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Bevel gear calculations

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Baldric19/11/2017 18:28:10
149 forum posts
11 photos

I am making a 3" Foden steam wagon and looking at the bevel gears in the differential having difficulty with some of the calculations, getting rather different answers depending on where I look.

The gears are 30 & 10 teeth 1/8" pitch, I am happy that the details from the drawing mean it is 12DP, and for the crown that gives a PD at the small end of 2.5".

​Using Steve Withnell's spreadsheet this says the gear cutter required for both the pinion and crown wheels is No.4 as the equivalent number of teeth is 32. Is this correct or is the calculation in C12 using Sin instead of cos?

Using this book I get the equivalent number of teeth as 94 & 11, a somewhat different answer. This book also says you could use the DP midway along the tooth rather than the small end.

I also have seen someone else building this has worked out the result as 142 & 16, but they used the DP at the outer end.

As my calculations end up with such a small number of teeth has anyone any thoughts on which method of calculation is best to use?

​Also a question about pressure angle, RDG stock cutters for 14.5 degrees, for the spur gears later 20 degrees is specified, the pressure angle for the bevel gears never seems to be mentioned. Is it OK to use 14.5 for both the bevel and spur gears rather than finding a sensibly priced UK supplier for 20 degree ones?

Thanks in advance.


Andrew Johnston19/11/2017 20:43:50
5173 forum posts
599 photos

To set the scene there are two basic ways of cutting bevel gears on a manual mill. Normal straight tooth bevel gears are designed using the DP at the outer face of the gear. Since the tooth form decreases towards the centre of the gear a normal involute cutter of the correct tooth form will be too wide to go through the inner tooth space. Special involute gear cutters, stamped "BEVEL", could be used. These had the correct tooth form for the DP at the outer edge, but were thin enough to go through the small end. These cutters do not seem to be available now.

The other method is the parallel depth form. These are designed using the DP at the inner face of the gear, and hence standard involute cutters can be used.

From what you say I assume that the differential bevels are parallel depth. I agree with the PCD calculation for the crown wheel. For 'proper' bevel gears the equivalent number of teeth used to determine the cutter number is the actual number of teeth divided by the cosine of the pitch cone angle. Thus we get the equivalent number of teeth as 94.8 for the crown wheel and 10.5 for the pinion.

However, the calculation for parallel depth gears seems to use a different method, as described in the book you link to. However, I get the same answers as before. The key is that the angle used for the parallel depth method is not the pitch cone angle, but 90° minus the said angle. I would ignore the reference to using the midpoint of the face width. Parallel depth bevels are an approximation, so I see no need to fine tune the cutter number.

It is perfectly acceptable to use a pressure angle of 14½° for all the gears. In all probability that is what the prototype would have used, as it is the older standard.

NB: I found the book linked to so useful I bought a paper copy. I first used it 40+ years ago to make some bevel gears using the BEVEL style involute cutters.


Baldric20/11/2017 12:16:22
149 forum posts
11 photos


​I should have mentioned that although the drawing does not show parallel depth gears that is what I will be using.
​Thank you for confirming my calculations, getting different answers form different places started to make me think I was doing something wrong.


Neil Wyatt20/11/2017 14:23:01
17304 forum posts
690 photos
77 articles
Posted by Baldric on 19/11/2017 18:28:10:

Using this book I get the equivalent number of teeth as 94 & 11, a somewhat different answer. This book also says you could use the DP midway along the tooth rather than the small end.

I also have seen someone else building this has worked out the result as 142 & 16, but they used the DP at the outer end.

There's not right or wrong DP, just a range of potentially suitable sizes 94:11 is very close to 142:16

As Andrew points out the DP also depends on which end of the tooth your method uses for the calculation.

I suggest you:

  • Decide the ratio you want. Set our your bevels as plain cones.
  • Choose sensible numbers of teeth to give a reasonable DP and approximate your ratio.
  • Recalculate based on using that DP.
Andrew Johnston20/11/2017 20:17:11
5173 forum posts
599 photos

There seems to be some confusion. The numbers 94/11 and 142/16 being bandied about are the equivalent tooth count, so that an appropriate involute cutter can be selected. They're not, in themselves, part of the design of the gears. They are only part of the practical issue of machining the gears. They are not related to the DP of the gears in any way, but to the ratios of the real number of teeth on the crown wheel and pinion.

The numbers are based on the diameter of the equivalent spur gear which is in the plane in which the tooth shape is defined. The important point to note is that this radius is not the same as the pitch circle diameter of the gears, but is an imaginary circle. The same idea is used for helical gears where the tooth shape is defined by the radius of the ellipse in the plane in which the tooth is defined.


Baldric21/11/2017 11:23:05
149 forum posts
11 photos


I didn't want to completely redesign the gears as thy fit within a cast part and as the model has been built by others the basic design should work, so in this came need to confirm the cutters to use.

From what I have read I think I have the correct calculations taking in to account the fact that this design of gear is an approximation for the average model engineer. I just need to get on and make them, first finish the arduino controlled rotary table to make my life easier.


Baldric01/07/2018 11:45:19
149 forum posts
11 photos

Just to provide results, I cut the crown wheels using a No.2 cutter and the pinions with a No.8, the depth was 0.180", the offset for the 2nd & 3rd cuts was 0.0625", the end results seem to work.

During the cutting I did have an issue with a crown wheel that meant I had to start again, I think the job moved on the table, so I made an adapter to hold my 4-jaw on the rotary table, this seemed far better. The attached photo shows this, along with use of an old drinks bottle held in place with a jubilee clip to contain the coolant.

2018-06-24 11.23.29.jpg

2018-06-29 18.16.07.jpg
2018-06-03 16.14.08.jpg

Andrew Johnston01/07/2018 11:52:29
5173 forum posts
599 photos

Looking good! thumbs up


Norman Rogers03/07/2018 11:20:48
17 forum posts
2 photos

Just picking up on one of Andrew's comments earlier in the thread, C R Tools in Sheffield (usual disclaimer) still list BEVEL gear cutters and so I bought those required for cutting my own differential gears.

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