Critical Speed of Ball Screw: How to Calculate It and Why It Matters

If a ball screw spins fast enough, the shaft itself starts to bow outward and whip around its own axis - even though no single component has failed. This is not a strength problem. It is a resonance problem, and it has a name: the critical speed of a ball screw.

For machine builders selecting a ball screw for a long-stroke gantry, a high-speed pick-and-place axis, or any application pushing toward higher feed rates, critical speed is one of the limits that decides whether a screw is usable at all - often before load capacity or backlash even become a concern.

This guide explains what critical speed is, how to calculate it with a practical formula, which variables actually move the number, and how it relates to the other speed limit every ball screw has: DN value.

What Is the Critical Speed of a Ball Screw?

Every rotating shaft has a natural frequency of lateral (sideways) vibration. Think of a jump rope: turn it slowly and it stays roughly straight, but past a certain speed it bows into a stable arc and starts whipping. A ball screw shaft behaves the same way.

Critical speed is the rotational speed at which the screw shaft's rotation frequency matches its natural bending frequency, causing the shaft to resonate. Once a screw operates at or near this speed, the result is:

• Severe lateral vibration of the shaft
• Increased noise
• Accelerated wear on the ball nut and bearings
• In extreme cases, permanent bending or shaft failure

Critical speed has nothing to do with how strong the screw material is. It is purely a function of geometry - shaft diameter, unsupported length, and how the ends are mounted. This is why two screws made from identical material can have very different critical speeds simply because one has a longer unsupported span.

How to Calculate the Critical Speed of Ball Screw

The standard practical formula for calculating ball screw critical speed (metric units) is:

nc = fc × (dr / Lc²) × 10⁷

Where:

• nc = critical speed (rpm)
• dr = root diameter of the screw shaft (mm) - this is the minor diameter, not the nominal/outer diameter
• Lc = unsupported length between bearing supports (mm)
• fc = end fixity factor, based on how the screw ends are mounted

End Fixity Factor (fc)

How the screw is supported at each end has a major effect on critical speed. The four standard mounting configurations and their typical fc values are:

Moving from a simple "supported-supported" configuration to "fixed-fixed" can more than double the allowable critical speed for the same shaft - without changing the screw diameter at all. This is one reason high-speed, long-stroke axes often justify the extra cost of fixed-end bearing supports.

Safety Margin: Critical Speed Is Not the Operating Speed

The calculated critical speed is a theoretical resonance point, not a usable operating limit. Industry practice is to keep the actual operating speed at 80% of critical speed or lower:

n_safe = nc × 0.8

Running close to 100% of critical speed leaves no margin for manufacturing tolerances, thermal expansion, or minor misalignment - any of which can shift the real resonance point slightly lower than the calculated value.

Worked Example

Take a ball screw with:

• Root diameter dr = 14.2 mm (typical for a nominal Ø16 mm rolled ball screw)
• Unsupported length Lc = 1000 mm
• End support: Fixed–Supported (fc = 15.1)

Step 1 - Calculate critical speed:

nc = 15.1 × (14.2 / 1000²) × 10⁷
nc = 15.1 × 0.0000142 × 10⁷
nc ≈ 2,144 rpm

Step 2 - Apply the 80% safety margin:

n_safe = 2,144 × 0.8 ≈ 1,715 rpm

So for this configuration, the screw should not be driven above roughly 1,715 rpm in continuous operation, regardless of what the motor or driver is capable of.

Now compare the same screw with a longer unsupported span - say a 2000 mm gantry axis instead of 1000 mm. Because Lc is squared in the formula, doubling the unsupported length cuts the critical speed to roughly one-quarter of the original value, not half. This is the single most common mistake in screw selection: assuming a longer-stroke machine just needs "the same screw, longer" when in fact the safe speed drops dramatically.

What Actually Changes Critical Speed

1. Unsupported length (Lc) - the dominant factor
Because Lc is squared in the denominator, length has by far the largest effect. A long-stroke axis is far more likely to be critical-speed-limited than load-limited. Adding an intermediate support, or simply specifying a shorter unsupported span in the mechanical layout, is usually the most effective way to raise the allowable speed.

2. Root diameter (dr)
A larger shaft diameter increases critical speed, but it also increases rotational inertia and the torque needed to accelerate the screw. Diameter should be increased deliberately, not as a default fix.

3. End fixity / support method
As shown in the table above, the mounting configuration at each end can shift critical speed by more than 6x between the weakest (fixed-free) and strongest (fixed-fixed) configurations. This is where the choice of bearing support housing on each end of the screw becomes part of the speed calculation, not just a mechanical afterthought.

4. Shaft straightness and alignment
The formula assumes a straight, properly aligned shaft. Bowed shafts, misaligned supports, or worn bearings will cause real-world vibration to appear at a lower speed than the calculated value predicts.

Critical Speed of Ball Screw vs. DN Value: Two Different Limits

Critical speed is often confused with another speed limit: DN value. They are not the same thing, and a ball screw selection should check both.

• Critical speed limits how fast the shaft can rotate before it resonates and whips.
• DN value (nominal diameter × max rpm) limits how fast the balls can circulate inside the nut before heat buildup and wear shorten the screw's service life.

A short, thick screw with strong end fixity might never approach its critical speed, but could still be limited by DN value. A long, slender screw is usually limited by critical speed long before DN value becomes relevant.

The rule for selection: calculate both limits, and use whichever allowable speed is lower.

Practical Takeaways for Selection

• Always check critical speed for any axis where stroke length exceeds roughly 600–800 mm, or where target speed is high relative to screw diameter.
• If critical speed comes out too low for your target feed rate, the most effective fixes - in order of typical cost-effectiveness - are: (1) shorten the unsupported span with an intermediate support, (2) upgrade the end fixity configuration with a more rigid bearing support, (3) increase shaft diameter.
• Don't evaluate critical speed in isolation. A screw selected purely on critical speed but paired with an undersized coupling or a support bearing rated for lower speeds will still create a weak point in the system.
• When in doubt, run the calculation at the design stage - not after the machine vibrates on the shop floor.

References

1. Thomson Industries Inc. "Ball Screw Assembly Critical Speed."

2. Engineers Edge. "Critical Speed Ball Screw and Lead Screws Formulas and Calculator."

3. Industrial Solutions Lab, University of North Carolina at Charlotte. "Ball Screw Selection Guide."

4. Rockford Ball Screw. "About Ball Screws" - Metric Catalog Introduction (End Fixity, Critical Speed, Column Load).

← Back to DLY Blog