M
Michael Press
Guest
This is a geometric discussion of how much the distance
between the chainwheel and the rear cogwheel changes as
the length of the chain is altered. The effects of
variations of chain pitch, worn cogs, deviations from
circularity, and deviations from concentricity are not
considered.
We know that changing the chain length by 1 inch
changes the distance between the chain wheel and the
rear cogwheel by about 1/2 inch. Is the change a bit
more or a bit less than 1/2 inch?
First some notation.
u distance between the centers of the cogwheels.
R radius of the chain wheel
r radius of the rear cog
l length of the chain.
p pitch of the chain.
If n gives the number of cogs then 2 * pi * r = n * p.
First we find l, given R, r, and u. Let O be the center
of the chain wheel, and Q be the center of the rear
cogwheel. Let the upper external tangent to the
cogwheels meet the chain wheel at A and the rear
cogwheel at B. AB is perpendicular to OA and to QB.
Draw a line parallel to AB through Q meeting OA at P.
Then OPQ is a right angle. Define a = |OP| = R - r. We
have
a^2 + |QP|^2 = u^2.
We see that |QP| = |AB| is the length of the straight
portions of the chain.
Next sin (angle PQO) = a/u. Define h = angle PQO.
Considering the run of chain above the line of centers
OQ, the chain where it is in contact with the chain
wheel subtends angle pi/2 + h, and where it is in
contact with the rear cogweel subtends an angle of
pi/2 - h. Therefore
l/2 = sqrt(u^2 - a^2) + R * ( pi/2 + h) + r * ( pi/2 - h)
= sqrt(u^2 - a^2) + a * arcsin a/u + (R + r) * pi/2. (*)
Differentiating and simplifying we get
dl/2 = u du / sqrt(u^2 - a^2) - (a^2 / x^2) * sqrt(1 - a^2 / u^2) du
= 1/sqrt(1 - a^2 / u^2) du - (a^2 / x^2) / sqrt(1 - a^2 / u^2) du
= sqrt(1 - a^2 / u^2) du .
So finally
du = 1/2 * 1 / sqrt(1 - a^2 / u^2) dl.
The factor 1 / sqrt(1 - a^2 / u^2) is greater than one.
Consider a 42/16 gearing where u = 16 inch.
a = R - r = 26 / (4 * pi) = 2.070
1/2 * 1 / sqrt(1 - a^2 / u^2) = 0.5042.
Removing one link shortens the distance between the
chain wheel and the rear cogwheel by approximately
0.5042 inch.
Equation (*) cannot be solved for u in terms of the
other variables. A non-linear solver code can be used
to get numerical solutions.
Here are results from such a computation for a 42/16
gearing. Links gives the number of rollers in the
chain; diff is the difference in the u values.
k = 1/2 * 1 / sqrt(1 - a^2 / u^2), the derivative from above.
index links u diff k diff - k
0 76 11.56
1 78 12.07 0.50785 0.50751 3.37601e-04
2 80 12.58 0.50720 0.50690 2.96363e-04
3 82 13.09 0.50663 0.50637 2.61664e-04
4 84 13.59 0.50613 0.50590 2.32243e-04
5 86 14.10 0.50568 0.50547 2.07122e-04
6 88 14.60 0.50528 0.50510 1.85536e-04
7 90 15.11 0.50492 0.50476 1.66878e-04
8 92 15.61 0.50460 0.50445 1.50663e-04
9 94 16.12 0.50431 0.50417 1.36502e-04
10 96 16.62 0.50404 0.50392 1.24075e-04
11 98 17.12 0.50380 0.50369 1.13125e-04
12 100 17.63 0.50358 0.50348 1.03435e-04
13 102 18.13 0.50338 0.50329 9.48294e-05
14 104 18.64 0.50320 0.50311 8.71587e-05
15 106 19.14 0.50303 0.50295 8.02987e-05
16 108 19.64 0.50287 0.50280 7.41445e-05
17 110 20.14 0.50273 0.50266 6.86072e-05
18 112 20.65 0.50259 0.50253 6.36108e-05
19 114 21.15 0.50247 0.50241 5.90905e-05
20 116 21.65 0.50235 0.50230 5.49908e-05
21 118 22.15 0.50225 0.50220 5.12635e-05
22 120 22.66 0.50215 0.50210 4.78672e-05
23 122 23.16 0.50205 0.50201 4.47657e-05
24 124 23.66 0.50196 0.50192 4.19276e-05
--
Michael Press
between the chainwheel and the rear cogwheel changes as
the length of the chain is altered. The effects of
variations of chain pitch, worn cogs, deviations from
circularity, and deviations from concentricity are not
considered.
We know that changing the chain length by 1 inch
changes the distance between the chain wheel and the
rear cogwheel by about 1/2 inch. Is the change a bit
more or a bit less than 1/2 inch?
First some notation.
u distance between the centers of the cogwheels.
R radius of the chain wheel
r radius of the rear cog
l length of the chain.
p pitch of the chain.
If n gives the number of cogs then 2 * pi * r = n * p.
First we find l, given R, r, and u. Let O be the center
of the chain wheel, and Q be the center of the rear
cogwheel. Let the upper external tangent to the
cogwheels meet the chain wheel at A and the rear
cogwheel at B. AB is perpendicular to OA and to QB.
Draw a line parallel to AB through Q meeting OA at P.
Then OPQ is a right angle. Define a = |OP| = R - r. We
have
a^2 + |QP|^2 = u^2.
We see that |QP| = |AB| is the length of the straight
portions of the chain.
Next sin (angle PQO) = a/u. Define h = angle PQO.
Considering the run of chain above the line of centers
OQ, the chain where it is in contact with the chain
wheel subtends angle pi/2 + h, and where it is in
contact with the rear cogweel subtends an angle of
pi/2 - h. Therefore
l/2 = sqrt(u^2 - a^2) + R * ( pi/2 + h) + r * ( pi/2 - h)
= sqrt(u^2 - a^2) + a * arcsin a/u + (R + r) * pi/2. (*)
Differentiating and simplifying we get
dl/2 = u du / sqrt(u^2 - a^2) - (a^2 / x^2) * sqrt(1 - a^2 / u^2) du
= 1/sqrt(1 - a^2 / u^2) du - (a^2 / x^2) / sqrt(1 - a^2 / u^2) du
= sqrt(1 - a^2 / u^2) du .
So finally
du = 1/2 * 1 / sqrt(1 - a^2 / u^2) dl.
The factor 1 / sqrt(1 - a^2 / u^2) is greater than one.
Consider a 42/16 gearing where u = 16 inch.
a = R - r = 26 / (4 * pi) = 2.070
1/2 * 1 / sqrt(1 - a^2 / u^2) = 0.5042.
Removing one link shortens the distance between the
chain wheel and the rear cogwheel by approximately
0.5042 inch.
Equation (*) cannot be solved for u in terms of the
other variables. A non-linear solver code can be used
to get numerical solutions.
Here are results from such a computation for a 42/16
gearing. Links gives the number of rollers in the
chain; diff is the difference in the u values.
k = 1/2 * 1 / sqrt(1 - a^2 / u^2), the derivative from above.
index links u diff k diff - k
0 76 11.56
1 78 12.07 0.50785 0.50751 3.37601e-04
2 80 12.58 0.50720 0.50690 2.96363e-04
3 82 13.09 0.50663 0.50637 2.61664e-04
4 84 13.59 0.50613 0.50590 2.32243e-04
5 86 14.10 0.50568 0.50547 2.07122e-04
6 88 14.60 0.50528 0.50510 1.85536e-04
7 90 15.11 0.50492 0.50476 1.66878e-04
8 92 15.61 0.50460 0.50445 1.50663e-04
9 94 16.12 0.50431 0.50417 1.36502e-04
10 96 16.62 0.50404 0.50392 1.24075e-04
11 98 17.12 0.50380 0.50369 1.13125e-04
12 100 17.63 0.50358 0.50348 1.03435e-04
13 102 18.13 0.50338 0.50329 9.48294e-05
14 104 18.64 0.50320 0.50311 8.71587e-05
15 106 19.14 0.50303 0.50295 8.02987e-05
16 108 19.64 0.50287 0.50280 7.41445e-05
17 110 20.14 0.50273 0.50266 6.86072e-05
18 112 20.65 0.50259 0.50253 6.36108e-05
19 114 21.15 0.50247 0.50241 5.90905e-05
20 116 21.65 0.50235 0.50230 5.49908e-05
21 118 22.15 0.50225 0.50220 5.12635e-05
22 120 22.66 0.50215 0.50210 4.78672e-05
23 122 23.16 0.50205 0.50201 4.47657e-05
24 124 23.66 0.50196 0.50192 4.19276e-05
--
Michael Press
