Gears — Complete Guide
A complete engineering reference for gears: gear geometry, a gear pair (ratio calculation), Lewis tooth strength, a rack and pinion, and planetary gearboxes — with live interactive calculators.
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Gears
Types · Geometry · Tooth Strength · Materials · Manufacturing · Planetary · Worm · Lubrication · Standards · Calculators
⚙️ Gear Types
Spur Gear
SimplestParallel
ShaftsParallel
Max Ratio1:7 in a single stage
Efficiency98-99%
LoadsRadial only — no axial!
AdvantageSimple · cheap · easy to manufacture and inspect
DrawbackNoisy · sudden contact · ↓contact ratio
UsesGearboxes · clocks · printers · transportation
▶ YouTubeHelical Gear
Common in IndustryThrust!
Helix Angle β15°-30° — common
Max Ratio1:10 per stage
Efficiency97-98%
Contact Ratioε ≥ 1.5 — smoother than Spur
⚠️ Axial LoadFa = Ft × tan(β) → a Thrust Bearing is required!
UsesAutomotive · industrial motors · transmissions
Double HelicalHerringbone — cancels axial load
▶ YouTubeBevel Gear
90° Direction Change
Straight BevelStraight teeth · easy to manufacture · noisy
Spiral BevelSpiral · quiet · ↑strength · Gleason/Klingelnberg
Zerol Bevelβ=0° spiral · ↓axial load
Max Ratio1:5 (straight) · 1:8 (spiral)
Efficiency96-99%
UsesDifferential · steering · rear final drive
Manufacturers: Gleason · Klingelnberg
▶ YouTubeHypoid Gear
AutomotiveEP Oil Mandatory
Difference from BevelPinion axis offset — doesn't intersect the Gear
Max Ratio1:10
Efficiency92-95% — lower than Bevel!
Advantage↓lower vehicle center · very quiet
⚠️ LubricationEP/GL-5 mandatory — very high sliding contact!
UsesAutomotive rear axle · LSD · Torsen Diff
▶ YouTubeWorm Gear
Self-Locking!Perpendicular
Max Ratio1:70 in a single stage!
Efficiency30-90% — depends on Lead Angle
Self-LockingLead angle < 5° → cannot be back-driven
⚠️ Heat↓efficiency = ↑heat. Cooling required!
Wheel MaterialBronze C932 · anti-galling
UsesHoists · valves · steering gearboxes
▶ YouTubePlanetary / Epicyclic
CompactCoaxial
ComponentsSun · Planet(s) · Ring (Annulus) · Carrier
Ratio per Stage1:3 to 1:12
Efficiency97-99%
AdvantageVery compact · distributed loads · coaxial
UsesServo · steering motor · ATM · printer · robotics
▶ YouTubeRack & Pinion
Rotary→Linear
PrinciplePinion rotates on a rack → linear motion
Feed Ratev = π × m × z × n / 60,000 (m/s)
Efficiency97-98%
AccuracyAGMA Q8-Q12 · DIN 4-8
Helical RackQuieter · recommended for high-speed CNC
UsesCNC · steering · lifts · 3D printers
▶ YouTubeInternal Gear
Annulus RingCompact
PrincipleTeeth inside the ring · rotates in the same direction!
Advantage↓center distance · ↑contact ratio · quiet
Minimum Teethz_ring - z_pinion ≥ 15 (to avoid Interference)
UsesPlanetary (Ring Gear) · epicyclic axle · ATM
▶ YouTube| Gear Type | Shafts | Max Ratio | Efficiency | Noise | Main Use |
|---|---|---|---|---|---|
| Spur | Parallel | 1:7 | 98-99% | ⭐⭐⭐ | General use |
| Helical | Parallel | 1:10 | 97-98% | ⭐⭐ | Automotive, industry |
| Straight Bevel | Intersecting | 1:5 | 96-98% | ⭐⭐⭐ | Simple direction change |
| Spiral Bevel | Intersecting | 1:8 | 97-99% | ⭐ | Automotive, aerospace |
| Hypoid | Intersecting+offset | 1:10 | 92-95% | ⭐ | Rear axle |
| Worm | Perpendicular | 1:70 | 30-90% | ⭐ | Hoists, Self-lock |
| Planetary | Coaxial | 1:12 | 97-99% | ⭐ | Servo, robotics |
| Rack & Pinion | Rotary→Linear | — | 97% | ⭐⭐ | CNC, steering |
📐 Gear Geometry
Basic Formulas
ISO 54 / DIN 867
dp = m × zPitch Diameter (mm)
da = dp + 2mAddendum Diameter
df = dp - 2.5mDedendum Diameter
db = dp × cos(α)Base Circle Diameter
h = 2.25mFull Tooth Depth
p = π × mCircular Pitch
a = m(z1+z2)/2Center Distance
i = z2/z1 = n1/n2Gear Ratio
Helical Gear Geometry
Normal vs. Transverse
mn (Normal Module)The specified module
mt = mn/cos(β)Transverse Module
αt = arctan(tan(αn)/cos(β))Transverse Pressure Angle
dp = mn × z / cos(β)Pitch Diameter
a = mn(z1+z2)/(2cos(β))Center Distance
Fa = Ft × tan(β)Axial Load from Helix
Typical β: 15°-30° · β>45° = Double Helical recommended
Contact Ratio
ε ≥ 1.2
Spur εε = (√(ra1²-rb1²) + √(ra2²-rb2²) - a×sin(α)) / (π×m×cos(α))
ε < 1.0❌ Prohibited — loss of contact!
ε 1.2-1.6✅ Regular Spur
ε 1.5-2.0✅ Helical — smoother
Rule↑ε = ↑smoothness · ↓noise · ↑strength
▶ YouTubeISO 54 Module Series
Series 1 — Preferred
Series 11 · 1.25 · 1.5 · 2 · 2.5 · 3 · 4 · 5 · 6 · 8 · 10 · 12 · 16 · 20
Series 21.75 · 2.25 · 2.75 · 3.5 · 4.5 · 5.5 · 7 · 9 · 11 · 14 · 18
Module SelectionAlways from Series 1! Otherwise sourcing issues
⚠️ DP (Diametral Pitch)American system — DP=25.4/m. Not compatible!
m=2 → dp=m×z=2×20=40mm. DP=12.7 ≠ m=2!
Undercutting
z_min = 17
z_min (20°)17 teeth ← without Undercutting
z_min (14.5°)32 teeth
z < z_minRoot cutting → ↓strength
SolutionProfile Shift (Correction) — x > 0
Profile Shift xx_min = (17-z)/17 for z < 17
Profile Shift changes the center distance! Recalculate a
▶ YouTubeBacklash
0.05-0.1 × m
BacklashClearance between meshing teeth
Typical Value0.05-0.1 × m (regular gear)
Zero BacklashSplit Gear · Spring-loaded · special Planetary
⚠️ Backlash=0Accelerated wear! Prohibited in regular gears
MobilityAnti-Backlash Gear — CNC · precision mechanisms
Too little Backlash = binding as the system heats up!
▶ YouTube💪 Gear Tooth Strength
Lewis Bending Stress
Lewis Formula 1892
σ = Ft / (b × m × Y × Kv)Bending stress [MPa]
Ft = 1000P / VP=kW, V=m/s → N
Y — Lewis Form Factorz=20: Y=0.308 · z=30: Y=0.340 · z=∞: Y=0.422
Kv — Dynamic Factor1.1 (precision) · 1.3 (regular) · 1.5+ (coarse)
σ_allow Steel200-400 MPa depending on material and condition
Lewis = a preliminary calculation. AGMA = more accurate!
▶ YouTubeAGMA Stress — Full Calculation
AGMA 2001-D04
σ_b = Wt / (b × mt) × KA × KV × Ks × KH × KB × YJBending
σ_c = ZE × √(Wt × KA × KV × Ks × KH / (dp1 × b × ZI))Contact
KA — Application1.0 (uniform) · 1.25 (light) · 1.75 (heavy)
KV — DynamicDepends on speed and AGMA Q accuracy
KH — Load Distr.Load distribution ← ↑b = ↑KH
AGMA.org · KISSsoft for full calculation
Hertz Contact Stress
↑Hard to Calculate
σ_H = ZE × √(Ft × KH / (b × d1 × ZI))MPa
ZE — Elastic Coeff.Steel/steel: ZE=191 √MPa
ZI — Geometry FactorDepends on i, α, x
σ_H,allow700-1500 MPa depending on material
PittingContact failure → pitting on the tooth surface
Contact stress = usually the limiting factor!
▶ YouTubeTangential Force Ft
Starting Point
Ft = 2T / dpT=torque [N·mm], dp=Pitch diameter [mm]
T = 9550 × P / nP=kW, n=RPM → T [N·m]
Ft = 1000P / VV=π×dp×n/60,000 [m/s]
Fr = Ft × tan(α)Radial Force
Fa = Ft × tan(β)Axial Force (Helical only)
Fn = Ft / (cos(α) × cos(β))Total Normal Force
Safety Factors
AGMA / ISO
SF_b (Bending)≥ 1.2 (industrial) · ≥ 1.5 (safety-critical)
SF_c (Contact)≥ 1.1-1.2
Pinion RulePinion 25% stronger than the larger gear
Life Ratioz1 × N1 = z2 × N2 (equal tooth cycles)
The Pinion rotates z2/z1 more times → reaches failure first!
Gear Failures
Failure Analysis
Bending FatigueRoot fracture · overload/fatigue
PittingHertz contact pitting · ↓surface · ↑noise
Scuffing / ScoringAdhesive welding · ↑speed · poor lubrication
Abrasive WearErosive wear · oil contamination
Plastic DeformationTooth bending · abnormal load
70% of gear failures = lubrication issues! → correct oil + cleanliness
▶ YouTube🔩 Gear Materials
Steel 4140 — AISI 4140 Q&T
Most Common
Composition0.40%C · 1%Cr · 0.2%Mo
HRC (Q&T)28-34 HRC (280-330 HB)
σ_bend_allow280-320 MPa
σ_contact_allow700-800 MPa
UseGeneral industrial gears · all-purpose
AdvantageAvailable · weldable · machinable
Steel 8620 — Case Hardened
Case HRC 58-63Automotive · Intensive
ProcessCarburize (0.8-1.2mm) + Quench
CaseHRC 58-63 · ↑↑contact durability
CoreHRC 25-35 · tough · ↓sudden fracture
σ_bend_allow380-420 MPa
σ_contact_allow1200-1400 MPa
⚠️ After HTGrinding is mandatory! AGMA Q10+
Steel 4340 Nitrided
HV 900+ Case↓Distortion
ProcessNitriding 500°C · 20-80h · 0.3-0.8mm
Advantage↓↓Distortion after HT! No grinding needed
CaseHV 900-1100 (≈HRC 68)
σ_bend_allow350-400 MPa
UsePrecision gears · aerospace · not repairable
Bronze C932 — Worm Wheel
Tin BronzeWorm Wheel Only
CompositionCu-Sn-Pb: 83%Cu · 7%Sn · 7%Pb · 3%Zn
HB60-80 HB
σ_bend80-100 MPa
Advantage↓friction coefficient with steel · ↓Scoring
⚠️ WormThe Worm must be hardened steel! (HRC 50+)
Plastic — Nylon / Acetal
Quiet · LightNo Lubrication
Nylon PA66Quiet · wear resistant · ↓loads
Acetal POM↑precision · ↓moisture absorption vs. Nylon
σ_bend40-60 MPa
AdvantageNo lubrication · light · anti-corrosion · quiet
⚠️ Drawback↑moisture swelling · ↓strength when hot · creep
Manufacturers: Igus (iglidur) · Ensinger
Cast Iron
GCI / SGI↓Vibration
GCI (Gray)180-250 HB · ↓noise · cheap
SGI (Ductile)250-350 HB · ↑strength vs. Gray
σ_bend GCI120-150 MPa
UseLarge slow gears · presses · gear racks
Advantage↑↑vibration damping · ↓cost · ↓wall thickness
| Material | HB/HRC | σ_bend (MPa) | σ_contact (MPa) | Use | Heat Treatment |
|---|---|---|---|---|---|
| 1045 N | 170-210 HB | 200 | 550 | Light gear | Normalizing |
| 4140 Q&T | 280-330 HB | 280-320 | 700-800 | All-purpose | Q&T |
| 8620 Case | HRC 58-63 | 380-420 | 1200-1400 | Automotive, intensive | Carburize+Q |
| 4340 Nitrided | HV 900+ | 350-400 | 1100-1200 | Precision, aerospace | Nitriding |
| Cast Iron GCI | 180-250 HB | 120-150 | 400-500 | Slow, racks | — |
| Bronze C932 | 60-80 HB | 80-100 | 280-320 | Worm Wheel | — |
| Nylon/Acetal | — | 40-60 | 80-100 | Quiet, no lubrication | — |
🏭 Gear Manufacturing & Quality
Hobbing
Most CommonContinuous
PrincipleA Hob (cutting worm) + gear rotate together
AccuracyAGMA Q6-Q9 · DIN 7-10
AdvantageFast · economical for mass production
LimitationNot suitable for direct internal gears
UseSpur, Helical, Worm gears · mass production
▶ YouTubeGear Shaping
Shaper Cutter
PrincipleA Shaper tool forms tooth by tooth
AccuracyAGMA Q5-Q8
AdvantageInternal gears · shouldered gears · Cluster Gear
DrawbackSlower than Hobbing
▶ YouTubeGear Grinding
AGMA Q10-Q13After Heat Treatment
MethodsForm Grinding · Generative (Niles/Reishauer)
AccuracyAGMA Q10-Q13 · DIN 3-6
RaRa 0.4-0.8μm
UseMandatory after Case Hardening · precision gears
▶ YouTubeAGMA Quality Grades
AGMA 2000-A88
Q3-Q5Coarse gears · agriculture · low speed
Q6-Q8Regular industrial gears · Hobbing
Q9-Q11High-speed gears · Shaving/Grinding
Q12-Q15High precision · aerospace · metrology
DIN 3962DIN 1 = most precise (reverse of AGMA!)
⚠️ AGMA ↑ = better. DIN ↓ = better!
Gear Inspection
AGMA / DIN
Single FlankChecking transmission profile error
Double FlankComposite Error · fast
CMM GearMeasuring Profile, Lead, Pitch, Runout
Span MeasurementMeasuring span W across teeth → module
Over PinDiameter measurement over a Pin → Pitch Diameter
Manufacturers: Gleason Metrology · Kapp-Niles
▶ YouTubeShot Peening
↑↑Fatigue Strength
PrincipleShot blasting → compressive stress at the tooth root
Improvement↑20-40% root fatigue strength
AMS 2432Aerospace Shot Peening standard
Almen StripShot Peening intensity — A/N/C
UseMandatory in aerospace gears · racing vehicles
▶ YouTube🌍 Planetary Gearbox
Planetary Components
Sun · Planet · Ring · Carrier
Sun GearCentral sun gear — usually the input
Planet Gears3-6 planet gears — equally spaced
Ring Gear (Annulus)Internal ring — usually fixed
CarrierHolds the planets — usually the output
EquationzR = zS + 2×zP · (zR + zS)/zS = i
i = (zR/zS) + 1 when Ring is fixed, Sun=input, Carrier=output
Planetary Gear Ratio
Choosing the Fixed Component
Ring Fixed (common)i = 1 + (zR/zS)
Sun Fixedi = 1 + (zS/zR)
Carrier Fixedi = -zR/zS (reverse direction!)
Typical ExamplezS=20, zR=60 → i=1+60/20=4:1
Typical Ratios3:1 to 10:1 per stage · up to 100:1 with two stages
Servo Planetary Gearbox
↑↑PrecisionArcmin
Backlash≤3 Arcmin (Standard) · ≤1 (Precision)
Torsional RigidityN·m/arcmin — ↑ = ↑positioning accuracy
Efficiency97% per stage · 94% for two
UseServo motors · robotics · 3D printers
▶ YouTubeHarmonic Drive
↑↑↑PrecisionZero Backlash
ComponentsWave Generator · Flex Spline · Circular Spline
Ratio1:30 to 1:320 in a single stage!
Backlash≤0 Arcmin! Less than 1 Arcmin guaranteed
Efficiency70-85% — relatively low
UseRobotics · aerospace · instrumentation · A/B axis
Manufacturers: Harmonic Drive LLC · HDS Japan
▶ YouTube🐛 Worm & Bevel Gears
Worm Gear — Parameters
Self-LockingHigh Heat
Number of Starts n11 (Self-lock) · 2 · 4 (↑efficiency)
Lead Angle γγ = arctan(n1 × m / d_worm)
Self-Lockγ < 5° → cannot be back-driven
Efficiency ηη ≈ tan(γ) / tan(γ + φ) · φ = arctan(μ)
Typical ηn1=1: 30-50% · n1=4: 70-90%
Ratio ii = z_wheel / n1
⚠️ (1-η) = heat. i>30 + Self-lock → very high heat!
Worm — Manufacturing and Materials
Steel/Bronze
WormSteel 4140 · Case HRC 55+ · Ground
WheelBronze C932 (Centrifugal cast) · HB 80
Manufacturing MethodWorm: Ground · Wheel: Hobbed
Running-InBreak-in required ← improves contact
⚠️ LubricantISO VG 220-460 + EP additive
▶ YouTubeSpiral Bevel — Gleason System
Gleason / Klingelnberg
GleasonTapered tooth depth · common in the US/Asia
Klingelnberg (Palloid)Uniform depth · common in Europe
Spiral Angle β35° (Gleason standard)
⚠️ IncompatibilityGleason ≠ Klingelnberg! Cannot be mixed
ManufacturingGleason 2/3 Face Milling · Phoenix
Manufacturers: Gleason · Klingelnberg
▶ YouTube💧 Gear Lubrication
Viscosity Selection
ISO VG
V > 25 m/sISO VG 46-68
V = 5-25 m/sISO VG 100-150
V = 1-5 m/sISO VG 150-220
V < 1 m/s (heavy)ISO VG 320-680
WormISO VG 220-460 + EP
HypoidGL-5 · EP absolutely mandatory!
Lubrication Methods
Splash · Circulating
Splash (Dip)V < 15 m/s · oil up to the lower gear's center
Spray/Mist15-50 m/s · spray nozzles
Forced CirculationV > 25 m/s · pump + filtration + cooling
GreaseSlow enclosed gears · NLGI 1-2
Splash QuantityOil up to the gear's center — no more! ↑heat
▶ YouTubeAdditives
EP Mandatory for Heavy Loads
EP (Extreme Pressure)Sulfur-Phosphorus · forms a protective layer
GL-1Mineral only · light-load Spur gears
GL-4Medium EP · medium Helical/Spur
GL-5Strong EP · Hypoid mandatory
⚠️ BronzeSulfur-based EP damages bronze! Worm → dedicated oil
Service Intervals
Per Manufacturer
Enclosed Gear (Splash)First change: 500h · then: 2,000-5,000h
Change PointTAN ↑3× the original value · water content >0.2%
Micron Filter<10μm filtration ← ↑3× oil life
Oil analysis: Tribology Israel
🏆 Standards & Manufacturers
International Standards
AGMA · ISO · DIN
AGMA 2001-D04Gear Tooth Strength Rating
AGMA 2000-A88Gear Classification — Q Grades
ISO 1328Cylindrical Gear Accuracy
ISO 6336Load Capacity — Bending + Contact
DIN 3962-3967Tolerances (1=precise, 12=coarse)
ANSI/AGMA 6034Practice for Worm Gears
Siemens / Flender — Germany
Heavy Industry
ProductsIndustrial Gearboxes · FLENDER · SIMOGEAR
UsesMining · paper · cement · ports
Neugart · Stöber · Alpha
Planetary Servo
NeugartPSBN · PLFE · WGN — planetary servo
StöberPOSIDRIVE · Servo + Planetary
Alpha GetriebeSP+ · High Precision · ≤1 arcmin
Gear Design Software
CAD · FEA
KISSsoftISO/AGMA/DIN · comprehensive · industrial
RomaxDriveline Analysis · automotive · aerospace
MASTASMT · gears + bearings + shafts
GearTeqSW/CATIA add-on · geometry
🧮 Gear Calculators
Gear Geometry Calculator
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Gear Pair Calculator
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Tooth Strength Calculator — Lewis
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Rack & Pinion Calculator
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Planetary Gearbox Calculator
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