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20120112, 09:41 PM  #1 
Unicyclist
Join Date: Jan 2012
Posts: 25

Is there a calculation comparing acceleration of one unicycle to another
I’m trying to figure out the important variables to consider when comparing different unicycle sizes. By reading through other posts, I’ve stumbled across: velocity, gear ratio, and acceleration.
Velocity When trying to compare one unicycle size to another, other posts talk about the relationship between wheel size, cadence and velocity (velocity = cadence X wheel circumference). I understand that for a given RPM a larger wheel is going to go faster. Gear Ratio I love this post with a table on gear ratios: http://195.66.135.134/forums/showthread.php?t=88333 Acceleration Other posts mention that in order to compare one unicycle size to another, it’s not just about velocity. You also need to account for how quickly a unicycle can accelerate/decelerate, which depends on wheel size, the mass of the wheel, and crank length. So, a relative acceleration calculation would be helpful (but it’s not as easy as the velocity calculation). Are there any posts that discuss how to calculate the acceleration? I’ve done a calculation, but would like to see other posts before I post something redundant. A quick five minute search did not uncover anything. What other variables are important? Is there anything else to be considered? Maybe a calculation on how easy it is for a given unicyle size to maintain a cruising velocity? I'm guessing this would be: Sum of Forces = mass X acceleration = Zero (because you are not accelerating at a constant velocity). So, the wind resistance + bearing friction + static friction force from the wheel on ground would have to equal the torqe from pedaling. The variables involved would be 1) mass of the rider+unicycle, 2) wheel size, 3) crank length, 4) drag coef of a bicyclist has to be posted somewhere, 5) coef of friction of tire on dirt or road is out there somewhere, 6) I'd ignore bearing friction. Sorry for geeking out on you! 
20120112, 10:23 PM  #2 
768  It's in your DNA
Join Date: Sep 2001
Age: 65
Posts: 9,101

By far the largest and most important variable, which you have not included in your list, is the rider. That one variable makes all the others essentially meaningless.
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20120112, 10:36 PM  #3 
Unicyclist
Join Date: Jan 2012
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Very true!! But, since I'm the only rider that I care about when trying to figure out what unicycle to buy, I cancel myself out...

20120112, 10:50 PM  #4 
Totally Doable
Join Date: Apr 2002
Location: Berkeley, CA
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You're overthinking this. A unicycle is not a physics problem; the number of variables in the realworld speed of a rider is far larger than can be calculated by formula.
Lighter unicycles accelerate faster (and you're always accelerating something, as long as your wheel is spinning). But if you measure the same rider in the same conditions 10 times, you'll get 10 different results on acceleration speed, and those results will be pretty significantly divergent. 
20120112, 11:13 PM  #5 
Unicyclist
Join Date: Jan 2012
Posts: 25

I completely agree, Tholub... I am totally over thinking this. Actually, I should be banished to the nerds' area of the forum.
At this point, I'm really just doing this for fun. Josh at Unicycle.com has me all set up with a new Drac 29". It's just that I enjoy theoretical problems and can't seem to let this go. I'm hoping there are some mechanical engineering profs. or physicists on the thread that can whip this out. Last edited by Danitz; 20120112 at 11:18 PM. 
20120112, 11:22 PM  #6  
Totally Doable
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Quote:


20120113, 12:58 AM  #7 
Unicyclist
Join Date: Jan 2012
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I’m not looking for any absolute acceleration values. I’m just trying to compare one unicycle size to another.
Angular form of Newton’s 2nd law is: T = I * α where T is torque, I is mass moment of inertia, α is angular acceleration. The torque is equal to the force your foot applies to the pedal times the crank arm length: T = F * r where F is force applied by your foot on the pedal, r is the crank length. So, F * r = I * α ===> F = I * α / r Ok, now here is where I think my logic may be off and could use some help. I think that in order to compare unicycle A to unicycle B, you would want to look at the acceleration of each unicycle for the same force applied by your foot on the pedal, F. So, Ia * αa / ra = Ib * αb / rb where Ia = mass moment of innertia of the wheel on unicycle A Ib = mass moment of innertia of the wheel on unicycle B αa = angular acceleration of unicycle A αb = angular acceleration of unicycle B ra = crank length on unicycle A rb = crank length on unicycle B In order to compare the relative accelerations of each unicycle rearrange the equation like this: αa/αb = Ib/Ia * ra/rb Now, the mass moment of inertia for a wheel (let's ignore the cranks and pedals just for simplicity) is equal to: I = m * R^2 where m = mass of the wheel R = wheel radius So, the final equation is: αa/αb = (mb * Rb^2) / (ma * Ra^2) * (ra/rb) = (mb/ma) * (Rb^2 / Ra^2) * (ra/rb) I used this equation to try to compare the Nimbus 29” Drac with the Nimbus 26” Muni: Unicycle A: (26” muni) Mass of the Nimbus 26” wheel, tire, and stock cranks = 4.3 kg (due to heavy tire) Stock crank length = 165 mm Radius of 26” wheel = 330.2 mm Unicycle B: (29” Drac) Mass of the Nimbus 29” wheel, tire, and stock cranks = 3.7 kg Stock crank lengths = 165 mm. Radius of 29” wheel = 368.3 mm So, αa/αb = (3.7/4.3) * (368.3^2 / 330.2^2) * (165/165) = 1.07 So, I think that this is saying that the 26” muni would only accelerate 7% faster than the 29” drac. Again, not quite sure about this logic. Last edited by Danitz; 20120113 at 01:03 AM. 
20120113, 01:07 AM  #8 
768  It's in your DNA
Join Date: Sep 2001
Age: 65
Posts: 9,101

I am positive about this logic.
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Greg Harper Destroying the climate by shutting down nuclear power plants, one by one, since 1979. Change is good. Bills are better. 
20120113, 01:56 AM  #9 
Totally Doable
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Location: Berkeley, CA
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If the 29er has a lighter rim and tire, it will accelerate faster than the 26", so even leaving aside the fact that the math you're doing is meaningless, your math is wrong.

20120113, 05:07 AM  #10 
North Shore ridin'
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The nerds' area is called unicyclist.com.
But to geek out a little, if you pretend all your comparison unicycles have the same type rim and tire (and spokes, cranks, pedals, etc.) it can add some meaning to your math. But it can't be translated to real unicycles since you usually won't get the same rim or tire in all those different sizes.
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20120113, 09:16 AM  #11 
is what it is
Join Date: Sep 2004
Location: hella Nor Cal
Age: 40
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Physicist here. I'd just like to point out that your calculations are meaningless until you include margins of error.
See: http://en.wikipedia.org/wiki/Propagation_of_uncertainty You ride them both then buy the one you like best. If you don't have the ability to ride both, then just buy one and save money for the other. Eventually you will own both. This is the first law of unicycling: As t approaches infinity, so does the number of unicycles you will own.
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20120113, 02:53 PM  #12 
XC Muni
Join Date: Jul 2008
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I'm no physicist, not even an engineer, but that doesn't keep me from posting
Okay, so you're a nerd, obviously bored at work/school, but really, we're talking about how fast a unicycle can accelerate? A turtle can accelerate faster than a unicycle! The funniest thing for me was after completing a twenty mile mtb race on my uni, I looked at my time and the mileage/terrain covered and realized that I could have run that same race faster than I rode it So, with my apprentice engineers hat I have a couple thoughts: A longer crank and more body weight/physcial strength, as well as having a handle that allows for increased leverage, these things will improve initial accelaration. All things held constant; same handle/crank size/tire design, a smaller and lighter uni will clearly be quicker to both start and stop. That said, being quick on a uni has never been my problem, I'm more inclined to be concerned with duration of sustained effort, overall endurance, change in skill level/ability when fatigued, etc... My goal is to build my skills and my muscles so that I can sustain longer rides at higher skill levels. So why would you be interested in acceleration of a uni? How about some studies regarding endurance riding and the effects of uni weight, crank length, wheel size, tire pattern, leg extension/bend, geared vs ungeared? This would all be very applicable and highly beneficial to me as a rider. 
20120113, 08:55 PM  #13 
Unicyclist
Join Date: Jan 2012
Posts: 25

Again, everyone, thanks for the feedback!!!
I don't care about the actual acceleration. I agree with you all. I also don't care about margins of error (not concerned about an absolute answer). Also, I've completely ignored friction at the bearing and at the wheel/ground contact. This all started as an exercise to help me decide between a 26" and 29" unicycle. Unfortunately, I don't have the time/patience to track down unicycles to actually try. So, I thought a little calculation may help me decide. As a result, I stayed up way to late the other night digging out my dynamics book and working on the equations above. (Being extra tired the next morning is probably why I couldn't ride straight: http://www.unicyclist.com/forums/showthread.php?t=91600 ) Here was my background logic. I know that wheel size has an obvious affect on velocity. I also know that crank length is the only economical control you have over gearing for a given wheel size. However, a statement was made on one other thread along the lines of, [it is not just crank length and wheel size that can cause one unicycle to feel so different than another. The mass of the wheel can also affect the feel.] So, my point was more to understand (or account for) the impact of wheel mass. I thought that looking at how easily a unicycle could accelerate was the way to do this. I think it was a useful exercise. With the same crank length, the 26" unicycle should accelerate faster. However, because of an extra heavy 3" wide tire on the 26" wheel, it was heavier than the 29". So, that offset the difference in wheel diameter and crank "gearing". My big assumption here is that more acceleration would provide more control over the unicycle. That's got to play into why it's easier for people to learn on a 20". Being a new rider with hardly any recent experience, this was just a guess. Now, as some of you have pointed out, it's time for me to forget about this and just go ride. Last edited by Danitz; 20120113 at 09:02 PM. 
20120113, 10:25 PM  #14 
Registered Unicyclist
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Location: Malvern, UK
Age: 48
Posts: 1,792

You're ignoring the elephants. A 20 is easier to learn on because it's lower (hence easier to get on and less scary when you're up there), because it's lighter (hence easier to throw around and less tiring) and because you go slower. To some extent, accelerating faster is a bad trait for learning as it results in less stability.

20120113, 10:34 PM  #15 
is what it is
Join Date: Sep 2004
Location: hella Nor Cal
Age: 40
Posts: 6,610

Then you must agree that 1 + 1 = 3.
Because, essentially, you could get such a result with large enough error bars... which your equation appears to have. A picture for illustration.
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