Does the screw stop at L10?

L10 is a 90%-reliability statistical life; reaching it raises failure probability, it does not jam instantly. Important equipment applies a 0.5–0.8× margin.

Is critical speed the same as nut speed?

n in the formula is screw-shaft speed; if the nut moves and the shaft is stationary, use the shaft-segment speed. Long screws usually rotate.

Each screw model page and the maker catalogue list Ca; this site shows it in the model parameter block.

Ball Screw Selection Calculation: Rated Life, Critical Speed, Buckling

How to size a ball screw: screen with rated life L10, critical speed Nc and buckling load Pc. Includes end-mount factors and a worked SFU2010 example covering diameter, stroke, speed and load.

Bigger is not always better: life must be adequate, speed must stay below critical, and the slenderness ratio must not buckle. The three formulas below handle 90% of first-pass sizing; verify against the real duty cycle afterwards.

1. Rated life L10

Rated life is the number of revolutions (or hours) that 90% of ball screws reach under a given axial load.

L = (Ca / Fm)³ × 10⁶ [rev]
Lh = (Ca / Fm)³ × 10⁶ / (60 × n) [h]

Ca: basic dynamic load rating of the screw pair (from the model page / catalogue); Fm: equivalent axial load; n: speed (rpm).

Equivalent axial load Fm

Fm = X × Fw + Y × Fr + Fp

Fw axial working load; Fr radial load (usually ≤0.1·Fa, otherwise Fm rises sharply); Fp preload-equivalent load; X, Y load coefficients. For pure axial drive with negligible radial force, Fm ≈ Fw + Fp.

2. Critical speed Nc

A slender screw resonates at high speed; operating speed should stay ≤ 0.8 × Nc.

Nc = (f × d1 / Lc²) × 10⁷ [rpm]

d1: screw shaft diameter (mm); Lc: distance between supports (mm); f: end-mount factor (table below).

Mounting	Factor f	Note
Fixed–Fixed	21.9	Highest stiffness, largest Nc
Fixed–Supported	15.1	Common layout
Supported–Supported	9.7	Medium stiffness
Fixed–Free	3.4	Cantilever, lowest Nc

3. Buckling load Pc

A compressed slender screw buckles; the max axial compressive load must stay below Pc.

Pc = π² × E × I / Le²
I = π × d1⁴ / 64 (mm⁴)
E = 2.06 × 10⁵ MPa Le = μ × Lc

Le effective length = end factor μ × support span Lc: fixed–fixed μ=0.5, fixed–supported μ=0.7, supported–supported μ=1.0, fixed–free μ=2.0.

4. Worked example: SFU2010

Given: shaft d1=20 mm, lead 10 mm, grade C7; stroke 800 mm, support span Lc=1000 mm, fixed–fixed (μ=0.5, f=21.9); axial load Fw=800 N, preload Fp≈0, speed n=1500 rpm; catalogue Ca=3.8 kN.

Item	Calc	Result
Rated life L10	(3800/800)³×10⁶	107×10⁶ rev
Life hours Lh	107×10⁶/(60×1500)	≈ 1190 h
Critical speed Nc	21.9×20/1000²×10⁷	≈ 4380 rpm → 1500 < 0.8×4380=3504 ✓
Buckling Pc	π²×2.06e5×(π×20⁴/64)/(0.5×1000)²	≈ 63.9 kN ≫ 0.8 kN ✓

Illustrative only — use the Ca, stiffness and Nc tables of the specific model catalogue for real sizing.

5. Selection checklist

First compute L10 from load; if short, enlarge diameter or pick a nut with higher Ca (more ball turns).
Then check Nc; if speed nears critical, shorten support span, enlarge diameter, or use fixed–fixed.
For compression, check Pc; if too slender, add an intermediate support or use a thicker shaft.
Finally close out by accuracy grade (C3/C5/C7) and the support unit (see the bearing load chart).

FAQ

Does the screw stop at L10?: L10 is a 90%-reliability statistical life; reaching it raises failure probability, it does not jam instantly. Important equipment applies a 0.5–0.8× margin.
Is critical speed the same as nut speed?: n in the formula is screw-shaft speed; if the nut moves and the shaft is stationary, use the shaft-segment speed. Long screws usually rotate.
Where do I get Ca?: Each screw model page and the maker catalogue list Ca; this site shows it in the model parameter block.

Ball Screw Selection Model Code Lead Accuracy Series Match Table Bearing Load Chart Full Model List

Formulas per JIS B 1192 and common ball-screw design practice; end-mount factors and material values are standard engineering figures.

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