First whirling speed of a loaded shaft, with its own weight included.
Or enter its mass below. Empty = bare shaft.
Empty = at mid-span.
First estimate only: bearings and housings are treated as rigid, so the true critical speed is lower than this result. Do not run at or above it. Shafts on machines that can injure people, or that run near this speed, need a rotordynamic analysis and a test.
Example Result
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The calculated result will appear here after you press Calculate.
Every shaft sags a little, and its centre of mass never sits exactly on its axis. At one particular speed the two effects line up: the unbalance force pushes the shaft outward as fast as it can spring back, and the deflection grows. That is the first critical, or whirling, speed. A simple way to estimate it comes from how far the shaft sags under its own loads: n_c = 946 ÷ √δ, with δ in mm and n in rpm. Enter the span, the diameter, the material and the loads, and the calculator gives the critical speed with the shaft's own weight included. The answer assumes rigid bearings, so the real figure will be lower.
Choose a shaft between two bearings, or an overhung shaft.
Enter the length and the diameter, and choose the material.
Enter the weight of each gear, pulley or rotor and where it sits. Leave them empty for a bare shaft.
For an overhung shaft, add the bearing span.
Add your operating speed to see how close it is.
Press Calculate. Use the 50% margin on a first estimate.
By O.E. — Partner, chairman of the board and CEO of a metal products manufacturer for 21 years; the company exports to 45 countries.
Drafted with AI assistance; calculations checked against primary sources and tested with the site's own code. About FlowUtil
Last reviewed
The speed does not depend on which way gravity acts: a vertical shaft has the same critical speed. Sag here is just a way to measure stiffness against mass.
| Symbol | Meaning |
|---|---|
| I | Second moment of area = π d⁴ ÷ 64 |
| δ | Sag under a central load = W L³ ÷ (48 E I) |
| n | (30/π) √(g ÷ δ) = 946 ÷ √δ (mm) = 187.7 ÷ √δ (in) |
| n_shaft | Bare shaft: (30/π) (π/L)² √(E I ÷ ρ A) |
| Dunkerley | 1/n² = 1/n₁² + 1/n₂² + … |
1A 500 N load at the centre of a Ø50 mm steel shaft simply supported over 800 mm.
δ = 500 × 0.8³ ÷ (48 × 200 × 10⁹ × 3.07 × 10⁻⁷) = 0.087 mm; 946 ÷ √0.087 = 3,208 rpm; shaft alone (12.3 kg) 9,291 rpm; 1 ÷ √(1/3,208² + 1/9,291²)
3,032 rpm
Leaving out the shaft's weight would overstate the result by 5.8%, and the error is on the unsafe side. In inches, a 2 in shaft over 32 in with 112 lbf at the centre sags 0.0034 in, giving about 3,240 rpm before the shaft's own weight.
Speed rises with the square of diameter and falls with span to the power 1.5. Shortening the span is usually the cheapest fix. Aluminium has a third of steel's stiffness and a third of its density, so a bare aluminium shaft has about the same critical speed as a steel one.
| Change from the example | Load-only critical speed |
|---|---|
| Diameter 40 mm | 2,053 rpm |
| Diameter 50 mm | 3,208 rpm |
| Diameter 60 mm | 4,620 rpm |
| Span 600 mm | 4,939 rpm |
| Span 1,000 mm | 2,296 rpm |
Find the sag and critical speed of each load alone, then combine them with Dunkerley's formula. The result sits below the true value, which makes it a cautious estimate. A load away from the centre sags less: 500 N at 250 mm from one bearing on the same shaft gives 0.064 mm and 3,733 rpm.
Published guidance varies: an SAE driveline guideline cited in the literature keeps operating speed below 75–85% of critical, and ball-screw makers recommend 80%. Those margins apply to a known critical speed. This calculator assumes rigid bearings, so the true value is lower, and a flexible housing can lower it a lot.
Treat this as a first estimate. Shafts on machines that can injure people, or that run near this speed, need a rotordynamic analysis and a test.
| Margin on the estimate | Example, 3,032 rpm | Use |
|---|---|---|
| 25% (run at 75%) | 2,274 rpm | once the figure has been confirmed by test or analysis |
| 50% (run at 50%) | 1,516 rpm | for a first estimate like this one |
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