Gears & Power Transmission — Engineering Reference
See also (Tier 3 family index): Gears Taxonomy
1. At a glance
A gear is a toothed wheel that transmits torque and angular motion between shafts through positive engagement of mating teeth. Unlike belts and chains, gears are kinematically constrained — there is no slip, no creep, and (within elastic limits) the input-to-output angular relationship is fixed by tooth count alone. Gears are the densest, most efficient, and most precise mechanical power-transmission element humanity has built; modern automotive transmissions, industrial gearboxes, robotic actuators, wind-turbine drivetrains, aircraft accessory gearboxes, and machine-tool spindles all run on gears.
The two metric/imperial systems that parametrise tooth size:
- Module m (metric) — m = d / z, the pitch diameter d (mm) divided by tooth count z. Standard ISO 54 modules: 0.5, 0.8, 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 32, 40, 50.
- Diametral pitch P_d (imperial) — P_d = z / d, teeth per inch of pitch diameter. Standard coarse pitches 2, 2.5, 3, 4, 5, 6, 8, 10; fine 12, 16, 20, 24, 32, 48, 64, 96, 120. Conversion: m (mm) = 25.4 / P_d, so P_d = 10 ⇔ m = 2.54 mm.
Core tooth geometry:
| Term | Symbol | Definition |
|---|---|---|
| Pitch circle | d | Imaginary rolling circle; gears in mesh have tangent pitch circles |
| Base circle | d_b | Circle from which the involute is generated; d_b = d · cos α |
| Addendum | h_a | Radial distance from pitch circle to tooth tip; h_a = 1·m (standard full-depth) |
| Dedendum | h_f | Radial distance from pitch circle to tooth root; h_f = 1.25·m |
| Whole depth | h | h = h_a + h_f = 2.25·m |
| Pressure angle | α | Angle between the line of action and the common tangent at the pitch point; ISO standard 20° |
| Helix angle | β | Tilt of the tooth relative to the gear axis (zero for spur) |
| Backlash | j_t | Tangential play at the pitch circle between mating teeth |
| Centre distance | a | a = (d₁ + d₂)/2 for external mesh; (d₂ − d₁)/2 for internal |
Pressure angles in service:
- 14.5° — legacy/obsolete; still found in some pre-1950 American machinery and clock-quality gears. Higher contact ratio, weaker tooth, more prone to undercutting.
- 20° — modern ISO/AGMA default. Best general-purpose compromise of strength, contact ratio, and undercutting margin.
- 25° — heavy-load duty (aerospace, off-highway). Stronger tooth root, lower contact ratio (often requires helical to compensate), more thrust on shafts.
The five selection drivers, in roughly the order they constrain the design:
- Shaft configuration. Parallel (spur/helical), intersecting (bevel), offset/non-intersecting (hypoid, worm), coaxial (planetary/harmonic/cycloidal).
- Ratio per stage. Spur/helical 1:1 to ~7:1 economically (up to 10:1 with care). Bevel ~1:1 to 5:1. Planetary 3:1 to 10:1. Worm 5:1 to 100:1. Harmonic 30:1 to 320:1. Cycloidal 6:1 to 200:1.
- Power capacity and life. Set by Hertzian contact-stress pitting (S_c) and root bending-stress fatigue (S_t). Computed per AGMA 2001-D04 / 2101-D04 (metric) or ISO 6336 parts 1-6.
- Backlash, transmission error, and noise. Positioning duty: arc-minutes spec. Power duty: noise in dB(A) and TE in micrometres.
- Efficiency, cost, envelope, mass. Spur/helical/planetary are efficient (96-99%); worm and harmonic dissipate heat that has to be designed for.
Where it sits in the design stack: gears live at the intersection of kinematics (ratios, contact path, conjugate action), mechanics-of-materials (Hertzian contact + Lewis bending), fatigue (subsurface pitting + root bending), materials-steel (case-hardened steels), bearings (EHL film thickness governs life), bearings (every gear shaft rides on bearings whose loads come from gear thrust and separating forces), and manufacturing (hobbing/grinding tolerances drive both noise and life).
2. First principles
2.1 Why involute teeth
A pair of mating gears must transmit uniform angular velocity ratio — if ω₂/ω₁ wavered as the contact moved along a tooth, the output would speed up and slow down each tooth pitch. The Fundamental Law of Gearing states that the common normal at the contact point must pass through a fixed point on the line of centres (the pitch point) for the velocity ratio to remain constant. Curves that satisfy this for any centre-distance are called conjugate.
Two conjugate-tooth families have ever seen industrial use:
- Cycloidal — used in clocks and watches and a few pre-1900 machine gears. Generated by tracing a point on a circle rolling on another circle. Conjugate only at the design centre distance — varies with any centre-distance error.
- Involute — generated by tracing a point on a string unwound from a base circle. Pressure angle remains constant along the tooth, conjugacy is preserved at any centre distance, and the resulting geometry is robust to assembly tolerance. Universal in all modern power transmission.
The involute is generated mathematically from the base circle d_b:
x = r_b · (cos θ + θ · sin θ)
y = r_b · (sin θ − θ · cos θ)
with θ the involute roll-angle parameter. The line of action is the common tangent to the two base circles, inclined at the pressure angle α to the common tangent of the pitch circles. All contact between mating involute teeth happens along this straight line. Force between teeth is normal to the tooth flank, i.e. along the line of action — this resolves into a tangential (torque-producing) component and a radial (separating) component on parallel-shaft spur gears.
2.2 Contact ratio
The number of tooth pairs simultaneously in contact, averaged over a rotation:
ε_α = (length of line of action) / (base pitch)
= (√(r_a1² − r_b1²) + √(r_a2² − r_b2²) − a · sin α) / (π · m · cos α)
with r_a = tip radius, r_b = base radius, a = centre distance. ε_α ≥ 1.0 is required for continuous transmission (one pair must engage before the previous disengages). Practical minimum ε_α ≥ 1.2 for power gears, ε_α ≥ 1.4 for quiet running, ε_α ≥ 1.6 for aerospace and noise-critical drives.
Helical gears add a face overlap ratio ε_β = (b · tan β) / (π · m_n) with b = face width, β = helix angle, m_n = normal module. Total contact ratio γ = ε_α + ε_β can easily reach 2.5-3.5, which is why helical gears run smoother and quieter than spur of the same module.
2.3 Undercutting and minimum tooth count
When teeth on the generating hob extend below the base circle of the gear being cut, they remove material from the dedendum — leaving an undercut that weakens the tooth root and lowers contact ratio. For a 20° pressure angle full-depth tooth cut by rack:
z_min = 2 · h_a* / sin² α = 2 · 1.0 / sin²(20°) = 17.097
So 17 teeth is the practical minimum for a standard 20° PA full-depth cut spur gear. Below that, the engineer chooses among:
- Profile shift (addendum modification, x-coefficient) — move the rack outward by x·m during cutting; the resulting tooth has thicker root and less undercut. Standard for pinions z = 12-16. Positive shift x = +0.3 to +0.6 is typical.
- Higher pressure angle — at 25° PA, z_min ≈ 12; at 30°, z_min ≈ 8. Less common because off-the-shelf cutters are 20°.
- Helical with low helix angle — virtual tooth count z_v = z / cos³β ≥ 17 even when actual z is lower.
2.4 Interference
Geometric interference occurs when the tip of one gear tries to engage the flank of the mating gear below the base circle, where no involute exists. The result is gouging, vibration, and rapid failure. Avoiding interference is the design constraint that drives the minimum-tooth-count rule. Tip relief (a small profile correction on the last 5-10% of tooth height) is a standard manufacturing remedy and also smooths the entry/exit impact that drives gear noise.
2.5 Force balance on a tooth
For a spur gear pair transmitting tangential force W_t at pitch radius r:
W_t = T / r (tangential — torque)
W_r = W_t · tan α (radial — separating)
W_n = W_t / cos α (normal to tooth, along line of action)
For helical gears (helix angle β, normal pressure angle α_n):
W_t = T / r
W_r = W_t · tan α_n / cos β (radial separating)
W_a = W_t · tan β (axial thrust)
The axial thrust W_a is what kills the bearings of a single-helical gear under heavy load — and the reason double-helical (herringbone) gears exist. Two opposed helices cancel thrust at the gear and leave only radial reaction at the bearings.
3. Practical math / design equations
3.1 AGMA bending stress (AGMA 2001-D04, metric units)
σ_F = (W_t · K_o · K_v · K_s · K_m · K_B) / (b · m_t · Y_J)
| Factor | Meaning | Typical |
|---|---|---|
| W_t | Tangential force at pitch line (N) | calculated from torque |
| K_o | Overload factor (driving + driven machine character) | 1.0 (smooth) to 2.75 (heavy shock both sides) |
| K_v | Dynamic factor (manufacturing-quality + speed) | 1.0 (precision, low speed) to 1.6+ (commercial, high speed) |
| K_s | Size factor | usually 1.0 (becomes >1 for very large gears) |
| K_m | Load-distribution factor (face-width-to-pitch-diameter, mounting, misalignment) | 1.1 (rigid, narrow) to 2.0+ (long overhang, wide gear) |
| K_B | Rim thickness factor (thin-rim gears flex) | 1.0 for solid; 1.0-2.0 for thin |
| b | Face width (mm) | typically 8·m to 16·m |
| m_t | Transverse module (mm) | m for spur; m_n / cos β for helical |
| Y_J | Bending geometry factor | 0.20 (z=12) to 0.50 (z=200+); tabulated AGMA 908 |
Allowable bending stress:
σ_F_allow = (S_t · Y_N) / (S_F · Y_θ · Y_Z)
with S_t = allowable bending stress (material + heat treat, AGMA Table), Y_N = stress-cycle factor (life), S_F = bending safety factor (1.25-2.0 typical), Y_θ = temperature factor, Y_Z = reliability factor (Y_Z = 0.85 for 99%, 1.0 for 90%).
3.2 AGMA pitting (contact) stress (AGMA 2001-D04)
σ_H = C_p · √( (W_t · K_o · K_v · K_s · K_m · C_f) / (b · d · I) )
with C_p = elastic coefficient (191 MPa^0.5 for steel-on-steel), d = pinion pitch diameter (mm), I = pitting geometry factor (0.05-0.20 typical, tabulated AGMA 908), C_f = surface-condition factor (1.0 for ground, 1.25+ for as-hobbed).
Allowable contact stress:
σ_H_allow = (S_c · Z_N · C_H) / (S_H · Y_θ · Y_Z)
with S_c = allowable contact stress (material + heat treat), Z_N = pitting stress-cycle factor, C_H = hardness ratio factor (≥1 when pinion harder than gear), S_H = pitting safety factor (1.0-1.5).
Engineering observation: pitting (Hertzian) typically governs the design of hardened-and-ground gear pairs, while bending typically governs softer/through-hardened pairs and any case-hardened pair with thin face width. Both must be checked.
3.3 ISO 6336 — broadly parallel
ISO 6336 (parts 1-6, latest revisions 2019-2022) uses essentially the same Lewis-bending and Hertz-contact formulations but with different factor names (K_A instead of K_o, K_Hβ instead of K_m, Y_F·Y_S instead of Y_J). The numerical results are within ~5% of AGMA for standard cases. Use ISO in European OEM contexts (automotive, wind), AGMA in North American industrial.
3.4 Planetary gear ratio (Willis equation)
For a simple planetary (sun S, planet P, ring R, carrier C) with tooth counts Z_S, Z_P, Z_R, and Z_R = Z_S + 2·Z_P:
(ω_S − ω_C) / (ω_R − ω_C) = −Z_R / Z_S (Willis fundamental equation)
Common operating modes (ratio = input speed / output speed):
| Fixed | Input | Output | Ratio i |
|---|---|---|---|
| Ring | Sun | Carrier | i = 1 + Z_R/Z_S |
| Carrier | Sun | Ring | i = −Z_R/Z_S (reversed) |
| Sun | Carrier | Ring | i = Z_R / (Z_R + Z_S) |
| Ring | Carrier | Sun | i = Z_S / (Z_R + Z_S) |
| Sun | Ring | Carrier | i = 1 + Z_S/Z_R |
3.5 Worm gear lead and self-locking
For a worm with lead angle λ and friction coefficient μ at the tooth contact:
tan λ = L / (π · d_w) L = lead = z_w · p_x d_w = worm pitch diameter
Self-locking when: λ < arctan μ (typically μ ≈ 0.03-0.08 oiled → φ ≈ 1.7°-4.6°)
Self-locking worms (single-start, λ < 4°) cannot be back-driven — useful for hoists and indexing tables. They are typically <50% efficient. Multi-start worms (λ > 10°) can be back-driven and run 80-90% efficient.
3.6 Worked example A — spur gear pair sizing
Given: Electric-motor input, gearbox driving a conveyor. Input power P_in = 5 kW (6.7 hp), pinion speed n₁ = 1750 rpm, ratio i = 4:1, target life 20 yr × 16 h/day × 250 day/yr ≈ 80 000 h. Loads: moderate (K_o = 1.25 for uniform input, light shock output). Use 20° PA spur, hardened-and-ground.
Step 1 — gear geometry. Trial m = 3 mm. Pick z₁ = 22 (above the 17-tooth minimum, fits a typical pinion stub). z₂ = i · z₁ = 88. Pitch diameters d₁ = 66 mm, d₂ = 264 mm. Centre distance a = 165 mm. Face width b = 10·m = 30 mm (within AGMA 8m-16m guidance).
Step 2 — tangential force.
T₁ = 60 · P_in / (2π · n₁) = 60 · 5000 / (2π · 1750) = 27.3 N·m (20.1 lb·ft)
W_t = T₁ / r₁ = 27.3 / 0.033 = 827 N (186 lbf)
Pitch-line velocity V = π · d₁ · n₁ / 60 = π · 0.066 · 1750 / 60 = 6.05 m/s (1190 ft/min).
Step 3 — dynamic factor K_v (AGMA, Q_v = 8 quality, commercial-hardened):
B = 0.25 · (12 − Q_v)^0.667 = 0.25 · 4^0.667 = 0.628
A = 50 + 56 · (1 − B) = 50 + 56 · 0.372 = 70.8
K_v = (A / (A + √(200·V)))^(−B)
= (70.8 / (70.8 + √(1210)))^(−0.628)
= (70.8 / 105.6)^(−0.628)
= 1.27
Step 4 — load distribution K_m. b/d₁ = 30/66 = 0.45; for rigid bearings + commercial mounting K_m ≈ 1.3.
Step 5 — bending stress (Y_J = 0.345 for z₁ = 22 meshing with z₂ = 88, AGMA 908):
σ_F = (W_t · K_o · K_v · K_s · K_m · K_B) / (b · m · Y_J)
= (827 · 1.25 · 1.27 · 1.0 · 1.3 · 1.0) / (30 · 3 · 0.345)
= 1707 / 31.05
= 55.0 MPa (7.97 ksi)
For AISI 4140 carburised + ground at HRC 58-62, S_t = 450 MPa per AGMA 2001 Table 3. Apply Y_N (life): at 80 000 h × 1750 rpm = 8.4 × 10⁹ cycles, Y_N ≈ 0.85. Reliability 99% → Y_Z = 0.85. Safety S_F = 1.5:
σ_F_allow = (450 · 0.85) / (1.5 · 1.0 · 0.85) = 300 MPa
σ_F = 55 MPa << 300 MPa allowable → bending is fine, by a wide margin.
Step 6 — pitting (contact) stress (I = 0.105 for external mesh, m_G = 4, 20° PA):
σ_H = C_p · √( (W_t · K_o · K_v · K_s · K_m · C_f) / (b · d₁ · I) )
= 191 · √( (827 · 1.25 · 1.27 · 1.0 · 1.3 · 1.0) / (30 · 66 · 0.105) )
= 191 · √( 1707 / 207.9 )
= 191 · √8.21
= 547 MPa (79.3 ksi)
For 4140 carburised, S_c = 1550 MPa. Z_N (life) at 8.4×10⁹ cycles ≈ 0.85. C_H = 1.0 (same hardness). Y_Z = 0.85. S_H = 1.2:
σ_H_allow = (1550 · 0.85 · 1.0) / (1.2 · 1.0 · 0.85) = 1291 MPa
σ_H = 547 MPa << 1291 MPa allowable → contact is fine.
Result: m = 3 mm, z₁ = 22, b = 30 mm, AISI 4140 carburised & ground at HRC 58-62 works comfortably for 5 kW at 1750 rpm input, 4:1 ratio, 80 000 h life. Could downsize to m = 2.5 mm if envelope mattered — recompute, expect σ_H to climb proportional to 1/√(d·b) and approach the limit. The pitting margin is usually the design constraint for hardened gear pairs.
3.7 Worked example B — planetary ratio
Given: Sun Z_S = 20, planet Z_P = 24, ring Z_R = Z_S + 2·Z_P = 68. (Geometric check passes.) Operating mode: ring fixed, sun input, carrier output.
From Table 3.4 row 1:
i = ω_S / ω_C = 1 + Z_R / Z_S = 1 + 68/20 = 4.40 : 1
If the sun spins 1000 rpm, the carrier spins 1000 / 4.40 = 227 rpm, same direction as the sun. The three planets carry the load — each planet sees only 1/3 of the input torque (modulo load-sharing factor K_γ ≈ 1.1-1.3 from manufacturing tolerance), so the planetary’s torque density is roughly 3× that of an equivalent spur-pinion-gear pair at the same envelope.
Common planetary ratios: single-stage 3:1 to 10:1 (theoretical limit ~13:1 for tight clearances). Multi-stage 30:1 to 300:1. Wittenstein alpha-series and Apex AB-series cover 3:1 to 100:1 in compact servo-motor mount packages.
3.8 Worked example C — harmonic drive for robot joint
Given: Cobot wrist joint. Peak output torque T_pk = 80 N·m (59 lb·ft) at start/stop, average T_avg = 25 N·m, max output speed ω_out = 20 rpm, required L_10 life = 10 000 hr at average load. Target ratio 30:1.
Harmonic Drive LLC CSG-series (cup-type, sealed, with cross-roller output bearing). From the CSG datasheet CSG-32-50:
| Rated torque T_R (cont.) | 137 N·m | | Peak torque T_max (start/stop, occasional) | 285 N·m | | Avg permissible torque T_avg_max | 191 N·m | | Ratchet torque (limit) | 569 N·m | | Rated input speed | 3500 rpm | | Max input speed | 7300 rpm | | L_10 life formula | L_10 = (T_avg_R / T_avg)^3 · 7000 hr |
Ratio choice 30:1 — available standard ratios for CSG-32 are 30, 50, 80, 100, 120, 160. Pick CSG-32-30. Recheck the same table for ratio-30:
| Rated torque T_R | 108 N·m | | Peak torque T_max | 220 N·m | | Avg permissible torque | 151 N·m | | Ratchet torque | 451 N·m |
Check 1 — peak torque: T_pk = 80 N·m ≤ T_max = 220 N·m. Margin 2.75×. ✓
Check 2 — ratchet torque: Should never see ratchet level in normal operation; emergency-stop check only. 80 N·m << 451 N·m. ✓
Check 3 — average permissible: T_avg = 25 N·m ≤ 151 N·m. ✓
Check 4 — L_10 fatigue life of the wave-generator bearing:
L_10 = (T_avg_R / T_avg)^3 · L_R
= (108 / 25)^3 · 7000
= (4.32)^3 · 7000
= 80.6 · 7000
= 564 000 hr
vs required 10 000 hr. 56× margin — could downsize to CSG-25 (smaller frame) and still meet life. Trade-off: smaller frame has lower torsional stiffness and higher no-load running torque relative to rating. For a wrist joint where dynamic positioning bandwidth matters, the larger frame is often the better choice.
Step 5 — no-load running torque (table at ω_in = 600 rpm, room temperature, CSG-32-30): T_NL ≈ 0.18 N·m at input shaft = 5.4 N·m referred to output. This is the joint’s static friction floor and matters when controlling small forces (cobot direct-teach, impedance control). For sub-1-N·m output-side resolution, consider a cycloidal (lower friction) or a backlash-cleared planetary.
Step 6 — input motor sizing. Output 20 rpm × 30 = 600 rpm input. Output 25 N·m / (30 · 0.85 efficiency) = 0.98 N·m at the input motor. Motor power at rated output: 25 · (20 · 2π/60) / 0.85 = 62 W continuous. A small servomotor — frameless 80-100 W class (e.g. Kollmorgen TBM-7615, Allied Motion MF0080).
4. Reference data
4.1 Gear families — capability comparison
| Family | Shaft layout | Ratio/stage | Efficiency/stage | Backlash | Speed | Cost |
|---|---|---|---|---|---|---|
| Spur | parallel | 1:1 - 7:1 | 98-99% | 5-30 arc-min | high | low |
| Helical | parallel | 1:1 - 10:1 | 98-99% | 5-20 arc-min | very high | low-med |
| Double-helical | parallel | 1:1 - 10:1 | 98-99% | 5-20 arc-min | very high | high |
| Straight bevel | intersecting | 1:1 - 5:1 | 97-98% | 5-20 arc-min | medium | med |
| Spiral bevel | intersecting | 1:1 - 8:1 | 97-98% | 3-15 arc-min | high | med-high |
| Hypoid | offset | 5:1 - 50:1 | 90-96% | 5-20 arc-min | high | med-high |
| Worm (single-start) | crossed | 5:1 - 100:1 | 30-50% | 3-15 arc-min | low | low-med |
| Worm (multi-start) | crossed | 5:1 - 60:1 | 70-90% | 3-15 arc-min | low | low-med |
| Planetary (1 stage) | coaxial | 3:1 - 10:1 | 96-98% | 3-20 arc-min | high | med |
| Planetary (2-3 stages) | coaxial | 10:1 - 1000:1 | 90-95% | 5-30 arc-min | high | med-high |
| Harmonic drive | coaxial | 30:1 - 320:1 | 80-85% | <1 arc-min | medium | high |
| Cycloidal (RV) | coaxial | 6:1 - 200:1 | 90-93% | <1 arc-min | medium | high |
| Rack & pinion | linear | n/a | 95-98% | 5-50 arc-min | high | low-med |
| Internal/annular | parallel | 1.5:1 - 7:1 | 98-99% | 5-20 arc-min | high | med |
4.2 Gear materials and heat treatment
| Material | Spec | Process | Surface HRC | Core HRC | S_c (MPa) | S_t (MPa) | Notes |
|---|---|---|---|---|---|---|---|
| AISI 1045 | UNS G10450 | Through-hardened | 28-35 | 28-35 | 700-900 | 200-260 | Cheap, low duty |
| AISI 4140 | UNS G41400 | Through-hardened | 30-40 | 30-40 | 900-1100 | 250-330 | Industrial workhorse |
| AISI 4140 | UNS G41400 | Induction-hardened | 50-55 | 30-35 | 1100-1300 | 300-400 | Limited case depth |
| AISI 4140 | UNS G41400 | Nitrided (gas/ion) | 60-65 | 30-35 | 1240 | 415 | Low distortion, no grind |
| 31CrMoV9 | DIN nitralloy | Nitrided | 62-68 | 32-38 | 1300 | 450 | Premium nitride |
| AISI 8620 | UNS G86200 | Carburised + ground | 58-62 | 30-40 | 1550 | 450 | Aerospace/automotive standard |
| AISI 9310 | UNS G93100 | Carburised + ground | 60-63 | 35-42 | 1700 | 480 | Aerospace premium (helicopter) |
| Vasco X-2M / Pyrowear 53 | aerospace | Carburised + ground | 60-63 | 38-42 | 1700+ | 520 | Hot-hardness, high-T life |
| Mn-bronze (C86300) | UNS C86300 | as-cast | n/a | 18-22 HRC | 550 | 140 | Worm wheel, large slow |
| Al-bronze (C95400) | UNS C95400 | as-cast | n/a | 19-23 HRC | 600 | 165 | Worm wheel, harder |
| POM (acetal) | Delrin 100 / 500 | injection molded | n/a | M85 (Rockwell) | n/a | 70 (yield) | Low-load, dry-running |
| PEEK | Victrex 450G | molded or machined | n/a | M99 | n/a | 95 (yield) | Chemical, high-T plastic |
| Nylon 66 + 30% GF | n/a | molded | n/a | M90 | n/a | 60 (yield) | Common timing gears |
S_c and S_t per AGMA 2001-D04 Tables 3 and 4 (grade 2 quality, 99% reliability, 10⁷ cycles).
4.3 Manufacturing processes
| Process | Geometry | Quality (AGMA Q) | Volume | Cost/part | Notes |
|---|---|---|---|---|---|
| Form milling (form cutter) | spur, helical, internal | Q5-Q7 | low (job-shop) | low | Low precision; one cutter per module/profile |
| Hobbing | spur, helical, worm | Q7-Q10 | medium-high | medium | Universal — one hob covers any z at fixed m, α |
| Shaping (Fellows/Sunderland) | internal, blind-shoulder spur, helical | Q7-Q10 | medium | medium-high | Required for internal gears (rings) |
| Broaching (internal) | splines, internal teeth (small) | Q9-Q12 | high | low at volume | Excellent splines; dedicated tooling |
| Skiving | internal gears | Q9-Q11 | medium | medium | Modern alternative to shaping; faster |
| Bevel generation (Gleason / Klingelnberg) | straight/spiral/hypoid bevel | Q8-Q11 | medium | high | Dedicated machines (5-axis) |
| Gear grinding (threaded wheel — Reishauer) | spur, helical (external) | Q11-Q14 | medium | high | Reishauer RZ, Gleason TAG/Genesis |
| Gear grinding (profile — Höfler/Niles) | spur, helical, internal | Q11-Q14 | low-medium | high | One flank at a time, slow, ultra-precise |
| Honing | spur, helical (finishing) | +1 grade | medium | medium | Removes 10-30 μm; improves finish, kills tool marks |
| Lapping (Gleason) | bevel sets | ”lapped match” | medium-high | medium | Run mating set together to seat |
| Powder metallurgy (sintered) | spur, helical, small bevel | Q6-Q9 | very high | very low | Small-module (m ≤ 2.5), automotive ancillaries |
| Plastic injection molding | spur, helical, small bevel | Q7-Q10 | very high | very low | Module 0.5-3, instrument/appliance |
| 5-axis milling | any geometry | Q6-Q10 | prototype | very high | Prototype and ultra-low volume |
| LPBF / DMLS (3D printed metal) | any | Q5-Q8 | prototype | very high | Internal cavities, lightweighting; finish often poor |
| EDM wire-cut | thin-section gears | Q6-Q9 | low | high | Hardened material, complex profile |
4.4 Quality grades — AGMA Q and ISO 1328
Legacy AGMA Q-numbers (AGMA 2000-A88, withdrawn 2003) ran 3 (very coarse) to 15 (highest precision). Modern AGMA 2015-1-A01 and ISO 1328-1:2013 use grade numbers 1 (highest) to 12 (lowest) — inverted from AGMA Q-numbers. Approximate equivalence:
| AGMA Q (legacy) | AGMA 2015 | ISO 1328 | Typical use |
|---|---|---|---|
| Q5 | A11 | 11 | Low-quality commercial; rough cast |
| Q7 | A9 | 9 | Industrial general; as-hobbed |
| Q9 | A7 | 7 | Industrial precision; finish-hobbed |
| Q10 | A6 | 6 | Automotive volume |
| Q11-Q12 | A5-A4 | 5-4 | Robotic actuators, ground gears |
| Q13 | A3 | 3 | Aerospace, machine-tool spindle drive |
| Q14 | A2 | 2 | Aerospace ultra-precision |
| Q15 | A1 | 1 | Master gears (metrology) |
Higher Q (legacy) = better; higher grade (modern AGMA 2015 / ISO 1328) = worse. Don’t confuse them. Quality grade affects K_v, transmission error, and noise — not directly the static strength.
4.5 Lubricant viscosity by pitch-line velocity (AGMA 9005-F16)
| Pitch-line velocity V (m/s) | ISO VG | AGMA grade | Use |
|---|---|---|---|
| < 5 | 460 | 7 | Slow industrial, splash bath |
| 5 - 10 | 320 | 6 | General industrial |
| 10 - 15 | 220 | 5 | Mid-speed industrial |
| 15 - 25 | 150 | 4 | Pumps, blowers, machine tools |
| > 25 | 100 | 3 | Spindle drives, high-speed |
EP (extreme-pressure, sulphur-phosphorus) additives required for σ_H > 1000 MPa (hardened gears) or any worm reducer. Synthetic PAO base for wide temperature range (-40 to +120 °C); synthetic PAG for wormwheel duty (lower friction, anti-scuff). AGMA 9005-F16 / ISO 12925 specify the testing requirements.
4.6 Backlash by application
| Application | Backlash spec | How achieved |
|---|---|---|
| Industrial gearbox | 0.05 - 0.20 mm tangential at PCD | Standard tolerance, no compensation |
| Robotic positioning (3rd-stage) | <5 arc-min | Q10-Q12 ground gears |
| Robotic positioning (joint output) | <1 arc-min | Harmonic drive or cycloidal RV |
| Servo-driven indexing table | <30 arc-sec | Anti-backlash duplex angular contact + ground gear |
| Optical scanner / metrology | <10 arc-sec | Preloaded duplex bearing + flexure-coupled gear |
| Watch/clock | n/a (cycloidal) | Cycloidal teeth, jewel bearings |
5c. Variants & topologies
5c.1 Spur and helical (parallel-shaft)
- Spur — teeth parallel to the axis. Pure rolling at the pitch point, sliding everywhere else along the tooth height. No axial thrust. Noisier than helical (impacts at tooth entry/exit). Industrial workhorse for moderate speeds (≤ 15 m/s pitch-line); typical OEM cost reference.
- Helical — teeth wrapped at a helix angle β = 8°-30° (typical 15°-20°). Gradual engagement spreads tooth load axially → quieter, higher load capacity, higher contact ratio. Generates an axial thrust force equal to W_t · tan β. Useful for high-speed and high-power; standard in automotive gearboxes (where bevel-helical or hypoid handles the final-drive right angle).
- Double-helical (herringbone) — two opposed helices on the same blank cancel axial thrust. Used in marine reduction gears, paper-mill drives, mining mill drives, large turbine-generator drives. Manufacturing harder (each helix may need a central groove for tool runout, unless cut with a Sykes shaper).
- Internal (annular) — teeth on the inside of a ring. Meshes with a pinion of smaller diameter for compact, high-contact-ratio drive. Required element of every planetary gearset (the ring). Cut by shaping (Fellows) or modern skiving.
- Rack and pinion — infinite-radius “gear” (the rack) mates with a normal pinion for rotary-to-linear conversion. Used in machine-tool linear axes, CNC gantry drives, automotive steering, large slewing structures. Backlash compensated by dual-pinion preload in precision drives.
5c.2 Bevel (intersecting and offset shafts)
- Straight bevel — teeth converge to the apex (like a section of cone). Conjugate motion if octoidal (true spherical involute) but most practical bevels are octoidal-approximate. Used for low-to-medium speed right-angle drives; differential side gears in older automotive axles.
- Spiral bevel — teeth curved in spiral form (Gleason “spiral”, Klingelnberg “palloid”, Oerlikon “cycloid”). Smoother engagement (analogous to helical→spur improvement), higher load, lower noise. Standard in modern automotive differentials and high-end industrial right-angle gearboxes.
- Hypoid — like spiral bevel but with offset pinion axis (pinion centreline doesn’t intersect ring-gear axis). Allows lower-mounted driveshaft in automotive (lower floor pan, deeper trunk); higher ratio per stage (up to 50:1 for indexing) but more sliding action → lower efficiency (90-96%), requires hypoid-specific EP additive (GL-5).
- Zerol bevel (Gleason) — spiral angle = 0° at midface; smoother than straight, no thrust complications of full-spiral.
5c.3 Worm
A worm (small-diameter helical-thread “screw”) drives a worm wheel (a gear with concave throat tooth profile that wraps the worm). Crossed shafts, typically 90°. Provides very high ratio per single stage (5:1 to 100:1) in a compact package, plus self-locking when lead angle < friction angle.
- Cylindrical worm — straight cylinder; standard.
- Globoidal worm (Cone-drive) — hourglass-shaped worm wraps the wheel for multi-tooth contact → much higher capacity and lower contact stress at the cost of manufacturing complexity. Cone Drive Operations Inc. (USA) is the dominant maker.
- Worm wheel materials — phosphor or aluminum bronze running against hardened-steel worm. The bronze wheel is the sacrificial element by design (cheaper to replace than the worm). Lubrication is critical — worm gears generate heat and a worm gearbox is often thermally limited, not torque-limited.
5c.4 Planetary (epicyclic)
Three concentric gear elements (sun, planets, ring) plus a planet carrier. Two of three (sun, ring, carrier) are kinematic input/output; the third is fixed. The classic “two-input-one-output” configuration also enables differentials and continuously variable transmissions.
- Simple planetary — sun + 3 (sometimes 4-5) planets + ring + carrier. Single stage 3:1 to 10:1; compound (planet-on-planet) reaches 12:1 in one stage.
- Multi-stage planetary — stages cascaded; e.g. three stages of 5:1 = 125:1. Standard high-ratio servo-actuator construction.
- Differential planetary — two ring gears with slightly different tooth counts engage one set of compound planets; produces enormous reduction (100:1 to 10 000:1) in one stage, but with low efficiency. Example: nutating-drive reducers.
- Standard makers: Wittenstein cyber-dynamic NP/NPL/NPS, Apex Dynamics AB/AD/AE, Neugart PLE/PFHE, Stober EZ/PE/PHK, SEW-Eurodrive R-series. Output torques 5 N·m (small servo) to 50 000 N·m (heavy industrial), backlash 1-30 arc-min depending on grade.
5c.5 Harmonic drive (strain-wave gearing)
Three concentric elements:
- Wave generator — elliptical cam with a ball bearing on its periphery. The input.
- Flexspline — thin-walled cup or hat with external teeth on its open end. The output (or fixed).
- Circular spline — rigid ring with internal teeth, two more teeth than the flexspline. The fixed (or output).
The wave generator deforms the flexspline into an ellipse; teeth engage at the two major-axis points and disengage at the minor-axis points. For each input revolution, the flexspline rotates by (Z_FS − Z_CS) / Z_FS in the opposite direction → Ratio i = Z_FS / (Z_CS − Z_FS) = Z_FS / 2 for the common 2-tooth-difference design.
Ratios available off the shelf: 30, 50, 80, 100, 120, 160, 200, 320 (single stage). Zero backlash by design (teeth always engaged at two points, preloaded by flexspline elasticity). Hollow shaft (cable routing). Standard makers: Harmonic Drive LLC (USA/JP) — CSG, CSD, SHG, CPL, HFUC, CSF series. Nidec (formerly Leaderdrive). Sumitomo Drive Technologies (HD-series, licensed).
Penalty: lower efficiency (80-85% peak, drops at low load), nonlinear torsional stiffness, soft-windup, ratcheting failure at gross overload, finite life dominated by flexspline fatigue.
5c.6 Cycloidal (RV, FA, MEGA)
A high-speed input shaft drives an eccentric cam that nutates one or two cycloidal discs. Each disc has lobes (typically 11-39) that engage pin/roller followers mounted in a fixed ring. The discs rotate slowly in the opposite sense; their motion is picked off by output pin/rollers that engage holes in the disc, translating disc rotation into output-shaft rotation. Net ratio is very high (6:1 to 200:1) in a compact axial-flat package.
- Single-stage cycloidal (Sumitomo Cyclo) — eccentric cam directly drives the cycloidal disc. 6:1 to 119:1.
- Two-stage RV (Nabtesco RV) — spur reduction stage drives the cycloidal stage; two cycloidal discs 180° apart cancel input shaft load. Ratios 30:1 to 200:1. Industry standard for the base joints of industrial robots (Fanuc, ABB, KUKA, Yaskawa).
- Standard makers: Nabtesco (RV-N, RV-C, RV-E, RV-S series), Sumitomo (Cyclo BBB, Cyclo 6000), Spinea (TwinSpin), Onvio (Twingear). Backlash <1 arc-min, shock tolerance 5× rated, efficiency 90-93%.
5c.7 Specialty topologies
- Magnetic gear — concentric rotors with rare-earth magnets coupled through a steel pole-piece array. Contactless, no lubrication, slip-limited torque. Niche applications (subsea, cleanroom).
- Strain-wave with floating circular spline — newer variants from Schaeffler PSC and Nidec Drive Technology; reduces friction and stick-slip.
- Roller-pinion drive (Nexen RPS) — rack with cylindrical pins meshes with a pinion with conjugate roller-engaging pockets. Zero backlash, long-rack linear duty; replaces precision rack-and-pinion in high-precision linear positioning.
6c. Selection criteria
Decision sequence, in order of constraint hierarchy:
- Shaft configuration. Parallel → spur/helical/double-helical. Intersecting → bevel. Offset → hypoid or worm. Coaxial → planetary/harmonic/cycloidal.
- Ratio required. ≤10:1 single stage → spur/helical or planetary. 10-50:1 → 2-stage planetary, worm, or cycloidal. 30-320:1 in compact package → harmonic drive. >100:1 → multi-stage planetary or cycloidal.
- Backlash budget. Power transmission (no positioning): standard hobbed acceptable. <5 arc-min positioning: ground gears + preload. <1 arc-min: harmonic or cycloidal. Direct-drive (torque motor, no gear) is the limit case.
- Efficiency budget. Continuous operation at high power → spur/helical/planetary (98%+). High ratio with continuous duty → cycloidal (90-93%) preferred over harmonic (80-85%) preferred over worm (30-90%, lead-angle dependent).
- Shock and overload tolerance. Robotic joint with collisions → cycloidal (overload spreads across many pins). Industrial → planetary with shear pin or torque limiter upstream. Harmonic ratchets at gross overload — protect with sensors or a slip clutch.
- Cost and availability. Off-the-shelf: planetary (cheapest), worm, harmonic, cycloidal (most expensive per N·m). Custom geometry (e.g. non-standard ratio) doubles or triples cost and lead time.
6c.1 Practical rules of thumb
- Robot wrist joint (small, low torque, high precision): harmonic drive. Zero backlash, hollow shaft, single-stage compact, ratio 80-160:1 typical.
- Robot base joint (large, high torque, shock-tolerant): two-stage cycloidal RV (Nabtesco). Higher efficiency than harmonic, far better shock tolerance, slightly more backlash, much higher torque density.
- Industrial conveyor / pump / fan drive: helical-planetary or helical-bevel from SEW-Eurodrive, NORD, Sumitomo. Cheap, robust, 30 000-100 000 h life standard.
- Mobile robot wheel drive: planetary in 1-3 stages with brushless motor input. Ratio 30:1 to 100:1 typical; off-the-shelf from Apex, Wittenstein, or built into the motor (gearmotor).
- Self-locking lift or hoist: single-start worm at λ < 4°. Sacrifices efficiency for back-drive prevention. Pair with appropriate brake regardless — never trust worm self-locking as a safety device.
- Right-angle drive: helical-bevel (Bonfiglioli A-series, NORD SK 92xxx) for general industrial; hypoid for automotive-style quiet/compact; spiral bevel for aerospace accessory gearboxes.
- High-ratio rotary index (slow, intermittent): single-start worm (cheap, self-locking) or two-stage planetary (efficient, back-drivable if needed).
- Backlash spec: state it in arc-minutes or arc-seconds, not as a percentage. The conversion isn’t well-defined and the value depends on gear pitch radius.
7c. Datasheet decoding
A typical industrial planetary servo-reducer datasheet line (Wittenstein alpha NP025S-MF1-10):
Frame size NP025 (output flange ~ 60 mm bolt circle, 25 = ID code)
Stages S (single stage)
Mounting MF1 (motor flange, NEMA 23 / 60-mm BLDC fit)
Ratio i 10:1
Rated torque T_2N 38 N·m
Max torque T_2max 67 N·m (occasional, < 1000 cycles/h)
Emergency torque T_2Not 110 N·m (single event; check after)
Backlash j_t ≤ 5 arc-min (standard); ≤ 1 arc-min (premium "P")
Torsional stiffness 8 N·m/arc-min
Rated input speed 3500 rpm; max 6000 rpm
Mass 2.5 kg
Efficiency η 97% (single stage at rated load)
L_10 life 20 000 h at T_2N
What each item really means:
- T_2N — continuous rated output torque; the design point for life calculation.
- T_2max — occasional peak (start/stop, max load); the cycle-count limit matters.
- T_2Not — emergency-stop torque, exceedable once before inspection.
- Backlash j_t — measured tangentially at the output shaft, in arc-minutes. Two grades are standard: “standard” (5-15 arc-min) and “low backlash” (1-3 arc-min) — the latter is 1.5-2× the price.
- Torsional stiffness — output torque per unit windup angle; matters for closed-loop control bandwidth.
- Rated input speed — thermally-limited continuous; max is short-duration.
- L_10 — bearing fatigue life of the output bearing under rated load; same statistical convention as bearings.
For harmonic drives, additional datasheet items:
- Ratcheting torque — overload limit beyond which the wave generator skips a tooth. Permanent damage.
- No-load running torque at room temp / cold start — significant for low-torque positioning duty.
- Hysteresis / lost motion — additional angular error under reversing load, separate from backlash.
- Torsional stiffness — usually three-segment: low-torque (soft), mid-torque (linear), high-torque (stiffening).
For cycloidal RV, additional datasheet items:
- Allowable moment on the output flange — critical when mounting a large arm with offset payload.
- Permissible axial / radial load on the output bearing.
- Lost motion as a separate spec from backlash (the residual hysteresis under reversal).
8c. Drive / interface and integration
8c.1 Mounting
Servo-reducers ship in three common envelopes:
- Motor flange + output shaft (most common) — motor bolts to the back, output is a keyed or splined shaft. Coaxial or offset configurations available.
- Motor flange + output flange — output is a precision-ground flange with bolt pattern; used when the reducer is the mechanical interface (robot wrist).
- Hollow shaft / hollow bore — wire routing through the centre of the joint; standard on harmonic drives and many cycloidals for robotic use.
The output bearing is integral to most precision reducers; cycloidal RV and harmonic drives have an internal cross-roller bearing capable of taking radial, axial, and moment loads simultaneously (see bearings §5c.1).
8c.2 Shaft connection
- Keyed shaft — DIN 6885 / ANSI B17.1. Cheap; slight backlash from key fit. Acceptable for power transmission, not for positioning.
- Spline (involute, parallel) — DIN 5480 / ANSI B92.1. Better torque density, less backlash than key but still finite play.
- Shrink disc / locking assembly (Ringfeder, Stüwe, Tollok) — frictional shaft-bore connection; zero backlash, transmits high torque, easy disassembly. Standard on industrial gearboxes.
- Direct bolted flange — output flange with bolt circle directly transmits torque via friction (NOT shear of the bolts). Standard on precision reducers; provides absolute zero backlash to the load.
- Bellows or disc coupling — between motor and reducer to absorb misalignment. Avoid for the reducer-to-load interface (introduces compliance).
8c.3 Lubrication interface
- Grease-for-life — sealed reducers (most servo planetary, harmonic, small cycloidal). 20 000-40 000 h grease life typical.
- Oil bath — industrial gearboxes, drain-and-fill at intervals (1000-5000 h). Vent breather plug standard. Check oil level before operation.
- Forced lube — large industrial (>50 kW) and high-speed gearboxes; integrated oil pump + cooler + filter. Lube quality monitoring is part of condition-based maintenance.
- Worm gearboxes are thermally limited — calculate heat dissipation (ambient + cooling fan vs power loss) and oversize accordingly. AGMA 9005-F16 / ISO 14635 cover the lube test methods.
9c. Real parts & sourcing
9c.1 Industrial reducers (general)
| Maker | HQ | Strength |
|---|---|---|
| SEW-Eurodrive | DE | Largest global; R/F/K/S series (helical, parallel-shaft helical, helical-bevel, helical-worm); huge SKU breadth |
| NORD Drivesystems | DE | Helical and helical-bevel block reducers; food/hygienic variants |
| Sumitomo Drive Technologies | JP | Cyclo cycloidal reducers (Buffalo NY plant); Paramax helical |
| Bonfiglioli | IT | A/HDP/300 series planetary; mobile/heavy-industry focus |
| Flender (Siemens) | DE | Large industrial — wind, mining, cement |
| Brevini (Dana) | IT | Heavy-duty planetary, marine |
| Hansen Industrial Transmissions | BE | Large industrial, paper mills, mining |
9c.2 Servo planetary
| Maker | Series | Backlash (standard / premium) | Power range |
|---|---|---|---|
| Wittenstein alpha | NP / NPL / NPS / SP / TP | 5 arc-min / 1 arc-min | 5 N·m to 5000 N·m |
| Apex Dynamics | AB / ABR / AD / AE / AF | 8 arc-min / 1 arc-min | 5 N·m to 4000 N·m |
| Neugart | PLE / PFHE / PSF / WPLE | 10 arc-min / 1 arc-min | 5 N·m to 1500 N·m |
| Stober | EZ / PE / PHK / PHQ | 5 arc-min / 1 arc-min | 5 N·m to 5000 N·m |
| Newstart (TS / VRT) | TS / VRT | 7 arc-min / 1 arc-min | 5 N·m to 500 N·m |
| Bonfiglioli MP | MP / TQ | 8 arc-min / 3 arc-min | 50 N·m to 5000 N·m |
9c.3 Harmonic drives
| Maker | Series | Notes |
|---|---|---|
| Harmonic Drive LLC (USA) | CSG / CSD / SHG / CPL / HFUC / CSF | Inventor; market leader; HD-LLC and Harmonic Drive AG (Germany) are licensees |
| Nidec Drive Technology | FB / FH / FS | Formerly Leaderdrive; aggressive pricing |
| Sumitomo (Fine Cyclo F2C) | F2C | Strain-wave variant |
| Cobalt Robotics | various | Aerospace/space-rated |
CSG-32-100 (example part number): C = cup-type, S = sealed, G = high-torque grade, 32 = frame size, 100 = ratio. 2024 pricing roughly USD 1500-4000 for an industrial CSG-25 to CSG-50.
9c.4 Cycloidal RV reducers
| Maker | Series | Notes |
|---|---|---|
| Nabtesco | RV-N / RV-C / RV-E / RV-S | Industry standard for robot base joints; ~60% global market share in industrial robot reducers |
| Sumitomo Drive | Cyclo 6000 / Cyclo BBB | Single-stage cycloidal heritage |
| Spinea | TwinSpin / DS / TS | Slovak; cobot-popular |
| Onvio | Twingear | Hollow-shaft compact cycloidal |
| Leader Harmonic / Nidec | CSF/CSG variants | Hybrid harmonic-cycloidal |
RV-40N-105 (example): RV = cycloidal, 40 = output torque ~ 400 N·m class, N = standard, 105 = ratio. 2024 pricing roughly USD 1500-5000 for industrial sizes.
9c.5 Bevel and worm
- Bevel & helical-bevel gearmotors: SEW-Eurodrive K-series, NORD SK 92xxx, Bonfiglioli A-series, Sumitomo Hyponic (hypoid right-angle).
- Worm gearboxes: Boston Gear (Altra), Cone Drive (globoidal), Bonfiglioli VF/W, NORD SK1S, Sumitomo Hedcon, Holroyd (industrial worm specialist).
9c.6 Counterfeit and grey-market risk
Industrial reducer brands are widely counterfeited, particularly in Asian secondary markets. The counterfeit rate on Nabtesco RV and Harmonic Drive parts from non-authorised online channels has been audited at 15-30%. Authorised distributors: Motion Industries, Applied Industrial Technologies, BDI (Bearing Distributors Inc.), Kaman Industrial, OEM-direct programs.
10c. Failure modes & derating
Per ISO 10825 / AGMA 1010-F14 (gear damage and failures — nomenclature and characteristics):
10c.1 Macro-pitting (Hertzian fatigue spalling)
The dominant high-cycle gear failure. Subsurface cracks initiate at the depth of maximum orthogonal shear stress (~0.3-0.6 mm below the tooth flank, depending on gear size), propagate parallel to the surface for millions of cycles, then turn upward and spall a flake. Pits enlarge progressively until tooth-flank load capacity is lost. The fatigue-life pillar of AGMA 2001 / ISO 6336 is set against this mode.
Mitigation: harder gears (S_c rises with surface hardness), better lubrication (κ > 1.5), no overload events (each overload above C_u contributes disproportionately to damage), avoid edge loading (use crowning).
10c.2 Micro-pitting (frosting)
Surface-initiated; clusters of micro-pits (5-50 μm) at the dedendum where sliding is highest. Looks like a matte band across the tooth flank. Caused by boundary lubrication (κ < 1) — typically thin EHL film + high asperity contact stress. Common on case-hardened gears running with too-light viscosity.
Mitigation: higher base-oil viscosity, anti-micropitting additives (specific MoDTC / sulphur-phosphorus chemistries — AGMA 9005-F16 specifies MP-rated lubes), smoother surface finish (superfinish reduces composite roughness).
10c.3 Scuffing (scoring, galling)
Adhesive wear: lubricant film fails locally, asperities cold-weld and tear. Produces radial scratches and material transfer from one flank to the other. Sudden, catastrophic — a scuff event can destroy a gear in minutes. Driven by flash temperature at the tooth contact:
T_flash = T_bulk + ΔT_flash
ΔT_flash ∝ μ · W_t · √V
(Blok flash-temperature theory, AGMA 925-A03). Susceptibility predicted by scuffing-resistance test FZG (DIN 51354) — lubes rated FZG ≥ 12 are standard for heavy-duty industrial gears.
Mitigation: EP additives (sulphur-phosphorus that react at high temperature to form sacrificial sulphide film), lower friction (smoother surface, better lube), tip relief (reduces contact stress at extremes of mesh), avoid running gears at flash temperature > 250 °C above bulk.
10c.4 Tooth bending fatigue (root crack)
Cyclic root bending stress σ_F initiates a fatigue crack at the point of maximum root stress (typically at the trochoidal fillet near the 30° tangent point per Lewis). Crack propagates through the root section until the tooth breaks off. Common in cyclic overloads, low-cycle high-amplitude duty (e.g. punch-press drives).
Mitigation: keep σ_F < S_t / S_F per AGMA 2001 (with appropriate Y_N for cycle count); shot-peen the root fillet to induce compressive residual stress (raises S_t by ~30-50%); generous fillet radius (avoid sharp-cornered root); use proper tooth profile (no undercut at root).
10c.5 Tooth fracture (overload)
Single-event ductile or brittle fracture from a force beyond static capacity. Differentiated from fatigue fracture by surface appearance (granular for brittle, fibrous for ductile, no beach marks). Often originates at a manufacturing defect (cutting mark, inclusion, decarb).
Mitigation: torque-limiting input (shear pin, slip clutch), cycloidal/planetary instead of harmonic for high-shock duty, sufficient static load margin.
10c.6 Wear
- Abrasive — hard contamination ground between teeth; uniform dull-grey appearance, measurable dimensional loss. Worsened by inadequate filtration.
- Adhesive (mild) — boundary lubrication; slow material transfer; precursor to scuffing.
- Polishing wear — paradoxically, gears that look “polished bright” have been wearing at sub-micropitting rate for long enough to remove material. Common on hobbed-only (un-ground) gears running clean.
10c.7 Spalling (subsurface case fatigue)
Distinct from macro-pitting: occurs in case-hardened gears when the case is too thin relative to operating Hertzian depth, and the subsurface shear-stress field reaches the case-core interface. Cracks initiate at that interface (a microstructural discontinuity), then propagate to the surface — popping out large flakes (millimetre scale) of case material.
Mitigation: case-depth design per ISO 6336-5 — total case depth ≥ 1.5-2× depth of maximum shear stress (typically 0.15·m for spur, 0.20·m for helical). Specify CHD (case hardness depth to 550 HV) on the part drawing.
10c.8 Plastic flow (cold-flow / hot-flow / rippling)
Tooth-flank material deforms plastically under overload — visible as rolled-over tips, rippled flanks, or root flow. Common on softer (through-hardened) gears under shock loading, or hardened gears at gross overload. Bronze worm wheels show plastic flow as part of normal break-in (a feature, not a defect; the gear conforms to the worm).
10c.9 Cracking and breakage
- Fillet cracks — root fatigue precursor; visible on careful inspection (magnetic-particle or dye-penetrant).
- Profile cracks (case crack-through) — case fatigue through to surface; case detachment.
- Tooth breakage — terminal event of any of the above.
10c.10 Derating: service factor K_o (AGMA 2001 Table 4)
| Driving machine | Driven machine: Uniform | Light shock | Moderate shock | Heavy shock |
|---|---|---|---|---|
| Uniform (electric motor, steam turbine) | 1.00 | 1.25 | 1.50 | 1.75 |
| Light shock (multi-cyl IC engine) | 1.25 | 1.50 | 1.75 | 2.00 |
| Medium shock (single-cyl IC engine) | 1.50 | 1.75 | 2.00 | 2.25 |
Apply K_o as the first multiplier on the tangential force. Most engineering errors come from undersizing K_o (“we’re driving a fan, K_o = 1”) — pumps with check valves slam, conveyors with material lumps shock, robots with collisions overshoot.
10c.11 Engineering judgement on derating
- Always check both bending and pitting. Pitting usually governs hardened ground gears; bending governs through-hardened gears with thin face. Don’t assume.
- Use K_v from the actual quality grade you intend to procure. Off-the-shelf “industrial” gears are Q8-Q10 (AGMA 2015 grade 9-7); precision robotic gears are Q11-Q14. Specifying Q12 when you’ll buy Q8 misrepresents K_v and overpredicts life.
- Pinion should be harder than the gear, typically by 30-60 HV. The pinion sees more cycles (z_gear / z_pinion times per gear cycle); higher hardness compensates. AGMA 2001 hardness ratio factor C_H ≥ 1 only when this rule is met.
- Specify case-depth on case-hardened drawings. CHD = case-hardness depth to 550 HV per ISO 6336-5. A common drawing error is to call out “carburised + ground” without specifying CHD; the heat-treat shop will hit some default that may or may not match your Hertzian-depth requirement.
- For harmonic drives, never design at peak torque. Design at average torque and check peak. Peak-torque cycles directly consume flexspline fatigue life.
- For cycloidal RV, specify allowable moment on the output flange in addition to torque. Robot arms with offset payloads can exceed moment limit at well below torque rating.
- For worm gearboxes, the thermal rating (continuous power dissipation capacity) is often lower than the mechanical rating. Check both. AGMA 6034 (worm gear rating) covers thermal sizing.
11. Cross-references
- bearings — every gear shaft rides on at least two bearings; gear thrust and separating forces define the bearing loads; preload, alignment, and lubrication interact strongly between gears and bearings
- fasteners-bolts — gearbox housings, output flanges, motor mounts, locknuts for shaft retention
- materials-steel — gear steels: AISI 1045 (through-hardened), 4140/4340 (medium-duty), 8620/9310 (carburised), 31CrMoV9 (nitrided), Pyrowear 53 / Vasco X-2M (aerospace)
- electric-motors — gearbox is always downstream of a motor; selection of motor + reducer is a coupled design problem (inertia matching, RMS torque, peak torque)
- mechanics-of-materials — Hertzian contact theory underlies pitting, Lewis bending theory underlies root fatigue
- bearings — EHL film thickness, viscosity selection, EP additives, AGMA 9005-F16 lube grades — gear life is dominated by lubrication
- vibration-dynamics — gear-mesh frequency (z·n/60), sidebands at shaft speed, transmission-error spectrum; condition monitoring of gearboxes
- machining — hobbing, shaping, broaching, grinding processes that produce gear geometry; tolerance grades and surface finish
- seals-taxonomy — gearbox housing seals (lip seals on input/output shafts) keep oil in and contamination out
- kinematics-dh (planned) — joint reduction ratios, torque/speed transformation, joint compliance from reducer windup
- motors-electric (planned) — integration of motor + reducer + bearing into a unit joint; harmonic/cycloidal selection trade-offs
- motors-electric (planned) — torque/speed/inertia matching between motor and reducer
12. Citations
- Shigley, J. E.; Mischke, C. R.; Budynas, R. G. “Mechanical Engineering Design,” 11th ed., McGraw-Hill, 2020. Chapters 13-15 cover spur, helical, bevel, and worm gear analysis; the AGMA equations are presented with worked examples in textbook form.
- Norton, R. L. “Machine Design: An Integrated Approach,” 6th ed., Pearson, 2019. Chapters 12-13 on gear geometry and stress analysis; clear treatment of contact ratio and undercutting.
- Dudley, D. W. “Dudley’s Handbook of Practical Gear Design and Manufacture,” 3rd ed. (Radzevich), CRC Press, 2016. The canonical practitioner reference — manufacturing, inspection, failure analysis, lubrication.
- Townsend, D. P. (ed.) “Dudley’s Gear Handbook,” 2nd ed., McGraw-Hill, 1991. Classic earlier edition; still authoritative on fundamentals.
- Litvin, F. L.; Fuentes, A. “Gear Geometry and Applied Theory,” 2nd ed., Cambridge University Press, 2004. The mathematical reference — involute geometry, conjugate action, tooth contact analysis (TCA).
- AGMA 2001-D04 (R2016) “Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth.” The North American rating standard.
- AGMA 2101-D04 (R2016) “Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth — Metric Edition.” Metric companion to 2001-D04.
- AGMA 908-B89 (R2019) “Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur, Helical and Herringbone Gear Teeth.” Source for J and I factors.
- AGMA 2015-1-A01 “Accuracy Classification System — Tangential Measurements for Cylindrical Gears.” Modern accuracy grades (replaces AGMA 2000-A88 Q-numbers).
- AGMA 1010-F14 “Appearance of Gear Teeth — Terminology of Wear and Failure.” Standard failure-mode nomenclature.
- AGMA 925-A03 (R2017) “Effect of Lubrication on Gear Surface Distress.” Flash-temperature scuffing analysis.
- AGMA 9005-F16 “Industrial Gear Lubrication.” Lube grade selection by gear type, speed, and load.
- AGMA 6034-B92 (R2017) “Practice for Enclosed Cylindrical Wormgear Speed Reducers and Gearmotors.” Worm rating including thermal.
- ISO 6336-1:2019 / -2:2019 / -3:2019 / -5:2016 / -6:2019 “Calculation of load capacity of spur and helical gears.” International (European-origin) rating standard, parts cover general method, surface pitting, tooth bending, material strength, and service life.
- ISO 1328-1:2013 “Cylindrical gears — ISO system of flank tolerance classification.” Modern grade definitions.
- ISO 10825:1995 “Gears — Wear and damage to gear teeth — Terminology.” International failure-mode nomenclature (parallels AGMA 1010).
- ISO 21771:2007 “Gears — Cylindrical involute gears and gear pairs — Concepts and geometry.” Definitional standard.
- DIN 3962-1 to -3:1978 “Tolerances for cylindrical gear teeth.” Historical German tolerance system (largely superseded by ISO 1328 but still cited in older OEM specs).
- DIN 3974-1, -2:1995 “Accuracy of worms and worm gears.”
- DIN 867:1986 “Basic rack tooth profiles for involute teeth of cylindrical gears for general engineering and heavy engineering.”
- Höhn, B.-R.; Stahl, K.; Tobie, T. et al. Various publications from FZG (Forschungsstelle für Zahnräder und Getriebebau, TU München). The world’s leading gear research institute; their FZG scuffing test, micropitting test, and tooth-root fatigue test rigs are the basis of the lubricant rating standards.
- Harmonic Drive LLC. “Engineering Data — CSG / CSD / SHG Series.” Product datasheets at harmonicdrive.net. Authoritative source for harmonic-drive sizing and life calculation.
- Nabtesco Precision Equipment. “RV Reducer Catalogue — RV-N / RV-C / RV-E Series.” Industry-standard cycloidal RV reducer specs.
- SEW-Eurodrive. “Gearmotor Catalogue — R / F / K / S Series.” Standard industrial-reducer reference with mounting, ratio, torque, and service-factor tables.
- Wittenstein alpha GmbH. “Servo-Reducer Catalogue — NP / NPL / SP / TP Series.” Robotic and machine-tool planetary reducers.
- Stadtfeld, H. J. “Gleason Bevel Gear Technology — Manufacturing, Inspection and Optimization.” Gleason Works, 2014. Bevel-gear specialist reference (Gleason is the dominant maker of bevel-gear cutting machines).