BIZOMAT MAKEOS · Mechanics
Mass Moment of Inertia: Cone, Torus, Ellipsoid and Shells
What is the mass moment of inertia and radius of gyration of this body?
Mass moment of inertia and radius of gyration of a solid cone, hollow sphere, thin spherical shell, torus, ellipsoid, thin ring about a diameter and thin-walled cylinder, with a parallel-axis offset.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Body and axis | choice | Body and axis; the option text names the dimensions. |
| Mass | kg | Mass of the body. |
| Dimension A | mm | First dimension as described. |
| Dimension B | mm | Second dimension where needed. |
| Dimension C | mm | Third dimension where needed (ellipsoid). |
| Parallel axis offset | mm | Distance from the body’s own axis to the rotation axis (0 if it rotates about its own axis). |
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: Body and axis
- What it is
- The body and the axis it spins about. Each option names the axis and what Dimensions A, B and C mean.
- Units and format
- Choose one of 8 options from the list: Solid cone about its axis (A = base dia); Solid cone about a base diameter (A = base dia, B = height); Hollow sphere about a diameter (A = outer dia, B = inner dia); Thin spherical shell about a diameter (A = dia); Torus about its axis (A = centre-line dia, B = tube dia); Ellipsoid about axis A (B, C = the other full axes); Thin ring about a diameter (A = dia); Thin-walled cylinder about its axis (A = dia).
- Example
- The form opens with “Torus about its axis (A = centre-line dia, B = tube dia)” selected as a worked example. Choose the option that matches your case.
- Get it right
- A cone about its own axis and about a base diameter give very different values; pick the axis it really turns on.
What this means: Mass
- What it is
- The mass of the whole body, from weighing, CAD mass properties, or volume times density.
- 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 10 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The moment of inertia is directly proportional to mass; the radius of gyration does not change.
- Get it right
- Use kg; grams must be divided by 1000.
What this means: Dimension A
- What it is
- The main size of the body, given as a diameter: the cone base, the sphere or shell outside, the torus centre line, the ring or the thin cylinder. For an ellipsoid it is the full axis it spins about.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 400 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter the diameter, not the radius; the tool halves it. For an ellipsoid A does not change the result.
What this means: Dimension B
- What it is
- The second size where the body has one: the height of a cone turned about a base diameter, the inner diameter of a hollow sphere, the tube diameter of a torus, or one of the other two full axes of an ellipsoid.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 60 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- For a hollow sphere it must be smaller than A. Bodies that do not name B ignore it.
What this means: Dimension C
- What it is
- The third full axis of an ellipsoid, at right angles to both A and B. Only the ellipsoid uses it.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter the full axis length, not the semi-axis.
What this means: Parallel axis offset
- What it is
- The distance between the body’s own axis and a parallel axis it actually rotates about, used to add the parallel axis term m d². Leave it at 0 if it spins about its own axis.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The added inertia grows with the square of the offset.
- Get it right
- The axes must be parallel; the offset is measured square to both.
Method and formulas
- Cone about axis 3mr²/10; about a base diameter m(3r²/20 + h²/10).
- Hollow sphere 2m(R⁵ − r⁵) ÷ 5(R³ − r³); thin shell 2mr²/3; torus m(R² + ¾r²) (R centre-line, r tube radius).
- Ellipsoid about axis a: m(b² + c²)/5 (semi-axes); thin ring about a diameter mr²/2; thin cylinder mr².
- Offset axis J = J₀ + m d².
- Example: a 10 kg torus of 400 mm centre-line diameter has J = 0.40675 kg·m².
Limitations
- Uniform density and ideal shapes.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size motors for rotating bodies of these shapes
- Find flywheel and rotor inertias
- Check CAD mass properties
Who it is for
- Automation engineers
- Design engineers
What you get
- Moment of inertia about the body axis and the offset axis
- Radius of gyration
Interpreting the result
- Combine with the reflected inertia tool to size a drive.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 228–240. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Moments of inertia of machine parts
In Inertia and Flywheels by the Numbers
What it covers: How to calculate the moment of inertia and radius of gyration of discs, rings, tubes, blocks and offset parts, and why the mass far from the axis is the mass that counts.
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