48610 Chap.14 Gears, Belts and Power Transmission
Gears, Belts and Power Transmission
Every machine that does something useful has to move power from a source to a point of use, usually changing the speed and torque on the way. Gears do this efficiently, usually above 95 per cent, and as a synchronous drive: the ratio between the two shaft speeds is exact, whether the drive steps the speed up or down, and nothing slips, and they are torsionally stiff and smooth. The whole theory follows from one condition.
For gears to mesh, the teeth at the contact point must be travelling at the same speed on both wheels, and since that speed is angular velocity times radius, the output speed is the input speed multiplied by the ratio of the radii, the diameters or, for teeth of the same size, the tooth counts. The same relationship holds for toothed belts and chains.
Torque runs the other way.
The tangential contact force is shared between the two wheels, and torque is that force multiplied by the radius at which it acts, so torque scales with radius while speed scales inversely. The torque ratio is therefore the reciprocal of the velocity ratio, which is the most commonly inverted relationship in the topic.
In a simple gear train, where each shaft carries one gear, intermediate idlers cancel out and change only the direction; in a compound train, where a shaft carries two rigidly connected gears, the intermediate counts do influence the ratio, which becomes the product of the driver tooth counts over the product of the driven ones.
Power ties it together: rotational work is torque times angle, power is torque times angular velocity, an ideal train conserves power, and a real one delivers the efficiency multiplied by the input.
What this chapter covers
- 01
Gears as a synchronous drive with an exact ratio
- 02
The pitch circle and the rolling cylinder analogy
- 03
The meshing condition: equal tangential speed
- 04
Ratio by radius, diameter or tooth count
- 05
Revolutions per minute to radians per second
- 06
Velocity ratio and torque ratio as reciprocals
- 07
Speeding up means losing torque, and the reverse
- 08
Simple trains and why idlers cancel
- 09
Compound trains and the product rule
- 10
Rotational work, power and efficiency
- 11
Synchronous against non synchronous drives, and slip
A single stage reduction with a power check
- +1Convert the input speed. Two pi times 1750 divided by 60 gives 183.3 radians per second.
- +1Apply the velocity ratio. The output speed is 183.3 times 18 over 54, that is 61.1 radians per second, or 583 revolutions per minute, and it turns the opposite way.
- +1Invert for torque. The output torque is 2.40 times 54 over 18, that is 7.20 newton metres for the ideal case.
- +1Check with power. The input is 2.40 times 183.3, which is 440 watts, and the output is 7.20 times 61.1, also 440 watts. They match because the tooth ratio appears once in the numerator and once in the denominator and cancels exactly.
- +1Apply the efficiency. The output power becomes 0.95 times 440, that is 418 watts. The speed is fixed by geometry and does not change, so the loss appears entirely in the torque: 418 divided by 61.1, which is 6.84 newton metres.
Key terms
- Pitch circle
- The locus of the contact point between two meshing gears, whose diameter is the size used in ratio calculations. Two meshing gears behave like cylinders of pitch diameter rolling without slipping.
- Meshing condition
- The requirement that the tangential speed of the teeth be the same on both gears, which generates every gear ratio relationship and also holds for toothed belts and chains.
- Velocity ratio
- Output speed divided by input speed, equal to the driving tooth count divided by the driven tooth count. It is a pure number and may be computed from speeds in any consistent unit.
- Torque ratio
- Output torque divided by input torque, equal to the driven tooth count divided by the driving one, and therefore the reciprocal of the velocity ratio.
- Simple gear train
- A serial chain of gears with a single gear on each shaft. Intermediate idlers do not alter the overall ratio; they bridge a distance and reverse the direction.
- Compound gear train
- A train with more than one gear rigidly mounted on at least one intermediate shaft, so the intermediate counts do influence the ratio, which becomes a product of the stage ratios.
- Synchronous drive
- A drive that transmits motion by mechanical engagement rather than friction, such as gears, toothed belts and chains, so that the ratio is exact and angular alignment is maintained.
Gears, Belts and Power Transmission FAQ
Which way does a gear ratio run?
Going from fewer teeth to more teeth slows the output and increases its torque, in the same ratio; going from more teeth to fewer does the reverse. The velocity ratio is the driving tooth count over the driven one and the torque ratio is its inverse.
A ratio quoted as a bare number such as three to one does not say which, so always state it with its direction, as a reduction or an increase, or better still give the output speed itself.
Do idler gears change the ratio?
Not in a simple train, where every shaft carries a single gear: each idler's tooth count appears once on the top and once on the bottom of the chain and cancels. Idlers exist to bridge a distance between input and output shafts and to reverse the direction of rotation.
In a compound train the situation is different, because at least one shaft carries two rigidly connected gears, and those intermediate counts do influence the ratio.
How does efficiency enter the calculation?
Only through the torque. The speed ratio is geometry, fixed by the tooth counts, so friction cannot change it. Losses can only appear in the force, and therefore in the torque, so the output power is the efficiency multiplied by the input power and the output torque is that power divided by the unchanged output speed.
A useful diagnostic follows: if a measured output speed differs from the calculation on a toothed drive, the cause is a wrong tooth count or a slipping friction belt, not efficiency.
When is a belt better than a pair of gears?
When the shafts are far apart, when you want the output turning the same way as the input, or when a little slip is acceptable or even useful as overload protection. A pair of external gears reverses the direction and needs the shafts close together; a belt keeps the direction and tolerates distance. A toothed belt gives the exactness of a gear with the reach of a belt.
A plain friction belt is the cheapest and the least predictable, and if your mechanism depends on a repeatable ratio that unpredictability is a design risk to record in the requirement list.
Assessment move
Convert once, at the top of the working, and keep every later line in radians per second, converting back to revolutions per minute only for the reader. Practise writing gear trains out mesh by mesh and cancelling the terms rather than trusting the general rule, because the product rule is easy to misapply when a shaft carries two gears.
Finish every transmission problem with the power check at both ends; it is the cheapest verification available in the subject. And for the project, work out the ratio and torque multiplication of whatever winder, lever or pulley your mechanism uses, because that calculation belongs in the report's table whether or not there is a gear anywhere in it.
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