EngineeringMechanical EngineeringPart 08 of 15
Spur and Helical Gear Fundamentals
Full gear rating runs to dozens of variables and charts. Fortunately the decisions that matter most in general design — tooth count, module and face width — can be made soundly from a small set of relationships.
- Module and proportions
- Hunting teeth
- Involute geometry
- Tooth force resolution
Executive summary
Australian Standard AS 2075 sets out the terminology; spur and helical gears dominate general mechanical power transmission. In a gear pair the smaller gear is the pinion, the larger the wheel. Normally the pinion drives, so the pair produces a speed reduction.
Spur gears have teeth cut parallel to the shaft axis. Helical gears have teeth cut at a helix angle — typically around 20 degrees for single helical, 30 to 35 degrees for double helical or herringbone — and one gear of a pair must be right-hand while the other is left-hand. Helical gears are quieter and inherently stronger, at the cost of an axial thrust the bearings must carry.
Gear trains and velocity ratio
- SimpleGears in line, one pair meshing at a time. Intermediate gears are idlers: they change direction and centre distance but not the overall ratio.
- CompoundTwo gears sharing a shaft, so ratios multiply. The standard way to obtain a large ratio without an enormous wheel.
- PlanetaryPlanet gears orbiting a sun gear within an annulus. High ratio in a compact, coaxial envelope.
- n, N
- teeth in the pinion and wheel
- d, D
- pitch circle diameters of pinion and wheel
Why very high ratios need a train
Consider a ratio of 125 from a single spur pair. A 17-tooth pinion demands a 2125-tooth wheel — 2142 teeth to cut, and a wheel pitch circle diameter measured in metres. The same ratio from a three-pair compound train, each pair at a ratio of five, needs about 306 teeth in total and a largest wheel a fraction of the size. Rule-of-thumb limits for a single pair are 1 to 5 for ordinary gear pairs and 5 to 60 for worm and wheel.
Tooth count and hunting teeth
It is impractical to cut too few teeth. To obtain a reasonable profile without undercutting, and a smooth transfer of load from pinion to wheel, a working rule is at least 17 teeth on a spur pinion and at least 14 teeth on a helical pinion at a 20 degree helix angle.
For even wear distribution, the same teeth should not re-mesh every revolution of the wheel. The ideal is that a given pair of teeth meet again only after the pinion has made as many revolutions as the wheel has teeth. That condition — called hunting teeth — is met when the tooth counts share no common factor.
| Pinion teeth | Wheel teeth | Velocity ratio | Pinion revolutions before the cycle repeats |
|---|---|---|---|
| 18 | 38 | 2.111 | 19 |
| 19 | 38 | 2.000 | 2 |
| 20 | 38 | 1.900 | 19 |
| 21 | 38 | 1.810 | 38 |
| 20 | 40 | 2.000 | 2 |
| 21 | 40 | 1.905 | 40 |
A 19:38 pair and a 21:38 pair sit within ten per cent of each other on ratio, but one re-meshes the same tooth pairs every two pinion revolutions and the other every thirty-eight. That is a very large difference in wear distribution for a very small difference in ratio. Where an exact ratio is genuinely required — a camshaft drive, for instance — hunting teeth cannot be provided and the trade-off must be accepted knowingly.
Involute geometry and clearances
Module and tooth proportions
Module is the single most important parameter in gear design. It is the pitch circle diameter divided by the number of teeth, and it must be identical for both gears of a pair.
Standard first-choice modules in millimetres are 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 32, 40 and 50. As module increases, tooth size increases: at a 200 mm pitch circle diameter, a 1 mm module produces 200 small teeth while a 10 mm module produces 20 large ones. Larger teeth carry more torque.
| Loading | Wheel face width W | Pinion face width |
|---|---|---|
| Relatively light | 8 M | Usually 5 to 10 per cent wider than the wheel, depending on assembly tolerances. |
| Moderate | 10 M | |
| Heavy | 12 M |
A comprehensive rating procedure is complex — the relevant standard for spur and helical gear rating runs to dozens of variables and charts. For preliminary work, module selection charts plotted against transmitted power and pinion speed are widely used, drawn for a nominal face width of 10 M and around 18 pinion teeth. Because helical gears are inherently stronger than spur gears, one standard module size below the chart value is generally satisfactory for a helical pair.
Worked example: proportioning a spur pair
A spur pair requires a reduction between 2.5 and 2.7. The pinion is proposed at 18 teeth with a 5 mm module.
| Pinion | Wheel | Ratio | Pinion revolutions before repeat | Verdict |
|---|---|---|---|---|
| 18 | 45 | 2.500 | 5 | Poor wear distribution. |
| 18 | 46 | 2.556 | 23 | Better. |
| 18 | 47 | 2.611 | 47 | Full hunting — selected. |
| 18 | 48 | 2.667 | 8 | Poor wear distribution. |
Taking 47 teeth: pinion pitch circle diameter 90 mm, wheel 235 mm, nominal centre distance 162.5 mm, addendum 5 mm, dedendum 6.25 mm, tooth depth 11.25 mm. At moderate loading the wheel face width is 50 mm and the pinion about 53.5 mm.
Efficiency and tooth forces
With well machined, well lubricated gears carried on rolling element bearings, efficiency is typically 95 to 96 per cent per gear pair. Overall efficiency of a train is approximately the product of the pair efficiencies — three pairs at 96 per cent gives about 88.5 per cent, which is a material loss worth carrying into the motor sizing.
The force between meshing teeth acts perpendicular to the tooth surface at the pitch point. Resolving it gives the loads the shaft and bearings must carry.
- Ft
- tangential force at the pitch point — this multiplied by pitch circle radius gives the transmitted torque
- Fs
- separating or radial force, acting along the line of centres and keeping the gears in mesh
- Fa
- axial thrust, present on helical gears only
- T
- transmitted torque, Nm
- d
- pitch circle diameter, m
- φ
- pressure angle, normally 20 degrees
- α
- helix angle
Worked comparison
A gear of 100 mm pitch circle diameter transmits 800 Nm at a 20 degree pressure angle.
The transverse resultant barely changes between the spur and helical cases. What changes is the appearance of a 5.82 kN axial thrust that did not exist before — a load the bearing arrangement must now locate and carry. Choosing helical gears for quietness without revisiting the bearing selection is a common and expensive oversight.
Design checklist
- Gear type chosen against noise, load, ratio and axial thrust tolerance.
- Velocity ratio within the rule-of-thumb limits for a single pair, or split across a compound train.
- Pinion tooth count at or above 17 for spur, 14 for helical at 20 degrees.
- Tooth counts checked for common factors, and hunting teeth provided where an exact ratio is not mandatory.
- Module selected from the standard first-choice series and identical for both gears.
- Addendum, dedendum and tooth depth derived from the module.
- Face width proportioned to the loading, with the pinion slightly wider than the wheel.
- Pressure angle stated on the drawing, not assumed by the manufacturer.
- Helical pair specified with opposite hands.
- Tangential, separating and axial forces calculated and carried into shaft and bearing design.
- Train efficiency carried into prime mover sizing.
- Final rating confirmed against the governing standard for any critical or high-power application.