Parallel-shaft compound reducers

The parallel family retains its single-stage spur/helical/double-helical pairs. Two- and three-stage constructions use external cylindrical pairs on three or four fixed parallel shafts. Each supports spur, helical and double-helical teeth. Legacy two_stage/three_stage keys remain spur; _helical and _double suffixes select the new teeth. These are nominal geometry and ideal kinematics, not rated gearboxes.

References and conventions

• Neugart, Multi-stage Gearbox, introductory spur direction statement and How is the gear ratio calculated?: external stages reverse rotation and their ratios multiply. Accessed 2026-11-03.
• Individual cylindrical pairs reuse the equations, sources and domains in calculations, including the current ISO 21771:2007 terminology, balanced profile shifts, working centre distance and usable involute limitations.

All axes are world +Z; length is mm, speed rpm, tooth counts integers. Positive rotation is counterclockwise viewed from +Z toward XY. All shafts are fixed in position; there is no rotating carrier. Let stage j have pinion teeth zp,j, wheel teeth zw,j, and the two shaft speeds nj−1 and nj. Equal pitch-line speed with external engagement gives:

zp,j * nj−1 + zw,j * nj = 0
ij = nj−1 / nj = −zw,j / zp,j
i = n0 / nN = (−1)^N * product(zw,j / zp,j)
nN = n0 / i

The preceding wheel and following pinion form a rigid compound on each intermediate shaft. For reverse operation the last shaft is prescribed instead: n0 = nN * i, and the reported input/output ratio is 1/i. The same equations hold at zero speed and for signed speeds. Efficiency does not change ideal RPM. Only overall assumed efficiency is specified; the report does not invent a per-stage loss distribution or per-stage mesh forces.

Helical and double-helical stages

KHK, ABCs of Gears Basic Guide B, printed pp4–5 equations (1.1)/(1.2), gives the external and compound speed relations; p19 equations (2.13)/(2.14) gives mt = mn / cos(beta) and d = z mt. KHK-USA, Making the Best Selection, p22, specifies opposite hands for mating parallel helical gears and adjacent opposite helices for herringbone gears. References accessed 2026-11-03.

Every active stage has its own signed pinion helix angle, normal module mn and normal pressure angle alpha_n. Its wheel uses the opposite angle. The next compound pinion may have a different stage angle. The existing normal-rack profile construction uses:

mt = mn / cos(beta)
alpha_t = atan(tan(alpha_n) / cos(beta))
rp = mn zp / (2 cos(beta)); rw = mn zw / (2 cos(beta))
a0 = rp + rw

Balanced shifts and added working centre distance retain their existing rules. A nonzero helix changes transverse diameters and interaxes, so layout envelope checks run again. The ideal tooth-count speed ratio is unchanged.

Axial tooth phase follows phi(z) = z tan(beta) / r, with z relative to the stage start. See Applied Mechanics, section 9.4, equations (9.4.1)/(9.4.2); its pitch diameter is translated here to the normal module convention above. Because zp kp + zw kw = 0 for slopes k=tan(beta)/r, the weighted mesh phase remains constant at every common axial section.

Double helices reuse the shared mirrored cylindrical surface. Joined sets every stage groove to zero and teeth meet at the mid-plane; Separated uses an independently adjustable positive groove per stage. Width includes the groove; each active half is (width − gap)/2. Matching halves and groove shoulders have the same axial phase function and opposite weighted slopes. Contact overlap is reported for one active half, including Joined:

epsilon_beta = ((width − gap)/2) abs(sin(beta)) / (pi mn)

The groove must leave more than 1 mm total tooth width under the existing pair policy. Separated halves retain the existing short transition to a continuous root hub. An equal-load assumption could balance opposing axial forces; this model does not establish that condition or calculate stage thrust/support loads.

Planar placement and tooth clocking

The first centre line points along +X. Centre-line turn · stage 2 rotates the A→B direction into B→C; the stage-3 turn rotates B→C into C→D. These are signed XY turns about +Z, not angles between shaft axes. Zero continues straight; +90° folds left and −90° folds right viewed from +Z. Turns accumulate.

For canonical working centre distance aj and cumulative angle theta,j:

Cj = Cj−1 + aj * (cos(theta,j), sin(theta,j))
theta,1 = 0
theta,j = sum(turn2 ... turnj)

Each stage has its own module, face width, pressure angle, backlash, balanced profile shift and centre-distance increase. The axial face bands advance along +Z with positive stage_gap between pairs. Intermediate shaft bores are shared by their two gears and connecting hub. The hub bridges the gap and must fit strictly below both tooth roots. Nominal shafts span their own gear bands plus the specified end extensions.

Pinions have a fixed zero tooth datum in the constructed compound. Wheel datum phi,w satisfies the external mesh phase relation:

zp * phi,p + zw * phi,w = (zp+zw)*theta + pi*(zw+1)  (modulo 2*pi)

The datum reduces to the existing single-stage phi,w = pi + pi/zw at zero centre-line angle and zero pinion phase. Differentiating recovers the shaft speed equation. Different tooth datums on a rigid compound are fixed offsets, not independent rotations. Ports carry explicit shaft and rigid-group identities; preview animation and RPM-only updates use these identities rather than names.

Layout validity and limits

Every stage runs the canonical external mesh checks. Nonmating gears, shafts and connecting hubs are also checked as full-rotation swept cylinders using tip radii and their actual axial spans. Unrelated cylinders must have either the requested axial or radial separation. Same-shaft interfaces and the two intentionally meshing gears in each stage are excluded from this envelope test. It conservatively rejects some potentially feasible interleaved layouts; it does not prove tooth conjugacy, manufacturing clearance or loaded contact. Discrete interference remains available and reports its own coverage limits.

For example, a 180° turn can fail because an extended shaft enters a previous gear's envelope. The angle itself is not forbidden: different axial spans can make the same turn admissible. Rejected browser generation retains the accepted worker source while the invalid draft blocks export in the document UI.

Compound wheels, hubs and pinions are exported as touching separate solids with common shaft metadata. There is no Boolean fusion or automatic CAD motion joint. Nominal bores/shafts do not define keys, splines, fits or torque capacity. Automatic catalog bearing seats and standard hardware currently cover the single-stage pair only; multistage constructions use manual shaft dimensions. The working-backlash solver likewise remains limited to single-stage pairs.

Independent checks

For tooth pairs 20/60 and 20/80, i=+12; at 1500 rpm the shaft speeds are 1500, −500, +125 rpm. Adding 20/60 gives i=−36 and output −125/3 rpm. Reverse prescribed output recovers the original input. At module 2, zero shift and centre increase, the first two interaxes are 80 and 100 mm; a +90° second turn gives centres (0,0), (80,0), (80,100). The three-stage default uses module 2.5 in its final pair (100 mm interaxis) to clear the preceding wheel with the default shaft overhangs. Mirroring turn signs preserves ratios and distances.

The KHK normal-system example gives d=60/cos(15°)=62.1165708246 mm for 30 teeth and normal module 2. The first default pair at 20° helix has interaxis 80/cos(20°)=85.1342217981 mm; zero helix recovers the old spur contours and interaxes. Tests also probe actual sections across both halves, closed moving surfaces and intentionally wrong phase controls.

Tests cover these Fraction benchmarks, signed/zero speeds, independent pair dimensions, phase relations, rigid compounds, nonmating envelope rejection, rate-only surface/pose preservation, finite searches, saved projects and shared STEP/STL/SVG/DXF exports. These checks do not establish strength, durability, bearing life or physical backdrivability.