Does the number of teeth of a chainring need to be prime?

[Skippy]

"Marz" <[email protected]> wrote in message
news:[email protected]...
>
> In answer to your question,"Does the number of teeth of a chainring
> need to be prime?"
>
>
> Only if you're going Turing.
>
>
>
> Sorry, couldn't help myself.
>
>
> Laters,
>
>
> Marz
>

Ouch! If that's the case, do Usenet posters pass 'the test'?

Skippy
E&OE & Apologies

 

[Michael Press]

In article
<[email protected]>,
[email protected] wrote:

> Michael Press wrote:
>
> > Nowhere did you apply the Chinese Remainder Theorem.
> >
> > For each chain roller to reach each cog it is sufficient
> > that the gcd(m,n) = 1, where m and n are the number of
> > cogs on each gear wheel.
> >
> > --
> > Michael Press
>
> Well, I did apply the CRT. I applied it to m and k where where k is the
> number of links in the chain, and m is the number of teeth of the
> chainring. This does not take shifting between chainrings into
> account.
>
> What you mention is helpful, because it shows the effect of shifting
> between chainrings.

The Chinese Remainder Theorem:

If gcd(m_1, m_2) = 1, then the congruences

x = c_1 (mod m_1, x = c_2 (mod m_2)

have a unique solution 0 <= x < m_1 * m_2.

--
Michael Press

 

[John Everett]

On 30 Jun 2006 00:15:58 GMT, [email protected] wrote:


>And why do you care? Besides, in machinery this is often called the
>"hunting tooth" gear box with prime number gear teeth. Ferrari ran
>44t-11t ring and pinion for years with no ill effect other than
>winning many races. Il Comandante Enzo didn't let theory overshadow
>practicality.

I'm sure you meant to say "Il Commendatori". ;-)


jeverett3<AT>earthlink<DOT>net http://home.earthlink.net/~jeverett3

 

[Friday]

Leo Lichtman wrote:
> <[email protected]> wrote: (clip) So 47.5 teeth is fine, showing that
> not only does the number of teeth not have to be prime, it doesn't even have
> to be an integer.
> ^^^^^^^^^^^^^^^^
> I would like to lower the gearing on my bike. I am too lazy and cheap to
> replace the chainring--can I just break off a few teeth? If I save these
> teeth in my saddle pack, would they be usable for emergency repairs on the
> road? ;-)
>
>

Well that makes sense, it'd save taking links out of the chain.

 

[Friday]

[email protected] wrote:
> I have noticed that the sizes of many common chainrings (in number of
> teeth) is a prime number and I have been asking myself why. I have come
> up with a theory and I would appreciate if someone could confirm it.
>
> My theory is based on an application of the Chinese Remainder Theorem
> (CRT). The CRT is a very old theorem. It is not as well known as
> Pythagoras Theorem, but I think it deserves a place up there next to
> it. For info on the CRT look it up on the WEB. I'll simply give the
> "biker's version" of it.
>
> Imagine you select a chain link at random and a chainring tooth at
> random. Obeserve which tooth the selected link meets (i.e., sits on top
> of) as you crank the chainring around. Will it ever meet the selected
> tooth? The CRT provides the answer:
>
> Every link in the chain will meet every tooth on the chainring if and
> only if the number of links in the chain and the number of teeth on the
> chainring are relative prime.
>
> Two numbers are relative prime if there is no number other than 1 that
> divides both. Given that the teeth wear on the chain links and the
> chain links wear on the teeth, by ensuring that the chain and chainring
> sizes (mesured in number of links and number of teeth respetively) are
> relative prime, the wear and tear is distributed evenly accross all the
> links and all the teeth. Imagine that a link would meet the same tooth
> every time it came around (which would happen if the chain size were a
> multiple of the chainring size) and that there is one bad link in the
> chain. Then the tooth that this link keeps meeting would wear
> differently from the other teeth. But if the sizes are relative prime
> this will not happen.
>
> If the chainring size is a prime number, then it becomes much easier to
> ensure that it is relative prime to the chain size. The chain size
> would need to be a multiple of the chainring size for them not to be
> relative prime. So this could be the reason for chosing a size such as
> 53 or 37.
>
> However, it is not always the case that the chainring size is a prime
> number. In compact drives the number of teeth is often an even number,
> and since the chain MUST have an even number of links, the two have a
> common divisor of at least 2. But since these bikes have derailers,
> shifting between the chainrings may move the chain an uneven number of
> links backwards or forwards relative to the chainring, so that over
> time one would still get the even distribution of wear between links
> and teeth. My guess is that for bikes that do not have derailers, the
> size of the chainring matters more.
>
> Similar arguments apply to the rear sprockets as well.
>
> Does this make sense? Can anyone provide some insight on this?
>
> Thanks
>
> JamesV
>

Not only that, but if you put your chain on in a figure 8 pattern you
can make your chain last twice as long because you're using both sides
of it. Why waste one side of the chain when you're paying for both
sides? Why hasn't Sheldon got this fact on his website?

Friday

 

[Sheldon Brown]

Jobst interjected:

> Hold it!
>
> If you are talking about a track bicycle with a fixed gear, forget it.
> The chain gets put on when you take the bicycle out of the car and
> that is random, regardless of number of teeth.

Not necessarily...I have a theory which posits that using even-sized
sprockets and carefully maintaining phase relations will lead to
improved chain/sprocket life.

See: http://sheldonbrown.com/chain-life

This works for all non-derailer systems, wheher fixed, free or internal
geared.

Speaking of fixed-gear drivetrains, some riders who use the drivetrain
instead of a brake fixate on the number of "skid patches" and its effect
on localized tire wear.

See: http://sheldonbrown.com/fixed.html#skid

Sheldon "Sprockets" Brown
+--------------------------------------------------------+
| As far as the laws of mathematics refer to reality, |
| they are not certain; and as far as they are certain, |
| they do not refer to reality. --Albert Einstein |
+--------------------------------------------------------+
Harris Cyclery, West Newton, Massachusetts
Phone 617-244-9772 FAX 617-244-1041
http://harriscyclery.com
Hard-to-find parts shipped Worldwide
http://captainbike.com http://sheldonbrown.com

 

[[email protected]]

"although I recall the sound of delivery trucks"
WW2?
the wailing heard from old brit BW WW era film when the sedan pulls off
iza comment on trans oil going to the war with civilians running on
thickend vegtable oil.
yes, no?

 

[[email protected]]

Although all of the rollers, and the inner plate projections that
locate the rollers wear, the wear is basically the same for all of the
rollers, so the center-to-center distance doesn't change on the rollers
connected by inner plates.
> +--------------------------------------------------------+
> | As far as the laws of mathematics refer to reality, |
> | they are not certain; and as far as they are certain, |
> | they do not refer to reality. --Albert Einstein |
> +--------------------------------------------------------+
guess ya gotta see it in person

 

[Luke]

In article <[email protected]>, Sheldon Brown
<[email protected]> wrote:

> Jobst interjected:
>
> > Hold it!
> >
> > If you are talking about a track bicycle with a fixed gear, forget it.
> > The chain gets put on when you take the bicycle out of the car and
> > that is random, regardless of number of teeth.
>
> Not necessarily...I have a theory which posits that using even-sized
> sprockets and carefully maintaining phase relations will lead to
> improved chain/sprocket life.
>
> See: http://sheldonbrown.com/chain-life
>
> This works for all non-derailer systems, wheher fixed, free or internal
> geared.
>
> Speaking of fixed-gear drivetrains, some riders who use the drivetrain
> instead of a brake fixate on the number of "skid patches" and its effect
> on localized tire wear.
>
> See: http://sheldonbrown.com/fixed.html#skid
>
> Sheldon "Sprockets" Brown

Sheldon, please comment on how your theory would mesh(!) with a FS/SS
setup utilizing, let's say, a 48t or 32t chainring and 16t cog. In this
situation the cog, along with being sync'ed with the chainring, will
always remain in the same position vis the downward (max-power) arc of
the pedalling cycle. Would this not contribute to uneven wear -- even
wear being preferable, no? -- of the drivetrain? Based upon that
supposition, I was given to believe that evenly divisible cog/chainring
combinations were to be avoided.

Luke

 

[Sheldon Brown]

>>I have a theory which posits that using even-sized
>>sprockets and carefully maintaining phase relations will lead to
>>improved chain/sprocket life.
>>
>>See: http://sheldonbrown.com/chain-life
>>
>>This works for all non-derailer systems, wheher fixed, free or internal
>>geared.
>>
>>Speaking of fixed-gear drivetrains, some riders who use the drivetrain
>>instead of a brake fixate on the number of "skid patches" and its effect
>>on localized tire wear.
>>
>>See: http://sheldonbrown.com/fixed.html#skid
>>
>>Sheldon "Sprockets" Brown
>
"Luke" asked:
>
> Sheldon, please comment on how your theory would mesh(!) with a FS/SS
> setup utilizing, let's say, a 48t or 32t chainring and 16t cog. In this
> situation the cog, along with being sync'ed with the chainring, will
> always remain in the same position vis the downward (max-power) arc of
> the pedalling cycle. Would this not contribute to uneven wear -- even
> wear being preferable, no? -- of the drivetrain? Based upon that
> supposition, I was given to believe that evenly divisible cog/chainring
> combinations were to be avoided.

I guess that's theoretically possible if you never remove the rear
wheel. If you were really worried about it, you could occasionally
remove the wheel and rotate the freewheel with respect to the chain.

I find it hard to believe this could be a real problem in practice.

I am inclined to believe that chain elongation precedes sprocket tooth
damage, and the phase relationshiop between the chain and rear sprocket
is unlikely to be constant.

I've worked on lots of tandems. If this were a problem I would expect
to see it on tandem synch chains, which are 1:1 gearing...but I have
never noticed any visible difference in tooth wear at different parts of
a tandem synch chainring.

If you were to use one of those combinations, uneven tire wear would be
more likely to be noticeable than uneven sprocket wear, in my opinion.

Sheldon "Sounds Like Overthinking Imaginary Problems To Me" Brown
+-------------------------------------------+
| It's easier to be original and foolish |
| than original and wise. |
| --Gottfried Wilhelm Leibniz |
+-------------------------------------------+
Harris Cyclery, West Newton, Massachusetts
Phone 617-244-9772 FAX 617-244-1041
http://harriscyclery.com
Hard-to-find parts shipped Worldwide
http://captainbike.com http://sheldonbrown.com

 

[Martyn Aldis]

In message <[email protected]>, Sheldon Brown
<[email protected]> writes
>Jobst interjected:
>
>> Hold it!
>> If you are talking about a track bicycle with a fixed gear, forget
>>it.
>> The chain gets put on when you take the bicycle out of the car and
>> that is random, regardless of number of teeth.
>
>Not necessarily...I have a theory which posits that using even-sized
>sprockets and carefully maintaining phase relations will lead to
>improved chain/sprocket life.
>
>See: http://sheldonbrown.com/chain-life
>

In a later post Sheldon wrote:

>I am inclined to believe that chain elongation precedes sprocket tooth
>damage, and the phase relationshiop between the chain and rear sprocket
>is unlikely to be constant.

This is where I'm not clear about the chain-life proposal as laid out in
the linked page. Does it suggest leaving one chain on until it's super
worn taking the chain wheel out with it?

I'm giving it a try on my hub gear bike, but not leaving a chain on
until it's fully worn but taking it off at about 1/32" per foot. I'm now
on my fourth chain on this system. When I get to 5 or 6 chains, I plan
to run through the set again taking another 1/32" step and so on. This
suits my bike as it means I don't have to move the wheel back far during
each chain's stint, and can stick with one brake block location for the
set of chains.
--

Martyn Aldis, e-mail [email protected]
==============================================================================

 

[[email protected]]

but I have
never noticed any visible difference in tooth wear at different parts
of
a tandem synch chainring.

no, but obviously that wear factor IS there but obliterated by the
other and major wear factors - amung them ignorance and sloth

but really has the all inner plates wear idea escaped me while i
consdier all rollers going elliptiucal at the same rate and
configuration?

interesting but compared to straight drive line desing numbers
realative to local and personal riding environment - not effectual in
common practice keeping the driveline clean

 

[[email protected]]

Sheldon Brown writes:

>>> I have a theory which posits that using even-sized sprockets and
>>> carefully maintaining phase relations will lead to improved
>>> chain/sprocket life.

>>> See: http://sheldonbrown.com/chain-life

>>> This works for all non-derailer systems, whether fixed, free or
>>> internal geared.

>>> Speaking of fixed-gear drive trains, some riders who use the
>>> drivetrain instead of a brake fixate on the number of "skid
>>> patches" and its effect on localized tire wear.

>>> See: http://sheldonbrown.com/fixed.html#skid

>>> Sheldon "Sprockets" Brown

> "Luke" asked:

>> Sheldon, please comment on how your theory would mesh(!) with a
>> FS/SS setup utilizing, let's say, a 48t or 32t chainring and 16t
>> cog. In this situation the cog, along with being sync'ed with the
>> chainring, will always remain in the same position vis the downward
>> (max-power) arc of the pedalling cycle. Would this not contribute
>> to uneven wear -- even wear being preferable, no? -- of the
>> drivetrain? Based upon that supposition, I was given to believe
>> that evenly divisible cog/chainring combinations were to be
>> avoided.

> I guess that's theoretically possible if you never remove the rear
> wheel. If you were really worried about it, you could occasionally
> remove the wheel and rotate the freewheel with respect to the chain.

> I find it hard to believe this could be a real problem in practice.

> I am inclined to believe that chain elongation precedes sprocket
> tooth damage, and the phase relationship between the chain and rear
> sprocket is unlikely to be constant.

> I've worked on lots of tandems. If this were a problem I would
> expect to see it on tandem synch chains, which are 1:1 gearing...but
> I have never noticed any visible difference in tooth wear at
> different parts of a tandem synch chainring.

Hold the phone! Chain wear is caused by chain articulation about the
link pins. The larger the sprocket the smaller that articulation is.
In fact it approaches zero with say a 52t - 52t transfer chain ratio
where the angle of chain articulation is 1/4 of that of a 13t
sprocket. Not flexing about a pair of zig-zags in a derailleur also
prevents adjacent grit in the chain link from entering the interface.
These chains and sprockets are noted for their low wear.

> If you were to use one of those combinations, uneven tire wear would be
> more likely to be noticeable than uneven sprocket wear, in my opinion.

Not at all, because the multiples would distribute wear over enough
long areas that you would not see it. Consider that in most commonly
used gears the rear wheel rotates three or more times for one crank
revolution.

> Sheldon "Sounds Like Overthinking Imaginary Problems To Me" Brown

That is a highly precise assessment of this whole thread. I concur.

Jobst Brandt

 

[Sheldon Brown]

I wrote:

..I have a theory which posits that using even-sized
>> sprockets and carefully maintaining phase relations will lead to
>> improved chain/sprocket life.
>>
>> See: http://sheldonbrown.com/chain-life
>>
>> I am inclined to believe that chain elongation precedes sprocket tooth
>> damage, and the phase relationshiop between the chain and rear
>> sprocket is unlikely to be constant.
>
Martyn Aldis wrote:
>
> This is where I'm not clear about the chain-life proposal as laid out in
> the linked page. Does it suggest leaving one chain on until it's super
> worn taking the chain wheel out with it?

Right. It's my belief that this approach would allow a single chain to
outlast a large number of chains replaced in the normal manner.

Sheldon "All Together Now..." Brown
+----------------------------------------+
| I never did a day's work in my life; |
| it was all fun. --Thomas Edison |
+----------------------------------------+
Harris Cyclery, West Newton, Massachusetts
Phone 617-244-9772 FAX 617-244-1041
http://harriscyclery.com
Hard-to-find parts shipped Worldwide
http://captainbike.com http://sheldonbrown.com

 

[John Thompson]

On 2006-06-30, [email protected] <[email protected]> wrote:

> Hold it!
>
> If you are talking about a track bicycle with a fixed gear, forget it.
> The chain gets put on when you take the bicycle out of the car and
> that is random, regardless of number of teeth.

But what if, like me, you ride your fixed gear bike on the road without
removing the rear wheel? I just hang it in the garage, and when I want
to go for a ride, I take it down and hop on! I noticed when I had an
even number of teeth on both the rear cog and chainring, only half the
chainring teeth would get dirty, presumably because this is where the
narrow part of each link pair lands, and by extension these teeth suffer
the most wear and tear. By using an odd number of teeth (either front or
rear) this was prevented and the wear spread over all the teeth.

--

John ([email protected])

 

[[email protected]]

On Sun, 09 Jul 2006 23:05:35 GMT, John Thompson
<[email protected]> wrote:

>On 2006-06-30, [email protected] <[email protected]> wrote:
>
>> Hold it!
>>
>> If you are talking about a track bicycle with a fixed gear, forget it.
>> The chain gets put on when you take the bicycle out of the car and
>> that is random, regardless of number of teeth.
>
>But what if, like me, you ride your fixed gear bike on the road without
>removing the rear wheel? I just hang it in the garage, and when I want
>to go for a ride, I take it down and hop on! I noticed when I had an
>even number of teeth on both the rear cog and chainring, only half the
>chainring teeth would get dirty, presumably because this is where the
>narrow part of each link pair lands, and by extension these teeth suffer
>the most wear and tear. By using an odd number of teeth (either front or
>rear) this was prevented and the wear spread over all the teeth.

Dear John,

A nice point!

Here's another solution:

http://www.sheldonbrown.com/harris/chains.html

Page down to the "New! 3/32" Half Links $3.95":

http://www.sheldonbrown.com/harris/images/half-link.jpg

For fatter chains, Sheldon also sells what he demurely calls "Offset
Links," but the price increase reveals what the lack of a picture
tries to hide:

http://www.sheldonbrown.com/harris/chains-wide.html

Page down to "Offset Links $4.95" for details on these porkers. A
greedy bike shop would demand 33% more for a half-link 33% wider, but
Sheldon increases the price only 25%.

I still carry a motorcycle half-link with cotter pin, more out
superstition and nostalgia than any real need.

Cheers,

Carl Fogel

 

[John Thompson]

On 2006-07-06, Sheldon Brown <[email protected]> wrote:

> Right. It's my belief that this approach would allow a single chain to
> outlast a large number of chains replaced in the normal manner.

But does it not also mean that if/when you *DO* need to replace a
drivetrain component, you will need to replace *ALL* the drivetrain
components (chain, ring, cog)? Might be an issue if you're running any
hard-to-replace parts (e.g. Campy 151mm BCD rings).

--

John ([email protected])

 

[Sheldon Brown]

>>Right. It's my belief that this approach would allow a single chain to
>>outlast a large number of chains replaced in the normal manner.

http://sheldonbrown.com/chain-life.html
>
John Thompson wrote:
>
> But does it not also mean that if/when you *DO* need to replace a
> drivetrain component, you will need to replace *ALL* the drivetrain
> components (chain, ring, cog)? Might be an issue if you're running any
> hard-to-replace parts (e.g. Campy 151mm BCD rings).

I believe this system is particularly well suited for hard-to-replace
parts, because it should make the parts last much longer.

It's actually very unusual to wear out chainrings on a non-derailer
system anyway, typically takes many tens of thousands of miles.

My Campy 151 chainring is running 1" block chain anyway...26/9.

Sheldon "Life Extension" Brown
+------------------------------------------+
| Athletic scholarships are a corrupting |
| cancer on the U.S. educational system. |
+------------------------------------------+
Harris Cyclery, West Newton, Massachusetts
Phone 617-244-9772 FAX 617-244-1041
http://harriscyclery.com
Hard-to-find parts shipped Worldwide
http://captainbike.com http://sheldonbrown.com

 

[[email protected]]

1/32?? holy cow!
if yawl let the chain wear wayout does not the chances for each unit
becoming too worn to engage properly go to an unacceptable level?
never stretch your luck and budget when replacing a chain - al the grate

 

[David Damerell]

Quoting Sheldon Brown <[email protected]>:
>I've worked on lots of tandems. If this were a problem I would expect
>to see it on tandem synch chains, which are 1:1 gearing...but I have
>never noticed any visible difference in tooth wear at different parts of
>a tandem synch chainring.

Do tandem sync chains wear significantly at all? I wouldn't be surprised
if the one on ours was bought by the original owner in the 80s...
--
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