A linear shaft that bent under load. The shaft was Ø20 mm hardened steel, 1000 mm long, supporting a ball bushing carrying 30 kg. The shaft was mounted on two end supports only (not a rail). At midspan, the shaft deflected 0.5 mm under the 30 kg load. The ball bushing bound — it couldn’t travel smoothly. The customer thought the bushing was bad. It was fine. The shaft was too flexible for the span. This is about linear shaft support and deflection limits.
The deflection calculation
A simply supported beam (shaft) with a concentrated load at midspan deflects:
δ = F · L³ / (48 · E · I)
Where F is the load (N), L is the span (mm), E is the Young’s modulus (200,000 N/mm²), and I is the second moment of area. For a solid circular shaft: I = π × d⁴ / 64. For Ø20 mm: I = π × 160,000 / 64 = 7854 mm⁴. F = 30 × 9.81 = 294 N. L = 1000 mm. δ = 294 × 10⁹ / (48 × 200,000 × 7854) = 294×10⁹ / 75.4×10⁹ = 3.9 mm. That’s way more than the 0.5 mm observed — the shaft wasn’t simply supported at the ends, it had a center support. With three supports (at 0, 500, 1000 mm), the effective span is 500 mm. δ = 294 × 125×10⁶ / (48 × 200,000 × 7854) = 36.8×10⁹ / 75.4×10⁹ = 0.49 mm. Matches the observation. But 0.5 mm deflection at the bushing location still causes binding.
The straightness requirement
A ball bushing needs the shaft to be straight within about 0.05 mm over the length of the bushing (typically 30 mm). If the shaft sags 0.5 mm at midspan, the local curvature over 30 mm is about 0.5 × (30/500)² = 0.0018 mm. That’s tiny — the bushing shouldn’t bind. But the real issue is the shaft’s natural vibration. At 30 kg moving at 0.5 m/s, the shaft vibrated at its natural frequency. The vibration caused the bushing to bind intermittently. The customer felt it as rough travel.
What was changed
1. Mounted the shaft on a supported rail. The Ø20 shaft was replaced with a linear rail (profile rail) mounted on an aluminum extrusion. The rail is rigidly supported along its full length. Deflection under 30 kg is 0.01 mm. The bushing travels smoothly. No vibration. No binding.
2. Added a center support to the round shaft. A cheaper fix: a support block at midspan. The effective span dropped to 500 mm. Deflection dropped to 0.5 mm. Still not ideal, but the vibration stopped. The system ran acceptably for 2 years. When the customer needed higher speed, it was upgraded to a profile rail.
3. Up-sized the shaft to Ø30 mm. I = π × 30⁴ / 64 = 39,761 mm⁴ (5x stiffer). With 1000 mm span: δ = 294 × 10⁹ / (48 × 200,000 × 39,761) = 0.77 mm. With center support: 0.096 mm. Close to acceptable. But Ø30 shaft blocks cost more than a profile rail. For new designs, the profile rail wins.
Round shaft vs profile rail
| Factor | Round shaft (unsupported) | Profile rail |
|---|---|---|
| Max span (30 kg load) | 300 mm | 1500 mm |
| Deflection at midspan | 1 mm per 500 mm span | 0.02 mm per 1500 mm |
| Speed limit | 0.3 m/s (whirl) | 5 m/s |
| Cost | $20/m | $100/m |
| Install difficulty | Easy (two end blocks) | Needs precision alignment |
The shaft support rule: round shafts need a support every 300 mm for loads over 10 kg. The binding bushing wasn’t defective — the Ø20 shaft over 1000 mm deflected 4 mm at midspan. For spans over 500 mm, use a profile rail. For light loads (under 5 kg), a round shaft on two end blocks is fine. The stiffness of a round shaft drops with the fourth power of diameter — upsizing from 20 to 30 mm gives 5x stiffness.
