BIZOMAT 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. |
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 diameter of the solid shaft, assumed the same along the whole length between the supports.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 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
- For the shaft alone the critical speed rises in proportion to the diameter.
- Get it right
- For a stepped shaft this simple model is approximate; use the diameter over most of the length.
What this means: Length between supports
- What it is
- The distance between the bearings, or for an overhung shaft the length from the fixed support to the free end.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 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
- For the shaft alone the critical speed falls with the square of the length.
- Get it right
- Measure between bearing centres.
What this means: Elastic modulus
- What it is
- The elastic modulus (Young’s modulus) of the shaft material, from the material data sheet. It sets the bending stiffness.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 1000 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.
- Get it right
- Enter GPa, not MPa: 1 GPa is 1000 MPa.
What this means: Density
- What it is
- The density of the shaft material: its mass per unit volume, from the material data sheet. It sets the shaft’s own mass.
- Units and format
- Enter a number in kg/m³: kilograms per cubic metre. The form accepts 0.000001 kg/m³ to 50000 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
- Enter kg/m³, not g/cm³: 1 g/cm³ is 1000 kg/m³.
What this means: Supports
- What it is
- How the bearings hold the shaft: simply supported (free to tilt) at both ends, fixed at one end and free at the other (overhung), or fixed at both ends.
- Units and format
- Choose one of 3 options from the list: Simply supported both ends; Fixed one end, free other (overhung); Fixed both ends.
- Example
- The form opens with “Simply supported both ends” selected as a worked example. Choose the option that matches your case.
- Get it right
- Real bearings lie between simple and fixed; choosing fixed gives a higher critical speed.
What this means: Disc mass (0 = none)
- What it is
- The mass of a single pulley, rotor or disc carried at mid-span, or at the free end of an overhung shaft. Zero calculates the bare shaft only.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0 kg to 1000000 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.
- Get it right
- Include hubs and fasteners of the disc. Several discs need a more detailed method.
What this means: Operating speed
- What it is
- The highest speed the shaft runs at in service. The tool compares it with the critical speed.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0.000001 rpm to 1000000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 750 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Include any overspeed or variable-speed range up to its top value.
What this means: Required separation margin
- What it is
- How far, as a percentage of the critical speed, the running speed must stay away from it on either side, from your design standard.
- 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 90%. 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 (25 for 25 %), not a fraction.
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.
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
What whirling is
In Whirling Shafts by the Numbers
What it covers: Why a spinning shaft bows into a rotating curve near one particular speed: the critical speed, the single-disc rotor, amplification of unbalance and why rotors settle above it.
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