KEVOS® MAKEOS · Strength
Shaft Critical Speed with Several Masses (Dunkerley and Rayleigh)
What is the first critical speed of this simply supported shaft carrying several discs or pulleys, and is the running speed safely clear of it?
First whirling (critical) speed of a simply supported shaft carrying up to three concentrated masses: the shaft’s own critical speed and each mass’s critical speed from its static deflection, combined by Dunkerley’s formula (a lower, conservative estimate; for one central mass it equals the critical speed tool), a Rayleigh estimate for the masses from their static deflection curve, and the running speed checked against your margin.
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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). |
| Bearing span | mm | Distance between the two bearings (simple supports). |
| Elastic modulus | GPa | Elastic modulus of the shaft. |
| Density | kg/m³ | Density of the shaft material. |
| Masses (0 = none) | ||
| Mass 1 | kg | First disc, pulley or impeller mass. |
| Mass 1 distance from the left bearing | mm | Its distance from the left bearing. |
| Mass 2 | kg | Second mass (0 if none). |
| Mass 2 distance from the left bearing | mm | Its distance from the left bearing. |
| Mass 3 | kg | Third mass (0 if none). |
| Mass 3 distance from the left bearing | mm | Its distance from the left bearing. |
| Operation | ||
| Running speed | rpm | Operating speed. |
| Margin from the critical speed | % | Separation you want between running speed and the critical speed. |
Method and formulas
- Shaft alone ω_s = (π ÷ L)² √(EI ÷ ρA).
- Each mass alone: influence coefficient α = a² b² ÷ (3EIL), ω_i = 1 ÷ √(m_i α).
- Dunkerley 1 ÷ ω² = 1 ÷ ω_s² + Σ 1 ÷ ω_i² (conservative).
- Rayleigh (masses only) ω² = g Σ m y ÷ Σ m y² using the static deflections y at the masses.
- Example: the shaft whirls at about 965.78 rpm (Dunkerley); running at 600 rpm is 62.126% of it.
Limitations
- Simple supports and a uniform shaft; overhangs and stepped shafts need a fuller model.
- Bearing and housing flexibility lower the real critical speed.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a fan or pump shaft with several impellers
- Find whether a speed increase approaches resonance
- Compare bearing spans for a line shaft
Who it is for
- Design engineers
- Rotating equipment engineers
- Maintenance engineers
What you get
- Shaft-alone and single-mass critical speeds
- Dunkerley and Rayleigh critical speeds
- Running speed against the margin
Interpreting the result
- Dunkerley under-predicts and Rayleigh over-predicts: the true first critical lies between them.
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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Open the Shaft Critical Speed with Several Masses (Dunkerley and Rayleigh) calculator
