BIZOMAT MAKEOS · Mechanics
Composite Rotor Inertia and Energy (Up to Three Shapes, Offsets, Stored and Released Energy)
For a rotor built up from rings, tori, cones, shells or tubes, what is its total moment of inertia, how much energy does it store at speed, and is the energy given up for my allowed slow-down enough for the duty?
Inertia and energy of a rotating assembly of up to three bodies: the moment of inertia of each body about its own axis or a diameter, moved to the rotation axis by its offset (as the more mass moments of inertia tool), the total inertia and radius of gyration, the kinetic energy at speed and the energy released for the allowed speed drop, against the energy each cycle of the duty needs.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Body 1 | ||
| Body and axis | choice | Shape of body 1 and the axis its inertia is taken about. |
| Mass (0 = unused) | kg | Mass of body 1 (0 to leave it out). |
| Dimension A | mm | Dimension A of body 1 (see the shape). |
| Dimension B | mm | Dimension B of body 1. |
| Dimension C | mm | Dimension C of body 1. |
| Offset of its axis from the rotation axis | mm | Distance between body 1’s axis and the rotation axis (parallel axis). |
| Body 2 | ||
| Body and axis | choice | Shape of body 2. |
| Mass (0 = unused) | kg | Mass of body 2. |
| Dimension A | mm | Dimension A of body 2. |
| Dimension B | mm | Dimension B of body 2. |
| Dimension C | mm | Dimension C of body 2. |
| Offset of its axis from the rotation axis | mm | Offset of body 2. |
| Body 3 | ||
| Body and axis | choice | Shape of body 3. |
| Mass (0 = unused) | kg | Mass of body 3. |
| Dimension A | mm | Dimension A of body 3. |
| Dimension B | mm | Dimension B of body 3. |
| Dimension C | mm | Dimension C of body 3. |
| Offset of its axis from the rotation axis | mm | Offset of body 3. |
| Duty | ||
| Speed | rpm | Running speed. |
| Allowed speed drop | % | How far the rotor may slow while giving up energy. |
| Energy needed per cycle | J | Energy the duty takes from the rotor each cycle. |
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.
Body 1
What this means: Body and axis
- What it is
- The shape of body 1 and the axis its own inertia is taken about. The list shows what dimensions A, B and C mean for that shape.
- 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
- Pick the axis that is parallel to the rotation axis; the offset is measured between those two axes.
What this means: Mass (0 = unused)
- What it is
- The mass of body 1. Set it to 0 to leave the body out.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0 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
- Its inertia rises in proportion to its mass.
- Get it right
- Weigh the part or take it from the CAD model with the correct material density.
What this means: Dimension A
- What it is
- The first dimension of body 1 as listed for its shape: base diameter of a cone, outer diameter of a sphere, centre-line diameter of a torus, diameter of a ring or tube, or the full axis A of an ellipsoid.
- 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.
- Effect on the result
- For most shapes the inertia rises with the square of this dimension.
- Get it right
- All dimensions are full diameters or full axes, not radii.
What this means: Dimension B
- What it is
- The second dimension of body 1: cone height for a cone about its base diameter, inner diameter of a hollow sphere, tube diameter of a torus, or a second full axis of an ellipsoid. Other shapes ignore 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 60 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored by the cone about its axis, thin shell, ring and tube; for a hollow sphere it must be smaller than A.
What this means: Dimension C
- What it is
- The third dimension of body 1, used only by the ellipsoid as its other full axis at right angles to A and B.
- 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
- Leave it at 0 for every shape except the ellipsoid.
What this means: Offset of its axis from the rotation axis
- What it is
- The distance between the axis body 1’s own inertia is taken about and the rotation axis of the assembly. The tool adds mass times offset squared.
- 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 rises with the square of the offset.
- Get it right
- Leave it 0 when the body is centred on the rotation axis; the two axes must be parallel.
Body 2
What this means: Body and axis
- What it is
- The shape of body 2 and the axis its own inertia is taken about. The list shows what dimensions A, B and C mean for that shape.
- 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 “Thin-walled cylinder about its axis (A = dia)” selected as a worked example. Choose the option that matches your case.
- Get it right
- Pick the axis that is parallel to the rotation axis.
What this means: Mass (0 = unused)
- What it is
- The mass of body 2. Set it to 0 to leave the body out.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0 kg to 10000000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Weigh the part or take it from the CAD model.
What this means: Dimension A
- What it is
- The first dimension of body 2 as listed for its shape: a cone base diameter, sphere outer diameter, torus centre-line diameter, ring or tube diameter, or ellipsoid axis A.
- 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 100 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter full diameters or full axes, not radii.
What this means: Dimension B
- What it is
- The second dimension of body 2: cone height (cone about its base diameter), inner diameter of a hollow sphere, tube diameter of a torus, or a second ellipsoid 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.
- Get it right
- Ignored by shapes that list only dimension A.
What this means: Dimension C
- What it is
- The third dimension of body 2, used only when body 2 is 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 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Leave it at 0 for every shape except the ellipsoid.
What this means: Offset of its axis from the rotation axis
- What it is
- The distance between the axis body 2’s own inertia is taken about and the rotation axis of the assembly.
- 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
- Leave it 0 when the body is centred on the rotation axis.
Body 3
What this means: Body and axis
- What it is
- The shape of body 3 and the axis its own inertia is taken about. The list shows what dimensions A, B and C mean for that shape.
- 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 “Solid cone about its axis (A = base dia)” selected as a worked example. Choose the option that matches your case.
- Get it right
- Pick the axis that is parallel to the rotation axis.
What this means: Mass (0 = unused)
- What it is
- The mass of body 3. Set it to 0 to leave the body out.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0 kg to 10000000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Weigh the part or take it from the CAD model.
What this means: Dimension A
- What it is
- The first dimension of body 3 as listed for its shape: a cone base diameter, sphere outer diameter, torus centre-line diameter, ring or tube diameter, or ellipsoid axis A.
- 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 150 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter full diameters or full axes, not radii.
What this means: Dimension B
- What it is
- The second dimension of body 3: cone height (cone about its base diameter), inner diameter of a hollow sphere, tube diameter of a torus, or a second ellipsoid 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.
- Get it right
- Ignored by shapes that list only dimension A.
What this means: Dimension C
- What it is
- The third dimension of body 3, used only when body 3 is 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 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Leave it at 0 for every shape except the ellipsoid.
What this means: Offset of its axis from the rotation axis
- What it is
- The distance between the axis body 3’s own inertia is taken about and the rotation axis of the assembly.
- 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
- Leave it 0 when the body is centred on the rotation axis.
Duty
What this means: Speed
- What it is
- The running speed of the rotor before it gives up energy to the duty, used for the stored energy and the energy released.
- 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 1500 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Stored and released energy rise with the square of the speed.
- Get it right
- Use the speed under load before the work stroke, not the no-load motor speed if they differ.
What this means: Allowed speed drop
- What it is
- How far the rotor may slow during each work cycle while it gives up energy, as a share of the running speed.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 0.1% to 90%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger drop releases more energy from the same inertia.
- Get it right
- Type the percentage itself; the drop your process tolerates comes from the machine or motor maker.
What this means: Energy needed per cycle
- What it is
- The energy the duty takes from the rotor in each cycle, for example the work of one punching, pressing or crushing stroke.
- Units and format
- Enter a number in J: joules. The form accepts 0 J or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 800 J as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter it in joules (N·m); 1 kJ = 1000 J.
Method and formulas
- Inertia of each shape about its axis (as the more mass moments of inertia tool), plus m × offset² (parallel axis).
- Total J and radius of gyration √(J ÷ m).
- Stored energy = ½ J ω²; released for a drop from ω₁ to ω₂ = ½ J (ω₁² − ω₂²).
- Example: J = 0.42263 kg·m² stores 5,213.9 J and gives up 990.65 J for a 10% drop.
Limitations
- Rigid bodies of uniform density; hubs, spokes and bolts add inertia not counted.
- Energy drawn faster than the motor can replace it.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the inertia of a built-up flywheel or rotor
- Check a rotor stores enough energy for a punch or shear
- Size a motor start from the total inertia
Who it is for
- Design engineers
- Machine designers
- Engineering students
What you get
- Inertia of each body and in total
- Stored energy and energy released
- Against the duty
Interpreting the result
- Mass at large radius dominates: a thin rim or torus gives far more inertia per kilogram than a solid disc.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 228–240, 182–186. 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
Built-up rotors and datasheet units
In Inertia and Flywheels by the Numbers
What it covers: How to add the parts of a built-up rotor into one inertia with its stored energy, and how to convert the inertias on motor and servo datasheets, including WR² and GD², without a factor-of-four error.
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