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I am writing a program to calculate chain lengths required for a particular sprocket pair when I noticed this Consider a 9 teeth sprocket paired with an identical 9 teeth sprocket (image reference) the pitch circle is green and the initial position of the chain (used to calculate the chain length and number of links) is shown in BLUE. The total length is 285.75mm which comes out to be 18 link chain for 15.875mm chain pitch.

Now I rotated the pitch polygon by 10 degrees (0.25 of the angle enclosed by 1 link which is 360/9 = 40deg) this is shown in orange. Then I drew the taut chain again (orange lines). Now if I measure the chain length, it is lesser than the calculated length before (by about 0.3mm)!

Why is this happening? Am I doing something wrong? This also means that if I had adjusted my centre distance to make the chain links a whole number in the second configuration, then as it rotates to the first configuration, the chain length increases..but we assume that the chain is inextensible Added another drawing for reference (pardon the messy nature). The green curves are the sprocket and chain in initial position, the magenta curves are the rotated configuration. As we can see, the pink line between the 2 sprockets cannot accommodate the 5 chain pitches and is interfering with the RH sprocket teeth. Practically, the chain slackness will take care of this change in length. My question is about the theoretical aspect. Is it impossible mathematically to wrap such a chain around sprockets without encountering any changes in link length during rotation?? Reference Image enter image description here

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You are not doing anything wrong.

If the chain is transmitting power, one side will be in tension and the other side will be slack, and the slack side can easily take up the changes in length.

Chains also "stretch" over time as the connections between the links wear.

If you don't want the chain to be flopping about on the slack side, you use a chain tensioner.

Of course if you have sprockets with more teeth, the size of the "problem" reduces.

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  • $\begingroup$ Practically yes, the slackness will take care of that. But is it mathematically impossible to wind a taut chain across such a sprocket arrangement assuming that the chain links are rigid? Is it just not possible without encountering changes in link lengths?? $\endgroup$ – Sherlock_Holmes Aug 5 '19 at 11:56
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A chain is designed to sit on the root between the teeth at all times while it is in contact with the sprocket, so it will not get lower during rotation as you assume in your drawing.

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  • $\begingroup$ But I have not assumed otherwise.. the pitch circle is not changing. so if the link is in contact, it will be sitting in the root only. No link in contact is sitting outside the root. The chain line went up because the left sprocket teeth rotated down while right sprocket teeth rotated up right? $\endgroup$ – Sherlock_Holmes Aug 5 '19 at 10:40

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