5.3 Gears, Belts, Chains & Rotational Relationships

Gears, belts, chains and rotational relationships

Separate direction from speed

Rotational questions usually contain two tasks:

1. determine which way the output turns;
2. determine how fast it turns relative to the input.

Solve them separately. A diagram can have the correct speed ratio but a different direction because of an idler or crossed belt.

Meshed gears

Two external spur gears in direct contact rotate in opposite directions. If Gear A turns clockwise, Gear B turns anticlockwise. Adding Gear C gives clockwise again. Each external mesh reverses direction, so an even number of meshes preserves the input direction and an odd number reverses it.

For two ideal gears:

speed of driver × teeth on driver = speed of driven × teeth on driven.

Thus:

N driven = N driver × T driver ÷ T driven.

If a 12-tooth driver turns at 600 rpm and drives a 36-tooth gear, the output is 600 × 12/36 = 200 rpm in the opposite direction. The larger gear turns more slowly.

In an ideal lossless model, slowing speed increases torque by the reciprocal factor. A 3:1 speed reduction ideally multiplies torque by three.

Idler gears

An idler sits between driver and output on its own shaft. It changes direction or spacing but cancels from the first-to-last speed ratio. For A driving B driving C:

NC = NA × TA/TB × TB/TC = NA × TA/TC.

The idler tooth count TB cancels. It still affects physical design and contact, but not the ideal overall ratio.

With three external gears there are two meshes, so output C turns in the same direction as input A.

Compound gear trains

In a compound train, two gears are rigidly attached to the same shaft. They rotate at the same angular speed. The ratio no longer cancels because one gear receives motion and its shaft-mate drives the next stage.

Example:

• Gear A: 10 teeth drives Gear B: 40 teeth.
• B shares a shaft with Gear C: 12 teeth.
• C drives Gear D: 36 teeth.

Stage 1 reduction = 40/10 = 4.

Stage 2 reduction = 36/12 = 3.

Overall reduction = 4 × 3 = 12.

If A turns at 1,200 rpm, D turns at 100 rpm. There are two external meshes, so D turns in the same direction as A.

Rack and pinion

A pinion gear engaging a straight rack converts rotation to linear motion. Track the velocity of the teeth at the contact point. If a gear above a rack rotates clockwise, the bottom contact point moves left, so the rack moves left in an ideal no-slip model.

Internal gears behave differently from two external gears: when a small external pinion meshes with teeth on the inside of a ring gear, the two rotate in the same direction. A worm and worm wheel usually provide a large speed reduction with shafts at right angles. Direction in a worm drive depends on thread handedness and viewing direction, so use arrows or thread details supplied by the diagram rather than a memorised universal direction.

Belts and pulleys

Ignoring slip:

N1D1 = N2D2,

where N is rotational speed and D is pulley diameter. A small driver turning a large driven pulley reduces speed.

• Open belt: pulleys turn in the same direction.
• Crossed belt: pulleys turn in opposite directions.

An idler pulley used only to tension the belt normally does not change the ideal speed ratio. Its placement can change belt routing and direction if the diagram explicitly shows contact sides, so trace the belt.

Chains and sprockets

For an uncrossed chain loop, driver and driven sprockets turn in the same direction. Tooth-count ratios work like gears:

N driven = N driver × teeth driver ÷ teeth driven.

A larger driven sprocket turns more slowly and ideally receives more torque. Chains reduce slip compared with friction belts because teeth engage the links.

Coaxial and connected wheels

Components rigidly fixed to the same axle have the same angular speed in rpm, although a larger radius has greater tangential speed at its rim:

v = ωr.

Two wheels in direct friction contact rotate oppositely and, without slip, have equal tangential speed at contact. Their angular speeds are inversely proportional to radius.

Common traps

• Assuming every larger gear means the entire train slows; identify which gear is driver at each stage.
• Multiplying by driven/driver when calculating speed instead of using driver/driven.
• Letting an idler alter the overall ratio.
• Forgetting that gears on the same shaft turn together.
• Counting gears rather than meshes for direction.
• Confusing angular speed with rim speed.

Use arrows for direction and a ratio equation for speed. Finally, apply the size check: a small driver moving a large output should reduce output rpm.