Friday, May 31, 2013

Aaannd... we're back.

Finished a very busy semester at school a few weeks ago, and took some time to relax and gather myself. Now, I'm back in the saddle and have a few exciting new projects coming up for the summer, more on those another time.

To make sure I enjoyed myself this summer (and to try to shrink my recently expanded mid-section) I purchased a new bicycle, a fixed gear bike.

I've certainly been critical of fixed gear bikes before, but I have no intention of riding on the street; my main issue with the 'fixie' fad is that these machines are laughably equipped to deal with urban cycling and its many stops and other vehicles. My intent is to take this bike to the exceptional (and lengthy) Bosque Trail here in Albuquerque and compete with myself for the fastest time on the trail. The benefit of a single speed setup is weight reduction, friction reduction and simplicity. I will put a large chainring on the front and focus all my energy on accelerating.
You might remember an interesting photo I put used the Scorchers article of a bicycle-policeman and his enormous chainring.

Arnold Kurth-bicycle police officer

This gave him an advantage in pursuing the speed-obsessed ragamuffins of the day. You can read more about mechanical advantage and gear reduction here.

The biggest chainring I've been able to find is a 54 tooth, installing a new chainring will necessitate a longer chain, and to find the length of the new chain will require... MATH! 
First, we need to know a few things:
A 54 tooth sprocket has a 27 inch circumference (half inch spacing between each tooth)
A 27 inch circumference has an 8.58 diameter (Use Pi)
The rear sprocket has 16 teeth, an 8 inch circumference and a diameter of 2.54 inches
The distance between the crank and the rear bearing in 17.25 inches

You'll have to forgive the crudeness of the sketch
If we look only at the chain we can start to divide it into different, easily calculable sections:

Seriously, drawings not my thing

The sprockets are only covered by chain for about half their circumference, and the chain has two sections running between each sprocket but we only need to calculate this once.
The sprocket measurements are easy, take the circumference and divide it by half. 
Rear = 4 inches of chain
Front = 13.5 inches of chain
To find the connecting length we can use the Pythagorean Theorem, which states that in a triangle with one right angle (90 degrees) the length of the opposite side will be the square root of the sum of each other length squared, or a2 + b2 = c2. To take advantage of this we need to identify a right-angled triangle in the setup, we know that the chain runs from the peak of the rear sprocket to the peak of the chainring (approximately), and this is the length we want to find (c2 or the hypotenuse, in mathematical terms). The bottom side of the triangle can go from the peak of the sprocket to a point inside the area of the chainring, this point is 1.27 inches directly above the center of the chainring. 1.27 is half the diameter of the rear sprocket, or the distance from the center of the sprocket to its peak.
The final side of the triangle is simply half the diameter of the chainring minus half the diameter of the sprocket, 8.58/2= 4.29, 4.29-1.27= 3.02.

Not to scale, duh
Now we can square 17.25 (17.25 x 17.25 = 297.56) and 3.02 (3.02 x 3.02 = 9.12) and add the products together (297.56 + 9.12 = 306.68) and find the square root of the sum (I don't know the notation for square root on this keyboard, but the answer is 17.51). Which means that the total chain length is 4 + 13.5 + 17.51 + 17.51 which equals 52.52 inches of chain. 
Now seems like a good time to mention that there are preprogrammed calculators available to find this for you, I used one to check my work:

But aren't you glad you went through all that?
The calculator got the same answer but was smart enough to round it up for functionality.
 There are a few reasons I chose to do the math myself, first being that I enjoy math and I tend to think about these things even if I don't need the new chain, second that someday I may not have a someone else's knowledge to draw upon, the internet and even books are not guaranteed to us and if someday they aren't around and you need to get something done you'll have to use your brain, and finally because as frustrating and boring as math can be (even I slept through it in school) that doesn't mean it doesn't apply to your life or that you can't find a way to make it fun. 
I'm going to do a few more like this, and I encourage anyone following at home to ask questions or point out where I could have simplified the process. 

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