When the axis of rotation of a spinning body is itself swivelled, a moment perpendicular to both axes arises that the bearings must carry: the gyroscopic moment MK = J·ωK·ωF. Because two angular velocities are multiplied, it can reach very large values even at a slow guiding motion and must not be neglected.
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When the axis of rotation of a spinning body is itself swivelled, a moment perpendicular to both axes arises that the bearings must carry: the gyroscopic moment MK = J·ωK·ωF. Because two angular velocities are multiplied, it can reach very large values even at a slow guiding motion and must not be neglected.
A rotor with J = 0.5 kg·m² spins at ωK = 3,000 rpm about its own axis. Its axis is simultaneously swivelled at ωF = 0.5 rad/s, perpendicular to the rotor axis. First convert the spin rate: ωK = 3,000·2π/60 = 314.16 rad/s. This gives MK = 0.5 kg·m² · 314.16 1/s · 0.5 1/s = 78.5 N·m. The bearings must carry this moment in addition, and it rotates with the guiding motion. The ratio is notable: at 0.5 rad/s – about one revolution in twelve seconds – the guiding motion is very slow, yet it still produces a substantial moment, because it is multiplied by a spin rate over 600 times larger.
The simplified form of the gyroscopic moment customary in machine dynamics is used. It holds strictly when the axes of spin and guiding motion are perpendicular to each other, and is additionally justified as long as the guiding angular velocity is much smaller than the spin rate – which is almost always the case in practice. If the axes are inclined and the guiding motion is not small, an additional term appears that depends on the difference of the principal moments of inertia and on the angle between the axes; that term is not included here. A symmetric rigid rotor with constant inertia and steady angular velocities is further assumed. Bearing elasticity, unbalance, self-weight, angular accelerations of either motion, nutation and precession as motions in their own right, and the question of how the moment distributes between the individual bearings – which requires the bearing spacing – are not covered.
A fast-spinning rotor resists having its own axis swivelled – with a moment acting across both motions. The bearings must carry this gyroscopic moment in addition, and it is regularly underestimated because it is the product of two angular velocities.
If a body spins at angular velocity ωK about its own axis while that axis is simultaneously swivelled at angular velocity ωF – the so-called guiding motion – a moment arises whose vector is perpendicular to both axes of rotation. It is called the gyroscopic moment. The index K stands for gyroscope (Kreisel) and the index F for guiding (Führung). For mutually perpendicular axes the form customary in machine dynamics applies: MK = J·ωK·ωF, where J is the rotor's mass moment of inertia about its own axis. The gyroscopic moment does no work – it accelerates nothing but loads the bearings alone.
A spinning bicycle front wheel held by its axle resists being tilted, veering surprisingly to the side and pushing noticeably against your hands. The faster it spins and the faster you tilt, the stronger the effect. That resistance is exactly the gyroscopic moment – pleasant on a bicycle, an additional rotating load in a bearing.
MK = J · ωK · ωF
Perpendicular axes of rotation: MK = J · ωK · ωFSpeed in radians per second: ω = 2π·n/60 with n in rpmInertia of a disc: J = ½·m·r²Bearing force from the moment: F = MK/a with the bearing spacing aVector form: the moment vector is perpendicular to both axes of rotation| Symbol / input | Meaning |
|---|---|
| Gyroscopic moment MK | The additional moment the rotor bearings must carry. Its vector is always perpendicular to both axes of rotation, which is why the bearing forces rotate with the guiding motion and are therefore alternating. It adds to all other bearing loads such as self-weight and unbalance, and it is not a drive torque: it does no work but only loads the bearings. |
| Mass moment of inertia J of the rotor | The mass moment of inertia of the rotating body about its own axis of rotation, i.e. the axis it spins about with ωK. The mass moment of inertia describes how strongly a body resists angular acceleration. Source: a formula collection for the basic shape, such as J = ½·m·r² for a disc, or a CAD mass calculation. For a symmetric rotor this is the principal moment of inertia about the axis of rotation. |
| Rotor spin rate ωK | The angular velocity at which the rotor spins about its own axis – in mechanical engineering usually given as a rotational speed in revolutions per minute. The index K stands for the gyroscope (Kreisel). Internally it is converted to radians per second: 3,000 rpm equals 3,000·2π/60 ≈ 314.2 rad/s. This conversion is the most common source of error in hand calculations. |
| Guiding angular velocity ωF | The angular velocity at which the rotor's own axis is being swivelled – the guiding motion. The index F stands for guiding (Führung). Examples: the yaw rate of a ship or vehicle carrying an installed rotor, or the swivel rate of a robot arm holding a spinning tool. This quantity is often small, but it is multiplied by the far larger spin rate – which is why the gyroscopic moment must not be neglected even at a slow guiding motion. |
J is the rotor's mass moment of inertia about its own axis of rotation, for example J = ½·m·r² for a disc or from a CAD mass calculation. ωK is the spin rate, usually given as a rotational speed in revolutions per minute. ωF is the angular velocity of the guiding motion, i.e. the swivel rate of the rotor axis – for instance the yaw rate of a vehicle or ship, or the swivel rate of a robot arm.
First establish which motion is the spin and which the guiding motion, and check that the two axes are approximately perpendicular. Then determine the inertia about the rotor axis and enter both angular velocities – the calculator accepts revolutions per minute and radians per second mixed and converts internally. The result is a moment, not a force: for the bearing forces, divide it by the bearing spacing and add the result with the correct sign to the remaining bearing loads from self-weight and unbalance.
A rotor with J = 0.5 kg·m² spins at 3,000 rpm while its axis is swivelled perpendicular to it at ωF = 0.5 rad/s. The spin rate corresponds to ωK = 3,000·2π/60 = 314.16 rad/s, giving MK = 0.5 · 314.16 · 0.5 = 78.5 N·m. The ratio of magnitudes is revealing: the guiding motion of 0.5 rad/s amounts to one revolution in about twelve seconds and is therefore very slow – yet it still produces nearly 80 N·m, because it is multiplied by a spin rate over 600 times larger. At a bearing spacing of 200 mm that corresponds to about 390 N of additional force at each bearing.
Both angular velocities enter linearly, so their product means that doubling both quadruples the moment. What matters in practice is the asymmetry of magnitudes: spin rates are typically a few thousand revolutions per minute, guiding rates often below one radian per second. That is precisely why it is tempting to dismiss the guiding motion as insignificant – yet the product remains substantial. And because the moment vector rotates with the guiding motion, the bearing forces are not static but alternating, which matters for bearing life calculations.
The calculation runs in kilograms, metres, seconds and therefore newton-metres internally. The inertia may be entered in kg·m² or kg·cm², the angular velocities independently in rad/s or rpm, and the moment in N·m, N·mm, lbf·in or lbf·ft. For hand calculations: a speed in revolutions per minute must be converted to radians per second with 2π/60, otherwise the result is wrong by a factor of 9.55.
Bearing loads of turbines, centrifugal pumps and flywheels in vehicles and ships during cornering or in a swell; tool spindles on swivelling machining heads and robots; gyroscopic stabilisers and ship stabilisation systems; inertial navigation and gyroscopic instruments; helicopter and drone rotors during pitch and yaw motions; balancing machines with a swivelling fixture.
The simplified form of the gyroscopic moment is used. It holds strictly for mutually perpendicular axes of rotation and is additionally justified as long as the guiding angular velocity is much smaller than the spin rate – practically almost always the case. For inclined axes with a non-small guiding motion an additional term appears that depends on the difference of the principal moments of inertia and on the angle between the axes; it is not included here. A symmetric rigid rotor with constant inertia and steady angular velocities is assumed. Bearing elasticity, unbalance, self-weight, angular accelerations, nutation and precession as motions in their own right, and the distribution of the moment between the individual bearings – which requires the bearing spacing – are not covered.
Common mistake: The most common error is a missing unit conversion: inserting the speed in revolutions per minute instead of radians per second makes the result wrong by a factor of 2π/60 ≈ 1/9.55. Second, the gyroscopic moment is often misread as a drive or braking torque – it does no work and need not be supplied by the drive but loads the bearings alone. Third, the guiding motion is neglected because of its small numerical value, although it is precisely its product with the high spin rate that creates the moment. And fourth, it is overlooked that the bearing forces rotate with the guiding motion and are therefore alternating – a material difference from a static load for bearing life.
Bearing loads of turbines, centrifugal pumps and flywheels in vehicles and ships during cornering or in a swell; tool spindles on swivelling machining heads and robots; gyroscopic stabilisers and ship stabilisation systems; inertial navigation and gyroscopic instruments; helicopter and drone rotors during pitch and yaw motions; balancing machines with a swivelling fixture.
J is the rotor's mass moment of inertia about its own axis of rotation, for example J = ½·m·r² for a disc or from a CAD mass calculation. ωK is the spin rate, usually given as a rotational speed in revolutions per minute. ωF is the angular velocity of the guiding motion, i.e. the swivel rate of the rotor axis – for instance the yaw rate of a vehicle or ship, or the swivel rate of a robot arm.
The simplified form of the gyroscopic moment is used. It holds strictly for mutually perpendicular axes of rotation and is additionally justified as long as the guiding angular velocity is much smaller than the spin rate – practically almost always the case. For inclined axes with a non-small guiding motion an additional term appears that depends on the difference of the principal moments of inertia and on the angle between the axes; it is not included here. A symmetric rigid rotor with constant inertia and steady angular velocities is assumed. Bearing elasticity, unbalance, self-weight, angular accelerations, nutation and precession as motions in their own right, and the distribution of the moment between the individual bearings – which requires the bearing spacing – are not covered.