Rack and Pinion — validation & limitations

What it calculates

• Drive force in all seven motion phases — accelerate, constant speed, decelerate (both directions) and hold — with gravity, guide friction and a working-stroke process force
• ISO 6336-based root bending (Method B) and pitting of BOTH pinion and rack, the rack as the exact z → ∞ limit of the gear engine
• Each member rated with its own material and its own life count — rack passes, never pinion cycles divided by an infinite ratio
• Reversing drives rated with the fully reversed bending factor Y_M 0.7
• Pitch diameter, feed per revolution π·d, pinion speed and torque, contact ratio and undercut check for straight and helical racks
• Two or more pinions sharing the load with a load-share factor
• Motor torque through the gearbox — peak and RMS over the cycle, with pinion, gearbox and rotor inertia and the inertia ratio
• Holding-brake torque for inclined and vertical axes
• Max feed force per pinion at the current size — compare it with a catalogue's permissible feed force only after matching K_A, safety factors, speed and life
• Emergency-stop and brake-engagement check as a static peak load, kept out of the motor sizing
• Dimensioned sketch of the pinion on the rack with the pitch line, module, pitch diameter, mounting height and tooth forces

Validation evidence — independent reference cases (2)

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.

• Peak drive force, feed per revolution, rack root and flank safety, motor torque

  Agrees

  Source: Hand derivation (research/rack-pinion/GOLDENS.md, case H) from Newton's law, involute geometry (DIN 3960), the ISO 6336-based rating (ISO 6336-2 pitting, ISO 6336-3 Method B root, as implemented) in the rack limit z -> infinity, and rigid-body inertia reflection

  Inputs: 250 kg, horizontal, mu 0.005, F_proc 400 N, 2 m/s, 3 m/s2, stroke 3000 mm; m 3, z 20, beta 19.5283 deg, b 30 mm, 16MnCr5 / 42CrMo4, Q7, K_A 1.25; i 5, eta 0.95

  Quantity	Reference value	Tolerance
  Peak drive force (forward acceleration)	1162.258 N	2e-6 relative
  Feed per revolution	200.000 mm	2e-6 relative
  Rack root SF / flank SH (reversing Y_M 0.7; worst over the speed range)	4.9088 / 1.7920 (SAFE)	2e-6 relative
  Motor peak / RMS torque	8.1308 / 4.0930 N·m	2e-6 relative

  tests/golden/REF-rack-pinion-horizontal-gantry.golden.test.ts

• Gravity-loaded vertical axis: peak force, hold torque and rack pitting

  Agrees

  Source: Hand derivation (research/rack-pinion/GOLDENS.md, case V) from Newton's law, the ISO 6336-based rating in the rack limit (as implemented), and the static hold torque m·g·r

  Inputs: 80 kg, vertical, 0.5 m/s, 2 m/s2, stroke 800 mm; m 2, z 24 spur, b 20 mm, 42CrMo4 pinion / C45E rack, Q8, K_A 1.25; i 10, eta 0.90

  Quantity	Reference value	Tolerance
  Peak drive force	944.532 N	2e-6 relative
  Hold torque at the pinion	18.8288 N·m	2e-6 relative
  Rack root SF	2.8991	2e-6 relative
  Rack flank SH (worst over the speed range)	0.9105 (FAIL)	2e-6 relative

  tests/golden/REF-rack-pinion-vertical-lift.golden.test.ts

Limitations

Not checked by this tool

• Rack fastening (screws, dowels, joints) and rack bar bending between its screws — Check the screws and dowels for F_t and the helical F_a with /bolts, and the rack as a beam between its screws with /beams
• Pinion shaft, key and bearings under the tooth forces — Check them with /shaft, /keys and /bearings using the F_t, F_r and F_a shown
• Gearbox service factor and motor duty / thermal rating — Only torque and speed are compared; check the gearbox rating and the motor duty cycle with the makers' data or /motors
• Face-load factor of an overhung pinion from shaft deflection — The open-gearing estimate assumes an offset straddle-mounted pinion; enter K_Hβ from a deflection analysis for an overhung one
• Scuffing, micropitting and wear of unhardened racks — Scuffing is a screening index only (ISO/TS 6336-20/21 not rated); wear of a soft rack depends on lubrication and the maker's data
• Special rack profiles (root fillet under 0.2·m) and rack joint pitch error — The rating assumes the basic rack profile; align butt joints with a joint gauge and check special profiles separately
• Backlash, positioning accuracy and axis stiffness — Not computed; use a dual-pinion electrical preload or a preloaded split pinion where backlash matters
• Load spectrum and electrical preload between pinions — Rated at the peak force (no Miner sum). A master-slave preload is not modelled and is NOT a process force: it loads one pinion in both directions; rate that pinion at F_t + preload in /gears