A ball screw that whined at 3000 RPM. The screw was Ø20 mm, lead 5 mm, length 1200 mm, supported at one end (fixed) and the other end free. The machine was a small XY table. At 3000 RPM (linear speed 250 mm/s), the screw vibrated audibly. At 2500 RPM, it was quiet. At 3500 RPM, it was louder but didn’t damage anything. The customer wanted 3000 RPM for cycle time. The issue was the screw’s critical speed — the screw was operating near its first bending mode. This is about ball screw critical speed and end support configurations.
The critical speed formula
The critical speed of a ball screw depends on the diameter, length, and end support configuration:
N_c = λ × (120 × 10⁹) × D / L²
Where D is screw diameter (mm), L is the unsupported length (mm), and λ is a factor based on end support: fixed-free = 0.36, fixed-supported = 1.56, fixed-fixed = 3.14.
For Ø20 mm, L=1200 mm, fixed-free: N_c = 0.36 × 120×10⁹ × 20 / 1200² = 0.36 × 24×10⁹ / 1.44×10⁶ = 0.36 × 16,667 = 6000 RPM. That should be fine. But wait — the formula above is for a solid shaft. A ball screw has a hollow core (for coolant) and the ball nut reduces the effective stiffness. The real critical speed is about 60% of the solid-shaft value: 3600 RPM. At 3000 RPM, the screw is at 83% of critical speed. That’s in the resonance danger zone (within 20% of critical).
The end support configurations
| Configuration | λ factor | Stiffness | Use for |
|---|---|---|---|
| Fixed-free (one bearing, other end free) | 0.36 | Lowest | Short screws (under 500 mm), light loads |
| Fixed-supported (one fixed, one floating) | 1.56 | Medium | General automation, medium length |
| Fixed-fixed (both ends duplex) | 3.14 | Highest | Long screws, high speed, precision |
What was changed
The fixed-free end was upgraded to fixed-supported. A simple bearing block (deep groove ball bearing) was added at the free end. The λ factor jumped from 0.36 to 1.56. The critical speed became 0.6 × 1.56/0.36 × 3600 = 9360 RPM. At 3000 RPM, the screw is at 32% of critical speed. No vibration. No whine. The cycle time target was met.
The DN value check
Beyond critical speed, the ball nut has a speed limit. The DN value (diameter × RPM) should not exceed 100,000 for standard ball nuts. For Ø20 mm at 3000 RPM: DN = 60,000. That’s within limits. At 5000 RPM: DN = 100,000 — the limit. Above that, the ball recirculation tubes can’t keep up. The ball nut overheats. The grease burns. The ball bearings fail.
For high-speed applications (over 3000 RPM on Ø20 screws), use a high-speed ball nut with larger recirculation tubes, or upsize the screw diameter. A Ø25 mm screw at 3000 RPM: DN = 75,000. Comfortable.
The critical speed rule: operate below 70% of N_c. The whining screw wasn’t defective — it was at 83% of critical speed on a fixed-free support. Upgrade to fixed-supported (add a bearing block). Check DN value: keep under 100,000. For long screws over 1000 mm, fixed-fixed is the only way to reach high RPM.
