Author Topic: Calculating cog diameter  (Read 420 times)

quixoticgeek

  • Mostly Harmless
Calculating cog diameter
« on: February 23, 2021, 04:42:21 pm »

For the gears used on bikes, be it on a cassette or a chain ring, is there an easy formula for calculating the diameter of the item, for a given tooth count? Trying to work out what I can fit in a given ultrasonic cleaner.

J
--
Beer, bikes, and backpacking
http://b.42q.eu/

citoyen

  • Occasionally rides a bike
Re: Calculating cog diameter
« Reply #1 on: February 23, 2021, 04:52:27 pm »
I don't actually know the answer but I would presume this is probably quite easy for maths boffins, given that the pitch and number of teeth are known variables.

A quick google turns up this:
https://rbracing-rsr.com/calcsprocketdiam.html
"The future's all yours, you lousy bicycles."

Re: Calculating cog diameter
« Reply #2 on: February 23, 2021, 04:53:34 pm »

For the gears used on bikes, be it on a cassette or a chain ring, is there an easy formula for calculating the diameter of the item, for a given tooth count? Trying to work out what I can fit in a given ultrasonic cleaner.

J

A very specific and unusual request, I couldn't help but investigate. The following might be a starter for 10:

http://www.machinehead-software.co.uk/bike/chain_length/sprocket_radius_table.html

Eddington: 129 miles

Re: Calculating cog diameter
« Reply #3 on: February 23, 2021, 04:55:23 pm »
(number of teeth) x 0.4 = approximate diameter in cm.

citoyen

  • Occasionally rides a bike
Re: Calculating cog diameter
« Reply #4 on: February 23, 2021, 04:56:32 pm »
A very specific and unusual request, I couldn't help but investigate. The following might be a starter for 10:

http://www.machinehead-software.co.uk/bike/chain_length/sprocket_radius_table.html

Fwiw, the values in that table seem to correspond to the values generated by the calculator in my link.
"The future's all yours, you lousy bicycles."

Re: Calculating cog diameter
« Reply #5 on: February 23, 2021, 06:45:09 pm »

For the gears used on bikes, be it on a cassette or a chain ring, is there an easy formula for calculating the diameter of the item, for a given tooth count? Trying to work out what I can fit in a given ultrasonic cleaner.

J

Ah, but is the diameter you want the tops of the teeth, the bottom or where the centreline of the chain runs?

Re: Calculating cog diameter
« Reply #6 on: February 23, 2021, 07:21:33 pm »
Measure the diameter of a known chainring or sprocket.
Each extra tooth adds half an inch divided by Pi = 4.04mm

Cudzoziemiec

  • Eating all the pies and drinking all the tea.
Re: Calculating cog diameter
« Reply #7 on: February 23, 2021, 07:25:24 pm »
O level maths alert: Isn't it number of teeth/2 = circumference in inches. Divide by pi to get diameter. That should be at mid point of chain, so add the height of the chain to get overall diameter. Convert into other cm, mm, barleycorns etc as desired.
Faster than a walk, slower than a train, often slightly higher than a person. (David Byrne)

Re: Calculating cog diameter
« Reply #8 on: February 23, 2021, 07:33:10 pm »
A cog is more like a regular polygon than a circle. The diameter of the enclosing circle of a regular polygon with half inch sides is 12.7/sin(pi/n) where n is the number of sides teeth.

Admittedly the difference is only 0.5 mm for 11 tooth and 0.1 mm for a 50 tooth (the polygon method is bigger), which will be less than the differences due to tooth design and wear.

Re: Calculating cog diameter
« Reply #9 on: February 23, 2021, 08:16:57 pm »
A cog is more like a regular polygon than a circle. The diameter of the enclosing circle of a regular polygon with half inch sides is 12.7/sin(pi/n) where n is the number of sides teeth.

Admittedly the difference is only 0.5 mm for 11 tooth and 0.1 mm for a 50 tooth (the polygon method is bigger), which will be less than the differences due to tooth design and wear.

 Being a luddite I don't know how much modern individual models vary but the difference between any post-indexing sprocket and a Sachs-Maillard or a IGH sprocket is fairly consequent. Measuring a known example of your preferred model and doing the maths to multiply up might be a safer way. In the end you are not interested in the diameters of a bunch of sprockets AIUI but only the tooth count of the largest one that is less than the opening of your cleaner. Other way to do it is to sneak into a shop with a suitable measuring instrument and measure them for real (or a cyclo-cross or mtb event perhaps).

Re: Calculating cog diameter
« Reply #10 on: Yesterday at 07:51:13 am »

For the gears used on bikes, be it on a cassette or a chain ring, is there an easy formula for calculating the diameter of the item, for a given tooth count? Trying to work out what I can fit in a given ultrasonic cleaner.

Ah, but is the diameter you want the tops of the teeth, the bottom or where the centreline of the chain runs?

The OP wants to know if a cassette or chain ring will fit in their ultrasonic cleaner.

Without having to cut the teeth off.   :o

Re: Calculating cog diameter
« Reply #11 on: Yesterday at 09:51:12 am »
A cog is more like a regular polygon than a circle. The diameter of the enclosing circle of a regular polygon with half inch sides is 12.7/sin(pi/n) where n is the number of sides teeth.

Admittedly the difference is only 0.5 mm for 11 tooth and 0.1 mm for a 50 tooth (the polygon method is bigger), which will be less than the differences due to tooth design and wear.
....which is significantly less than the 7mm or so width of the chain you have omitted.

4 x (number of teeth + 2) would be a good rule of thumb.

Re: Calculating cog diameter
« Reply #12 on: Yesterday at 11:48:17 am »
.............Without having to cut the teeth off.   :o

 ;D That's an assumed requirement.

Re: Calculating cog diameter
« Reply #13 on: Yesterday at 12:09:42 pm »
....which is significantly less than the 7mm or so width of the chain you have omitted.

The polygon gives you to the centre of the chain pins between the teeth, which is the only thing that's exactly calculable, since the actual length of the teeth is up to the designer.

(there is a calculable maximum length the teeth can be for the chain links to still be able to roll on and off, but since bicycle sprockets are never that pointy I can't be arsed looking it up or deriving it)

Re: Calculating cog diameter
« Reply #14 on: Yesterday at 01:18:59 pm »
....which is significantly less than the 7mm or so width of the chain you have omitted.

The polygon gives you to the centre of the chain pins between the teeth, which is the only thing that's exactly calculable, since the actual length of the teeth is up to the designer.

(there is a calculable maximum length the teeth can be for the chain links to still be able to roll on and off, but since bicycle sprockets are never that pointy I can't be arsed looking it up or deriving it)
I get that, I was just questioning worrying about fractions of a mm difference between a triacontatetragon and circle when you have quantity orders of magnitude larger to estimate.

Re: Calculating cog diameter
« Reply #15 on: Yesterday at 01:36:08 pm »
I hadn't worked out exactly how little difference it'd make until I'd composed my pedantic comment, and I'm not going to waste an opportunity to be right on the internet (for a useless value of right).