Calculate gyroscopic torque, spin rate, precession rate, or moment of inertia from any three values with unit conversions in rad/s, rpm, N·m, or lbf·ft.
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Gyroscopic Torque Formula
The gyroscopic torque, also called gyroscopic couple, is found from spin-axis inertia, spin rate, and precession rate. This scalar model assumes perpendicular spin and precession axes, steady precession, spin-dominated angular momentum, and negligible nutation. For a general axis angle θ, the torque magnitude includes sin(θ).
- τ = gyroscopic torque or couple
- I = moment of inertia about the spin axis
- ω = spin rate of the rotating body
- Ω = precession rate
When solving for a missing value, the same formula is rearranged as needed:
The calculator converts all inputs to base SI units before applying the formula: moment of inertia in kg·m², angular speeds in rad/s, and torque in N·m. If you choose rpm, lbf·ft, or lbm·ft², the value is converted first, then the final answer is converted back to your selected output unit.
- Calculate torque: enter moment of inertia, spin rate, and precession rate.
- Calculate moment of inertia: enter torque, spin rate, and precession rate.
- Calculate spin rate: enter torque, moment of inertia, and precession rate.
- Calculate precession rate: enter torque, moment of inertia, and spin rate.
Common Unit Conversions for Gyroscopic Torque Calculations
| Quantity | From | To base unit |
|---|---|---|
| Moment of inertia | 1 lbm·ft² | 0.0421401101 kg·m² |
| Angular speed | 1 rpm | 0.104719755 rad/s |
| Torque | 1 lbf·ft | 1.355817948 N·m |
| Change in input | Effect on gyroscopic torque |
|---|---|
| Double the moment of inertia | Torque doubles |
| Double the spin rate | Torque doubles |
| Double the precession rate | Torque doubles |
| Any one factor is zero | Torque is zero |
Example Calculations
Example 1: Calculate gyroscopic torque
Suppose a rotor has a moment of inertia of 0.25 kg·m², a spin rate of 300 rad/s, and a precession rate of 2 rad/s.
The gyroscopic torque is 150 N·m.
Example 2: Calculate precession rate
Suppose the torque is 80 N·m, the moment of inertia is 0.5 kg·m², and the spin rate is 400 rad/s.
The precession rate is 0.4 rad/s.
FAQ
What is the difference between spin rate and precession rate?
The spin rate, ω, is how fast the object rotates about its own spin axis. The precession rate, Ω, is how fast that spin axis changes direction. Both are angular speeds, but they describe different motions.
Why do angular speeds need to be in rad/s?
The formula τ = I * ω * Ω is dimensionally consistent when angular speeds are in radians per second. If you enter rpm, it must be converted to rad/s before the formula is applied.
Can gyroscopic torque be negative?
In full vector form, gyroscopic torque has direction and can be positive or negative depending on the chosen coordinate system. This calculator uses nonnegative scalar magnitudes. It does not determine vector direction, a unique net force, or bearing reactions.
