Ball Screws — validation & limitations

What it calculates

• ISO 3408-5 L10h fatigue life — loaded-track equivalent axial load with preload lift-off at 2^1.5·F_pr (2.83× preload force)
• Critical (whirling) speed with end-fixity factors for fixed-fixed, fixed-floating and fixed-free mounting (beam-theory method)
• Euler column buckling load and safety ratio for the screw root diameter
• Static safety factor (C0 / loaded-track load) on the ISO 3408-5 static load rating
• Drive torque and heat generation including preload drag, with screw efficiency
• ISO 3408 accuracy grades C0–C10 with travel-error (e_t) and worst-case positioning error over stroke

Standards this tool applies

• ISO 3408 — L10h life on the ISO 3408-5 relation (preloaded track split per the ball-screw maker catalogue method), critical whirling speed, Euler column buckling, static safety factor on the catalogue static rating, C0–C10 travel-error reference (JIS B 1192 naming), drive torque and heat generation. ISO 3408 Parts 1–4 are not implemented.

Validation evidence — independent reference cases (3)

Each case runs the tool's engine on a worked example whose values come from a published source or a hand derivation from the cited equations, and an automated regression test asserts the engine against those values within the stated tolerance. "Conservative" means the engine is known to sit on the safe side of the reference and the test asserts that side. Sources are cited; their text is not reproduced.

• Euler column buckling load and verdict bands

  Conservative

  Source: Euler column buckling, Shigley 11e §4-14 eq. 4-42; Roark 8e Table 15.1; a ball-screw maker's 'permissible axial load' catalogue method (eta1 = 2 fixed-supported)

  Inputs: d_r 26.4 mm, L 1200 mm, E 206 000 N/mm2; fixed-fixed, fixed-pinned, fixed-free; loads 3, 5, 9 kN

  Quantity	Reference value	Tolerance
  P_cr fixed-fixed	134 663.6 N	1e-6 relative
  P_cr fixed-free	8 416.47 N	1e-6 relative
  P_cr fixed-pinned (catalogue eta 2)	67 331.8 N (exact eigenvalue 68 872 N)	1e-6 vs catalogue; below exact, within 2.5 %
  Buckling ratio fixed-free at 9 kN	106.93 % (FAIL)	1e-4 absolute

  Conservative simplification: For fixed-pinned ends the engine uses the catalogue factor 2 instead of the exact 2.0457, so its buckling load sits about 2.2 % below the exact eigenvalue.

  tests/golden/REF-ballscrews-buckling.golden.test.ts

• Critical (whirling) speed of the screw shaft

  Conservative

  Source: Euler-Bernoulli beam vibration, Roark 8e Table 16.1; a ball-screw maker's 'permissible rotational speed' catalogue method (0.8 safety factor, root diameter)

  Inputs: d_r 26.4 mm, L 1200 mm, E 2.06e5 N/mm2, gamma 7.85e-6 kg/mm3; fixed-fixed, fixed-supported, fixed-free

  Quantity	Reference value	Tolerance
  n_c fixed-fixed	4013.02 rpm	0.15 %
  n_c fixed-supported	2765.51 rpm	0.15 %
  n_c fixed-free	630.66 rpm	one-sided (engine <= reference), within 1.5 %
  Speed ratio at 3000 rpm fixed-fixed	74.76 %	0.15 percentage points

  Conservative simplification: The engine carries the catalogue's rounded speed factor of 3.4 for fixed-free ends (hand value 3.44), so its critical speed reads about 1.2 % below the exact value.

  tests/golden/REF-ballscrews-critical-speed.golden.test.ts

• Static safety factor f_S = C0a / Fa_max and its verdict

  Agrees

  Source: Static safety f_S = C0a / Fa,max with the basic static axial load rating C0a (ISO 3408 series); ball-screw maker catalogue static safety factor

  Inputs: C0a 20 kN, no preload, Fa 4 kN, peak 0 / 12.5 / 25 kN, C 30 kN, 300 rpm, d_r 26.4 mm, L 300 mm

  Quantity	Reference value	Tolerance
  f_S, no peak	5.000 (no flag)	1e-9 relative
  f_S, peak 12.5 kN	1.600 (CHECK)	1e-9 relative
  f_S, peak 25 kN	0.800 (FAIL)	1e-9 relative
  L10h	23 437.5 h	1e-6 absolute

  tests/golden/REF-ballscrews-static-safety.golden.test.ts

Limitations

Conservative simplifications

• Euler column buckling load and verdict bands — For fixed-pinned ends the engine uses the catalogue factor 2 instead of the exact 2.0457, so its buckling load sits about 2.2 % below the exact eigenvalue.
• Critical (whirling) speed of the screw shaft — The engine carries the catalogue's rounded speed factor of 3.4 for fixed-free ends (hand value 3.44), so its critical speed reads about 1.2 % below the exact value.

Not checked by this tool

• Support (end) bearing life and rating for the axial load — Size the fixed-end bearings with /bearings or the maker's support-unit rating
• Support-bearing and housing stiffness in the axial rigidity — The stiffness panel covers shaft and nut only; add the support-unit stiffness from the maker
• Torsional stress and wind-up of the screw shaft at the drive journal — Not computed; check the journal section with /shaft or /torsion
• Nut flange bolts and nut housing — Check the flange bolts for the peak axial load with /bolts
• Duty-cycle averaging (cubic-mean load and mean speed) — One constant load and speed is rated; enter the cubic-mean load of your cycle as the axial load
• Lubrication interval and sealing against contamination — Not assessed; follow the ball-screw maker's lubrication and seal recommendations