The Screw That Whined at 3000 rpm
We chose a ball screw for a high-speed linear axis: Ø20 mm, 10 mm lead, 2 m long. The motor was sized for the torque. At 3000 rpm, the screw started to whine — a high-pitched noise, and the axis vibrated. We dropped the speed to 2000 rpm; the whining stopped. The problem: the screw’s critical speed. A long, slender screw (Ø20 mm, 2 m) has a natural frequency. Running it above the critical speed excites resonance. The screw whines and vibrates. The mistake was choosing a long, thin screw for high speed without checking the critical speed. We either shortened the unsupported length (added a center support), upsized the screw to Ø32 mm, or limited the rpm.
Ball screw critical speed is the rpm at which the screw resonates. Exceed it, and the screw whines and fatigues. This article covers the calculation.
What Is Critical Speed?
A rotating shaft (the ball screw) has a natural frequency. When the rpm matches the natural frequency (or a multiple), the shaft resonates — it vibrates strongly (like a jump rope spinning). Below the critical speed, the shaft runs smoothly. Above it, it vibrates and fails.
The critical speed depends on:
- Screw diameter (d): Thicker = stiffer = higher critical speed.
- Unsupported length (L): Shorter = stiffer = higher critical speed.
- Support method: Fixed-fixed (both ends supported rigidly) is stiffer than fixed-supported or supported-supported.
The Critical Speed Formula
The critical speed (rpm) for a ball screw is:
N_c = λ × (d / L²) × 10⁷
Where:
- N_c is the critical speed (rpm)
- d is the screw diameter (mm)
- L is the unsupported length (mm, between supports)
- λ is the support factor: fixed-fixed = 21.9, fixed-supported = 12.1, supported-supported = 4.8
For d = 20 mm, L = 2,000 mm, fixed-supported (λ = 12.1): N_c = 12.1 × (20 / 4,000,000) × 10⁷ = 12.1 × 0.000005 × 10⁷ = 12.1 × 50 = 605 rpm. That’s very low! The screw can’t run above 605 rpm (fixed-supported, 2 m long).
For d = 32 mm, L = 2,000 mm, fixed-fixed (λ = 21.9): N_c = 21.9 × (32 / 4,000,000) × 10⁷ = 21.9 × 0.000008 × 10⁷ = 21.9 × 80 = 1,752 rpm. Better, but still limited.
For d = 25 mm, L = 1,000 mm (shorter span, center support), fixed-fixed: N_c = 21.9 × (25 / 1,000,000) × 10⁷ = 21.9 × 0.000025 × 10⁷ = 21.9 × 250 = 5,475 rpm. Plenty.
The critical speed rule: Check N_c before choosing the screw. The screw that whined was Ø20 mm, 2 m long, fixed-supported (N_c = 605 rpm) running at 3000 rpm. That’s 5× over. Either shorten the span (center support), upsize the diameter (Ø32), or use fixed-fixed supports. Run the screw at under 80% of N_c (safety).
Step 1: Support Method (Article 61)
The support method (fixed-fixed, fixed-supported, etc.) dramatically affects the critical speed. Fixed-fixed (both ends rigid) is 4.5× stiffer than supported-supported. For a long, high-speed screw, use fixed-fixed (article 61). The fixed end takes the thrust; the supported end handles radial only.
Step 2: Center Support (Intermediate Bearing)
For a long screw (over 1.5 m), add a center support (an intermediate bearing in the middle of the screw). It halves the unsupported span (L becomes L/2). Since N_c scales with 1/L², halving L quadruples N_c. A center support is the cheapest way to raise the critical speed.
But the center support must be aligned (it adds complexity). And the screw must have a groove (a relieved section) for the center support. For very long screws, a center support is standard.
Step 3: DN Ratio (For Rotary Motion)
For a rotating screw (the screw turns, the nut moves), the DN ratio (diameter × rpm) is another limit:
DN = d × N
Where d is the screw diameter (mm) and N is the rpm. The lubrication (grease) limits DN. Typical limit: DN < 70,000–100,000 (for grease). For d = 20 mm, N = 3,000 rpm: DN = 60,000 (OK). For d = 20 mm, N = 5,000 rpm: DN = 100,000 (at the limit). Above this, use oil-air lubrication (article 88).
Step 4: Choose Based on Application
| Screw Length | Speed | Recommendation |
|---|---|---|
| <1 m | High (3000 rpm) | Standard (critical speed not an issue) |
| 1–1.5 m | High | Upsize diameter (Ø25–32), fixed-fixed supports |
| >1.5 m | High | Center support, or use a linear motor (no rotating screw) |
| Any length | Low (<1000 rpm) | Critical speed not an issue |
A Critical Speed Checklist
- What is the screw diameter? (d)
- What is the unsupported length? (L, between supports?)
- What is the support method? (Fixed-fixed?)
- Calculate N_c = λ × d / L² × 10⁷?
- Is the operating rpm under 80% of N_c?
- If not: upsize d, shorten L (center support), or change supports?
- What is the DN ratio? (d × N < 70,000?)
- For grease: is the DN OK?
- Is there a center support? (For long screws?)
- Does the screw whine at the operating speed? (Test?)
The Bottom Line
Ball screw critical speed is the rpm limit for long, slender screws. The screw that whined at 3000 rpm was Ø20 mm, 2 m long, fixed-supported (N_c = 605 rpm). Run at under 80% of N_c. For long, high-speed screws, upsize the diameter, use fixed-fixed supports, or add a center support. Check the DN ratio for lubrication. The screw that runs quietly at high speed wasn’t the longest one — it was sized for critical speed.
