Gears Taxonomy — Family Index

A Tier 3 family-index covering the principal gear types, their geometry, manufacturing, accuracy systems, and where each fits in real machinery. Companion to [[Engineering/gears-power-transmission]] (Tier 2 rating + design math) and [[Engineering/Tier3/bearings-taxonomy]].

1. At a glance — categorize by shaft arrangement

Gears classify naturally by the relative orientation of the shafts they connect:

• Parallel shafts — axes coplanar and parallel.
  • Spur, helical, double-helical (herringbone), internal (ring), rack-and-pinion.
• Intersecting shafts — axes coplanar and cross at a point (commonly 90°).
  • Straight bevel, miter (1:1 bevel), Zerol bevel, spiral bevel.
• Non-parallel, non-intersecting (skew) shafts — axes neither parallel nor intersecting.
  • Worm-and-wheel, hypoid, crossed-helical, spiroid, face gear.
• Specialty / non-classical — geometry breaks the simple shaft-pair model.
  • Planetary (epicyclic), harmonic drive (strain-wave), cycloidal / RV, magnetic gear.

The shaft-arrangement axis interacts with the second-order axis of tooth profile — involute, cycloidal, Wildhaber–Novikov — and the third-order axis of manufacturing process (hobbed, shaped, ground, sintered, molded). Sections 8–11 cover those.

2. Parallel-shaft gears

2.1 Spur gears

Teeth straight and parallel to the axis. The geometric baseline of involute gearing.

• Pressure angle: 20° is the modern standard (ANSI/AGMA 1012, ISO 53). 14.5° survives in legacy replacement work; 25° appears where higher beam strength is needed and undercut tolerance allows.
• Sizing: module m (ISO, mm) or diametral pitch P (AGMA, in⁻¹); m = 25.4 / P.
• AGMA quality typically Q6 to Q12 for industrial work; Q10–Q12 hobbed-and-ground.
• Cheapest gear to cut; tolerates wide centre-distance variation.
• Disadvantage: full tooth engages and disengages at once → noise, impact, lower contact ratio (~1.4–1.8 typical).
• Usage: low/medium-speed gearboxes, hand tools, fixtures, gear pumps.

2.2 Helical gears

Teeth wrap helically at helix angle β (typically 15° to 30°; up to 45° in specialty designs).

• Contact ratio splits into transverse + face (axial overlap) → total ratio commonly 2.5–4.0; engagement is gradual, hence quiet and smooth.
• Higher load capacity than spur at equal face width.
• Penalty: axial thrust load proportional to tan β; bearings must absorb it.
• Used everywhere quietness matters: automotive transmissions (helical until the synchronizer-to-final), HVAC reducers, machine-tool spindles.

2.3 Double-helical and herringbone

Two opposing-helix halves on one blank — axial thrusts cancel internally.

• Double-helical: a relief groove separates the two halves; allows tooth grinding.
• Herringbone: continuous Vee with no clearance gap; cut by a special Sykes generator or Citroën chevron-shaper (André Citroën’s original 1898 patent gives “Citroën” its chevron logo).
• Used in marine reduction gears, large turbo-compressor bull gears, paper-mill drives, cement-mill open gearing.

2.4 Internal (ring) gears

Teeth point inward from an annular ring. Mate with an external pinion of smaller pitch diameter.

• Shorter centre distance, higher contact ratio, lower sliding losses than equivalent external pair.
• The “ring” of a planetary gearset is always an internal gear.
• Cut on a Fellows-type gear shaper (the hob cannot reach inside a closed ring).

2.5 Rack-and-pinion

Rack = infinite-radius gear, straight tooth row.

• Converts rotary motion to linear (rack-and-pinion steering, machine-tool axes, gantry positioning).
• Stitched rack sections of 0.5–2 m are joined and ground for long travel.
• Helical rack (e.g., Atlanta, Güdel) gives smoother engagement than straight cut for high-speed gantries.

3. Intersecting-shaft (bevel) gears

Pitch surfaces are cones meeting at the shaft intersection point.

3.1 Straight bevel

Teeth straight along the cone face. Two parallel cutting systems dominate historically:

• Gleason (USA) — face-mill, single-indexing.
• Klingelnberg / Oerlikon (Germany / Switzerland) — face-hob, continuous-indexing palloid system.

A 1:1 bevel pair on 90° shafts is a miter gear.

3.2 Zerol bevel

Curved teeth like a spiral bevel but at zero spiral angle. Curve gives smoother engagement without producing thrust direction reversal; cut on Gleason machines.

3.3 Spiral bevel

Teeth curved and oblique at a spiral angle (commonly 35°; range 25–45°).

• Higher capacity and lower noise than straight bevel; engagement is gradual along the curved tooth.
• Found in automotive differentials (light-duty), aircraft turboprop reduction gearboxes, helicopter main rotor drives.

3.4 Hypoid

Strictly a skew-shaft gear (shafts offset, not intersecting) but historically grouped with bevels because the tooth-cutting machinery and pitch geometry (offset hyperboloidal pitch surface) are bevel-derived.

• Offset E between pinion and ring axes typically 30–50 mm in passenger cars; allows a larger, stronger pinion and lower driveline tunnel.
• Used in rear-wheel-drive and 4WD automotive rear axles (e.g., Dana, Eaton, GKN Driveline, AAM ring-and-pinion sets).
• Sliding action requires hypoid-rated GL-5 lubricant.

4. Non-parallel, non-intersecting (skew-shaft) gears

4.1 Worm gear

Worm = a screw; mating wheel = a partial helical gear “throated” around it.

• Starts: single, double, triple, quad (occasionally hexa-start).
• Reduction per stage: 5:1 to ~100:1 (compare to 6:1 typical for spur/helical single stage).
• Self-locking when friction angle exceeds lead angle. Lead angle ≤ ~5–6° on hard-steel-on-bronze with normal lubrication usually self-locks (treat as a safety convenience, never as a brake — vibration can defeat it).
• Material pairing: hardened/ground steel worm (16MnCr5, 20MnCr5, 18CrNiMo7-6, often case-carburized to HRC 58–62) + bronze worm wheel — phosphor bronze CuSn12Ni-C, CuSn12, or nickel-aluminium bronze for higher loading.
• Efficiency drops with ratio: 80–95% at low ratios, 50–70% at 50:1+.
• Vendors: David Brown (Bonfiglioli), Cone Drive, Sumitomo, Flender worm units.

4.2 Crossed-helical (screw) gears

Two helical gears on shafts at an angle, neither parallel nor intersecting.

• Point contact rather than line contact → very low load capacity.
• Used for light-duty motion transfer (timing gears in old machines, instrument drives).

4.3 Spiroid

A patented (Illinois Tool Works, 1950s) hybrid between worm and hypoid; pinion is a tapered conical thread, gear is a face-type ring.

• Higher ratio per stage than hypoid, higher capacity than worm.
• Found in actuator drives (e.g., Spiroid by ITW, now part of Power Solutions International / Spiroid Gearing).

4.4 Face gear

Pinion is a spur or helical; gear is a flat (or near-flat) toothed disc. Used in helicopter accessory drives and some torpedo-tube actuation; bridges parallel-axis cutting practice with right-angle geometry.

5. Planetary / epicyclic gearsets

A coaxial arrangement of sun (central external gear), planets (3–6 typical) carried on a carrier (arm), and ring (internal annular gear).

5.1 Configurations and ratios

Let S, P, R be tooth counts (with R = S + 2P).

• Carrier fixed, sun → ring (or ring → sun): pure star arrangement, ratio = R/S.
• Ring fixed, sun input, carrier output: i = 1 + R/S. Most common single-stage reducer mode (3:1 to ~10:1).
• Sun fixed, ring input, carrier output: i = 1 + S/R. Less common.
• Compound planetary (Ravigneaux, Simpson) — multiple sun/ring sets share a carrier → multiple speeds with simple clutching (automatic transmissions).

5.2 Why use a planetary stage

• Load shared among 3+ planet meshes → high torque density.
• Coaxial input/output (no offset).
• Reductions 3:1 to ~10:1 per stage; two- and three-stage units reach 100:1–400:1.
• Vendors (industrial): Wittenstein alpha (alpha SP/TP/cyber series), Nidec Shimpo (VRT, VRL), Apex Dynamics (AB, AE series), Stober (PHQ, PHK), Onvio, Neugart, Harmonic Drive HPG/CSG line (also a planetary product family).

6. High-ratio specialty drives

6.1 Harmonic drive (strain-wave gear)

Invented and patented by C. Walton Musser in 1957 (US Patent 2,906,143); commercialized in Japan as Harmonic Drive (United Shoe Machinery / Hasegawa Gear Works, today Harmonic Drive Systems Inc. and Harmonic Drive LLC under Sumitomo Heavy Industries).

Components:

• Wave generator — elliptical plug with thin-section ball bearing on its outer race.
• Flexspline — thin-walled cup with external teeth that flexes elliptically.
• Circular spline — rigid internal gear with two more teeth than the flexspline.

Operating principle: the wave generator deforms the flexspline so its teeth mesh with the circular spline at the two long-axis lobes only. Each full rotation of the wave generator advances the flexspline by the 2-tooth difference, giving very high reduction in a single coaxial stage.

• Ratios 30:1 to 320:1 single stage.
• Zero backlash (preloaded interference engagement).
• High repeatability (arc-seconds class), low ratcheting torque, but limited stiffness in torsion.
• Dominant gearing for collaborative-robot joints and 6-axis industrial-robot wrists (UR3/5/10/16, ABB YuMi, KUKA LBR iiwa wrist stages, Doosan M-series).

6.2 Cycloidal drive / RV gear

Eccentric cam drives one or two cycloidal disc(s) against a fixed ring of pins; output picked off through follower pins or rollers in the disc.

• Tooth contact is rolling on cycloidal profile; many pins share load simultaneously (~50–70% engagement) → high shock capacity.
• Two-stage RV combines an input-side spur reduction with a cycloidal output stage (Nabtesco’s “Rotary Vector” architecture).
• Ratios 30:1 to ~200:1 single stage; RV units reach 250:1.
• Vendors: Nabtesco RV (industry leader for industrial-robot base/shoulder joints since 1986; FANUC, Yaskawa, KUKA, ABB use Nabtesco on axes J1–J3), Sumitomo Cyclo (Cyclo 6000, Cyclo BBB5), Spinea TwinSpin (Slovakia, integrated bearing-reducer), Onvio (Vigel), and Leader Drive (China).

6.3 Comparison: harmonic vs cycloidal

Property	Harmonic	Cycloidal RV
Reduction (single stage)	30:1–320:1	30:1–200:1
Backlash	~0 arcmin	~1 arcmin
Torque density	High	Very high
Shock capacity	Moderate	High
Torsional stiffness	Lower	Higher
Typical robot axis	Wrist (J4–J6)	Base / shoulder (J1–J3)
Mass	Light	Heavier
Efficiency (rated load)	60–80%	70–90%
Lifetime at rated	~10,000 h	~6,000–20,000 h
Output bearing	Sometimes integrated (e.g. CSG-2UH unit)	Usually integrated crossed-roller

A practical rule from industrial robot integrators: if the axis needs high stiffness, shock tolerance, and torque (large arms, big payloads), choose cycloidal RV. If the axis needs compact mass, near-zero backlash, and high reduction in one stage (wrist joints, surgical robots, cobots), choose harmonic.

7. Magnetic gears

Torque transmitted by interacting magnetic fields between rotor pole pieces — no tooth contact, no lubrication, no wear in the gear pair.

• Slip-torque limit (rather than mechanical fatigue) defines maximum load; overload simply slips and re-engages.
• Excellent isolation: hermetic separation between drive and driven sides (subsea, sterile bioreactors, semiconductor wet processes).
• Torque density approaches conventional gearing (1–100+ kN·m/m³ for modern designs) but at higher cost.
• Vendors: Magnomatics (UK, magnetic pseudo-direct-drive marine and wind), KEB (clutches/couplings), MagnaDrive (couplings).

8. Tooth profiles

8.1 Involute (universal)

Generated by unrolling a taut string from a base circle; the involute curve is the path of any point on that string.

• Property: meshing involutes deliver constant angular-velocity ratio regardless of small centre-distance error → the “conjugate action” of involute gears is robustness to assembly tolerance.
• Standard for all modern power gearing. Cut by hob, shaper, or generator-grinder.

8.2 Cycloidal

Tooth flank is a section of an epicycloid + hypocycloid pair.

• Found in clocks and watches (small modules; low load; high efficiency at very small tooth counts where involute would undercut).
• Cycloidal drive (Section 6.2) uses cycloidal tooth on the disc but with pin engagement, not cycloidal-on-cycloidal meshing.

8.3 Wildhaber–Novikov (circular-arc)

Convex tooth on pinion + concave tooth on gear, both circular arcs. Developed by Ernest Wildhaber (1923 US patent) and reduced to practice by Mikhail Novikov (1956 USSR).

• Contact ratio drives entirely on axial overlap, not transverse — so only helical implementation works.
• High contact area and load capacity in theory; very sensitive to centre-distance error in practice.
• Used in Soviet-era helicopter transmissions and a few specialty Western applications.

9. Materials and heat treatment

9.1 Through-hardened (medium-carbon)

• AISI 1045, 4140, 4340 → 28–35 HRC.
• Cheaper, used for slow-speed industrial and machine-tool gears.

9.2 Case-hardened (carburized)

The mainstream for power gearing:

• AISI 8620, 9310, 4320 (USA); 16MnCr5, 20MnCr5, 18CrNiMo7-6 (DIN/EN).
• Carburized to 0.8–1.5 mm case depth; surface HRC 58–62, core HRC 30–40.
• Followed by gear grinding or shaving + honing to restore profile distortion from heat treat.

9.3 Nitrided

• AISI Nitralloy 135M, 31CrMoV9.
• Surface hardness HV 700–900 (HRC ~60–65), but shallow case (~0.3–0.5 mm) — limited to medium loads.
• Distortion is much lower than carburizing → low-distortion grinding optional.

9.4 Induction-hardened

Selective surface hardening on flame- or induction-heated medium-C steel (1050, 4140). Common on large mill gears and slewing-ring teeth.

9.5 Bronzes (for worm wheels)

• CuSn12-C (G-CuSn12), CuSn12Ni-C (G-CuSn12Ni), CuSn5Pb20-C for high-speed sliding.
• Aluminium-bronze CuAl10Ni for very high load with sacrificial wear.

9.6 Polymers

• POM (Delrin), PA66, PA6, PEEK, UHMW-PE for small low-load gears.
• Quiet, self-lubricating, corrosion-immune; limited torque, limited temperature.
• 3D-printed gears (FDM PETG/PA, SLS PA12, MJF PA12) used in prototypes and low-duty production.

10. Quality / accuracy systems

Three systems coexist; conversion is approximate, not bijective.

• AGMA 2000-A88 (legacy): Q-numbers Q3 (rough) to Q15 (high precision); higher = better.
• AGMA 2015-1-A01 (current): grades A1 (best) through A12 (worst); inverse of legacy AGMA.
• ISO 1328-1:2013: grades 0 (theoretical perfect) through 12 (rough); lower = better. Grade 5 ≈ aerospace; grade 7–8 ≈ industrial; grade 10–12 ≈ rough commercial.
• DIN 3962/3963: similar to ISO 1328 with grades 1–12.
• JIS B 1702-1:1998: aligned with ISO 1328.

Rule of thumb:

• Hobbed, no further finishing: ISO 8–10.
• Hobbed + shaved or honed: ISO 6–7.
• Ground (Reishauer, Maag/Niles, Höfler, Klingelnberg HRGS): ISO 3–5.

11. Manufacturing processes

11.1 Generating processes (most accurate)

• Hobbing — workhorse for external spur, helical, worm; high productivity, ISO 8–10 as-hobbed.
• Gear shaping (Fellows) — only practical method for internal gears and for gears with a blind shoulder; ISO 7–9.
• Gear grinding — finishing process after hardening. Profile (form) grinding (Höfler, Klingelnberg) or generating-thread grinding (Reishauer); ISO 3–5.
• Honing — fine surface finish, low material removal; corrects helix and profile errors after heat treat.
• Shaving — pre-hardening generating finish; ISO 6–7 if undisturbed by HT.
• Skiving (gear skiving) — high-productivity hard-finishing alternative to grinding for internal gears; Gleason, EMAG, Reishauer offerings ~2010s onward.

11.2 Forming and other processes

• Milling — disc cutter per module/tooth-count band; rough only.
• Broaching — limited to short gears.
• Powder-metal sintering — high-volume low-load auto parts (timing-belt sprockets, oil-pump gears); near-net.
• MIM (metal injection moulding) — small precision steel gears.
• Plastic injection moulding — POM, PA, PEEK gears.
• Additive manufacturing — DMLS / SLM for metal; SLS / MJF / FDM for polymer.

12. Selection heuristics

Constraint	First-choice gearing
Parallel shafts, low cost, low/med speed	Spur
Parallel shafts, high speed or quiet	Helical
Parallel shafts, very high power, no thrust on bearings	Herringbone / double-helical
Rotary → linear	Rack-and-pinion
Right-angle, light/medium duty	Bevel (straight or spiral)
Right-angle, auto rear axle	Hypoid
Right-angle, very high reduction, self-locking acceptable	Worm
Coaxial, high reduction, high torque density	Planetary (single or multi-stage)
Coaxial, high reduction, zero backlash, precision (robot wrist)	Harmonic drive
Coaxial, high reduction, high shock load (robot base)	Cycloidal RV
Sealed / no-contact transmission	Magnetic gear
Automotive transmission (multi-ratio)	Helical + planetary + synchronizer (manual) or planetary + clutch packs (automatic)
Quiet residential reducer	High-contact-ratio helical

13. Backlash and accuracy targets

• Commercial spur/helical gearing: 0.1–0.5 mm circumferential backlash (size-dependent).
• Precision ground gears: 0.01–0.05 mm.
• Planetary servo reducer (Wittenstein alpha SP+): 1–6 arcmin.
• Cycloidal RV: ~1 arcmin.
• Harmonic drive: essentially zero (specification often quotes hysteresis < 1 arcmin instead).

For motion-control applications, the lost-motion specification (peak-to-peak under bidirectional rated torque) is more meaningful than static backlash.

14. Major vendors

14.1 Robotics gearing

• Harmonic Drive LLC / Harmonic Drive Systems Inc. — CSG, CSF, SHG, SHF, HFUC series; the de-facto industry standard for cobot and wrist drives.
• Nabtesco — RV-E, RV-N, RV-C series for industrial robot base and shoulder joints since 1986.
• Sumitomo Heavy Industries — Cyclo 6000, Fine Cyclo F-series (precision robotic).
• Spinea — TwinSpin (integrated cycloidal + crossed-roller bearing).
• Leader Drive (China) — RV competitor, growing share in domestic Chinese robot OEMs.

14.2 Industrial precision planetary

• Wittenstein alpha (Germany) — alpha SP, TP, cyber.
• Nidec Shimpo / Nidec Drive Technology — VRT, VRL.
• Stöber — PH, PHQ, PHK.
• Apex Dynamics — AB, AE, AT series.
• Neugart — PLE, PLPE, PSN.
• Onvio (Vigel) — high-end planetary.

14.3 Large industrial gearboxes

• Flender (Siemens) — bevel-helical, parallel-shaft.
• Bonfiglioli — modular bevel-helical and planetary.
• SEW-Eurodrive — modular helical, helical-bevel, helical-worm, planetary (MOVIGEAR integration).
• David Brown Santasalo — heavy-industrial.
• Renold — worm and planetary.

14.4 Automotive transmissions

• ZF Friedrichshafen — 8HP and 9HP planetary automatics, dual-clutch units.
• Aisin — torque-converter automatics for Toyota and others.
• Allison Transmission — heavy-truck and off-highway planetary automatics.
• GKN Driveline / Dana / AAM — hypoid axle ring-and-pinion sets and limited-slip differentials.

15. Cross-references

• [[Engineering/gears-power-transmission]] — Tier 2: ISO 6336 / AGMA 2001 tooth-rating math, lubrication, efficiency.
• [[Engineering/Tier3/bearings-taxonomy]] — Tier 3: shaft-support side of every gearbox.
• [[Engineering/Tier3/couplings-taxonomy]] — Tier 3: connecting gearbox output to driven machine.
• [[Robotics/manipulator-design]] — where harmonic and cycloidal RV gears live in robot architecture.
• [[Robotics/motors-electric]] — torque/speed source matched to gear reduction.
• [[Engineering/bearings]] — EP/EHD oil regimes, GL-5 for hypoids, worm-wheel grease.

16. Citations

• Shigley, J. E., Mischke, C. R., Budynas, R. G., Nisbett, J. K. Shigley’s Mechanical Engineering Design, 11th ed., McGraw-Hill, 2020 — Chapters 13–15 on gear geometry, force analysis, and rating.
• ANSI/AGMA 2001-D04, Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth, American Gear Manufacturers Association.
• ISO 6336 (parts 1–6):2019, Calculation of load capacity of spur and helical gears, International Organization for Standardization.
• ISO 1328-1:2013, Cylindrical gears — ISO system of flank tolerance classification.
• ISO 53:1998, Cylindrical gears for general and heavy engineering — Standard basic rack tooth profile.
• Townsend, D. P. (ed.). Dudley’s Gear Handbook, 2nd ed., McGraw-Hill, 1991.
• Litvin, F. L., Fuentes, A. Gear Geometry and Applied Theory, 2nd ed., Cambridge University Press, 2004.
• Musser, C. W. Strain Wave Gearing, US Patent 2,906,143, 1959 (filed 1957).
• AGMA 925-A03, Effect of Lubrication on Gear Surface Distress.
• DIN 3962-1:1978 / DIN 3963:1978, Tolerances for Cylindrical Gear Teeth.
• JIS B 1702-1:1998, Cylindrical gears — ISO system of accuracy.
• AGMA 933-B03, Basic Gear Geometry.
• AGMA 923-B05, Metallurgical Specifications for Steel Gearing.
• AGMA 6011-J14, Specification for High-Speed Helical Gear Units.
• ISO 6336-5:2016, Calculation of load capacity — Strength and quality of materials.
• Wildhaber, E. Helical Gearing, US Patent 1,601,750, 1926 (filed 1923).
• Maitra, G. M. Handbook of Gear Design, 2nd ed., Tata McGraw-Hill, 1994.
• Stadtfeld, H. J. Gleason Bevel Gear Technology, Gleason Works, 2014.
• Radzevich, S. P. Theory of Gearing: Kinematics, Geometry, and Synthesis, 2nd ed., CRC Press, 2018.

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End — Gears Taxonomy Tier 3 family index.