Topic: Hey math wizards! Help with gearing, please

[Patrick]

Many years ago I did a very thorough assessment of my mathematical skills. Based on that assessment, I determined that it would be in my best interest to become a writer, so I could avoid math.

So I’m asking for some help from the math wizards out there (I’m thinking engineers here) to help me calculate gearing so as to keep original ratio.

I am restoring a K0750. I need to replace the sprockets. Stock gearing on this bike was, I understand, a 17-tooth front sprocket and 48-tooth rear. Easy enough. But here’s where the math comes in. I will be running a 16-inch rear wheel in place of the 18-inch stock wheel. So it appears the circumference of the wheel will be smaller and the rear wheel will turn more frequently, affecting the gearing. I want to keep the gearing stay as close to original as possible.

Now I took math through high school and even some calc in college. As a matter of fact, my university to this day holds something of festival to commemorate the day I decided not to take any more math. They do, however, miss the comedy my answers would often provide.

Here’s how I calculate it:

The original wheel had a diameter of 457.2mm (18 inches times 25.4). The original tire was 4.0 inches (101.6mm) wide. I have no idea what the ratio of height to width of the original tire was, so I assuming height to be 90 percent of width, thereby assuming the originals to be 101/90-18. Ninety percent of 101.6mm is 91.26mm. To get total diameter I add the height of the tire twice, or 182.52mm, to the diameter of the wheel. So total diameter would be 639.2mm (182.52*2) + 457.2. As Circumference equals Diameter times Pi, the circumference of the original wheel and tire should be 639.2*3.1415 =  2008.04mm.

The new wheel is 16 inches (406.4mm) and the tire I am mounting is a 130/90-16. The height of the tire would be .9*130 or 117mm. So to get total diameter I add 90 percent of 130 mm twice (117mm*2=234mm) to allow for the tire on both sides of the rim. The diameter would be 406.4mm + 234mm = 640.4mm. Circumference would then be 640.4*3.1415= 2011.81mm.

If my math is correct, it appears I should keep original gearing to retain original performance.

God, my head hurts.

Is this correct?

Patrick

[pae]

Sounds good to me. As you say, you've lost 2 inches on the rim diameter, but with the same profile (90) on a wider tyre you get those 2 inches back again in increased tyre wall height. So total diameter is the same (or near as dammit)

Now go back to your university and tell them you're doing just fine as you are

regards, Phil

[KB02]

I agree, only I did find a couple of errors in your math:

18" rim:
Your 90% ratio of 4" tire would be this: 4*25.4 = 101.6  101.6*0.9 = 91.44
You will need to add this number twice to you rim diameter of 457.2 or 475.2 + 91.44 + 91.44 = 640.08

640.08 * 3.14159 = 2010.87

16: rim:
math looks good, except I get 640.4 * 3.14159 = 2011.87

The differnce being:
2011.87 - 2010.87 = 1mm   or roughly the thickness of a dime.

I think stock gearing will be fine (Assuming the 4" tire has a 90% side wall ratio)

[upperlake04]

This downloadable gearing calculator might help..
 http://www.dropbears.com/u/utilities/gearing.htm

[Patrick]

Y'know, KB02, I got 91.44 for that first calculation the first time I did it too. Then, double checking my work, I got the other number. Urg, Microsoft. It was the Windows calculator I used....

I was very surprised the circumferences came out so similar. I guess that 16-inch wheel isn't going to drop the back end at all.

And a gearing calculator. I should have figured....

Thanks ya'll.

Patrick