A line shaft that vibrated at 1200 RPM. It was a 30 mm diameter steel shaft, 1800 mm long, supported by two bearings at the ends. It drove a series of pulleys. At 1200 RPM, the shaft whined loudly and vibrated 0.5 mm. At 900 RPM or 1500 RPM, it was quiet. The customer thought the bearings were bad. They were fine. The shaft was running at its critical speed. This is about shaft deflection and critical speed.
The critical speed
Every shaft has a speed at which it naturally resonates. Below that speed, the shaft runs smooth. At the critical speed, even tiny imbalances grow into large vibrations. Above the critical speed, the shaft self-centers and runs smooth again (if you can accelerate through the critical speed quickly). The critical speed for a simply supported shaft:
N_c = (30/π) · √(E · I / (m · L⁴)) · constants
A simpler formula for a steel shaft:
N_c ≈ 1.2 × 10⁸ · D / L²
Where D is diameter (mm) and L is bearing span (mm). For D=30 mm, L=1800 mm: N_c = 1.2×10⁸ × 30 / 1800² = 3.6×10⁹ / 3.24×10⁶ = 1111 RPM. That’s right at the operating speed of 1200 RPM. The shaft was running through its critical speed. The vibration at 1200 RPM confirmed it.
What I changed
1. Added a center support bearing. I added a third bearing at the midpoint of the 1800 mm shaft. The effective span dropped from 1800 mm to 900 mm. New critical speed: N_c = 1.2×10⁸ × 30 / 900² = 3.6×10⁹ / 8.1×10⁵ = 4444 RPM. That’s 3.7x the operating speed. The shaft ran smooth at 1200 RPM. The vibration disappeared.
2. Upsized the shaft to 40 mm. For a new design, I’d use a 40 mm shaft. N_c = 1.2×10⁸ × 40 / 1800² = 1481 RPM. Still close to 1200. I’d need 50 mm: N_c = 1852 RPM. With 50 mm shaft, the critical speed is 1.5x operating speed. That’s the rule: keep operating speed below 50% of critical speed (or above 150% if accelerating through).
3. Changed the drive speed. The simplest fix: change the operating speed from 1200 to 900 RPM (or 1500). At 900 RPM, we’re below the critical speed (1111) with margin. The shaft ran smooth. But the process required 1200 RPM for product flow. So the center bearing was the fix.
The deflection check
Critical speed is related to shaft deflection. A shaft that deflects 1 mm under its own weight at midspan has a low critical speed. I check the deflection:
δ = 5 · w · L⁴ / (384 · E · I)
For a 30 mm steel shaft, 1800 mm span: weight per meter = π × 15² × 7850 × 10⁻⁹ = 5.55 kg/m. I = π × 15⁴ / 4 = 39,794 mm⁴. E = 200,000 N/mm². δ = 5 × 5.55×9.81 × 1800⁴ / (384 × 200,000 × 39,794). This gives about 0.8 mm. A shaft that deflects 0.8 mm at midspan is too flexible. The critical speed confirms it. I add the center support to reduce deflection to 0.05 mm.
The operating speed rule
| Operating speed | Critical speed should be | Why |
|---|---|---|
| Below critical (subcritical) | 2x operating speed | Margin below resonance |
| Above critical (supercritical) | 0.5x operating speed | Accelerate through quickly, run above |
| Variable speed | Avoid resonance band entirely | Don’t run within ±20% of N_c |
The shaft I design: keep operating speed at least 2x below critical speed. The whining shaft wasn’t bad bearings — it was 1200 RPM at 1111 RPM critical speed. Add a center support or upsize the diameter. For variable-speed drives, map out the critical speed and program the drive to skip through resonance quickly.
