BIZOMAT 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. |
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.
What this means: Shaft diameter
- What it is
- The outside diameter of the shaft between the bearings, treated as the same all along the span.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 2000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Stiffness rises with the fourth power of diameter, so a thicker shaft raises the critical speed strongly.
- Get it right
- For a stepped shaft this is only an approximation; use the diameter over most of the span and treat the result with care.
What this means: Bearing span
- What it is
- The distance between the centres of the two bearings, which the tool treats as simple supports. Every mass must sit within this span.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 20000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1500 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A longer span lowers the critical speed sharply.
- Get it right
- Measure between bearing centres, not to the shaft ends.
What this means: Elastic modulus
- What it is
- The elastic (Young’s) modulus of the shaft material, which sets how much the shaft bends under the rotating masses.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 400 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher modulus makes the shaft stiffer and raises the critical speed.
- Get it right
- Enter it in GPa, not MPa; take it from the material data sheet.
What this means: Density
- What it is
- The density of the shaft material, used for the shaft’s own mass when the tool finds the critical speed of the bare shaft.
- Units and format
- Enter a number in kg/m³: kilograms per cubic metre. The form accepts 0.000001 kg/m³ to 20000 kg/m³. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 7850 kg/m³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the density of the shaft material only; the masses of discs and pulleys go in the mass inputs.
Masses (0 = none)
What this means: Mass 1
- What it is
- The mass of the first disc, pulley, gear, impeller or rotor mounted on the shaft, including its hub.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0 kg to 100000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A heavier mass lowers the critical speed.
- Get it right
- Use 0 if there is no mass. Weigh the part or take it from the CAD model or supplier data.
What this means: Mass 1 distance from the left bearing
- What it is
- Where the first mass sits along the shaft: the distance from the left bearing centre to the centre of the mass.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 20000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 500 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A mass near mid-span lowers the critical speed more than one near a bearing.
- Get it right
- Measure to the centre of the mass, and keep it between the bearings or the tool reports an error.
What this means: Mass 2
- What it is
- The mass of a second part mounted on the shaft, such as another pulley, coupling half or impeller, including its hub.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0 kg to 100000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 15 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A heavier mass lowers the critical speed.
- Get it right
- Use 0 if there is no second mass; its position is then ignored.
What this means: Mass 2 distance from the left bearing
- What it is
- Where the second mass sits: the distance from the left bearing centre to the centre of the second mass.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 20000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1000 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Measure from the same left bearing as the other masses.
What this means: Mass 3
- What it is
- The mass of a third part mounted on the shaft, including its hub, when the shaft carries three masses.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0 kg to 100000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A heavier mass lowers the critical speed.
- Get it right
- Use 0 if there is no third mass; its position is then ignored.
What this means: Mass 3 distance from the left bearing
- What it is
- Where the third mass sits: the distance from the left bearing centre to the centre of the third mass.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 20000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 750 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Only used when the third mass is above 0.
Operation
What this means: Running speed
- What it is
- The speed the shaft runs at in service, compared with the combined (Dunkerley) critical speed.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0.000001 rpm to 100000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 600 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- For a variable-speed drive, check the whole speed range, not just the normal speed.
What this means: Margin from the critical speed
- What it is
- The separation you want between running speed and the critical speed, as a share of the critical speed, applied both below and above it.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 0% to 80%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 25% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Type the percentage itself, not a fraction. Running inside the band either side of critical is reported as near critical.
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.
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
Critical speeds of real shafts
In Whirling Shafts by the Numbers
What it covers: Estimating the first critical speed of a shaft carrying several masses: Dunkerley's lower bound, Rayleigh's energy method, how close each is to the exact answer, and how margins are defined.
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