KEVOS® 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). |
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.
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