Enter the moment of inertia, spin rate, and precession rate into the calculator to determine the gyroscopic torque (gyroscopic couple).

Gyroscopic Torque (Couple) Calculator

Enter any 3 values to calculate the missing variable

Gyroscopic Torque Formula

The following equation is used to calculate the gyroscopic torque (also called the gyroscopic couple or gyroscopic moment) for a spinning body undergoing precession.

\tau_g = I\,\omega\,\Omega
  • Where Ï„g is the gyroscopic torque (N·m)
  • I is the moment of inertia about the spin axis (kg·m²)
  • ω is the spin angular speed (rad/s)
  • Ω is the precession angular speed (rad/s)

To calculate the gyroscopic torque, multiply the moment of inertia by the spin angular speed and the precession angular speed.

What is Gyroscopic Torque?

Definition:

Gyroscopic torque (often called a gyroscopic couple) is the torque associated with changing the direction of a spinning object’s angular momentum. For a rotor with angular momentum L, an applied torque causes L to change direction according to Ï„ = dL/dt.

A gyroscope typically consists of a rapidly spinning rotor mounted in bearings (often with gimbals) so it can change orientation. When an external torque acts on the rotor (for example, due to gravity acting at an offset), the gyroscope responds by precessing, and the magnitudes are commonly related by τg = I·ω·Ω.

How to Calculate Gyroscopic Torque?

Example Problem:

The following example outlines the steps and information needed to calculate gyroscopic torque.

First, determine the moment of inertia about the spin axis. In this example, I = 0.25 kg·m².

Next, determine the spin rate. In this case, ω = 3000 rpm, which is ω = 3000 × (2π/60) = 314.159 rad/s.

Then, determine the precession rate. In this case, Ω = 2 rpm, which is Ω = 2 × (2π/60) = 0.209440 rad/s.

Finally, calculate the gyroscopic torque using the formula above:

τg = I·ω·Ω

τg = (0.25 kg·m²)(314.159 rad/s)(0.209440 rad/s)

τg = 16.493 N·m

FAQ

What is the significance of the precession rate in calculating gyroscopic torque?
The precession rate (Ω) is the rate at which the spin axis changes direction. In the common relationship τ = I·ω·Ω, the required/associated gyroscopic torque is directly proportional to Ω. Faster precession requires a larger torque for the same rotor inertia and spin rate.

How does the moment of inertia affect gyroscopic torque?
The moment of inertia (I) measures how the rotor’s mass is distributed about the spin axis. For the same spin rate and precession rate, a larger I produces a larger gyroscopic torque because τ is directly proportional to I.

Can gyroscopic torque be negative, and what would that indicate?
Torque is a vector quantity. Depending on the sign convention and coordinate system, the computed value can be negative, indicating the torque direction is opposite the chosen positive direction. The magnitude is typically discussed as |τ| = I·|ω|·|Ω|.