There is another page which is a step by step guide to making a spur gear.
for making large spur gears see “Making large spur gears using a rotary table”
Making spur gears
One of the most common ways of making spur gears is by using a dividing head on a milling machine. The dividing head makes it possible to cut a gear with almost any number of teeth. It is possible to use either a horizontal milling machine or a vertical milling machine
Either way the workpiece is held using a dividing head.
see page on “dividing head”
see page on “holding a workpiece using the dividing head”
The whole point of using a dividing head is that it is designed for dividing a circle into an equal number of parts. When gear cutting this determines the number of teeth to be cut.
For a truly comprehensive explanation of gears the reader is referred to “Gears and Gear Cutting” by Ivan Law.
This leaves one more choice. This is whether a horizontal or a vertical milling machine is available. Given the choice the horizontal system is more rigid and would enable the user to cut bigger gears or cut any gear faster than would be possible on a vertical milling machine.
Most home machinists do not have horizontal milling machines but have a vertical one. In this case the gear cutter is held on a stub arbor held in the vertical spindle’s socket. In this case, the cutter cuts into the workpiece in the middle of the front side.
The cutters for gear cutting are designed for use on a horizontal arbor. If a vertical milling machine is being used then the same cutter is used but it is mounted on a stub arbor. This is then fitted to the vertical head.
Use of spur gears
Spur gears can be use to transfer power. In doing this they can also be used to shift the axis from where the power comes from to any other axis parallel to the first. They can also be used to convert the speed of one shaft to a different speed on the other. Of course the power stays the same but the torque changes.
Where two shafts are connected by gears then the relative speeds of rotation will be directly proportional to the diameters of the gears. If a gear with, say, 20 teeth turns one with, say, 40 teeth one turn of the first will only make half a turn of the second. If one gear is rotating one way then the next gear will rotate the other way. If there is a third gear between the other two, it will not affect the speeds but it will reverse the direction of rotation of what was the second shaft.
Where two spur gears are in mesh with each other, if one is bigger it is the gear and the other is the pinion. A rack can be seen as being a small part of a spur gear with an infinite radius. Where a spur gear is used with a rack it is the pinion.
Spur gears – parameters
There are still a few key features of gear we need and have to be defined.
For two gears of any size (i.e. for any number of teeth) to mesh properly there are several conditions that must be fulfilled:
A Metric or Imperial
The dimensions of the gear will be either imperial of metric. Standard sizes of metric or imperial gears cannot be used with imperial gears and vice versa. Used means two gears whose teeth are engaged properly.
B Tooth form
For gears to work properly the teeth have to roll over each other. Otherwise they would rub and this would lead to rapid wear and friction. Also this shape must have the property that as one gear rotates the other always rotates at every point in the circle at the same rate. This is known as constant velocity. There are two common shapes that have this quality the first is the cycloidal. This used (a very long time ago) to be popular but is now superseded by the involute form. Watchmakers still use a variant of the cycloidal form.
C – Size of teeth
For two gears, with any number of teeth, to mesh properly the size of the teeth on both gears must be the same.
The sizes of metric and imperial gears are defined differently.
In the imperial system tooth size is called diametrical pitch or DP. The diametrical pitch is the number of teeth a gear has for each inch of its pitch circle diameter. For example a gear of 1 inch pitch circle diameter and 10 teeth would be 10DP.
In the metric system the tooth size is MOD. The MOD of a gear is the pitch circle diameter in mm divided by the number of teeth.
D – Pressure angle
Where two gears touch each other the angle relative to a normal on the pitch circle diameter to the tooth face at this point can vary. In the past the fashion was for a pressure angle of 14.5º. The current fashion is for 20º. For special applications this can be even higher.
For involute gears to mesh properly it is necessary for the teeth on both gears to have the same pressure angle.
Because in the past 14.5º was the standard this was often not marked on a cutter. Because modern cutters have different pressure angles they are usually marked as having a pressure angle of 20º. Consequently a cutter that is not marked with a pressure angle is probably a 14.5º one.
E – Depth of cut
The depth the cutter needs to cut is always marked upon the gear cutter. It includes an allowance of 10% at the bottom of the cut so the top of one lot of teeth will not clash with the bottoms of the other gear. This makes no difference to the user – just cut to the depth marked on the cutter. Because of this is is essential that the gear blank has an outside diameter that is correct because this is the reference point for determining the depth of cut.
On some English cutters made for cutting metric gears the depth of cut can be marked in inches!
The above variables define a tooth of a certain size. If this is fixed the only other variables are directly related to the number of teeth
X – Number of teeth
The number of teeth is important since this determines the gearing obtained with other gear of this or other numbers of teeth.
Y – Pitch circle diameter
Though a gear has an outside diameter this is not as important as the pitch circle diameter. This is the distance between the two centers of two identical gears meshed together. It is roughly the distance from the middle of a tooth on one side of the gear to the middle of a tooth on the other side.
Z – outside diameter
The gear will have an outside diameter. This is the size the blank has to be turned down to. Some gears, for example 16DP will have pitch circle diameters and outside diameters that are both multiples of 1/8 inch. But, this is an exception. Since gears are usually defined by their tooth size and the number of teeth these set the outside diameter. The result is usually a very non-standard size.
Outside diameter = (pitch circle diameter + 2) / diametrical pitch
Cutters for cutting spur gears
It might be thought that a cutter that was made for a tooth size would make any gear with that size tooth. But, unfortunately, for any given size of tooth, pressure angle etc the shape of the tooth will vary depending upon the number of teeth on the gear being made.
Cutters for gear cutting are covered on another page “cutters for horizontal milling”
This means that to be able to cut a gear with any number of teeth but for a given size of tooth a set of eight cutters is required. In order to cope with this it is necessary to have a cutter for each of a range of teeth numbers.
Browne and Sharpe first devised this system in 1865. Their cutters are:
Cutter no from to
8 12 teeth 13 teeth
7 14 16
6 17 20
5 21 25
4 26 34
3 35 54
2 55 134
1 135 rack
(These only apply to involute gears. One snag with cycloidal gears is that the number of cutters needed is much larger.)
It will be noticed that there are no cutters for cutting less than 12 teeth. As the number of teeth gets this low a true involute form starts to clash at the bottom of the tooth against the top of the other tooth. Cutters for small numbers of teeth are modified to prevent this. What happens is that the base of the tooth is thinned. To do this for even less teeth would need so much thinning that the strength of the teeth would be compromised.
It is interesting to note that a rack, which is just a part of a gear of infinite radius, has sides to the teeth that are straight lines. These sides just happen to be at an angle that is the same as the pressure angle for this tooth. This is a property of the involute tooth form.
It will be seen elsewhere on this site that similar cutters are used for making bevel gears. The technique recommended somewhere else on this site for producing bevel gears produces what are known a parallel teeth. These can be cut with the standard cutters as used for spur gears. For making bevel gears where the teeth vary in both height and width special cutters are needed. They look just like the cutters for making spur gears but they are marked “bevel” and are not suitable for cutting spur gears.
Marking of gear cutters
The bore of the cutter is never marked. Those for use on milling machines will usually be 1 inch and upwards. But smaller bores sizes are used for gears cutters used for making gears for clocks and watches.
The outside diameter of cutters with a 1inch bore is usually not more than 75mm so they will fit all but the smallest milling machine.
Most cutters will be marked with:
the size of the tooth – mod or DP,
the number of the cutter in the Browne and Sharp set
the range of teeth sizes it can cut,
the depth of cut needed,
the pressure angle (not always.)
Buying gear cutters
It should be clear that given the number of parameters which define a gear cutter the number of different gear cutters available is very large. If one buys gear cutters on the off chance they might be useful and one bears in mind how seldom any gear cutter is ever going to be used then the conclusion has to be that it is a waste of time buying gear cutters unless there is good reason to believe they will be used or they are very good value.
If you are the sort that just cannot help collecting things on the off chance they might one day come in useful then rather than buy any cutter simply because it is cheap make a point of collecting ones that you are most likely to use. The first set of cutters to collect has to be those needed to make gears of the form of the change wheels on your lathe.
If you think you might need others go for relatively standard sizes. An example of this are the sizes taken from the catalogue of one well-known supplier of gears. These are:
imperial DP 4, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64 (smallest)
Metric MOD 0.5, 0.7, 0.8, 1.0, 1.25, 1.5, 2, 2.5, 3, 4 (biggest)
Spur gears – Mathematics
Many books on cutting gears are full of maths. But there is a simple solution to this. There are several suppliers of gears who provide catalogues with details of every gear they supply. Suppose you need a change gear to match some you already have. For any gear you have from a set of change gears it will have a certain number of teeth and a precise outside diameter. In the catalogue there will only be one gear that fits these two parameters be it metric or imperial. Having found it you will now know whether it is imperial or metric, and the size of the tooth.
There is a slight chance this is a fluke. If the same procedure is done with a second gear from the set of change gears, if it confirms the first, then it is probably right.
Though a manufacturer’s catalogue will tell us the pressure angle we cannot tell what the gear we have is. If it is British made and old then it is probably 14.5°. If it is continental or far eastern of any modern date it is probably metric and the pressure angle will be 20°.
If the pressure angles are wrong the gears will not mesh properly. But for the use in gear trains on lathes and milling machines this is not a problem because they will be in use for very little time, the loading will be limited and the spacing is easily adjustable.
Knowing this we can then look up in the same table the outside diameter of any gear for any number of teeth directly.
Apart from all of these parameters there are others that will affect the gears possible performance.
Material the gear is made from:
Length of the tooth
Fitting to a shaft
Making the workpiece
The first step is to make a blank. It is important that the outside diameter of this is as accurate as possible. It might seem odd because when the gear is in use the outside does not touch anything. The importance of the outside diameter is that it is used as the reference from which the depth of cut is taken.
When making gears they will be fitted into some sort of gearbox. The distance apart of the axes of the axle the gears are on will need to be a very precise distance apart. If a gear is undersize, it will introduce backlash between the gears. If the gears are oversize the gears might not fit together at all or, if they do, they might not rotate freely.
It is essential that when the gear blank is being made on the lathe the axis of the axle or the arbor or the mandrel it is on is coaxial with the axis of the spindle.
When the blank has been finished it is moved with its axle/arbor/mandrel onto the dividing head. Again, the axle,arbor/mandrel must be coaxial with the axis of the dividing head.
Whilst turning the outside of the blank to the required diameter it is worth putting a small chamfer on the two corners.
A mandrel between centers will probably be accurate naturally. But an arbor or axle held in a three jaw chuck might be out. This error will be a few thou. This might be acceptable on a large gear but could be serious with small gears. In this case it would be better to use a four jaw chuck and align it properly.
Not many gears are useful without them being on a shaft or being able to fit on a shaft. either it has to be cut with a shaft as one piece or it will need a hole in it. Both of these have to be made whilst the workpiece is being turned.
Holding the gear for turning on the lathe
A gear has to fit a shaft accurately. Usually it has to be fixed in some way to the shaft. The easiest way is to machine the blank so the gear has a boss on it. This is drilled and tapped to take a grub screw to hold it in place. A better arrangement is to have two grub screws. These should be at 90º round to each other on the boss. The shaft should have flats milled on it where the screws touch it.
The next simplest way is to cut a keyway on the shaft, and a keyway in the bore of the gear and put a key in when fitting the gear to the shaft. The simplest type to get right easily is a Woodruff key. This is covered elsewhere. This can be done with the simplest of slotting tools. A good fit is essential.
The ultimate way is to put splines on the shaft and matching splines inside the bore of the gear. This is a bit more than most home workshops can manage.
The center hole is center drilled, drilled and reamed or bored on the lathe. It is essential that this hole is concentric with the outside of the gear. If it is not there might be backlash when the gear is in one position but it will stick in the opposite position.
The gear blank can be made on the end of a bar and then parted off. In this case the blank would then be mounted on an arbor or mandrel.
Holding the workpiece on the milling machine for cutting the teeth
There are three ways of holding the workpiece.
This is covered in “Dividing head – holding the workpiece”.
It can be held on a mandrel.
Having done this it is possible to turn the outside to the required diameter by turning between centers on the lathe. If this is done it is possible to transfer the blank whilst it is still on the mandrel straight onto the dividing head and tailstock though the dog might need changing. The “wider” end should be at the dividing head. Cutting should then be done towards the dividing head.
It is also possible to hold the workpiece with an arbor with a shoulder and parallel shank and hold the workpiece on with a washer and nut arrangement.
This needs to be bolted very tightly. It only works because there are no forces attempting to turn the workpiece round. All of the cutting forces are parallel to axis of the arbor. It is possible to use a drop of Loctite 603 to hold it from rotating. After the gear has been cut the gear and the arbor can be heated up and the glue will loose its grip.
If just one thin gear is to be made it needs a dummy blank on the side the cutter is cutting towards to support it. The outside of the dummy needs to be at least as large as the outside diameter of the gear being cut. Thin gears cannot be held properly on a tapered arbor. Neither can very wide ones.
It is also possible, especially with smaller gears for the axle of the gear and the gear to be part of the same piece of metal. In this case the end of a bar is turned down to the outside diameter of the gear. But a length of the bar is turned so it is concentric with the end but this part is for holding it in a chuck.
If several thin gears are needed they can all be cut altogether in one go. Then they are parted off to the thicknesses required.
Alignment of the dividing head
The dividing head must be aligned along the x-axis of the milling machine.
The dividing head must be horizontal.
If using a tailstock it must be at the right height
If using a mandrel it must lie along the x axis and be horizontal
If in horizontal mode, the cutter must be above the blank and the center line of the cutter must be in line with the axis of the spindle of the dividing head in the vertical plane.
If horizontal milling:
Then the y axis should be locked. this should remain locked for the entire job.
If vertical milling
If in vertical mode the cutter must be in the center of the side of the workpiece.
Setting and using the dividing head
The way the the dividing head is set up and used determines the number of teeth that will be cut.
The handle on the dividing head has to turn, say, 40 times to go through one circle. But one turn of the handle can be divided further by using the holes in the ring of holes selected on the division plate.
To see how this is worked out see “Dividing head – dividing the circle”
The result will be that for most numbers for each tooth (ie gap) to be cut it is need so many turns of the handle and for the indexing pin to pass so many spaces on the appropriate ring of holes. The number of spaces is set by using the sector arms.
When starting it is important that the index pin is in the top hole in the ring being used and the sector arm are set so the left arm is touching the pin as shown.
fig sector arm at the start
At the start the pin will be in a hole touching the left arm.
The dividing head’s shaft is locked. The tooth is cut.
The pin is then turned to the hole to the left of the right sector arm. The pair of sector arms are then rotated so the left sector arm is now touching the pin. This moves the pin by the number of holes between the sector arms. If the pin has to also move through another one or more complete turns then these can now be done. But however many complete turns are done the pins will still end up in the same hole.
fig sector arms moved
fig final position of the indexing pin
Often, where the pin only moves by so many holes it is quite easy to keep track of things. It can be seen by looking at the workpiece whether the pin has been moved.
Where more turns are needed one way of keeping track is to roughly mark the position of each tooth (ie gap) with a felt tip pen before cutting begins. Before every cut the position of the tooth and the cutter are checked.
Where the teeth cannot be cut in one pass – horizontal mode
If the teeth are cut in one pass then the whole process of making the gear is done with the milling table at just one height. This means the milling table can be locked at the right height for the whole process. If the teeth are cut in two passes then the table is locked, height wise for the first pass. It is then raised and locked for the second pass.
Where the teeth cannot be cut in one pass – vertical mode
The depth is determined by the movement in the y (ie in/out, direction. Between the first pass and second pass the table is moved in the in/out direction. But is locked the rest of the time, ie whilst cutting.
In both of the above cases, because of the shape of each tooth, ie the gap, to cut about half the metal in one go means the first round of cuts should be about two thirds of the required depth leaving the rest for the second round of cuts.
Cutting the gear
Fit the gear blank on an arbor in the chuck on the dividing head.
Rotate the dividing head through one circle without the cutter touching the workpiece. This is to check that there is nothing that is going to compromise the job before it is finished. Check that the cutter can cut all the way from one side of the gear through to the other side. Check the cutter does not clash with the chuck or the dog if using a mandrel.
In this photo the cutter is cutting away from the dividing head. It should be towards the dividing head though this means it has to rotate backwards.
When cutting a gear on a horizontal machine the cutter will normally turn so that it cuts towards the dividing head. When using a mandrel the wide end of the mandrel should be at the dividing head end and cutting should be towards the dividing head. Here, it is this way because it is easier to see what is happening.This is true whether the dividing head is left handed or right handed.
The dividing head should be set so the sector arms are spaced as needed for the number of teeth being cut. The whole gear cutting process starts with the dividing head set to zero.
As the dividing head is driven round it can happen that you lose the position. The trick to prevent this is to have a starting point that is reproducible. The way this can be done is before anything else is done, is to set the dividing pin to the zero position on the dividing plate. Then mark the indexing ring as closely as possible to the indexing pin. If the surface is black it can be marking by using Snowpake.
One problem that can cause confusion is where the number of holes is greater than the number in the current circle. This will occur whenever the number of teeth is less that the number of turns needed to turn the spindle by one whole circle. For example, if the movement for each tooth is x turns of the dividing handle and then y spaces on the dividing plate.
The dividing head should always be rotated one way when one tooth has been cut and you are moving to the next tooth. If it should happen to you go too far you must come back past the position required and then move forwards to the position required. This is to allow for any play in the dividing head. During the cut the dividing head should be locked so it cannot rotate.
If during the cutting of a gear the cutter starts at the wrong point the whole job is usually completely screwed up. The main cause of this problem is interruptions in the cutting process. To avoid this happening it is best to cut gears only when there is time to complete the whole job in one go. It also helps if there is no-one else around who might interrupt the process. Mobiles must be turned off.
It is very desirable to cut all of the teeth in one pass if at all possible. If the teeth have to be cut in more than one pass then it is best to cut all of them to one depth with the table locked. Then the process is repeated for a new depth. To do this the axis is unlocked and then locked again for the new depth. This continues till the right depth is reached. This reduces the possibility of confusion or error. For each tooth, whilst it is being cut, the dividing head spindle should be locked.
Backlash in the dividing head when making spur gears
The backlash on most dividing head can be adjusted. Very often it cannot be completely got rid of without making it difficult to turn. When making a spur gear the cutting forces are neutral, they do not tend to move the workpiece one way or the other. But the workpiece might be free to rotate slightly.
Ideally, every time the dividing head is rotated, before cutting the workpiece should be pushed against it. The shaft of it should then be locked. Then the cutting is done. When this has finished the milling table should be moved so the cutter is in the starting position. Notice that doing this the cutter will move through the cut without any effect on it.. Then the spindle can be unlocked.
This is quite different to the process used when making a helical gear.
The end of cutting a gear
The job starts with cutting the first tooth, i.e., the first gap. If there are n teeth to be cut, then when the last tooth, i.e., the last gap, is cut the gear wheel will not have done a complete circle. At this point it is useful to check everything by turning just one more tooth position. If everything has gone to plan then the indexing pin should be back on the zero position on the dividing plate. And, of course, this tooth, i.e. gap, should have already been cut and the cutter should fit perfectly into this gap.
If it does not it could be that:
A the indexing was not done properly,
B the workpiece slipped in the chuck,
C the milling table was not locked where ever possible,
D backlash in the dividing head was not always taken up properly
Or, of course, any one of many other reasons.
If the workpiece is mounted on a mandrel it should be removed from it using an arbor press.