BIZOMAT MAKEOS · Power Transmission
Ball Screw Critical Speed and Buckling
Is this ball screw below its critical speed and buckling load?
Critical (whirling) speed and Euler buckling load of a ball or lead screw from root diameter, unsupported length and end fixity, against the operating speed and axial load.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Root diameter | mm | Root (minor) diameter of the screw from the catalogue. |
| Unsupported length | mm | Distance between bearings, or from the bearing to the nut at its furthest position. |
| End mounting | choice | Bearing arrangement at the screw ends. |
| Elastic modulus | GPa | Modulus of the screw steel. |
| Density | kg/m³ | Density of the screw material. |
| Maximum operating speed | rpm | Highest screw speed in service. |
| Maximum compressive axial load | N | Largest compressive load on the screw. |
| Allowed share of critical speed | % | Maximum share of critical speed your design rule or the screw maker allows (often 80%). |
| Allowed share of buckling load | % | Maximum share of buckling load your design rule allows (often 50%). |
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: Root diameter
- What it is
- The root (minor) diameter of the screw shaft at the bottom of the ball groove, from the screw catalogue. The tool treats the shaft as a plain rod of this diameter.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 21 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Critical speed rises in proportion and buckling load with the fourth power of this diameter.
- Get it right
- Use the root diameter, not the nominal screw diameter.
What this means: Unsupported length
- What it is
- The unsupported length of the screw: between the bearings, or from the bearing to the nut at its furthest position, whichever governs the case being checked.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1200 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Critical speed and buckling load both fall with the square of length.
- Get it right
- Use the worst case, the longest unsupported span in service.
What this means: End mounting
- What it is
- How the screw ends are held by bearings: fixed means a bearing pair that resists bending; supported means a single bearing that allows rotation; free means unsupported.
- Units and format
- Choose one of 4 options from the list: Fixed – fixed; Fixed – supported; Supported – supported; Fixed – free.
- Example
- The form opens with “Fixed – supported” selected as a worked example. Choose the option that matches your case.
- Get it right
- Choose what the mounting really provides; a single radial bearing is supported, not fixed.
What this means: Elastic modulus
- What it is
- The elastic modulus of the screw shaft material, from the material data sheet or screw maker.
- 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 206 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter it in GPa, not MPa.
What this means: Density
- What it is
- The density of the screw shaft material, from the material data sheet.
- 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.
- Effect on the result
- A denser material lowers the critical speed; buckling load is unaffected.
- Get it right
- Enter kilograms per cubic metre, not grams per cubic centimetre.
What this means: Maximum operating speed
- What it is
- The highest speed the screw turns in service, compared with the allowed share of critical speed.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0 rpm to 1000000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2000 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the screw speed in rapid traverse, not the motor speed if there is a belt ratio.
What this means: Maximum compressive axial load
- What it is
- The largest axial force that pushes on the screw in compression, compared with the allowed share of buckling load.
- Units and format
- Enter a number in N: newtons. The form accepts 0 N or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Include acceleration and process forces; tension loads do not cause buckling.
What this means: Allowed share of critical speed
- What it is
- The highest share of the calculated critical speed you allow in service, from your design rule or the screw maker.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 1% to 100%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 80% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the allowed speed line; the critical speed does not change.
- Get it right
- Type the percentage itself, not a fraction.
What this means: Allowed share of buckling load
- What it is
- The highest share of the calculated buckling load you allow in service, from your design rule or the screw maker.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 1% to 100%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the allowed load line; the buckling load does not change.
- Get it right
- Type the percentage itself, not a fraction.
Method and formulas
- I = π d_r⁴ ÷ 64; A = π d_r² ÷ 4.
- Critical speed n = (λ² ÷ 2πL²) √(EI ÷ ρA) × 60, with λ = 4.730, 3.927, π or 1.875 for fixed–fixed, fixed–supported, supported–supported and fixed–free.
- Buckling P = π²EI ÷ (KL)², with K = 0.5, 0.7, 1 or 2.
- Example: a 21 mm root, 1200 mm fixed–supported screw has a critical speed of 2,750.4 rpm and a buckling load of 27.508 kN.
Limitations
- Uniform shaft at the root diameter; the nut and support stiffness affect real values.
- Nut life, DN limit and drive torque are separate checks.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a ball screw axis before ordering
- Choose end bearing arrangement
- Find the longest screw for a speed
Who it is for
- Design engineers
- Maintenance engineers
- Automation engineers
What you get
- Critical speed and allowed speed
- Buckling load and allowed load
- Utilisation of each
Interpreting the result
- If the speed is too high, use a larger diameter, a fixed–fixed mounting, a shorter length or a rotating nut.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2561–2568. 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
Rods in machines: cylinders and screws
What it covers: Buckling of hydraulic cylinder rods and screws: the length and mounting that count, why the rod must survive the relief valve's force rather than the duty, and the same check for ball screws.
Method explained here · Library collection
Power screws and linear motion: leadscrews, ball screws, linear guides and cams
In Machine Elements and Mechanical Design
What it covers: Turning rotation into accurate straight-line motion is central to machines and automation. How to size leadscrews and ball screws, choose guides and actuators, and design cam motion.
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