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# 1.13 Gears and gearing

Gears and gearing (Figure 7) are a feature of practically all machinery and are by no means confined to food mixers. The function of gearing is to transmit rotary motion and power from one place (for example, a motor) to another (in the case of the food mixer, the tools doing the mixing), usually with changes in speed, direction or both. You'll find gears and gearing in all types of powered transport (including bicycles), in factory machinery and in many household items, from electrical drills to cameras. The gears can either be driven directly from wheel to wheel (by friction or interlocking teeth) or remotely (by belt or chain).

Figure 7 A selection of gears: top left wooden gears from a windmill; bottom left steel gears from heavy machinery; top right gears manufactured using 'micromachining' on a microscopic scale; bottom right a theoretical gear built up at an atomic level

So why do we need gears and gearing?

If you've ever ridden a bike, you'll know that it's easier to cycle uphill in a low gear – less effort is needed to turn the pedals. The penalty is that you appear to go more slowly – you need more turns of the pedal crank to cover a particular distance. The situation is reversed when going downhill, so you change to a higher gear. There is a pedalling speed at which your legs can operate most efficiently and comfortably, and the purpose of the gears is to allow your legs to work at that optimum speed.

The same principle applies to an electric motor or car engine. You can't do a hill start in a car in fourth gear, and travelling along a motorway at 70 mph in first gear is unfriendly to the engine!

Let's look at the working of gears in a bit more detail. Figure 8 shows two wheels with their rims in contact. Friction ensures that turning one will cause the other to rotate – they'll act as a pair of gearwheels. If there is no slipping between the two as they move, then, at their contact point, the velocity (v 1) of the rim of gear 1 must equal the velocity (v 2) of the rim of gear 2, i.e.

But the two wheels have different diameters. If the velocity of their rims is the same, they must be rotating at different rates. The smaller wheel will complete more than one revolution as the larger wheel turns one revolution, simply because its circumference is smaller.

Rate of rotation is usually expressed either as the number of revolutions in a given time (e.g. revolutions per minute, or rpm) or in terms of a quantity called the angular velocity. Angular velocity is the number of degrees turned in a given time, like 'ordinary' velocity is the amount of distance covered in a given time. Angular velocity is conventionally symbolised by the Greek letter ω ('omega'), and its units are degrees per second (as long as the angles are expressed in degrees).

Figure 8 Two wheels in contact

The circumference of a circle with radius r is equal to 2. A wheel with a larger radius will clearly have a larger circumference. If it is being driven from another wheel, then the larger the wheel being driven, the slower it will turn.

In order to work out the angular velocity, we need to work out how many degrees are turned through in a given time. This is all very well, but it would mean that we would always have numbers that are difficult to manipulate cropping up in calculations of angular velocity. So we use another system, which is to describe a circle as sweeping out a number of radians.

You can think of a radian as being just like a degree, but rather than there being 360 of them in a circle, there are 2π of them. This may sound complicated, but it has the huge advantage that it makes the maths easier!

Because the circumference is 2πr, and the wheel turns 2π radians in the same time that the circumference is 'moved' this distance, the angular velocity in radians is simply:

For our two wheels with radii r 1 and r 2, we have that

and

Since at the point of contact v 1 = v 2, combining these two expressions gives

The fraction r 1/r 2 defines the gear ratio of this particular system. A similar thing applies to gears with teeth, where the teeth interlock to turn the wheels, the ratio being N 1/N 2 where N is the number of teeth on each wheel. The lower the gear, the lower the value of the gear ratio. Notice also that directly-driven gearwheels like those in Figure 8 rotate in opposite directions to one another.

Gear ratios are the same for indirectly-driven gearwheels (Figure 9), whether by belt (r 1/r 2), or by chain (N 1/N 2). However, as you can see, both gears rotate in the same direction here.

Figure 9 Indirect gearing

Not all manufacturing processes can be used sensibly to make gearwheels, so occasionally we'll look at the manufacturing aspects of some of the other parts which go into making the complete food mixer.

Figure 10 An exploded view of the train of gears in the food mixer

Figure 10 shows an exploded view of the gear train from the food mixer in Figure 6. You can see that this is a fairly complex assembly of intermeshing parts. The complexity arises because not only does the mixing tool spin on its own axis but the axis itself also moves around a circular 'orbit' in the bowl of the mixer. In addition, this particular gear train 'gears down' the motion from motor to tool by a factor of 20. But don't worry about the details of Figure 10. We're going to concentrate on the simplest gearwheel in this assembly, which is known as the planet gear. A photograph of this and its associated static ring gear is shown in Figure 11.

Figure 11 The ring and planet gears