Post by Steve FreidesPost by Mower ManWrong. It does. And it's so obvious as to beggar belief. The slack in
the chain is utterly irrelevant, too.
Let us consider what's happening at a point in the rotation of the
chainrings where it's clearly not round. Are you suggesting there is a
different amount of pedal travel in order to advance the chain at the
rear wheel by one link? That's the crux of the issue - you are arguing,
I believe, that the amount of pedal travel varies throughout a pedaling
circle as the shape of the chainring changes - because if it doesn't,
then there is no difference.
The pedal still travel over a perfect circle but not the chain.
So for a given travel of the pedal, the chain travel a different
amount or if you prefer, to achieve a given amount of travel of the
chain, the amount of travel needed by the pedals varies.
Post by Steve FreidesWhat really could make a difference is if the shape of the chainring
effectively changed the gear ratio during a single pedal revolution. If
that happened, then we'd be talking about something tangible, the
reduction of force required by a lower gear at the point the rider's
legs were weakest. Now that sounds like it could be truly useful.
We are talking about definite physical, mechanical differences. This
is a hard fact.
OK, last try at explaining this:
The precise maths for the elipse of ovoid require inifite series and
usenet is maybe not the ideal place to go there so please allow me to
try to simplify by using a square.
1- chain, teeth, strap, pulley, gear, etc
Some peoples have been focusing in teeth too much.
Can we all agree that any of the following is equivalent:
Front = 40 teeth Rear = 20 teeth
Front = 400 teeth Rear = 200 teeth
Front = 4000 teeth Rear = 2000 teeth
Front = 4 million teeth Rear = 2 millions teeth
Front = 4 billion teeth Rear = 2 billion teeth
There's no difference in the gearing for any of the above.
So for the sake of simplicity, I'll continue this with a case of a
front rig with 8000 teeth with each teeth being 0.1mm.
2- The pedal move along a circle.
I think we all agree with this.
3- The chain rig is solidly attach to the pedal. Rotating the pedal
10 degrees rotate the chain rig 10 degrees around its point of
rotation regardless of the shape.
4- The chain is always in contact with the front part of the rig so it
will follow the perimeter of the rig.
5- Circular chain rig:
my 8000 teeth chain rig is 800.0 mm in circumference, it has a radius
of 127.3mm and a diameter of 254.6 mm (i.e. 800.0/Pi)
If you rotate the pedal 1/16th of a rotation (22.5 degree), the rig
will rotate 22.5 degree and the chain will be pulled by the length of
an arc of 22.5 degree with a radius of 127.3mm which is 50.0mm. As
this is a perfect circle, this is also 1/16th of the total
circumference. This will pull 500 teeth on the chain.
Since this is a perfect circle, all possible segments of 22.5 degree
are identical to any other segment.
6- Square chain rig
Now let's replace our circle chain rig with a square one.
This is a square with each side 200.0mm long.
(bad ascii art for orientation reference:
___
| . |
|___|
The perimeter of the square is 4*200.0mm = 800.0mm. This is the same
perimeter as our circular chain rig. This square also has 8000
teeth and a full 360 deg rotation with pull 8000 teeth on the chain.
If you draw a vertical or horizontal line from the center of the
square to the edge, it will be 100.0mm
If you draw a diagonal line to the corner, it will be 141.4mm
(i.e. ( 1/sin(45) ) * 100.0mm )
The pedal however are still going around in a circle. So what happens
if we compute the length of a partial rotation of the square. The
simplest way to compute this is to draw right-angle triangle and use
trigonometry.
Let's try 45 degrees: We can draw that this will go from the
horizontal to the 45 deg angle and visually we can se that this will
be 100.0mm. Trigonometry also shows that this is 100.0mm
(tan(45)*100.0mm) as expected.
What about 22.5 degree: From 0 to 22.5 degree, the length of the
perimeter section is tan(22.5) * 100.0 = 41.4mm.
The length of the section from 22.5 to 45 degree is:
(tan(45)*100.0mm) - (tan(22.5)*100.0mm) = 58.6mm
22.5 degree rotation centered over the center of the flat bit:
The length of the section from -11.25 to +11.25 is:
(tan(-11.25)*100.0mm)+(tan(11.25)*100.0mm)= 39.8mm
22.5 degree rotation centered over the corner:
The length of the section from 33.75 to 56.25 is:
((tan(45)*100.0)-(tan(33.75)*100.0)) * 2 = 66.36mm
So basically, if we turn the pedals 22.5 degree (along the circular
movement of the pedal), the amount the chain will be pulled depends on
the position of the square.
0 to 22.5 deg => 41.4mm =~ 414 teeth
22.5 to 45 deg => 58.6mm =~ 586 teeth
45 to 67.5 => 58.6mm =~ 586 teeth
67.5 to 90 => 41.4mm =~ 414 teeth
-11.25 to 11.25 deg => 39.8mm =~ 398 teeth
11.25 to 33.75 deg => 46.9mm =~ 469 teeth
33.75 to 56.25 => 66.4mm =~ 664 teeth
56.25 to 78.75 => 46.9mm =~ 469 teeth
There is still exactly 200.0mm per 90 degree but some subsections result
in more mm. There's still exactly 2000 teeth per 90 degree rotation and
8000 teeth per 360 degree.
7: elipse, ovoid or any other shape:
The same principle applies. The maths just become a bit more
complicated and you may need infinite series to get the exact result.
Visually, it is noticeable looking at:
Loading Image...that the speed of travel of the point on the circle is constant but the
speed of travel of the point on the elipse varies, sometime being
faster and sometimes slower. This is also what happens would happen
to a chain on an elipse.
If you want more details, you could look at papers such as:
http://www.noncircularchainring.be/pdf/Biomechanical%20study%20chainrings%20-%20release%202.pdf
The purely mechanical section of this paper is irrefutable. They
compute the angular velocity of various chainrig using AutoCAD and
MATLAB. The biomechanical section where they produce an equation to
model the human is where there is room for interpretation.
Some manufacturers also have literature on their site on the subject.
Yannick
(sorry about the long message but this is not pantomine, this is
science and facts)