I lack about twenty credit hours for a mechanical engineering degree........
I believe a sprocket should have teeth
Anyways.....
I pirated this but I did remember something about this way back in college days.....
About Chain Wear: This is about the impact of sprockets number of teeth on chain / kit wearing.
Hypothesis:
The wear of the chain depends on the frequency the same link hits the same sprocket tooth. The more often the hit, the faster the wear.
Principles:
We must calculate how many chain revolutions are necessary for a given link to hit a given tooth.
A chain link can hit a given sprocket tooth from once per chain revolution (higher wear, a given link always hits the same tooth at each chain revolution) to once per a number of revolution equal to the number of sprocket teeth (lower wear, a given link hits the another tooth at each revolution and it takes "number of teeth" chain revolutions to hit aga the given tooth).
Example: Faster wear - the same links always hits the same tooth at each chain revolution
Chain:123456789
Tooth:123123123 1st rev (tooth 1 hits links 1,4,7 during the fist chain revolution)
Tooth:123123123 2nd rev (tooth 1 hits the same link during the second revolution
Example: Longer wear - it takes 'number of teeth' revolutions to hit again the same tooth
Chain:123456789
Tooth:123412341 1st rev (tooth 1 hits links 1,5,9)
Tooth:234123412 2nd rev (tooth 1 hits links 4,
Tooth:341234123 3rd rev (tooth 1 hits links 3,7)
Tooth:412341234 4th rev (tooth 1 hits links 2,6)
Tooth:123412341 5th rev : equal to 1 rev : it takes 4 rev to hit again the same tooth
Calculus:
The same tooth hits the same link at revolution number R when R x [number of links] is divisible by [number of teeth]
Alternatively, using a spreadsheet, the following will work: Teeth / GCD(Links, Teeth), where GCD is the Greatest Common Denominator
Data:
Links Front Sprocket (teeth) Rear Sprocket (teeth) Comments
14 15 16 17 18 43 44 45 46 47 48 49 50 51 52
108 7 5 4 17 1 43 11 5 23 47 4 49 25 17 13
110 7 3 8 17 9 43 2 9 23 47 24 49 5 51 26 Classic Stock (16/47)
112 1 15 1 17 9 43 11 45 23 47 3 7 25 51 13 GS/Dakar Stock (16/47)
114 7 5 8 17 3 43 22 15 23 47 8 49 25 17 26
116 7 15 4 17 9 43 11 45 23 47 12 49 25 51 13
118 7 15 8 17 9 43 22 45 23 47 24 49 25 51 26
120 7 1 2 17 3 43 11 3 23 47 2 49 5 17 13
Table: Number of revolutions necessary for the same tooth to hit the same link
Color Meaning
Worst Possible Score
Best Possible Score
Conclusions:
The stock 47 is very good scored because it takes 47 chain rev for the same tooth hits the same link (highest theoretic score) - this is regardless of the number of chain links. (compare this to the 4 rev of a 48 or 17 of a 52).
The stock 16 has a score of 8 rev (on the classic) which is not that bad because I think know it's hard to use the 17 (lack ofspace) On the GS though, the 16T has a score of 1 rev (the worst possible!) - you would be better off going to the 15T
Anyway the 16 wears faster than the 47, but that is no surprise because it is much smaller (top theoretic score for the 16 is 8 rev - see detailed observations below).
The 15 scores bad with only 3 rev for the same tooth hits the same link. Comparing with the 16, can we say the wear is more than 2.5 higher? And the kit will last 2.5 less time?
Detailed Observations: Here are some more detailed observations based on the table above...
Even numbers of teeth on sprockets will never achieve an optimum wear - This is due to the even number of teeth in the chain, and the nature of mathematics behind this. Thus a 16 tooth sprocket can only achieve a maximum of 8!
A prime number of teeth will always achieve the optimum wear pattern - see the 17T front and 47T rear sprockets
The 15T on a 112 link chain is an odd case - the remainer of 112 divided by 15 is a prime number, and thus results in an optimum theoretical wear.
A help to make the kit last longer (on a classic) could be to manually shift the chain along 1Tooth every 1000km! On the GS you are probably better off using a 15 or 17T sprocket.
The other thing that always springs ot mind is the number of teeth. Even numbers of teeth have stress patterns across them that let the wheel flex as it turns (gap opposite gap). Odd numbers of teeth avoid this, but can sort of ripple on the way round (divisible by 3, gap opposite tooth on each pair of 3 tooth groups). Prime numbers of teeth avoid even this. Flexing and rippling let the side-plates of the chain contact the teeth with more force, rather than the rollers. 47 is a prime number so BMW got their sums right. 16 is even. 17 would be better.>
Link to the whole artic.
http://faq.f650.com/FAQs/SprocketsOtherSizesFAQ.htm#Section%202:%20Modification%20and%20Theory
