BIZOMAT MAKEOS · Mechanics
Gyroscopic Couple and Bearing Reactions
What gyroscopic couple does this spinning rotor exert when its axis is turned, and what load does it put on the bearings?
Gyroscopic couple of a spinning rotor whose axis precesses (turns) at a given rate, from the rotor’s moment of inertia, spin speed and precession speed, and the resulting pair of bearing reactions.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Rotor | choice | Shape used to estimate the moment of inertia, or enter it. |
| Rotor mass | kg | Mass of the rotor. |
| Rotor outer radius | m | Outer radius of the rotor. |
| Moment of inertia (entered) | kg·m² | Polar moment of inertia of the rotor about its spin axis. |
| Spin speed | rpm | Speed of the rotor about its own axis. |
| Precession speed (axis turning rate) | rpm | Rate at which the spin axis itself is turned (for example a slewing or turning rate). |
| Bearing span | mm | Distance between the two bearings carrying the rotor. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Rotor
- What it is
- How the rotor’s moment of inertia about its spin axis is found: from mass and radius as a uniform solid disc or a thin ring, or typed in directly.
- Units and format
- Choose one of 3 options from the list: Solid disc; Thin ring; Enter the moment of inertia.
- Example
- The form opens with “Solid disc” selected as a worked example. Choose the option that matches your case.
- Get it right
- Real rotors with hubs, webs and spokes lie between the disc and ring cases; enter the inertia from CAD for an accurate result.
What this means: Rotor mass
- What it is
- The mass of the spinning rotor, used with the outer radius when the rotor is treated as a solid disc or a thin ring.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0.000001 kg to 10000000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The inertia, and so the gyroscopic couple, rises in proportion.
What this means: Rotor outer radius
- What it is
- The outer radius of the rotor, from the spin axis to the rim. For a thin ring the whole mass is taken as concentrated at this radius.
- Units and format
- Enter a number in m: metres. The form accepts 0.000001 m to 100 m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.2 m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The inertia, and so the couple, rises with the square of the radius.
- Get it right
- Radius, not diameter: halve a measured diameter.
What this means: Moment of inertia (entered)
- What it is
- The polar mass moment of inertia of the rotor about its own spin axis, used only when the rotor option is set to enter the moment of inertia.
- Units and format
- Enter a number in kg·m²: kilogram square metres (mass moment of inertia). The form accepts 0.000001 kg·m² to 100000000 kg·m². Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.4 kg·m² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Take it from CAD mass properties or the supplier, and check it is about the spin axis rather than a diameter.
What this means: Spin speed
- What it is
- The rotor’s speed about its own axis, for example the motor, flywheel or wheel speed.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0.000001 rpm to 1000000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3000 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Angular momentum and the gyroscopic couple rise in proportion.
What this means: Precession speed (axis turning rate)
- What it is
- The rate at which the spin axis itself is swung round, such as a vehicle turning, a crane slewing or a boat pitching.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0.000001 rpm to 100000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The couple and both bearing reactions rise in proportion.
- Get it right
- To convert a turning rate in degrees per second to rpm, divide it by 6.
What this means: Bearing span
- What it is
- The distance along the shaft between the centres of the two bearings carrying the rotor; the gyroscopic couple is reacted there as two equal and opposite forces.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 300 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The bearing reactions are inversely proportional to the span; the couple itself is unchanged.
Method and formulas
- Angular momentum H = J ω_s.
- Gyroscopic couple = J ω_s Ω_p, about the axis perpendicular to both the spin and precession axes.
- Each bearing carries ± couple ÷ span, in opposite directions.
- Example: a 20 kg disc at 3000 rpm turned at 10 rpm exerts 131.59 N·m.
Limitations
- Steady precession at right angles to the spin axis.
- Add the gyroscopic reactions to the weight and other bearing loads.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check bearing loads on a spindle mounted on a slewing or turning machine
- Estimate gyroscopic loads on vehicle or ship rotors
- Size a gyroscopic stabiliser
Who it is for
- Design engineers
- Mechanical engineers
- Engineering students
What you get
- Gyroscopic couple
- Bearing reaction pair
- Angular momentum
Interpreting the result
- The couple grows with both speeds; slowing the turning rate is often the easiest way to reduce it.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2998. Topic reference; no source tables or recommended settings are reproduced.
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