KEVOS® MAKEOS · Strength
Shaft Critical (Whirling) Speed
At what speed will this shaft whirl, and is the running speed clear of it?
First critical speed of a uniform solid shaft with simple, fixed–free or fixed–fixed supports, optionally with a single disc (Dunkerley), checked against the operating speed.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Shaft diameter | mm | Diameter of the shaft (uniform). |
| Length between supports | mm | Distance between bearings, or overhang length for the fixed–free case. |
| Elastic modulus | GPa | Elastic modulus from the material data sheet or an approved materials database (steel is typically about 200 GPa). |
| Density | kg/m³ | Density of the shaft material (steel about 7850 kg/m³). |
| Supports | choice | How the bearings restrain the shaft; real bearings lie between simple and fixed. |
| Disc mass (0 = none) | kg | Mass of a single pulley, rotor or disc at mid-span (or at the free end when overhung). |
| Operating speed | rpm | Highest running speed. |
| Required separation margin | % | How far the running speed must stay from the critical speed, from your design standard. |
Method and formulas
- Shaft alone: ω = (βL ÷ L)²√(EI ÷ ρA), with βL = π (simple), 1.875 (fixed–free) or 4.730 (fixed–fixed).
- Disc on a massless shaft: ω = √(k ÷ m) with k = 48EI/L³, 3EI/L³ or 192EI/L³.
- Combined (Dunkerley): 1/ω² = 1/ω_shaft² + 1/ω_disc²; n = 60ω ÷ 2π.
- Example: a 40 mm shaft over 1500 mm with a 20 kg disc has a critical speed of about 1,092.9 rpm.
Limitations
- Dunkerley’s method underestimates the critical speed slightly (conservative).
- Bearing and support stiffness lower the critical speed; gyroscopic effects are ignored.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check long line shafts, rollers and spindles
- Decide on intermediate bearings
- Screen a drive before detailed rotor dynamics
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Critical speed of the shaft alone
- Critical speed with the disc (Dunkerley)
- Operating speed as a percentage of critical
Interpreting the result
- Running above critical is possible but the shaft passes through resonance on start-up and shut-down.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 197, 302. Topic reference; no source tables or recommended settings are reproduced.
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