Finite rack and pinion

Three tooth types share the same geometry and motion/export pipeline: spur, helical and double helical. Double helices can join or be separated by a relieved groove, with a continuous pinion core and rack base. The pinion is the rotary input. Linear input/reversed member roles are not offered in this version.

Reference equations and frames

Primary reference: KHK, Calculation of Gear Dimensions, section 4.1(3), Table 4.5, Figure 4.3 and equation 4.2 for spur racks; section 4.3(3), Table 4.13 and equation 4.18 for normal-system helical racks. Official technical reference.

Lengths are millimetres, internal angles radians, UI pressure/helix angles degrees, input speed RPM. Normal module mn and pressure alpha_n are shared:

mt = mn / cos(beta)
alpha_t = atan(tan(alpha_n) / cos(beta))
pt = pi * mt
rp = mt * pinion_teeth / 2
travel_per_revolution = pt * pinion_teeth = 2*pi*rp
x = rp * pinion_angle
linear_speed_mm_s = 2*pi*rp * input_rpm / 60

The rack pitch line is world Y=0, teeth point toward +Y and travel is along +X. The pinion shaft is +Z through (0,rp). Right-hand positive pinion rotation drives positive rack translation. The lower pinion surface has matching axial trace x(z)=rp*phi(z), where phi(z)=z*tan(beta)/rp. The derived rack has the opposite conventional hand to the pinion: matching traces incline in the same world direction at their common contact region. Do not independently assign rack hand. Double helices reverse this slope in the second working half.

This is a rotary-to-linear conversion. No dimensionless reduction ratio, rack output RPM or output torque is reported. Nominal output force is input torque divided by reference radius, with the selected assumed efficiency. It is not an allowable load or linear-guide friction model.

Tooth and blank construction

Pinion working flanks reuse the existing cylindrical involute and trimmed rounded rack-tool root envelope at zero profile shift. The finite rack has straight transverse flanks with nominal sharp corners, addendum mn, dedendum 1.25mn and the specified base thickness below the roots. Reference pair backlash j is split: pinion reference tooth arc and rack reference tooth thickness each decrease by j/2. Rack tooth half-width at height y is pt/4 - j/4 - y*tan(alpha_t).

Every requested rack tooth is complete. Fixed rectangular end planes include additional untoothed lands accommodating axial sweep; total physical base length is (rack_teeth+1)*pt + abs(helix_sweep). The rack tooth count controls length in whole pitches; base thickness does not add guides, shafts, holes or hardware.

Explicit, corresponding axial sections follow the pinion working helices and its optional relieved central transitions. Rack sections follow the same axial levels, trace displacement and relief fraction. The groove relief is nonworking nominal geometry. In each pinion interval, the underlying defined section is R(phi0+f*delta_phi)*(q0+f*delta_q). Its second derivative norm is bounded by Rmax*delta_phi^2 + 2*abs(delta_q)*abs(delta_phi). Subdivision limits the axial vertex-trace chord error to 0.003 mm using that bound. This does not bound total involute profile sampling error, whole triangulated working surfaces or manufacturing accuracy. Transverse pinion profiles retain their existing finite 0.015 mm adaptive chord criterion. Surface budgets remain 250,000 faces per part.

Compatibility and finite stroke

Pointed or overlapping rack teeth, inadequate pinion bore walls, unusable pinion contact, insufficient engagement and exhausted mesh budgets are refused. The rack-tip approach coordinate must be positive, and its contact radius must stay above the pinion's actual trimmed involute start. The nominal transverse contact ratio is the available line-of-action length divided by transverse base pitch and must exceed one. No strength or elastic-contact result is implied.

The conservative symmetric travel excludes the horizontal active pinion footprint, the full helix sweep and one rack pitch from each end of the reference toothed interval. It is a supported preview stroke, not a mechanical end stop. Both members stop together at its boundary. Reversing RPM continues from the same angle/displacement; zero RPM retains the current position. Initial pose returns to nominal geometry. Programmatic poses outside the working stroke are rejected. The finite rack has no rotary repetition period; interference analysis samples its bounded working stroke instead. Such discrete tests do not certify continuous clearance at every parameter and position.

Complete physical linear guides, mounting, pinion shaft/bearings, retention, lubrication and end stops in CAD. The generator exports two blanks; supports and their quantities are not inferred.

Independent benchmark and tests

KHK Table 4.13 uses mn=2.5, alpha_n=20°, beta=10°57′49″, z=20: reference diameter 50.92956 mm, transverse pressure 20.34160° and approximately 160 mm per pinion revolution. With pitch-line height 27.5 mm, mounting distance is 52.965 mm. The spur default mn=2,z=24 gives 48*pi mm/rev and 30.159289474462 mm/s at 12 RPM.

tests/test_rack_geometry.py compares these independent published examples, solves the literal common-normal relative-velocity equation on an independently constructed involute, compares rack flank positions/normals, and exercises actual shared moving meshes with wrong-direction collision controls. It also covers the zero-helix limit, closed solids, bores/end planes, signed/stationary RPM, retained geometry/continuous poses, invalid geometry, projects, worker parity and standalone STEP/STL/profile exports. Native and browser checks are separate from these equations and sampled-solid tests.