Gear Types, Gear Ratio and Gear Trains with Spur Gear Calculations
Gears transmit rotation between shafts with an exact speed ratio, no slip and high efficiency. They reduce motor speed for conveyors, multiply torque in a car gearbox, and turn a corner in a hand drill. To design or check a gear drive you need to know which gear type suits the shaft layout, how tooth size is specified, and how speed and torque change through a train. This article covers all three with a complete spur gear example.
Common Gear Types
| Type | Shaft arrangement | Advantages | Limitations | Typical use |
|---|---|---|---|---|
| Spur | Parallel | Simple, cheap, no axial thrust | Noisy at high speed | Machine tools, clocks, gear pumps |
| Helical | Parallel (or crossed) | Smooth and quiet, higher load capacity | Axial thrust needs thrust bearings | Automobile gearboxes, reducers |
| Double helical / herringbone | Parallel | Thrust cancels out | Costly to make | Large turbine reducers, rolling mills |
| Bevel | Intersecting, usually 90 degrees | Changes direction of drive | Accurate mounting needed | Differentials, hand drills |
| Worm and worm wheel | Non-intersecting, 90 degrees | Very high ratio in one stage, can be self-locking | Low efficiency due to sliding | Hoists, gate operators, tuning pegs |
| Rack and pinion | Rotary to linear | Simple linear motion | Backlash in precise work | Steering, CNC axes, lifts |
Tooth Size and Geometry
In metric practice tooth size is given by the module m, the pitch circle diameter divided by the number of teeth. Two gears can mesh only if they have the same module and the same pressure angle, which is 20 degrees for most modern gears.
Module m = d / T (mm) Pitch diameter d = m * T Circular pitch p = pi * m Addendum = 1.0 m Dedendum = 1.25 m Outside diameter da = d + 2m = m (T + 2) Centre distance C = (d1 + d2) / 2 = m (T1 + T2) / 2 Gear ratio i = T2 / T1 = N1 / N2 Torque M = 60 P / (2 pi N) Tangential force Ft = 2 M / d Radial force Fr = Ft * tan(phi)
Standard first-preference modules include 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8 and 10 mm. Very small pinions with 20 degree full-depth teeth can suffer interference and undercut; for this reason designers usually keep the pinion at about 17 to 18 teeth or more unless the teeth are profile-shifted.
Worked Example: Spur Gear Pair
A 2 kW motor running at 1440 rpm drives a 20-tooth pinion meshing with a 60-tooth gear. Module is 3 mm and pressure angle 20 degrees. Gear efficiency is 98 percent.
Pinion pitch diameter d1 = 3 * 20 = 60 mm Gear pitch diameter d2 = 3 * 60 = 180 mm Centre distance C = (60 + 180) / 2 = 120 mm Outside diameters 66 mm and 186 mm Circular pitch p = pi * 3 = 9.42 mm Gear ratio i = 60 / 20 = 3 Output speed N2 = 1440 / 3 = 480 rpm Input torque M1 = 60 * 2000 / (2 pi * 1440) = 13.26 N m Output torque M2 = 13.26 * 3 * 0.98 = 39.0 N m Tangential force Ft = 2 * 13.26 / 0.060 = 442 N Radial force Fr = 442 * tan 20 = 161 N
The tangential force drives the gear and is used to check tooth bending stress (Lewis equation) and surface stress. The radial force pushes the two shafts apart and must be carried by the bearings, together with the tangential force.
Simple and Compound Gear Trains
Simple train with an idler
In a simple train each shaft carries one gear. If a 30-tooth idler is placed between the 20-tooth pinion and 60-tooth gear, the overall ratio is (30/20) x (60/30) = 3, unchanged. The idler only reverses the direction of the output and fills a larger centre distance.
Compound train
In a compound train, two gears are fixed on the same intermediate shaft, so the ratios multiply. This gives large reductions in a compact space.
Stage 1: driver 20 T -> driven 60 T ratio 3 Stage 2: driver 15 T -> driven 45 T ratio 3 Overall ratio = 3 * 3 = 9 Input 1440 rpm -> output 160 rpm Output torque (ideal) = 9 * input torque
With two external meshes the output turns in the same direction as the input. Each external mesh reverses direction, so an odd number of meshes reverses the output.
Common Mistakes
- Trying to mesh gears of different modules or pressure angles.
- Counting an idler gear in the ratio. It changes direction only.
- Using the outside diameter instead of the pitch diameter for force and centre distance calculations.
- Ignoring axial thrust when replacing spur gears with helical gears.
- Choosing a worm drive for a high-efficiency application; its sliding contact wastes power.
Summary
Choose the gear type from the shaft layout, size teeth with the module, and remember that speed divides and torque multiplies by the tooth ratio. Pitch diameters give centre distance and tooth forces, and compound trains multiply stage ratios for large reductions. With these relations you can lay out a reducer and pass the forces on to shaft and bearing design.