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Rack meshes

Standalone rack-and-pinion geometry, forces, ISO 6336 limit ratings, tooth root FE, static safety and reversed bending.

  • ISO 6336-1:2019
  • ISO 6336-2:2019
  • ISO 6336-3:2019
  • ISO 6336-5:2016

Try this example in the app

The Rack meshes module rates a pinion and a straight rack as a standalone pair. Give either pinion torque in N·m or axial rack force in N, with the other field empty. It returns contact and root stresses, fatigue and static safeties for each member, contact ratios, lead, backlash, forces and stiffness. Member mounting, the rack beam, the yoke and steering duty are separate tracks.

Geometry and forces

Enter normal module, pinion teeth, normal pressure angle, pinion helix angle, crossing angle, face width and pinion profile shift. The rack tooth's addendum, dedendum, root radius and datum thickness are relative to the normal module. The default thickness is π/2; changing it changes the equivalent rack shift. The optional pinion thickness is its normal reference-circle thickness divided by the normal module. Empty uses its profile shift. The pinion cutter's root radius and dedendum are separate from the rack's actual root geometry.

The rack tooth angle is βr = Σ + β − 90°. Lead is π mn z₁ / cos βr, in mm per pinion revolution. Forces on the pinion are Ft = 2000 T / d₁, Fr = Ft tan αt and Fa = Ft tan β. Axial rack force is Ft cos βr / cos β; its cross force is Ft sin βr / cos β. The reported rack velocity and pinion power satisfy power balance.

Lift moves the rack away from tight mesh for the entered tooth thicknesses, shortening the contact path by lift / sin αt and opening backlash 2 lift tan αt. A thinner pinion changes that tight-mesh datum distance; to hold the actual distance fixed, increase lift by the normal thickness reduction divided by 2 tan αn. The contact ratio requirement applies to both transverse and total contact ratios at the entered lift. Interference, pointed teeth and fillets that do not fit are errors with a geometry hint.

An optional symmetric variableRatio.lead table holds [absolute displacement, lead] pairs. It interpolates linearly and holds the end lead beyond the table. position chooses the rated displacement. Constant lead gives the plain rack results exactly. A varying lead uses a local equivalent rack: the unchanged base circle sets the local transverse pressure angle and base pitch. Root factors retain the nominal normal section. This is an approximation, not a generated variable flank; validate the manufactured flank separately. Local transverse pressure angles must stay within 15°–35°. variableRatio.maxSlope is empty unless you enter a slope limit.

Rating method and its range

Contact and root ratings use ISO 6336-2/-3:2019 Method B with the wheel taken to infinite tooth count. The stiffness uses the same infinite-tooth device as ISO 6336-1:2019, 9.3.2.6 b). The dynamic factor Kv of method B is held to at least 1 and, outside the main resonance range, to at most 2 (ISO 6336-1:2019, 6.3 and 6.2.6), as for a gear pair; a mesh loaded at 100 N/mm or more whose Kv is set to 2 is warned. The rack's form and stress-correction factors come from a straight flank and circular fillet, with the critical section at the 30° tangent. No rack-specific standard rating is claimed.

ISO 6336's validated range is normal pressure angle 15°–25°, helix angle up to 30° and transverse contact ratio 1.0–2.5. A mesh outside those ranges still rates and carries an outside ISO 6336's validated range warning. Below transverse contact ratio 1, the root load is at the tip and the transverse load-distribution factor is 1. The rating cannot establish uninterrupted engagement when the total contact ratio is below 1.

Both members get a finite-element root comparison when their geometry can be meshed. The rack FE contains three straight teeth with circular fillets, held 3.5 modules below the root plane; the pinion uses its virtual spur tooth. The FE finds the tensile peak instead of fixing a 30° section. Outside the validated range, FE Y_FS is shown first, followed by ISO YF·YS. An unavailable FE result is stated, and is never replaced with a pass. These are normal-section models, not a three-dimensional contact analysis.

No published rack worked example is held. Validation rests on gear-limit convergence and hand-calculation tests for contact ratio, lift, forces, rack root factors and Annex B YM, plus a rack FE comparison. A 10,000,000-tooth gear wheel converges within a relative 0.00001 for the tested spur mesh (z₁ 20) and helical meshes (z₁ 18, x₁ 0.3, β 15°; z₁ 12, x₁ 0.5, β 20°; z₁ 17, x₁ 0.1, β 10°), including the rack's YF, YS, ZD, root and contact stresses. The rack's ISO-style accuracy deviations default to the pinion's grade values because ISO 1328 has no rack tolerance table; enter measured deviations when known.

Life, reversed bending and static loads

Select each member's heat-treatment group in Material groups for life and reversed bending. Induction/flame hardening with a hardened tooth root uses IF, distinct from case hardening Eh. Each material can also carry heatTreatment: surfaceHardened selects IF, caseHardened Eh, throughHardened V, nitrided NT and nitrocarburized NV. The member pair's materialKinds overrides those selections. An omitted or unspecified treatment keeps the legacy boolean default (Eh if surfaceHardened is true, V otherwise); Annex B adds a note to confirm it. Material names do not select the physics. The app, each result side's materialKind and the report show the group used. For IF, Annex B Table B.1 gives endurance M = 0.4 and static M = 0.6; at R = −1.2, YM = 0.660377 and YM,static = 0.769231. Eh uses static M = 0.7, giving YM,static = 0.825243.

Pinion cycles are 60 × rpm × life hours. Set rackLoadCycles to the cycles at the rack tooth being assessed. Without it, the rating assumes the pinion cycle count for each rack tooth; this is not stroke counting and can underestimate a short reversing duty. Tooth-by-tooth stroke counting belongs to the duty track.

reverseBending: "annexB" applies ISO 6336-3:2019 Annex B YM from R = −1.2 × reverseLoadRatio (lower flank load / higher flank load). The endurance and static knees use their own material coefficients, with logarithmic interpolation between knees. A zero lower-flank load gives YM = 1. For a case-hardened tooth, endurance M = 0.8 − 0.15 YS and static M = 0.7. The method reads the sign inconsistencies in B.1 and B.3 as negative R: at R = −1, YM = (1 + M) / 2. This interpretation is explicit; the 2006 Annex B has not been checked. reverseBending: "iso6336-5" applies the 0.7 factor of ISO 6336-5:2016, 5.3.3 to endurance only: static strength keeps factor 1, with logarithmic interpolation between the knees. "none" gives YM = 1.

Static safeties use the material life curves' static knees at the entered peak rack force, or the rated force if no peak is entered, with application factor 1. Dynamic and distribution factors are recomputed at that load. Requirements are off by default: enter minimum root, contact, static and lifted-contact-ratio values for your application. The example's requirements are template values, not standard values.

CLI and agents

axm new rack.axm.json --template rack-mesh
axm rack-mesh add rack.axm.json --name "Second mesh"
axm rack-mesh update rack.axm.json RM1 --patch '{"torque":null,"rackForce":1500,"rackLoadCycles":1000000}'
axm solve rack.axm.json --json
axm rack-mesh remove rack.axm.json RM2

The project list is rackMeshes[], with RM identifiers. Pure commands are addRackMesh, updateRackMesh and removeRackMesh; MCP uses the same commands through axm_apply. axm set rack.axm.json rackMeshes.RM1.faceWidth 25 follows the same command path. The app's changes and lead-table edits support Undo.