KEVOS® MAKEOS · Mechanics
Stacked Solid Centre of Gravity (Cylinders, Frustums, Cones, Hemispheres, Tipping Angle)
For a vessel, hopper or column made of stacked round solids (or the material filling them), where is its centre of gravity, what does it weigh, and at what tilt would it tip over its base?
Centre of gravity of an upright stack of up to three round solid parts — cylinders, frustums, cones and hemispheres, either way up — from the bottom upwards: the volume and centroid height of each part (as the centroid of shapes tool), its mass from its density, the combined mass and centre of gravity height including any height of legs or base below, and the angle of tilt at which the centre of gravity passes over the edge of the base, against your least acceptable tipping angle.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Part 1 (from the bottom) | ||
| Shape | choice | Shape of the bottom part. |
| Diameter D₁ | mm | Diameter of part 1 (base, or top for a cone point down). |
| Diameter D₂ (frustum top) | mm | Top diameter of part 1 (frustums only). |
| Height (cylinders, frustums, cones) | mm | Height of part 1. |
| Density | kg/m³ | Density of part 1 (of the contents for a filled vessel). |
| Part 2 (from the bottom) | ||
| Shape | choice | Shape of part 2, sitting on part 1. |
| Diameter D₁ | mm | Diameter of part 2. |
| Diameter D₂ (frustum top) | mm | Top diameter of part 2 (frustums only). |
| Height (cylinders, frustums, cones) | mm | Height of part 2. |
| Density | kg/m³ | Density of part 2. |
| Part 3 (from the bottom) | ||
| Shape | choice | Shape of part 3, sitting on part 2. |
| Diameter D₁ | mm | Diameter of part 3. |
| Diameter D₂ (frustum top) | mm | Top diameter of part 3 (frustums only). |
| Height (cylinders, frustums, cones) | mm | Height of part 3. |
| Density | kg/m³ | Density of part 3. |
| Base | ||
| Height of the bottom of the stack above the base | mm | Height of legs or plinth below the stack. |
| Base width (smallest, across the tipping edge) | mm | Width of the base across the edge it would tip over. |
| Least tipping angle accepted | ° | Smallest tilt the stack must survive without tipping. |
Method and formulas
- Centroid heights: cone h ÷ 4 from its base; frustum h (R² + 2Rr + 3r²) ÷ 4(R² + Rr + r²) from the large end; hemisphere 3r ÷ 8 from the flat face (as the centroid of shapes tool); cylinder h ÷ 2.
- Mass = volume × density; CG height = Σ m z ÷ Σ m plus the base height.
- Tipping angle = atan ((base width ÷ 2) ÷ CG height).
- Example: 1,649.3 kg with its CG 1,892.9 mm up; it tips at 17.588°.
Limitations
- Solid, uniformly dense parts; for a vessel, add the shell as separate parts or masses.
- Static tilt; wind, sloshing and acceleration reduce the margin.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the CG of a full hopper on legs
- Check a column or vessel is stable on its base
- Estimate the weight of a filled bin
Who it is for
- Design engineers
- Riggers
- Plant engineers
What you get
- Volume, mass and centroid of each part
- Combined mass and CG height
- Tipping angle
Interpreting the result
- Raising a full hopper on taller legs lowers the tipping angle quickly; widening the leg spread restores it.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 221–227, 66–76. Topic reference; no source tables or recommended settings are reproduced.
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