Sprockets Equation for changing wheel size and maintaining original top speed

Discussion in 'Transmission / Drivetrain' started by Ollie, Feb 14, 2014.

  1. Ollie

    Ollie Member

    Hello all, I'm switching to 24" wheels because I have a really strong set from a dirt jump bike that I want to use. My more maths-literate friend has made an equation to calculate what size rear sprocket will be required:

    (pi x [radius of smaller wheel]^2)/(pi x [radius of larger wheel]^2) = [ratio]

    [no. of teeth on original sprocket] x [ratio] = [no. of teeth on new sprocket]

    24" x 24" = 576
    3.14159265359 x 576 = 1809.55737
    26" x 26" = 676
    3.14159265359 x 676 = 2123.71663
    1809.55737 / 2123.71663 = 0.85207101
    44t x 0.85207101 = 37.4911244 (dunno if you can find a 37t sprocket, so round down to a 36t or up to 38t)

    Hope y'all find it useful! :)

  2. FurryOnTheInside

    FurryOnTheInside Well-Known Member

    A 40T sprocket on a 24" nominal wheel would be equivalent to your 44T on your old 26" nominal wheel.
    I say nominal because it's not really 24" or 26" and I haven't taken tyre size into account in the calculation.

    I used Sheldon Brown's online gear calculator, inputting a 100T front sprocket and 100RPM in both cases, then 44T rear sprocket with 26" nominal wheel size, which gave a road speed of 17.6mph. Then I input a 24" nominal wheel size, and a range of rear sprocket sizes from 44T down to 35T and I looked at the road speeds with each. The 40T came out as approximately matching the 17.6mph road speed of the 26" wheel.
  3. Ollie

    Ollie Member

    Hey Furry, I've used Sheldon Brown's calculator before, it's been a great help. The only thing it couldn't do was account for different tyre heights, so this equation uses the total diameter of rim and tyre. My calculation ended up rounding down to 40t, so no different from Sheldon's anyway. :)

    My friend also gave me a second equation to work out what size sprocket is required to increase the speed by a given amount:

    [lower original speed] / [higher target speed] = [ratio]

    [original no. of teeth] x [ratio] = [target no. of teeth]

    Example (30mph with a 44t sprocket to 40mph):
    30 / 40 = 0.75
    44 x 0.75 = 33
  4. butterbean

    butterbean Well-Known Member

    It's much more simple than that. Many people fail to realize that the rear wheel counts as a pulley when calculating gear ratio. With pulleys, 1" of diameter = 1 mph. A 2" smaller wheel will give you roughly 2mph less top speed, and if you are really that worried about such a negligible difference, you will need a sprocket with 2 less teeth. But since they don't make odd-sized sprockets, you will probably have to go down at least 4 teeth.
    Ollie likes this.
  5. FurryOnTheInside

    FurryOnTheInside Well-Known Member

    Actually Sheldon's gear calculator has just about every common wheel and tyre size including the old obsolete ones.. it lacks the new "fat bike" tyres like 3"-5" though.

    But more importantly we now have three different answers between the three of us! :p

    No no no no no! xD That is to say, I disagree..
    42T is just not right, nor is 36T, 37T or 38T lol! It is definitely 40T to get the same speed on a 24" wheel as you had with the 44T on your old 26" wheel if you use the same size tyre.

    It may be as simple as
    26 x 100 = 2600
    2600 / 24= 108.33
    so a 26" wheel has 108.33% of the diameter AND 108.33% of the circumference of a 24" wheel.
    So therefore the sprocket on a 26" wheel needs to have 108.33% the number of teeth that a 24" wheel has, to have the equivalent gearing.
    44 / 108.33 = 0.40616
    0.40616 x 100 = 40.616

    40.616 x 108.33% = 44

    But Sheldon calculates for us so we can't possibly make mistakes.. like assuming a 26" wheel is really 26" diameter with it's tyre on, or a 24" wheel is really 24" diameter with it's tyre on. Some leeway must be expected for different load or tyre pressure differences which can vary from trip to trip and affect tyre drop, which affects a wheel's true "roll out" circumference of course.. but that's getting way too nerdy. :p
    Ollie likes this.