Lego

About Gears in General

 

Why Gear Wheels have Teeth

There are a few ways in which one wheel can be made to turn another:  by simple friction, by rope, belt or chain transmission, and by gear teeth.

At any one time in such a transmission there is one wheel that drives and one that is driven (though of course in the same pair these roles may be reversed at certain times)

Simpe friction of smooth wheels touching at their rims is unusable if the axles are at a fixed distance:  the slightest wear will diminish the friction, uneven wear is worse, and in any case the force needed to turn the driven wheel must be less than the friction, otherwise slipping and hence wear will result.  The only way simple friction can work is if at least one of the axles can move and is pushed in the direction of the other, thereby maintaining the friction even after some wear has happened.  Such mechanisms exist but are rather rare.

Belts and ropes present the same problem of friction, but the contact area is much larger and the amount of friction can be maintained by a third wheel that keeps the belt or rope taut.  These belt transmissions were widely used in industry and can be seen on mopeds and in cars where they are used for driving the alternator and other accessories.

Chains are just a special case of gear wheels:  the wheels must have teeth and no slip is possible.  Chain transmissions are used in bicycles and motorbikes as well as in cars to drive the camshafts.

Gears have teeth to ensure there is no slippage.

But there are two different reasons to avoid slip:  one is to avoid wear when power is transmitted, the other is to ensure that the rotation angle is precise over many rotations when information is transmitted.

To see how these two are different, consider first a transmission by belt or rope:  under good conditions there is no slip and no wear, but the number of turns made by the driven wheel is only approximately known from the number of turns made by the driving wheel.  The diameters of the wheels are not precisely known and the effects of the expansion and contraction of the wheels, belts and ropes also influences the ratio of turns.  Over a more or less long time it is impossible to predict the number of turns of the driven wheel even if the number of turns of the driving one are known.  But as long as the power is transmitted, this drift of relative angle is not important.

In a clockwork however the hour hand must turn exactly once for each twelve times the minute hand turns, and this must hold as long as the clock functions:  information is transmitted, not power.  In an internal combustion engine the position of the cams (and hence the valves) must have a precise relation to the position of the crankshaft, and this relation must hold over all turns the engine makes while it runs.  It would lead to catastrophic failure if there were even the smallest drift.  Here too, information is transmitted ("the crankshaft is now at this position, the intake valve must be opened")

In both cases putting teeth on the wheels avoids bad results.  For camshafts chains or toothed belts are also used, but the principle is of course the same.

How many Teeth?

This question can be answered only by distinguishing between transmission of power and transmission of information.

Transmitting Power

In a car's gearbox the power of the engine is transmitted to the wheels, and because of the power characteristics of the internal combustion engine, different ratios should be available, hence a gearbox.  The precise transmission ratios are not important.  But because all the power passes through the single pair of teeth that is engaged at any one moment, it is important that tooth wear is kept to a minimum.  I will not go into the very important details of the shape of the teeth, but one consideration seems obvious:  the teeth should wear as evenly as possible.

The best way to spread the wear evenly is to make teeth contact each other evenly, i.e. it is bad if the same tooth on one wheel always meets the same other tooth on the other wheel.  The smalles defect on one tooth will always rub on the same other tooth instead of being spread over all other teeth.

For that reason gears in power transmission always have mutually prime numbers of teeth.  For example, if a transmission ratio of about 0.5 is desired, you will not see wheels with 16 and 32 teeth, you may see 17 and 33.  That gives a ratio of 0.515… which is close but ensures that over time all teeth rub against all other teeth.  This choice of mutually prime numbers explains why gear ratios in cars have these strange values that you may have read in motoring magazines.

Transmitting information

Here we care much more about the precision of the rotation angles after an indefinite number of turns than we do about wear.  Mechanical clocks and watches are the prime example, but of course the driving of the camshaft in a car has the same requirement of always keeping two axles in synchronisation.  In fact camshaft drives are a rare example where power and information are transmitted in the same drive.  It is also the reason why you have to replace or service that drive often during the lifetime of your car:  there is much more wear because the teeth always have to engage with the same other teeth.

Transmit power using mutually prime numbers of teeth

Transmit information using the desired number of teeth

Lego Gear Teeth numbers

Lego gears do not have a great need for transmitting power.  The distances between their axles are however limited by the possible ways in which Technics bearing bricks can be combined.  This is a more important constraint than transmitting either information or power.  More in the page on Lego gears, their dimensions, their meshing and the meshing calculator.