When Pi doesn't matter...

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Manc33
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When Pi doesn't matter...

Postby Manc33 » 21 Sep 2019, 11:49pm

If you take the diameter of a circle and multiply it by 3.14159 it gives the circumference.

What if you count the teeth on a cassette sprocket and divide that by 3.14159... you'll get the diameter, right?

Wrong!

You need to divide the teeth by about 4.1

Examples:

[mm/teeth]

45.3 / 11 = 4.118181818181818
49.6 / 12 = 4.133333333333333
53.3 / 13 = 4.1
57.5 / 14 = 4.107142857142857
61.5 / 15 = 4.1
73.6 / 18 = 4.088888888888889
86.3 / 21 = 4.10952380952381
97.7 / 24 = 4.070833333333333
113.9 /28 = 4.067857142857143

That's ruddy strange but it all checks out. The average of the above is 4.09952902 so let's just call it 4.1

This means you can work out how much the upper jockey wheel will need to go towards (or away from) the cassette, if going from one low sprocket to another.

I worked out a 46t sprocket is about 12mm bigger (all around it) than a 40t. (diameter is 24.6mm difference)

Sorry to bore you with it :lol:

Why is this not working out to 3.14159? It's not like it's a small difference, needing to divide teeth by 4.1 :shock:

I guess it's to do with the teeth having spaces between them and them being counted, as opposed to it being a continuous circumference measurement. :oops:

You know now anyway it's 4.1 :P
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NATURAL ANKLING
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Re: When Pi doesn't matter...

Postby NATURAL ANKLING » 22 Sep 2019, 1:23am

Hi,
But you are using a unitless number = teeth.

Try using the sprocket PCD.
http://www.tsubaki.ca/pdf/E206-236_Spro ... ection.pdf
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Tigerbiten
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Re: When Pi doesn't matter...

Postby Tigerbiten » 22 Sep 2019, 3:19am

It's because you're mixing metric and imperial units.
The tooth pitch is 1/2 inch and you want the diameter in mm.
So you need first need to convert inches to mm before dividing by pi.
1 tooth x0.5x25.4/3.14= 4.045 mm.
2 teeth x0.5x25.4/3.14= 8.090 mm.
.............
That gives you an increase in diameter of 4.045 mm per tooth.

Or ...
Tooth count divided by pi gives you the radius in inches.

Easy ........ :D

Bez
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Re: When Pi doesn't matter...

Postby Bez » 22 Sep 2019, 10:56am

Let's take one of your examples, the 24t sprocket.

97.7mm is about 3.85".

24t is 24 1/2" links, so the circumference is (approximately*) 12".

12" divided by 3.85" is…

;)

* NB for bonus detail, you need to measure the pitch circle diameter (PCD), and that's not quite what you might expect it to be because it needs to join the vertices of a regular polygon with each side being 1/2", rather than a circle whose circumference is the number of teeth times 1/2". And it's going to be hard to manually measure the PCD.

Manc33
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Re: When Pi doesn't matter...

Postby Manc33 » 22 Sep 2019, 12:13pm

At least I know my vernier calipers work :lol:
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DaveReading
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Re: When Pi doesn't matter...

Postby DaveReading » 22 Sep 2019, 1:13pm

Bez wrote:And it's going to be hard to manually measure the PCD.

True, but you can calculate it for a given sprocket size with some fairly trivial geometry.

Bez
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Re: When Pi doesn't matter...

Postby Bez » 22 Sep 2019, 1:18pm

Yeah, of course, it’s a regular polygon. But what I meant was that whatever the OP’s measured, it’s probably not the PCD.

Manc33
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Re: When Pi doesn't matter...

Postby Manc33 » 25 Sep 2019, 10:51am

Does anyone know what the exact number is for this calculation, I am going to make another calculator for it. I put the figure in as 4.09952902 but it's a guess and it can start to go 1mm out here and there on the diameter of larger sprockets/chainrings.

What's the point of a calc like that?

You might want this answer to work out exactly how much you need to raise or lower a front mech before swapping to a bigger/smaller chainring, or you might want to know how much your b-tension screw needs adjusting on a rear mech if you plan on swapping to a cassette with a larger/smaller low sprocket. It might save the hassle of buying the part to find it cannot be used because your front or rear mech cannot be adjusted enough to suit the new chainring or cassette sprocket.

For example some rear mechs can have the b-tension screw removed completely and still not have the upper jockey wheel close enough to the low sprocket, some can have the b-tension screw turned around to max it out and still not clear the low sprocket properly. Some frames might have low bottle cage bolts that could get in the way if you wanted to put a much bigger chainring on with a derailleur and finally, some frames tubing might start tapering out towards the bottom bracket meaning a small outer might not be viable with the front derailleur needing to be clamped where the tubing is wider than the derailleur clamp.

If it's going to be a close call, such a calc could save the hassle of buying the bike part to find you can't make it work on your setup.

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Last edited by Manc33 on 25 Sep 2019, 11:36am, edited 1 time in total.
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mattheus
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Re: When Pi doesn't matter...

Postby mattheus » 25 Sep 2019, 11:12am

Mr Manc;
Don't forget you can always calculate things like this as ratios
i.e. if you go from a 20T cog to a 30T cog, then you can be certain that the diameter will be 1.5 times as big (i.e. increased by 50%).

This saves you needing to know all the details, and will work whatever the system(e.g. something that isn't a standard modern bicycle cog)

Manc33
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Re: When Pi doesn't matter...

Postby Manc33 » 25 Sep 2019, 11:41am

Of course, I was overthinking it, but the diameter in mm of sprocket A is still going to be needed to get the mm difference when going to sprocket B. It's probably close enough just using teeth *4.1 = diameter. Then to get an increase it's half of that, the above calc is all finished :mrgreen:
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Tigerbiten
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Re: When Pi doesn't matter...

Postby Tigerbiten » 27 Sep 2019, 1:48pm

Manc33 wrote:Does anyone know what the exact number is for this calculation, I am going to make another calculator for it. I put the figure in as 4.09952902 but it's a guess and it can start to go 1mm out here and there on the diameter of larger sprockets/chainrings.

It's ........
"Tooth Count" multiplied by "Tooth Size" divided by Pi = Diameter in inches.
Then just convert inches to mm.
So just use a tooth count of 1 for the increase in diameter per tooth.
1 x 0.5 / 3.14159 x 25.4 = 4.04254 mm per tooth.

Luck ..... :D