_speed for the driver shaft speed n_in (aliases n_in, rpm_in, omega_in, driver_speed, nin) and gear_ratio for the positive pair ratio GR (aliases ratio, gr, R, i, reduction_ratio). Both must be finite numbers; gear_ratio must be strictly positive within the published absolute range so the division remains a finite IEEE double. Optional precision (aliases decimals, dp) chooses how many decimal places to expose for numeric fields, defaulting to twelve and capped at fifteen so JSON goldens stay stable across runtimes. String forms of numbers are accepted when they match a strict finite decimal pattern with optional scientific notation; embedded spaces, unit suffixes, and non-numeric tokens fail with a clear invalid_input message naming the field. On success the response echoes the rounded inputs, publishes output_speed as n_in divided by GR, velocity_ratio as one over GR, transmission as reduction, overdrive, or 1:1, formula and message as English summary strings, and steps as the substitution log you can assert in tests or show next to a sketch. Failed validation—missing fields, non-finite numbers, zero or negative gear_ratio, or values outside absolute limits—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item. Pair this endpoint with a gear-ratio calculator when you start from tooth counts, or with a torque-ratio check when you need ideal torque multiplication; this page answers only output shaft speed from known n_in and GR.

Where gear output speed fits in design, study, and software

Students meet gear-pair output speed when a machine-design course introduces spur gears, when a lab asks for driven RPM from motor nameplate speed and a stated reduction, or when a kinematics problem needs n_out before power or efficiency estimates. Product and education tools use the rewrite when a learner types n_in and GR and the UI must show n_out with a clean formula string. On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split—free interactive exploration and a priced, identical API—matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result.