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Helical milling – gear trains


gear trains

If we know the lead required this enables us to work out the gear ratio needed to produce this lead. The overall ratio between the movement of the milling table and the rotation of the workpiece is determined by the pitch of the lead screw, the gearing between the lead screw and the auxiliary input, and the gearing in the dividing head.

Suppose the pitch of the lead screw is 5mm and the ratio between the drive on the dividing head and the rotation of output of the dividing heads. This is usually 40:1.  If the gearing between the leadscrew and the drive to the dividing head is 1:1 then one turn of the lead screw will move the table by 5mm and 40 turns will move it by 200mm – the lead is 200mm. The workpiece will have turned once.

In this situation we still need two gears, one on the leadscrew and one on the drive into the driving head. The one on the leadscrew is the “driver” and the one on the dividing head is the “driven”.

It does not matter how many teeth there are on each. What matters is simply the ratio of the number of teeth on one gear to the number on the other. In this case this number must be the same for a ratio of 1:1.

Furthermore the two gears must be physically big enough to mesh with each other. **** This second condition is unlikely to ever be met with just two gears so it becomes necessary to introduce a third gear to fill the gap. The size of this does not affect the ratio but it does reverse the direction of rotation of the dividing head. Such a gear is known as an “idler”. If the dividing head rotates one way we will get, say, a left-handed helix. If it rotates the other way we will get a right-handed helix.

If one idler reverses the handedness of the thread, then two idlers will reverse it back. That is, they cancel each other out so long as they are the same size. But they can still be useful in making the other gears “fit”.

Fig.  driver, idler, idler and driven gears

A gear train with one driver and one driven gear seldom gives enough choice over the ratios possible. This choice can be increase by replacing the idler with a pair of gears linked together as shown in Fig.

Fig.  gear train with four gears

In this arrangement the second and third gear are fitted to the same stub axle. They are both keyed to this axle so they turn together at the same speed. If the first gear is a driver then the second is driven. The third becomes another driver and the fourth is another driven.

If the direction needs to be reversed then an idler could be introduced between the first driver and the first driven or between the second driver and the second driven.

The overall ratio, R, will be:

driver 1 x driver 2


driven 1 x driven 2

The ratio for the gear train is simply a ratio. This can be expressed as one number divided by a number or simply a number where the divisor is 1. It might, for example, be 1.220.

This does not, in any way, tell us what gears we need. We can convert it to whole numbers and it would be:




This is simply a ratio so we can divide by top any bottom by any suitable factor and the ratio stays the same.

In this case

1220                      610                305                  61

—-                =          —        =         —         =        —

1000                      500                250                  50

When it has been reduced to the simplest possible fraction we look for the factors that exist in both top and bottom. The 61 is a prime number and so has no factors. Fifty has 5, 5 and 2 as factors.



5 x 5 x 2

The 61 can only be implemented by one gear with 61 teeth. So in this example the driver, 61 can only drive one driven, 50.

We can also multiply top and bottom by any number and the ratio will stay the same. For example:

61 x 20


5 x 5 x 2 x 5 x 2 x 2

This can give drivers of 61 and 20 but the bottom factors could be split in several ways. One of these would be 25 and 40.

The traditional ways of computing a gear ratio are unbelievably tedious to use. The solution is the computer. This requires a program. One suitable program is “Hobnail” written by Gareth Evans. This was actually written for computing the gearing for hobbing helical gears. But can be used for this way of making helical gears – one tooth at a time.

Using Hobnail for computing gear trains for cutting helical gears

Hobnail is covered later under hobbing helical gears.

Gear trains – practical points

It is essential that the gears are locked to each shaft, unless they are idlers, with a key. It is essential that where there is only one gear on a shaft that it cannot slide sideways and so become disengaged or fall off the end of the shaft.


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