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Old 2017-05-24, 05:36 PM   #31
Engineer on a Unicycle
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Originally Posted by tholub View Post
The energy doesn't "come back".
That kind of answer would fail your physics exam.

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It's in the wheel because you put it in there. And if you want to get the wheel moving faster again, you have to put more energy into it. Accelerating from 2 m/s to 4 m/s requires exactly the same amount of energy every time.
And decelerating it returns exactly the same amount of energy, every time.

When you fail to grasp basic physics, you're left with only personal experiences - those are valid, but they don't generalize; they are feelings, not explanations.

What's actually slowing the wheel each cycle is a fractional version of the same principle that allows you to "bowl" a car tire and have it continue some distance up a hill as it slows, converting rotational and kinetic energy of the velocity of the center of mass, to the potential energy of being higher on the hill.

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It seems particularly odd from a scientific perspective that you provide empirical evidence that cycling is faster than running, and then use that evidence as the basis for an assertion that running is faster than cycling.
You missed that one was about a sub 20% hill and the other about a 33% hill - part of the whole point was trying to identify the crossover in advantage. There may also be an issue of duration, though I hadn't commented on that.
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Old 2017-05-24, 05:42 PM   #32
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Originally Posted by Engineer on a Unicycle View Post
That kind of answer would fail your physics exam.



And decelerating it returns exactly the same amount of energy, every time.

When you fail to grasp basic physics, you're left with only personal experiences - those are valid, but they don't generalize.
That's a pretty rich assertion.

Call me back when you can point to a single real-world example that matches your fantasy version of the way that physics is working here. Maybe once you get beyond Physics 101 you'll learn that you have to first comprehend the system you're analyzing before you can start making assertions about it.
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Old 2017-05-24, 10:55 PM   #33
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Very interesting discussion! Thanks guys. I think Engineer on a unicycle makes some very good points and from my own experience on using physics to build the best wheel possible the numbers don't lie, it was a much better wheel. So as long as his numbers and theories hold up then I think they probably have something.

My own experience is that unicycles climb easier than bikes but I need to compare against a decent rider. I also had a theory that perhaps it's the ability to hold onto the handle and pull yourself directly down into the cranks while on a bike you can stand and put weight in the pedals but can pull against something to drive into the pedal. You have the handlebars but where they are located is not as useful for pulling yourself into the pedals.
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Old 2017-05-25, 01:24 AM   #34
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Originally Posted by Engineer on a Unicycle View Post
In fact, they do. The research that went into that comment found a lot of indications of 17 lbs plus, with under 15 considered to create tradeoffs, yes, under 14 lbs is possible. It's also possible to make a carbon unicycle for an ideal hill, removing the strength reserve normally there for all-around riding.
You didn't check my garage for your research - I have a 15lb road bike hanging there, no tradeoffs at all involved, it's all quite standard stuff and nothing particularly exotic. You won't get a practical unicycle (suitable for riding at speed uphill) weighing 7lb, let alone 5lb. Though it's a moot point anyway, because even if you were getting your claimed 10lb advantage, that's only ~6% of a typical light rider/cycle combo - and this weight saving appears to be the only possible advantage for the unicycle which has been proposed. Yet there are far larger inefficiencies involved in riding a unicycle, some of which you touched on without really considering them properly.

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Originally Posted by Engineer on a Unicycle View Post
To argue that this is an efficiency loss you have to argue that what is slowing you is a wasteful mechanism like muscle braking
...
Translating to Fargo at an apparent 33% grade, to keep going for an hour plus the rate of travel straight up the face of the hill would be a bit under 4 miles per hour; that's faster than people appear to do it, but already slow enough stability may indeed start to need specific effort rather than fairly passive micro-steering.
Because the point seemingly being missed by both of you is that there is a lot of energy expended in balancing on a unicycle - your muscles are doing a lot of work which doesn't help your progress up the hill at all. You appear to be looking at the problem as if you only have to mechanically roll the wheel up the hill without doing any balancing (though even then the weaving is significant - there's probably more than 6% inefficiency right there). It's easy for experienced riders to underestimate the amount of work we do to balance - even when it is seemingly effortless or "passive" there are still a lot of micro adjustments going on, and those certainly aren't passive as far as your muscles are concerned.
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Old 2017-05-25, 01:30 AM   #35
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Originally Posted by Pinoclean View Post
My own experience is that unicycles climb easier than bikes
You're comparing you on a unicycle to you on a bike? I find that extremely hard to believe - as already pointed out, all the evidence suggests that simply doesn't happen. It may be that you're going slower up the hill than you realise on your unicycle compared to the bike (it's worth pointing out that the typical "gear" on a unicycle is way lower than that on a typical bike - the bottom gear on the 15lb bike I own is ~40", yet I can get up some pretty steep hills on that which I'd be walking even with a much lower geared unicycle).

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I also had a theory that perhaps it's the ability to hold onto the handle and pull yourself directly down into the cranks while on a bike you can stand and put weight in the pedals but can pull against something to drive into the pedal. You have the handlebars but where they are located is not as useful for pulling yourself into the pedals.
It's certainly not that - because when riding a bike (or indeed a unicycle) efficiently uphill you won't be using your upper body strength at all - that's wasted energy. The limiting factor given proper gearing isn't your strength.
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Old 2017-05-25, 02:23 AM   #36
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Originally Posted by aracer View Post
You didn't check my garage for your research - I have a 15lb road bike hanging there, no tradeoffs at all involved, it's all quite standard stuff and nothing particularly exotic. You won't get a practical unicycle (suitable for riding at speed uphill) weighing 7lb,
I specifically mentioned the existence of 14 and 15 lbs bikes. But you haven't considered taking that lightweight road bike and building a unicycle with the same technology. You'll be leaving more than half of it in hacksaw trimmings on the ground - about the only thing that would end up weighing more is the saddle and tire, but there's only one of those. For comparison an off-the-shelf Nimbus E-sport weighs 8.5 lbs, and to my knowledge incorporates no carbon composites at all.

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The limiting factor given proper gearing isn't your strength.
Indeed, it is not, and that's a critically important point.

Let's take 5 watts per kilogram of body mass, a reasonable hour+ endurance output number for a fit two-wheeled athlete spinning cycle cranks (since they talk about this kind of data all the time) and look at what this means in terms of hill grade where a cadence can be maintained, when considering only the power expended lifting the body mass up the hill - the weight of the cycle, friction losses, and the small power used in moving horizontally are not considered. Those simplifications probably make the table substantially less accurate at low grades and larger wheels.

"Gear Inches" is a traditional term for the effective size of a geared bicycle wheel; it's also of course directly equivalent to the size of an actual unicycle wheel, though some unicycles have rolling diameters a little different from their commonly named size.

What should immediately jump out from this is that Fargo at 33% is too steep for a typically fit rider to maintain optimal endurance output near the low end of that cadence range at 80 rpm even on a 16 inch unicycle. And a 16 inch unicycle is probably impractically small to ride in real world conditions without wasting a lot of energy on balance tasks.

Essentially, this means that anyone riding Fargo strait up on a unicycle (or a bike without their rear chainring near twice the size of their front) is not using a proper gear for endurance efficiency - instead, they're in a strength challenge mode.

However, in riding switchbacks across the hill as often seen, the energy expended in climb per unit time can be lowered, potentially putting the rider back in an endurance rather than strength mode, and increasing the travel speed such that balance can be maintained more by low-cost microsteering rather than by balance-specific exertions.

Looking at the table below, my suspicion is that a hill in the 20% range ridden on a 24 inch wheel is likely where a unicycle shines most - it looks like it can be ridden fast enough to be "rolled" rather than "stepped" and most of the exertion is still going to climb, so weight matters a lot. (Past posts indicate that while the grade is in this range, the poor surface quality of the Mt. Washington Auto Road presents quite a few challenges to unicyclists itself)
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Old 2017-05-25, 05:05 AM   #37
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Originally Posted by Engineer on a Unicycle View Post
Looking at the table below, my suspicion is that a hill in the 20% range ridden on a 24 inch wheel is likely where a unicycle shines most - it looks like it can be ridden fast enough to be "rolled" rather than "stepped" and most of the exertion is still going to climb, so weight matters a lot. (Past posts indicate that while the grade is in this range, the poor surface quality of the Mt. Washington Auto Road presents quite a few challenges to unicyclists itself)
Speaking as someone who has ridden enormous numbers of 20% grades, on 24" unicycles, 29" unicycles, and bikes: Bikes are faster, and it's not close. There's no human being who can ride up a 20% grade on a unicycle and "roll" it. Have you tried it? Do you have video of someone trying it? Do you have any evidence other than your own assertion that it should be ideal?

For most unicyclists, a 20% grade is in the "I can't ride that" category. For strong climbers, it's an extremely strenuous and slow climb.

Maybe the problem is that you took Physics 101, but never took the lab where you learn how to test hypotheses. Here's the way the lab report should start:

Objective: Given that we observe that bicycles are faster than unicycles on hill climbs, despite the lower mass of unicycles, we intend to test for possible explanations. We will test both physiological and mechanical explanations. Physiological explanations could include differences in ability to provide force to the pedals due to requirements for balance. Mechanical explanations could include differences in translation of force input to forward momentum.

Methods: (you can take it from here)
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Old 2017-05-25, 05:12 AM   #38
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Whew!

Now I'm out of popcorn, so I feel obligated to post something.

Uh.....

I have no ammo from Physics class, since I never took one. All my physics knowledge comes from real-world experiences, and watching a lot of sports and circus. Some things are obvious enough, while others are subtle and probably invisible to someone not familiar with the activity.

Mr. Engineer, I know you're riding on some hills because Manhattan and the Bronx are not flat. But you're a relatively new rider, and if your concentration has been on covering distance, you probably haven't "studied" uphill riding much yet. What is your engineering background, btw? We get all sorts of interesting people in unicycling, with interesting skills and knowledge.

Tom Holub has a ton of hands-on experience with uphill riding, and I'm pretty sure he's right about a bicycle being basically the more efficient vehicle for going uphill under any conditions. As you can see he likes to nerd-out on some of these topics and not yield while he still thinks he can explain something.

Unicycles only have one wheel. They do not track in a straight line, even if you try really hard. An efficient climber learns how to minimize this, but doing so involves expending energy with each pedal "step" to keep the wheel pointed straight. I pendulum my non-seat-holding arm as part of that. On a bike you only need to make some tiny steering inputs to keep yourself pointed in the most efficient direction up the hill.

Engineer's descriptions of riding up the hills seem to involve a constant spin being maintained. Naturally this would be the most efficient way to ride, but your spin starts to break down as the slope gets steeper. As you hit a hill with speed, you can spin into it. but if it gets steep, that spin will slow (unless you have a little wheel), and after that you will be "lugging", with power pushes alternating between little gaps of power as the pedals pass through vertical. As it gets steeper, it becomes more of a series of big pushes, that might have to include little micro-rests between steps. This is a good way to ride up a steep hill that's long.

Anyway, all of that is pretty inefficient. Too bad we can't downshift. 36" wheels are horrible for riding up anything steep. We only put up with them because they're so much better than smaller wheels when the going is flat or downhill.

I've been to Fargo Street once, at the 2014 CA Muni Weekend. I brought my 24" Miyata with 125mm cranks (a "Track" uni), and my 26" Roger Davies carbon Muni with 150mm cranks. I'm not sure which one I ended up using, but I think it was the Muni because it's really light, and had more leverage. I couldn't get very far going straight up (and I consider myself a pretty good uphiller), and switched to zig-zagging to avoid a dismount. I did eventually stop, more than once, to "replenish oxygen" before I made it to the top.

We don't really get to spin "efficiently" up things that are considered steep, unless we're riding relatively small wheels. If I were to seriously challenge Fargo Street, with conditioning and training, I think I might opt for a very light 700c wheel with "medium" cranks. Medium means I think I'd end up with something not real long, and definitely not very short. For that street.

The guy who has the unicycle record (far as I'm aware) for Mt. Diablo, Greg Drummond I think was his name, used a Coker that was super stripped down and lightened. I think he went crazy with a drill on it, though I haven't seen the uni myself. Coker (or any other 36") tires are notoriously heavy, but I think he preferred that to a smaller wheel. I have also ridden a 36" up Mt. Diablo. The higher half of the ride is mostly fairly steep, and much of it is accomplished half a revolution at a time. Then there's that last bit, which is somewhere north of 20% and I could only make it up that a little bit at a time (many stops).

Anyway, the key points I guess I'm making are that unicycles generally don't get "spun" up long slopes that are steep; it usually ends up being a series of pushes; how hard depending on the grade. Also that, because unicycles don't go in a straight line, a unicyclist has to use a bunch of additional energy just to try to keep an efficient line.

Also, with my armchair physics knowledge, every time people apply "textbook" physics arguments to unicycles here, they inevitably seem to make assumptions or leave out small details that aren't actually that small. All the mechanics of riding a unicycle must be hellishly messy in mathematical format.

Anyway, thanks for that fascinating discussion, and feel free to pick my post apart now. I am now armed with a bag of M&M's!
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Old 2017-05-25, 05:45 AM   #39
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Originally Posted by johnfoss View Post
Engineer's descriptions of riding up the hills seem to involve a constant spin being maintained. Naturally this would be the most efficient way to ride, but your spin starts to break down as the slope gets steeper.
That's specifically why I calculated the grades where a fit cyclist (which is to say, not me!) could maintain spinning, based on commonly used numbers for their (not my!) sustainable power output. Reduce them a bit if you like, the point wasn't to show what is possible, but rather what isn't - to explain why the medium-wheel steep-hill climbs we see are out of the spinning regime.

Once you're out of that regime and "stepping" the wheel (and I personally am at far, far lower slopes), you have a nice personal strength challenge, but you're no longer riding efficiently. Likely you're better off walking, except for the lacking feeling of accomplishment.

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Anyway, the key points I guess I'm making are that unicycles generally don't get "spun" up long slopes that are steep; it usually ends up being a series of pushes
That may be tradition and/or because the travel speeds and small wheels for a spinning ascent are insufficient for low-cost stability, but from a sustainable energy output perspective, it's unwise. I think anyone wanting to accomplish the proposed "everesting" would have to find a course and wheel where they could stay in a smooth spinning, seated regime. And yes, it sounds a lot like riding a century up a hill on a 24, which doesn't sound fun at all.

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Also, with my armchair physics knowledge, every time people apply "textbook" physics arguments to unicycles here, they inevitably seem to make assumptions or leave out small details that aren't actually that small. All the mechanics of riding a unicycle must be hellishly messy in mathematical format.
The primary physics dispute concerned the beneficial side of the big wheel flywheel effect, something already well established in threads here more than 10 years ago, and intuitively observable in bowling bowl in a halfpipe type experiments.
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Old 2017-05-25, 10:13 AM   #40
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Originally Posted by Engineer on a Unicycle View Post
I specifically mentioned the existence of 14 and 15 lbs bikes. But you haven't considered taking that lightweight road bike and building a unicycle with the same technology. You'll be leaving more than half of it in hacksaw trimmings on the ground - about the only thing that would end up weighing more is the saddle and tire
I did, even if I wasn't explicit enough about it for you. That bike is all standard off the shelf parts, so let's apply the same principle to the unicycle. In the same way as for many issues on this thread you're dismissing the important point in passing - there's a pound and a half extra right there in the saddle and tyre, quite significant when you're suggesting a 7lb uni for that 8lb saving. It simply doesn't scale in quite the way you seem to think.

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when considering only the power expended lifting the body mass up the hill - the weight of the cycle, friction losses, and the small power used in moving horizontally are not considered. Those simplifications probably make the table substantially less accurate at low grades and larger wheels.
or for unicycles, where the power used for things other than climbing the hill isn't so insignificant!

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increasing the travel speed such that balance can be maintained more by low-cost microsteering rather than by balance-specific exertions.
It's still not as low cost as you seem to assume.

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What is your engineering background, btw? We get all sorts of interesting people in unicycling, with interesting skills and knowledge.
I know you weren't asking me, but if we're throwing around authority, I have an engineering degree. I specialised in electronics and computers, but also studied mechanics and structures, so have what could reasonably be described as degree level knowledge of those. Most importantly though, my training taught me how to correctly apply critical thinking, and not to ignore the important issues for the sake of simplification...

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Unicycles only have one wheel. They do not track in a straight line, even if you try really hard. An efficient climber learns how to minimize this, but doing so involves expending energy with each pedal "step" to keep the wheel pointed straight. I pendulum my non-seat-holding arm as part of that. On a bike you only need to make some tiny steering inputs to keep yourself pointed in the most efficient direction up the hill.
...
Also, with my armchair physics knowledge, every time people apply "textbook" physics arguments to unicycles here, they inevitably seem to make assumptions or leave out small details that aren't actually that small. All the mechanics of riding a unicycle must be hellishly messy in mathematical format.
Right there even without any formal qualifications you get to the heart of the issue.

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Originally Posted by Engineer on a Unicycle View Post
That's specifically why I calculated the grades where a fit cyclist (which is to say, not me!) could maintain spinning, based on commonly used numbers for their (not my!) sustainable power output.
But the issue is that on such a hill where you can maintain a spin on a unicycle (and even with such a spin you're still using lots of energy on micro balancing however good you are) a bicycle would already be going faster than a typical unicycle on the flat. Because such a gradient is way lower than you seem to assume. In order to maintain a spin, the force put into each pedal stroke has to be low enough that you're not outputting that max power you mention. On a bicycle you can just mash the pedals around with no control input.
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Old 2017-05-25, 02:40 PM   #41
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But the issue is that on such a hill where you can maintain a spin on a unicycle (and even with such a spin you're still using lots of energy on micro balancing however good you are) a bicycle would already be going faster than a typical unicycle on the flat.
No they wouldn't - on a steep hill the bicycle's speed is limited by the work required to lift the mass up the hill vs the cyclist's sustainable power output. They will be going at speeds quite ordinary for a unicycle on a flat, and their gear inches will be in the unicycle range. That is not by itself enough to demonstrate that a unicycle is more advantageous, but it immediately puts things into much more direct comparison than they would be on flat terrain. And bicycle race results on 20% grade in the ~8 mph range bear that out (this is after all where they're barely beating the foot runners).

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Because such a gradient is way lower than you seem to assume
There could be errors in my figures, but the first thing I did when I determined them was check against the actual uphill bike race on Mt. Washington, and they are fairly closely for that - just under a mile of climb in just under an hour. A quick look at videos shows mostly a steady spin, with relatively little out of saddle time.

In terms of what could be done on a unicycle, there seems to be a lack of data, as people don't seem to be trying these rides on unicycles with the gearing and crank lengths that would theoretically make a spinning ascent similarly possible, or on terrain that would minimize balance disruptions.

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In order to maintain a spin, the force put into each pedal stroke has to be low enough that you're not outputting that max power you mention. On a bicycle you can just mash the pedals around with no control input.
The power I spoke of was the maximum sustainable power over time, and the goal would be to use that, since it is what the cyclist is supposedly rated at producing for an hour or more duration.

But that's a power for efficient spinning, so the goal was to avoid things like standing on the pedals - and yes, that means the force has to be reasonable, hence a low gear or small wheel is required. The assumption from the bike world inherited here is that maximum sustainable output is achieved by spinning lots of repetitions at limited force, rather than a few at high force.

One thing I want to calculate next is the effective "winch radius" of a pedal powered elevator that would lift the rider upwards at the same rate as the vertical component of those in the table. If this where the same as the crank length, you'd need your full weight on the pedal (and for the full cycle, not just the horizontal peak) so standing and pulling on a bar would be required. If it's below half the crank length, things are more reasonable. Bike cranks of course tend to be a little longer.

And yes, the simplification of looking only at the vertical component of the effort looses validity on the more shallow grades - on the steepest it presents a sort of best-case limit, as things can only get worse when the other losses are included.
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Old 2017-05-25, 03:21 PM   #42
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This is a very interesting discussion, and I'm hoping it doesn't devolve.

Subtle comments which impugn the intelligence/integrity/motives of the other participants aren't helpful to this inquiry; they prevent both sides from thinking clearly and therefore become impediments to getting at the truth.
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Old 2017-05-25, 03:57 PM   #43
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Here are the "winch" numbers you'd need to exert 5 W per lifted kilogram lifting yourself on a pedal powered elevator (they also apply for exerting the same power output when riding on a flat, it's just the "task" is no longer as directly related to the theoretical energy density of the rider).

Cadence vs Winch radius in inches vs % lifted weight pushed on 170's
Code:
30        6.39     96%
40        4.80     72%
50        3.84     57%
60        3.20     48%
70        2.74     41%
80        2.39     36%
90        2.13     32%
100      1.91     29%
The implication here is that turning 170's at 30 rpm, you'd need to push a pedal with an average of 96% of the lifted weight (body + cycle) throughout the entire cycle, to achieve theoretical sustainable power. That's standing on the pedal, yanking on the bar, and then some - clearly inefficient even if achievable at all.

But at 70 RPM you'd only need to exert an average of 41% of lifted weight, and at 80 RPM only 36%. Somewhere in there (actually regardless if on a hill or a flat) the force exerted becomes reasonable for seated spinning, at least on a bike.

The really meaningful question is then, if (on a perfectly smooth paved hill, without wind, etc) a unicycle with the same mechanical advantage could also be spun.

A number of people have argued "no" - but the actual attempts seem to generally be made on unicycles which do not have comparable mechanical advantage in the selected crank vs wheel size, so they don't really answer that question.
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Old 2017-05-25, 09:29 PM   #44
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Originally Posted by Engineer on a Unicycle View Post
but the actual attempts seem to generally be made on unicycles which do not have comparable mechanical advantage in the selected crank vs wheel size, so they don't really answer that question.
An interesting question might be why nobody is using the mechanical advantage which is "theoretically optimal" in events where they're trying to go as fast as possible.

Though let's do the numbers anyway - 5W/kg equates to a climb rate of 0.5m/s, so on the 20% grade you think is optimal that's 2.5m/s speed (5.6mph) and on a 29er with ~2.3m rolling diameter that's a cadence of 65rpm, so only just below what you seem to think is optimal. We're already not far off what people do use to climb hills.

So let's do it from the other direction - when I last did a hill climb I did it on a 29er with 150 cranks, so if we assume 80rpm is optimal (which doesn't seem unreasonable - it's a comfortable climbing cadence on a bike, and going from the 170s to the 150s you're only raising the ratio of lifted weight to 41%) then your speed is 3.07m/s (6.9mph) and the required gradient for 5W/kg is ~16%. So there we have a typical uni used for hill climbing on a typical gradient which meets your optimal mechanical advantage - yet it's still significantly slower than the bikes. Have we put the "they're not using the right gear ratio" to bed?

FWIW the hill climb I did was about a 10% gradient and that was something I could pedal up at a reasonable cadence without being reduced to half revs or zig zagging. I tried the 127 hole, but it was too much of a struggle - I'd be very surprised if I could have happily climbed a 16% gradient smoothly on that setup. I admit I'm not the most skilled uni rider, though I'm not that bad at climbing and I think the only bike I beat was a woman with her child in a kiddy seat - the people who I kept pace with on bike rides were way faster than me.

Because we come back to the same point, you can do all the calcs you like on this, but you're still wasting energy balancing and wobbling on a unicycle.
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Old 2017-05-25, 10:50 PM   #45
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Originally Posted by aracer View Post
So let's do it from the other direction - when I last did a hill climb I did it on a 29er with 150 cranks, so if we assume 80rpm is optimal (which doesn't seem unreasonable - it's a comfortable climbing cadence on a bike, and going from the 170s to the 150s you're only raising the ratio of lifted weight to 41%) then your speed is 3.07m/s (6.9mph) and the required gradient for 5W/kg is ~16%. So there we have a typical uni used for hill climbing on a typical gradient which meets your optimal mechanical advantage - yet it's still significantly slower than the bikes. Have we put the "they're not using the right gear ratio" to bed?
I really appreciate your taking time to run through a scenario. However this still strikes me as a bigger wheel and shorter cranks than the unicycle likely to make it's best possible showing in comparison to a bike with matched gearing, nor are you making a comparison between equivalent setups to begin with.

If we want to argue strictly about the benefits of a unicycle configuration vs. a bike configuration, then I think that the unicycle's best advantage will be on a hill where the bike has to use gear reduction (gear inches smaller than wheel size), probably down around 24. The unicycle should have a wheel matching those gear inches, and be using cranks as long as the bike's.

Otherwise, if the bicyclist is spinning comfortably in an endurance mode, and the unicyclist straining, it's not really a comparison of one wheel vs. two, but rather most prominently a comparison between the power output achievable with two different gear ratios.

In terms of the validity of comparison, there's no way you get a bike vs. unicycle speed difference out of my tables, because they make no consideration of which is being ridden - the only way you get out a speed difference predicted by those tables is to place the two riders in different places on them with different gear ratios.

Quote:
Because we come back to the same point, you can do all the calcs you like on this, but you're still wasting energy balancing and wobbling on a unicycle.
Only once you have the two riders on the same gearing and crank setup could argued losses to unicycle-unique challenges be measured, and presumably we're trying to pick a regime where those are likely to be as minimal as possible.
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Last edited by Engineer on a Unicycle; 2017-05-25 at 11:05 PM.
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