math problem involving gear inches: help?

wayover13

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Apr 9, 2013
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WI USA
I need some help doing some mathematical calculations for my delta trike (20 inch drive wheel). It has a complex drivetrain, so calculating the gearing is not very straightforward: it has front chainrings and 2 intermediary sprockets between the chainrings and the sprocket on the rear wheel, one on either end of a jackshaft. The below graphic, taken from documentation provided by the trike's manufacturer, gives a visual overview (note that the graphic shows a gear cluster both at the rear wheel as well as one in the middle of the bike, while my bike has only the one gear cluster in the middle, having but a single sprocket at the rear wheel).
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Here is the formula I was given by the manufacturer for figuring gear inches on this bike: ratio of front chainring to middle gear cluster sprocket, times ratio of other two sprockets, times 20 (diameter in inches off rear wheel). So, if I have a 36 tooth chainring going to an 18 tooth sprocket on the middle gear cluster, and and a 24 tooth sprocket going to a 20 tooth sprocket on the rear wheel, the formula would be 36/18 x 24/20 x 20 = 40 (gear inches). So, to begin with, does this seem a sensible way to figure gear inches for a complex drivetrain like this?

I need to resolve this matter soon because I'm unhappy with the current gearing system and am planning to change things. I've decided that, while I will keep the triple chainring configuration on the front, I want to use an internally geared hub on the rear wheel and do the bulk of shifting there: this would, of course, involve making both intermediary sprockets on either end of the jackshaft single sprockets. That will make calculations even more tricky, so I'm trying to develop an alternate formula that will help me to decide which size sprockets to use.

The IGH I plan to use is the Shimano Alfine 8. It has been suggested to me that a possible way to figure gear inches would be as follows. Use an on-line gear calculator (I've been favoring http://gear-calculator.com) to come up with high and low gear-inch figures for the chainring sizes I have (46/36/24) in combination with the sprocket on the Alfine hub--say, a 20 tooth sprocket. I am then to multiply that gear-inch figure by the number of teeth on the larger of the two intermediary sprockets, and then divide that result by the number of teeth on the smaller of the two intermediary sprockets.

As an example, according to the gear calculator, the 24 tooth chainring in combination with a 20 tooth sprocket on an Alfine hub spoked into the 20 inch wheel would result in a 13.7 inch low gear. Were one of the intermediate sprockets to have 28 teeth and the other 22 teeth, the resulting gear inches, according to the revised formula, would be 17.4. Does that sound right?

As may be clear, my math skills are not very advanced. This matter becomes especially confusing to me because quantifiers are being mixed--numbers of inches on the one hand and numbers of teeth on the other. Input on this matter will be appreciated.
 
Everything in the chain line has a ratio, from one segment to the next. Each ratio multiplies with the next.

So, I don't see any reason not to use the formula hte manufacturer provides (which is really the same as what you suggest at the end). If you like, you can check with http://sheldonbrown.com on the gearing pages, for some explanations and such.

The same method can be used with your new gearing just as well, except you don't have to calculate the ratio inside the IGH; it's already given for each gear. You just need the ratio for the jackshaft output to the IGH input on your new sprockets.


(I don't like math, so I don't use gear-inches or any of that; if I don't like the gearing on something, then I experiment: if I need something faster I increase the front sprockets or decrease the rear. If I need something with more torque I decrease the front or increase the rear. If I have multiple stages, then these changes can happen in any stage, or be spread among them. )


Before you pick the input sprocket to the IGH, make sure you look at the torque input limits on it, so you don't exceed what it can take. (especially in gears other than 1:1, IGHs can be less strong than you'd expect).

If the new jackshaft sprockets the same size, then you can simply eliminate them from the calculations, since the ratio is 1:1.

If not, then AFAIK you should be able to use the average of the two as if it was a single sprocket. (say one is 28t, and the other is 26t, then call it a 27t)
 
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