Deriving a unicycle crank length formula?

Having been bred an engineer, I spent some time thinking about this and
I offer the following. What we want to know is how will a given crank
on a given wheel feel in terms of power and control. Where feel means
many things as mentioned in the strings referenced--control, speed,
torque, rotational diameter, power, knee pain...etc.

Work equals force times distance. For different wheel sizes on
different rims to feel the same, will essentially require that the rider
be accomplishing the same amount of work.

Each wheel size has a different diameter, when circumference is
calculated...Pi times diameter, you can obtain a number for each wheel
size. Then by multiplying by 2.54 you can determine the circumference
in centimeter.

20� diameter times Pi equals 62.80 in or 159.51 cm
24� diameter times Pi equals 75.36 in or 191.41 cm
26� diameter times Pi equals 81.64 in or 207.37 cm
29� diameter times Pi equals 91.06 in or 231.29 cm
36� diameter times Pi equals 113.04 in or 287.12 cm

By accomplishing a ratio of circumference divided by crank length in
centimeters, you can determine a ratio for comparison. Using this
approach each crank length and wheel diameter can be compared in a grid.
I hope the chart works on the internet...

Wheel -- 20 24 26 29 36

Crank

85 -- 188 225 244 272 338

110 -- 145 174 189 210 261

114 -- 140 167 182 203 252

125 -- 127 153 166 185 230

140 -- 114 137 148 165 205

150 -- 106 127 138 154 191

165 -- 97 116 126 140 174

170 -- 94 113 122 136 169

175 -- 91 109 118 132 164


These numbers provide a referenced baseline for the work of a given
crank/wheel combination for the same distance output. Similar numbers
between columns will represent similar feel in terms of unicycle
performance--given that each of us has a different comfort level and
range of comfortable skill. Note that a 20 inch with a 125 crank gives a
rating of 127...which compares in terms of work and feel to a 24 inch
with a 150 (similarly a 127 work rating). Likewise a 26 inch with a 165
crank should feel about the same in terms of work but recognizing that
the larger rotation of the longer cranks has a subjective impact.

As I look at it, the ratio of around 125 to about 175 covers the range
of most riders comfort zones. The higher number being harder to pedal
with a smaller pedal arc and the opportunity for greater speed. The
lower number makes it easier to pedal a give distance due to increases
lever arm resulting in probably lower speeds due to larger pedal arcs.
As the crank to wheel size ratio goes down to about 100, the torque
skyrockets and traction probably becomes the limiter...although large
pedal arcs may be less comfortable. In the opposite direction, as the
ratio goes up...the opportunity for speed due to shorter pedal arcs
increases with a penalty in power and hill climbing ability.

Note that AspenMike, who as I recall, rides 175s on his 36 inch Coker,
carries a tougher pedal stroke than someone who rides a 20 inch unicycle
with 110s--making his performance on the Iron Horse all the more
remarkable. This chart should give riders the chance to decide how
cranks compare on different wheel sizes--recognizing that every one of
us will have a comfort zone on this chart of varying width based on
skill and strength.

I bought a 24 inch Torker LX with 150 cranks (work ratio 127) and
quickly went to 114s (work ratio 140) which are more comfortable in my
riding situation. On the KH29XC I bought with 29 inch wheel and 150
cranks (work ratio 154), I also want to optimize for speed so I bought a
set of 125 cranks (work ratio 185). By comparison, the 29 inch with
150s starts me at a higher work ratio than my previous riding situation
(harder cranking, less control, less power) and my decision to learn to
ride the same unicycle with only 125s is a commitment train to even
less control and power but this is offset by the ability for a better
spin and higher speed.

Hope this helps. Any subjective opinions on the field application of
this rough science is welcome.

Carey


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Having been bred an engineer, I spent some time thinking about this and
I offer the following. What we want to know is how will a given crank
on a given wheel feel in terms of power and control. Where feel means
many things as mentioned in the strings referenced--control, speed,
torque, rotational diameter, power, knee pain...etc.

Work equals force times distance. For different wheel sizes on
different rims to feel the same, will essentially require that the rider
be accomplishing the same amount of work.

Each wheel size has a different diameter, when circumference is
calculated...Pi times diameter, you can obtain a number for each wheel
size. Then by multiplying by 2.54 you can determine the circumference
in centimeter.

20 in diameter times Pi equals 62.80 in or 159.51 cm
24 in diameter times Pi equals 75.36 in or 191.41 cm
26 in diameter times Pi equals 81.64 in or 207.37 cm
29 in diameter times Pi equals 91.06 in or 231.29 cm
36 in diameter times Pi equals 113.04 in or 287.12 cm

By accomplishing a ratio of circumference divided by crank length in
centimeters, you can determine a ratio for comparison. Using this
approach each crank length and wheel diameter can be compared in a grid.
I hope the chart works on the internet...

Wheel -- 20 24 26 29 36

Crank

85 -- 188 225 244 272 338

110 -- 145 174 189 210 261

114 -- 140 167 182 203 252

125 -- 127 153 166 185 230

140 -- 114 137 148 165 205

150 -- 106 127 138 154 191

165 -- 97 116 126 140 174

170 -- 94 113 122 136 169

175 -- 91 109 118 132 164


These numbers provide a referenced baseline for the work of a given
crank/wheel combination for the same distance output. Similar numbers
between columns will represent similar feel in terms of unicycle
performance--given that each of us has a different comfort level and
range of comfortable skill. Note that a 20 inch with a 125 crank gives a
rating of 127...which compares in terms of work and feel to a 24 inch
with a 150 (similarly a 127 work rating). Likewise a 26 inch with a 165
crank should feel about the same in terms of work but recognizing that
the larger rotation of the longer cranks has a subjective impact.

As I look at it, the ratio of around 125 to about 175 covers the range
of most riders comfort zones. The higher number being harder to pedal
with a smaller pedal arc and the opportunity for greater speed. The
lower number makes it easier to pedal a give distance due to increases
lever arm resulting in probably lower speeds due to larger pedal arcs.
As the crank to wheel size ratio goes down to about 100, the torque
skyrockets and traction probably becomes the limiter...although large
pedal arcs may be less comfortable. In the opposite direction, as the
ratio goes up...the opportunity for speed due to shorter pedal arcs
increases with a penalty in power and hill climbing ability.

Note that AspenMike, who as I recall, rides 175s on his 36 inch Coker,
carries a tougher pedal stroke than someone who rides a 20 inch unicycle
with 110s--making his performance on the Iron Horse all the more
remarkable. This chart should give riders the chance to decide how
cranks compare on different wheel sizes--recognizing that every one of
us will have a comfort zone on this chart of varying width based on
skill and strength.

I bought a 24 inch Torker LX with 150 cranks (work ratio 127) and
quickly went to 114s (work ratio 140) which are more comfortable in my
riding situation. On the KH29XC I bought with 29 inch wheel and 150
cranks (work ratio 154), I also want to optimize for speed so I bought a
set of 125 cranks (work ratio 185). By comparison, the 29 inch with
150s starts me at a higher work ratio than my previous riding situation
(harder cranking, less control, less power) and my decision to learn to
ride the same unicycle with only 125s is a commitment train to even
less control and power but this is offset by the ability for a better
spin and higher speed.

Hope this helps. Any subjective opinions on the field application of
this rough science is welcome.

Carey


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Didn't mean to put that in twice.

Carey


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As you said, there have been many, many threads discussing crank length
throughout the years here on this forum. My personal belief is that
there is no way to quantify all of the variables involved and derive a
useful formula for this.

I looked at the site you listed about bicycle crank length, and to be
honest, I disagree with the writer's basic premise even for a bicycle.
If I were to follow his formula, I would have 188mm cranks on my bike.
I can't imagine how uncomfortable that would be even if I could find a
crankset that long! I use a 165mm crankset on my fixed-gear, and 170mm
on my commuter. I often wish I could find a crankset with much shorter
cranks to experiment with on my bicycles. On a bicycle, leverage isn't
nearly as big a factor as it is on a unicycle, since gearing is easily
manipulated.

A standard unicycle always has a 1:1 gear ratio. Every time your feet
make a circle, the wheel makes a circle. The only way to manipulate
gear/inches is to change the size of the wheel.

Optimum crank length is based more on individual strengths and
preferences. I prefer fairly short cranks on my unicycles, but many
stronger riders prefer even shorter cranks. I use 125mm on my Coker,
and 110's on my 29er. I don't have a 24x3 muni anymore, but when I did
I discovered that I didn't like the 170mm cranks that came on it, and
that I could ride it much better using 150's. With 170's I felt like my
knees were trying to hit me in the chin!

This is just one old mans humble opinion, but I don't think that there
is any formula that could determine an ideal crank length for a unicycle
or a bicycle. Individual preference, strength, skill level and the
terrain that you ride determine what crank length is "right".

Chuck


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Chuck,

Your preferences chart out at 210 and 230, the top right portion of the
chart--in my book a power rider! A work ratio of over two hundred looks
to me like it would take strength as well as skill.

Carey


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Roger from unicycle.com UK worked up a spreadsheet a few years ago
somewhat along those same lines, Carey. I don't know if it is still
available on his website, but it is fun to play with. It calculates the
relationship between wheel size, crank length, speed, cadence and
footspeed.

I've got it somewhere, I'll try to find it if you are interested.

Chuck


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On Sat, 24 Sep 2005 18:08:19 -0500, "Carey" wrote:

For different wheel sizes on
different rims to feel the same, will essentially require that the rider
be accomplishing the same amount of work.

Using such a table is a fruitful approach, and it is useful, in that
if I look up to different set-ups with the same work ratio and imagine
riding them, they would indeed feel about the same. However, I don't
buy the logic on which the table is based (i.e. your copied statement
above). In the first place, the actual work done is larger than you
assume, because both legs exert force on the pedals, and moreover it
happens in a sort of erratic way (in the time domain), as required to
keep balance. But more importantly, I think that a skilled rider
cruising along a horizontal path will not be bothered about power, as
the required power output is generally lowish. He'll be more concerned
with control and log movement.

It now happens that if you take the reciprocal of each "work ratio" in
your chart, you end up with numbers which are proportional to what
others have called "gearing ratio". One definition of gear ratio (by
Mikefule I think) is crank length divided by wheel radius, both in the
same length units. E.g. for a 20" unicycle with 5" cranks (a very
common combination), the gearing ratio is 0.5. For a 24" unicycle with
6" cranks, the gearing ratio is also 0.5. And indeed (I think) these
two combination feel about the same, or let's say, you can't make them
feel more similar by changing cranks. The reason for that, I think, is
that gear ratio determines how easy a unicycle can be controlled, and
not so much that the required power is the same. This breaks down at
the point where too high crank lengths make pedaling uncomfortable in
itself.

All in all, I think that gear ratio would be a good indicator of
unicycle "feel" with various wheel sizes and cranks. Also, the numbers
nicely vary between 0 (pedals at axle like a BC wheel) and 1 (pedals
at circumference of wheel), and are therefore easily understandable.

Klaas Bil - Newsgroup Addict
--
"Unicycling is like glue: you have to stick with it, and it's not to be sniffed at - Mikefule"

Klaas Bil,

I think you are right. The work ratio I mention is an imprecisely
defined approximate for comparison reasons only, with the ratio
numerator and denominator randomly selected but consistently applied.
The logic of a 0 to 1 scale is a good approach. In fact, it sounds like
the gear ratio you mention is a similar ratio that works from only
diameter and crank length...which makes sense since Pi is a constant in
the equation. It would be a comparable estimate to my effort--and I
notice, makes the same point about a 20 with 125s and a 24 with 150s.

All this data aside, what I know is that the 150s on my 24 are very
powerful and precise but too much arc, the 114s made it smooth and fast,
and I can still control it well. The 150s on the 29 feel good now but
as I get better on it I will move to 125s. Kind of scary as I UPD'd off
the front last night at pretty high speed and was barely able to keep my
feet.

I am looking for speed that makes me immune to mosquito attacks here.

Chuck--If you can find that chart, I'd be interested in it. Sounds like
you guys have thought about this in depth before. My approach was to
give me a feel for where I would start with cranks on different wheel
sizes, given that I knew what I liked on one.

Carey


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On Sun, 25 Sep 2005 09:49:50 -0500, "Carey" wrote:

Chuck--If you can find that chart, I'd be interested in it. Sounds like
you guys have thought about this in depth before. My approach was to
give me a feel for where I would start with cranks on different wheel
sizes, given that I knew what I liked on one.

I've seen Roger's chart (it was a spreadsheet) but not saved it. I
have made one in Dutch that is roughly similar - you can have it if
you want. But you can make one yourself easily, the calculations
involved are very simple.

Klaas Bil - Newsgroup Addict
--
"Unicycling is like glue: you have to stick with it, and it's not to be sniffed at - Mikefule"

Chuck Webb wrote:
*Roger from unicycle.com UK worked up a spreadsheet a few years ago
somewhat along those same lines, Carey. I don't know if it is still
available on his website, but it is fun to play with. It calculates
the relationship between wheel size, crank length, speed, cadence and
footspeed.

I've got it somewhere, I'll try to find it if you are interested.

Chuck *

That spreadsheet of Roger’s is useful and it’s here;

http://www.unicycle.uk.com/Cranklength.xls


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