Help with bevel gear calculations

[ZanParticipant @zan]

I am struggling with bevel gear calculations for the od of the pinion ( and wheel). It’s the first gears I have cut for over 50 years and to coin an expression. “ it’s doing my ed in”. Perhaps those with superior knowledge can help here   I’m showing the calcs for the pinion, when the problem is found here all should be easy to follow up for the wheel the cutters are made to clicksring’s “tools glorious tools” utube   A long but interesting task!

It’s for a governor  of 32 dp  with 16 and 32 teeth.

Inner pitch dia. = n/dp = 16/ 32 = 0.5

Slope  angles are  ( pinion ) 26.34. And  (wheel) 64.26 degrees

For reference I am using Lammas data from ME Oct 1991 p 447 and Laws book p 106

Using each of these I get an od for the pinion of. 0.696  (Law)  0.756  (Lammas). Which is correct!

Law is not fully clear about his calculations for the large end dia., it took a while for me to realise the last number in his calculations is in fact the difference between the PCD and the od  of the small end. That is the addendum x 2 and that he uses the cone length in the first expression.

Addendum is thus 1/dp x sin26.34 x 2.  Or .0.0312 x0.4436 x2.  = 0.0137 (on single tooth),  0.027 on diameter

Inner cone length is the same to both authors at 0.563 and face length of 1/3 = .0.75 total

This gives me

(Law)…..[(cone length)x sin26.34×2 ] + addendum. = (.75 x .4436 x 2)+ .027 = 0.696

 

Lammas does it differently using outer pitch dia.  ( 1.4 x ipd ) = 0.7.     So

Od= opd + (2 x cos 63.24 / dp) =  0.7 + (2x .896 / 32)  = 0.756

Now that’s a big difference for such a small gear.  Both should be the same     Help would be appreciated

[Les RileyParticipant @lesriley75593]

If you message me an email address I can send you a spreadsheet that does it all for you.

It is based on Ivan Law’s book method and saves all the confusion and head-scratching.

You can play about with different DP s and tooth numbers until you get the blank sizes you need to fit into your job.

Les.

[Zan]

On 31 October 2024 at 14:55 Zan Said:

Slope  angles are  ( pinion ) 26.34. And  (wheel) 64.26 degrees

 

Od= opd + (2 x cos 63.24 / dp) =  0.7 + (2x .896 / 32)  = 0.756

What is the total angle of the gears? 64.26 + 26.34 does not add up to 90.0 (that is not necessarily a mistake, but is unusual).

Where does 63.24 come from?

Are both of these simply typos. in your post or something more serious?

I was under the impression that the slope angles of a 1:2 pair at 90 degrees would be arctan(0.5)=26.56 and arctan(2.0)=63.43 but this may be irrelevant to your own construction.

[ZanParticipant @zan]

Ah thanks dc3!  a typo, I got the numbers from Lammas, he shows a 2:1@ 26.34 and 63. 26,but that would not account for the differences on the pinion between Lammas and Law

 

lammas states the angles are for “Pitch  &shaft angles. True bevel gear, parallel depth bevel gear” perhaps your angle is for  true bevel gears?

[Zan]

On Zan Said:

perhaps your angle is for true bevel gears?

The angles come from the two pitch cones. If the shafts are at 90 degrees, the angles must be complementary.

In your reply above, you have provided a third angle. So now we have 64.26, 63.24 and 63.26 to choose from.

Parallal depth is a method of manufacture (cutting) not anything related to calculating dimensions. We had some discussion of this here, and the original (100 year old) parallel depth references are provided:

Uniform Depth Bevel Gear,,,”MODULAR” not “DP”

It seems to me to be dangerous to use numbers from someone else’s work without knowing how they have been derived. Is there any online, downloadable version of the Lammas document so we can all see it and play bevel bingo?

See also:

https://www.daycounter.com/Calculators/Bevel-Gear-Calculator.phtml

https://www.otvinta.com/bevel.html

[JasonBModerator @jasonb]

I’ve often heard it said that Law has an error in his formula. Would need to check but it is something like using Cosine rather than sine. Works OK for his 45deg pair but wrong for anything else.

[JasonB]

On JasonB Said:

I’ve often heard it said that Law has an error in his formula.

Thanks for the reminder. See:

Cutting parallel tooth gears

[DC31k]

On DC31k Said:

On JasonB Said:

I’ve often heard it said that Law has an error in his formula.

I think so too. Also, it doesn’t help for clarity that he uses 20 DP and 20 teeth in the example.

[ZanParticipant @zan]

Problem solved!

after a lot of trawling through links and numerous web searches the conclusion is the faulty calculations in Laws explanations. It seems that his calculations, which without a lot of study is difficult to follow could possibly arise from the simple fact that cos and sin 45 are in fact identical and the calculations as shown do not readily apply to ratios which are not 1:1

Lamas  Me 1991  pp 446 gives a clearer and longer set of calculations.   DK3, yes thanks for pointing out my typos( moral, do not use mobile for long posts] although a difference of only 17 seconds as calculated makes about 1 thou difference and without a lot of faff with setting the dividing head up using my sine bar and slips is a waste of time and effort ( esp with shop made cutters).    However, the Lammas calculations are far too long to show here, but they worked out exactly the same as those from Mr R.S.Minchin from ME   15/11 1964  (is he unwin?]

provide exactly the same numbers as Lammas so I show those below for others to use . He also shows the best diagram I have seen to show the reason for the truncated tooth profile at the big end of a parallel cut bevel gear

Thanks also go to Les Riley, who very promptly sent an .xls  calculator which yielded almost instantly the same numbers as my protracted calculations!  5 mouse clicks and 3 numbers……

Note in stage 8 should read 2 cos y / dp

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