The Ball Screw That Ran Away With Itself
We selected a Ø25 mm ball screw for a horizontal axis: 800 mm travel, 10 kg load, 1 m/s top speed. The L10 life calculation was fine — about 20,000 hours. On the floor, the screw whined loudly above 1.5 m/s and started to vibrate. We’d checked the life but not the critical speed. The screw’s DN value (diameter × speed) was over the limit. At 1 m/s with a 20 mm lead, the screw spun at 3,000 RPM. The critical speed for an 800 mm unsupported length Ø25 screw is about 2,500 RPM. We were over it. The screw was whipping. We shortened the support (added a middle bearing block) or went to a coarser lead (lower RPM for the same linear speed). The whine stopped. The mistake was sizing for life but not for the screw’s physical limit.
Ball screw selection has two independent checks: the L10 life (how long the balls last) and the critical speed (how fast the screw can spin without whipping). Skip either, and the screw fails prematurely. This article runs both numbers.
The Two Checks: Life and Speed
A ball screw has two limits:
- L10 life: How long the ball nut lasts (fatigue). Based on the load and speed.
- Critical speed (Nc): The RPM at which the screw resonates (whipping). Based on the diameter, length, and bearing support.
The screw must satisfy both. A screw that meets the life but exceeds the critical speed vibrates. A screw that meets the critical speed but is undersized for load wears out.
Step 1: L10 Life Calculation
Like bearings, the ball screw has a dynamic load rating C (from the catalog). The equivalent load P determines the life.
L₁₀ = (C / P)³ × 10⁶ revolutions
For ball screws, life is in meters (not revolutions). Convert: L₁₀ (meters) = L₁₀ (rev) × lead / 10⁶.
Equivalent Load P
The load on the screw varies through the cycle. Calculate the average:
P = ³√[(P₁³ × n₁ × t₁ + P₂³ × n₂ × t₂ + …) / (n₁ × t₁ + n₂ × t₂ + …)]
Where P₁, P₂ are the loads at different points in the cycle, and n₁, n₂ are the RPMs at those points. For a simple axis that accelerates, cruises, and decelerates, the average load is close to the continuous load.
For a 10 kg load on a horizontal axis: P ≈ m × g × μ = 10 × 9.81 × 0.005 (ball screw friction) = 0.49 N. That’s tiny. But if the axis presses with 500 N during operation, P = 500 N during that portion. The equivalent load depends on the process forces, not just the weight.
Example: Ø25 mm Screw
A Ø25 mm ball screw with 20 mm lead has a dynamic rating C ≈ 9,000 N. If the equivalent load P = 500 N (pressing): L₁₀ = (9000/500)³ × 10⁶ = 18³ × 10⁶ = 5,832 × 10⁶ revolutions. In meters: 5,832 × 10⁶ × 0.02 / 10⁶ = 116,640 meters. At 1 m/s, that’s 116,640 seconds = 32 hours? No — that’s 116,640 meters of travel. At 1 m/s, that’s 116,640 seconds = 32.4 hours. That seems short. But wait, the pressing happens for only 10% of the cycle. The rest is no-load travel. The equivalent load is lower. With P ≈ 200 N average: L₁₀ = (9000/200)³ × 10⁶ × 0.02 = 45³ × 10⁶ × 0.02 = 911,250 × 10⁶ × 0.02 = 18,225,000 meters. At 1 m/s, that’s 18,225,000 seconds = 5,062 hours. About 2 years of 8-hour days. Acceptable for most machines.
Step 2: Critical Speed (Whipping)
The screw is a long shaft supported at the ends. At a certain RPM, it resonates (like a jump rope). This is the critical speed Nc.
Nc = λ × (d × 10⁷) / L²
Where d is the screw root diameter (mm), L is the unsupported length (mm), and λ is a support factor (20.5 for fixed-fixed, 15 for fixed-supported, 3.4 for fixed-free).
For a Ø25 mm screw (root diameter ~21 mm), L = 800 mm, fixed-supported (λ=15): Nc = 15 × (21 × 10⁷) / 800² = 15 × 2.1 × 10⁸ / 640,000 = 15 × 328 = 4,920 RPM. Wait, that’s high. Let me recalculate. The standard formula: Nc (RPM) = (λ × 10⁷ × d) / L². For d=21, L=800: Nc = 15 × 21 × 10⁷ / 640,000 = 315 × 10⁷ / 640,000 = 4,921 RPM. But the screw was whipping at 3,000 RPM. Why?
Because the actual support was not fixed-supported. The inboard bearing was a simple deep groove ball bearing (not angular contact), which doesn’t truly fix the shaft. The effective λ was lower (more like 7 for a poorly supported end). Nc = 7 × 21 × 10⁷ / 640,000 = 2,297 RPM. That matches the whipping at 3,000 RPM. The bearing arrangement matters as much as the screw size.
| Screw Diameter | L = 500 mm | L = 800 mm | L = 1200 mm |
|---|---|---|---|
| Ø15 mm (root ~12) | 7,200 RPM | 2,800 RPM | 1,250 RPM |
| Ø20 mm (root ~17) | 10,200 RPM | 4,000 RPM | 1,800 RPM |
| Ø25 mm (root ~21) | 12,600 RPM | 4,900 RPM | 2,200 RPM |
| Ø32 mm (root ~28) | 16,800 RPM | 6,500 RPM | 2,900 RPM |
(Values are approximate for fixed-fixed mounting, λ=20.5.)
Step 3: The DN Value
A quick rule of thumb: the DN value (screw diameter × speed) must stay under a limit.
DN = d × n
Where d is the nominal diameter (mm) and n is the RPM. Standard ball screws: DN under 100,000. High-speed screws: DN under 150,000.
For our Ø25 mm screw at 3,000 RPM: DN = 25 × 3,000 = 75,000. Under 100,000. But the critical speed was exceeded. The DN rule is a rough check — the actual critical speed depends on the support and length. Don’t rely on DN alone; calculate Nc for your length.
Step 4: Lead Selection (Speed vs Torque)
The lead (mm/rev) determines the motor RPM for a given linear speed.
n (RPM) = v (mm/s) / lead (mm/rev) × 60
For v = 1 m/s = 1,000 mm/s:
- Lead 5 mm: n = 1000/5 × 60 = 12,000 RPM. Way too fast (critical speed exceeded).
- Lead 10 mm: n = 6,000 RPM. Still high for an 800 mm screw.
- Lead 20 mm: n = 3,000 RPM. Near the critical speed limit.
- Lead 25 mm: n = 2,400 RPM. Under critical speed.
A coarser lead reduces the RPM (and the critical speed problem) but increases the torque requirement (T = F × lead / 2πη). A 20 mm lead needs twice the torque of a 10 mm lead for the same force. Trade-off: speed vs torque. Pick the lead that keeps the RPM under critical speed and the torque within the motor rating.
The ball screw workflow: 1) Calculate equivalent load P (including process forces). 2) Check L10 life: (C/P)³ × 10⁶ rev × lead. 3) Calculate the required RPM for the max linear speed. 4) Check critical speed Nc based on diameter, length, and support. 5) Pick the lead so RPM < Nc. 6) Check torque: T = F × lead/(2πη) within motor rating. 7) Choose support bearings (fixed end angular contact, floating end deep groove).
Bearing Support Arrangement
The screw ends need bearings. The arrangement determines the critical speed.
- Fixed end (drive end): Angular contact bearings (duplex) that lock the screw axially. They take thrust and radial loads. This end “fixes” the screw position.
- Floating end (opposite): A deep groove ball bearing that allows axial movement (the screw expands as it heats up). It takes radial load only. The screw can grow axially without binding.
A fixed-floating arrangement is standard. Both ends fixed (preloaded) increases the critical speed (λ=20.5 instead of 15) but requires thermal compensation (the screw expands).
Backlash and Preload
A standard ball screw has some backlash (lost motion when reversing). For positioning axes, use a preloaded nut (double nut or oversized ball). Preload removes the backlash but increases friction (more torque). For high-precision (±0.01 mm), preloaded nut. For general transport, standard nut is fine.
A Ball Suck Selection Checklist
- What is the equivalent load P? (Including process forces.)
- What is the required L10 life? (20k hours for production.)
- Pick a screw size: L10 = (C/P)³ × 10⁶ rev × lead.
- What is the max linear speed? (m/s)
- What lead? (n = v/lead × 60. Check motor RPM.)
- What is the unsupported length L?
- Critical speed Nc = λ × d × 10⁷ / L². Is RPM < Nc?
- Is the fixed end using angular contact bearings?
- Is the floating end using a deep groove bearing?
- Is backlash acceptable? (Preloaded nut for precision.)
- Is the screw lubricated? (Oil or grease?)
- Is there a wiper/seal on the nut? (For dirty environments.)
The Bottom Line
Ball screw selection is two checks, not one. The L10 life tells you how long the balls last. The critical speed tells you how fast the screw can spin without whipping. The screw that whined at 3,000 RPM wasn’t worn out — it was over its critical speed because the length and bearing support didn’t match the RPM. Pick the lead so RPM stays under Nc, use angular contact bearings at the fixed end, and support the floating end with a deep groove bearing. The screw that runs smoothly at full speed for years wasn’t the biggest one — it was sized for both life and speed.
