makeinu

I think you guys are getting distacted. Obivously energy is always conserved. Apart from certain loss processes (drag, friction, etc) energy will always remain within the system.

However, the body is primarily restricted by the power it can deliver, not the energy. It doesn't take any more energy to move an elephant up and down a ladder than it does a paperclip (zero in both cases), but your body will waste a lot of energy in order to muster the necessary power to move the elephant.

Furthermore, it's better to retain control over acceleration/deceleration with your legs. If you invest energy in getting a large wheel up to speed then you risk losing that energy if you need to brake, but if you keep the energy in your muscles then you don't risk as much. On average the higher risk should translate to more energy lost due to braking.

Speedo

Even more different if (since this is in folding bikes) the biker folds the bike puts it on his back and carries it up the hill. But in any of those cases examining the energy is useful. It's just a technique, it reveals relationships in the system. How the rider experiences the system, may be misleading. It's subjective.


Well, no and yes. None of the friction and drag forces are "constant". Bearing friction will be proportional to rotation rate, and hence speed. Aerodynamic drag is proportional to the square of the relative air speed. I think that tire rolling resistance is proportional to speed, but don't know for sure. But, yes I agree that on level ground, in no wind, if you stop applying power they will cause the bike to slow down.

Okay, we are at the point where we agree that the bike is "up to speed". In getting the bike up to speed I paid a penalty for rotating vs non-rotating mass. Now I'm up to speed, and I deliver power to the system in a non-constant way. In the subjective way that I experience this, unless I am piston pedalling up a hill, I don't experience this as pulses. Why not? Well, we need to look at (sorry ) energy. The inertia of the system, that is, the energy stored as kinetic energy smooths the motion over the non-constant input power. We aren't really in a steady state, but have small variations around a given speed. With each power pulse the energy delivered is distributed between heat (friction, and drag) and kinetic energy (we speed up a bit). In the between pulse period, the kinetic energy goes to heat (friction and drag) and we slow down a bit.

All you need to recognize is that kinetic energy is kinetic energy. It can be stored as forward motion, and it can be stored in a rotating object. It doesn't matter which. Consider the example of the bike with massive flywheel wheels. At speed, with each power pulse I would put most of the energy delivered into rotational kinetic energy. Very little would go into translational kinetic energy, so I would experience very little speed change, but I have stored the same amount of energy. Between pulses the very slightest of slowing the the flywheel will deliver enough energy to overcome friction and drag. So I would experience very little slowing between pulses. Note, though, that there is not more power going into the system then there is in the light wheel case.

Note to all respondees: please don't get on my case as advocating giant flywheel wheels. I'm not. They are just a specific example that is useful because they are so extreme!

Speedo

Uh, sorry, but you don't seem to be clear on energy, power, or the realtionship between them.

To move a mass M, up a ladder, increasing it's height by h, in a gravitational field with gravitaional acceleration g, will take energy M*g*h. So, if an elephant is a mass M and a paperclip is mass m, then the difference in energy required would be (M-m)*g*h.

Power is energy per unit time. It is the rate of energy delivered, or extracted from a system. You are right in that a rider will be limited in how much energy he can deliver per unit time. His power does have some finite limit.

Braking is an issue. In the other discussions I was arguing that in near steady state, you get the energy out of the system, in overcoming drag and friction, that you put in. If you hit the brakes, then all that energy goes to heat. But remember, hitting the brakes you give up kinetic energy from ALL sources. That includes the translational kinetic energy. The translational kinetic energy of a 150 pound biker on a 25 pound bike dwarfs the rotational kinetic energy in the rotating wheels. If you are doing nothing but start and stop, your wheels are a small part of your worries!

Speedo

dalmore

They most certianly are constant in the sense that they are not pulsing.

It's obvious that you are not getting it. The bike is never at a constant speed. Never. It's either speeding up or slowing down.

makeinu

Yes, it reveals precisely that you are leaving out the most important energies. It reveals that smaller wheels, like gearing, weight, and almost everything else about a bicycle are not meant to reduce mechanical losses, but biomechanical losses. It reveals that we are better off considering the power than the energy.

It smooths the pulse at the output. The input is still very much experiencing pulses, which affects the sources ability to generate energy with minimal losses. Matching the load seen at the input to the source is the most important factor and it is one you are completely ignoring. It's the entire reason for cycling in the first place!

Yes, there is more power going into the system then in the light wheel case. Assuming constant gearing, every turn of the pedals needs to provide energy to the flywheel in addition to linear acceleration of the bike. The linear acceleration of the bike is not magically reduced to compensate. The speed remains the same. The additional energy stored in the flywheel comes from your legs. As you keep saying, you get it back later, but your legs don't care about later. You can't convince your legs to let you jump into outer space by promising that you won't walk for the next 10 years.

Speedo

If what you say below is true, that the bike is never at a constant speed then they must be pulsing. They are proportional to the speed, or the square of the speed. As the speed changes, they change.


No, no, yesterday was Valentines Day. I did!

I'm not claiming that the bike is at constant speed. I'm only saying that if you look at what happens as the bike speeds and slows with the pedal pulsing there is energy that goes into kinetic energy on the speed up, that comes out on the slow down. If the bike had no kinetic energy, then it would stop dead as soon as no power was put in. But the kinetic energy of the bike goes toward overcoming friction, rolling resistance and aero drag, so the bike continues to move forward (slowing as it goes as the kienetic energy is turned to heat). Some of that kinetic energy disappated is the rotational kinetic energy of the wheels.

Speedo

makeinu

What do I need to clarify? As you seem to know, power is energy per unit time. The body is unconstrained in the energy it can produce, but it is constrained in the power it can produce. Producing a large amount of power makes you tired. Producing a large amount of energy only makes you tired if it corresponds to a large amount of power.

That's just to go up. As you argued in the case of the wheel, the energy is returned when you come back down. However, the energy lost by the source (your body) due to heat, etc is much greater with the elephant. The reason more energy is lost is because the maximum power is greater when lifting the elephant.

The energy required to go up the ladder is proportional to the maximum power. That is why you intuitively considered only the energy required to go up the ladder and not the total energy at the end of the day. Formally this is the maximum power.

Likewise, in the case of bicycle wheels, we need to consider the energy required to get up to speed (not the total energy at the end of the journey) because this is proportional to the peak power provided by your legs.

Not only is the power limited, but, more importantly, higher power results in greater losses. Your body is less efficient when its required to deliver too much power.

dalmore

I'm not sure where the disconnect in our terminology blies. But the air causes aerodynamic drag does not vanish for an instant while I'm not appliying force on the pedal. That air is still resisting the forward movement of the bike and causing it to slow. If that's not "constant" perhaps "consistent" is a better word.



Exactly!! And because of this the rotating mass is ALWAYS having more of an effect that non-rotating mass. After all you did a fine job of proving that back in Post number 1

Speedo

Consistent I'll buy. It will change a bit around some mean value in the same way that the speed will change a bit around a mean value.


I think that you are just freaked out by the idea that energy can go into and out of a rotating mass. Let's look at somebody pushing a sled. No rotating mass. I'm trying to maintain my sled at a constant speed. I have to overcome drag, and aero drag. With each push I add a little kinetic energy to the system: we go faster. Between pushes that kinetic energy is disapated through runner drag and aero drag, and I slow down a bit. Is this shocking? Are we surprised? No! It's inertia. Momentum. Take your pick of terms. It was the fact that I had the kinetic energy that allowed me to move forward between pushes. The momentum of the package, me and the sled, carried me along.

Okay, I'm on the bike. I'm "at speed". "At speed" is near constant, but will vary with how I am pushing on the pedals. With each push on the pedals I add kinetic energy. I am adding kinetic energy not just in translational kinetic energy, but in rotational kinetic energy of the wheel. There is not only translational momentum, but angular momentum as well. Just as in the sled case, it is the momentum that carries us forward between pushes. The fact that some of your push energy went into angluar momentum means that at that it did not go into translational momentum, so your speed increase at that push was not as much at the push. BUT that angular momentum will now help carry you through the non-push interval, and your speed will not decrease as much.

When you are trying to maintain speed, what you put into angular momentum in little pulses, comes back to you, helping carry you forward, between the pulses. You don't lose because of it.

Speedo

Energy is important as an analysis tool to understand the inter-relationships between the parts.



The input is the cyclist adding energy in the most efficient way they can manage. I'm only showing that that energy, once added is not squandered by rotating masses.



No, it's not magic. It's physics.

Speedo

You know, at some point above I introduced the concept of massive flywheel wheels to help with the idea of angular momentum, and energy stored in the rotating mass. They were just an example. I'm not recommending that people go out and put massive wheels on their bikes. I'm sorry for any misunderstanding.

As I noted earlier:

I'm always happy to concede that light is better than heavy. My only claims are that:

1) When accelerating the bike it is, in fact, true that a pound in the wheels is like two on the bike.
2) After the bike is up to speed, there is no continuing penalty due to the fact that the mass of the wheel is rotating.

dalmore

ok I guess I see what you are saying on this point ... For the sake of clarity say I'm averaging a constant speed of 20 mph by bouncing between 21 and 19 mph. You are saying that the rotational energy change when dropping from 21 mph to 19mph is equal to the rotational energy change of going back from 19 to 21 so it's a net zero game. ok, I can swallow that.

BUT that is not to say that I agree that rotating mass is the same as non rotating mass except when accelerating. Only that I don't have the math skills to argue the point.

makeinu

Not when you leave out the most important energies, such as the energy lost by the body at a given power level.

Which is more efficient with the smaller wheel.

This goes without saying. There is no electricity involved. So the only way to squander energy is by heat. The wheel obviously isn't going to heat up merely because it's rotating. So the energy will be squandered in the brakes via the excess momentum and in the heating of your muscles due to the increased power requirement.

The linear acceleration certainly isn't physically reduced to compensate either. At a given cadence, the bike will only go slower if you reduce the gear. Otherwise you will have to pedal harder, just like you have to pedal harder with a heavier bike, just like you have to pedal harder when going up hill. Pedaling harder and deviating from optimal cadence both waste more biomechanical energy. Therefore using a bigger wheel wastes more biomechanical energy.

I'm not saying that you're recommending that people go out and put massive wheels on their bikes. I'm saying that the fact that larger wheels do not themselves dissipate energy is irrelevant to the conclusion that a bike with larger wheels will waste more energy.

Yeah, but you are so occupied with analyzing the energy that you're missing the reason why light is better than heavy. Your missing the most important inter-relationships which you claim your energy analysis is so helpful to understand.

Ok, but the bike will most likely never be "up to speed". Most of the time riding is probably spent pedaling/accelerating and braking.

jur

I'm with you on this one Speedo.

Looks like it is going to require a detailed simulation to decide it. I have a simulator, just need the math to put into it. I'll see if I can squeeze in the time. Or maybe you can furnish a starting point?

We will need equations for:
forward motion, with drag from wind resistance;
model of torque supplied by legs on pedals, I am thinking a sine wave with variable amplitude and offset may do here;
angular momentum converted to linear momentum as a function of wheel diameter, with wheel moment of inertia variable;
maybe some drag from bearings;

That will probably do.

Then the sim should show if the terminal velocity for a steady state torque model for cases of different wheels is different.

[edit] An approach from conservation of energy is going to be most reliable and easy.

Speedo

Nowhere have I said that the rider would have to provide more power then they otherwise would. I have no idea where you've gotten that idea from.

The rider outputs his normal power level. A bike with relatively heavier wheels acclerates more slowly, so takes longer getting to speed. Once at speed the rider is again providing the same amount of power, and riding the same speed. You can say that the overall consumed slightly more energy because the rider out put the same amount of power and thus lost time in the accelerations, or preserved the acceleration time by outputing more power. In either case it is not a dramatic difference.



Which smaller wheel? The only thing I've said about a smaller wheel is that before I ever looked at this I would have guessed that a smaller wheel got something of a free ride. The analysis shows, and various other posters have confirmed, that the wheel diameter alone doesn't make a difference. But a smaller wheel may naturally be a lighter wheel, and I'm always happy to concede that lighter is better.



Which increased power requirement?


Aside from my slight aside above who said anything about a bigger wheel size?

[QUOTE=makeinu]I'm not saying that you're recommending that people go out and put massive wheels on their bikes. I'm saying that the fact that larger wheels do not themselves dissipate energy is irrelevant to the conclusion that a bike with larger wheels will waste more energy.[/QUOTETE]

Do you think that we are having a discussion about wheel diameter? I don't really have anything to say about wheel diameter.


Please go back and re-read the thread. I really don't have any particular argument about wheel diameter. I would guess that the specific issue of large vs small diameter is going to be lost in the specifics of the particular wheels. i.e. the rim mass, the tire mass, the tire characteristics, number and type of spokes ...

If you are spending all of your time accelerating and then braking then you have my sympathy. But if that's the case the energy in and out of the wheel angular momentum will be overwhelmed by the linear momentum swings.

Speedo

makeinu

I got the idea that the rider would have to provide more power from the physics of the situation. If you fix the power provided by the rider then the rider's candence must be reduced to compensate for the reduced acceleration. Alternatively you can fix the cadence and the power provided by the rider must be increased to compensate for the heavier wheels. Either way the biomechanical efficiency is decreased.

As we've already established. The relevant difference between larger and smaller wheels is the difference in rotating weight, which is what this whole thread is about. All else being equal larger wheels are heavier wheels and smaller wheels are lighter wheels.

Then why do you keep going on about the mechanical energy being independent of the rotating mass? Either the total mechanical energy is important and, since rotating weight does not affect the total mechanical energy, rotating weight is unimportant or the total mechanical energy is irrelevant and lighter is better. Lighter is better because the conservation of energy you keep talking about is irrelevant.

Obviously small changes in weight will give small changes in performance regardless of whether the weight is linear or rotating, but braking and accelerating must be awfully popular among most cyclists. Why else would light weight and low momentum be considered so important?

Speedo

Why? A slowly accelerating rider is going through the same range of speeds as the fast accelerating rider just a lower rate. The range of strategies for matching available power to load (standing, shifting gears, ...) are available to be used by any rider.



The thread is about rotating vs non-rotating mass. Some posters, yourself, included have made that claim. I would agree that if I were out to make the lightest wheels possible, I'd go for smaller rims, but that doesn't mean that small wheels must be lighter. On my small wheels I run wider tires and the rim itself is wider. I may have to try measuring, but I would not immediately conclude that my 406 wheels (and tires) are lighter than my 700Cs.

Also, there must be some other things coming into play, otherwise we's all be riding those bikes that use skateboard wheels! Time trial bikes use smaller wheels than 700C. Maybe that's the optimum. I don't really have a strong opinion one way or the other in the big vs small wheels.


Please point me to where I said that.

Total mechanical energy is important. The thing I've been blathering on about is that riding at steady speed the mechanical energy that goes into rotating the mass doesn't hurt you. This is even true when steady speed is (as pointed out by Dalmore) not truly steady.

Amen to your first point.

As to your second, well, my party line has been lighter is better. How light? Damned if I know. I have a friend who insists on spending thousands of dollars every year to have the latest lightest widget. I suppose he's got the dosh, and likes to bike, so what else is he going to spend it on? I have other friends who will ride nothing but 25 year old bikes that are as much as 6 or 7 pounds heavier than a middle of the line new bike. I must admit that I learned much about the mechanics of bikes to puncture Mr. Lightbike's claims about his nanogram wheels!

Speedo

I don't think it will convince anyone who isn't ready to be convinced. The model is going to be a set of diferential equations. People will have to buy the model as well as the idea of numerically integrating the equations to get a result.

On the other hand, I've never been one to let practical considerations stand in the way of writing equations! I take the challenge!

I am going away this weekend. Maybe next week I can put something together. Do you want equations? Are you going to put this into Matlab or something similar? I have Matlab and can write code that can be integrated by their differential equation solver.

If Dalmore and makineu (I think they are the last who haven't been driven away by the pedantry) are interested we can post the pieces here.

Speedo

makeinu

True, but when you isolate each parameter you get the same answer: Heavier bike makes a tougher ride for the cyclist and every gram on the wheel counts double.

Good point, but if your 406s aren't lighter than your 700Cs then I wouldn't say it means that smaller wheels aren't necessarily lighter. It means that wider wheels are heavier or aluminum is lighter than steel, etc. Every physical statement must be qualified with "all else being equal". If all else isn't equal then almost anything could happen.

What's there to simulate? Everything is analytically solvable. It's just boils down to how large the various effects are when subject to real world nonidealities.

Speedo

I wondered if that were true. It might be doable if one assumed constant input power, but one of the questions to be addresssed was what happend in the case of non-constant input. The pulsing pedal stroke case. It is a periodic input, maybe Fourier analysis. Not being very smart, I often resort to numerical integration.

If you have a solution feel free to contribute it!

Speedo

jur

I have PSpice, an electronic circuit simulator. But it can take stuff like laplace transforms equations directly and produce graphical output from those. I use it extensively.

The diff. equations can be easily converted to Laplace (s-domain) format. But perhaps it is easiest, instead of constructing a diff eq model and putting that in, to construct a model with all the various pieces connected with a block diagram format.

makeinu

I'm an electrical engineer not mechanical. I don't have time to struggle with figuring out what the right mechanical equations should be (if you remember my first post in this thread I didn't even know whether the equation you gave for rotational inertia should be in radians or not), but if you write down the differential equations then I will solve them.

However, I still think the most important factor is how efficiently the body can deliver power for different loads. Biomechanical losses are very tricky to analyze.

Fear&Trembling

As things currently stand 700c wheels are more often than not lighter than 20"/16" wheels - this is because of the r&d that goes into racing. Look at some of the weights on weight weenies (if you are really that sad!).

If you want to see heavy rotating masses check out this guy and his machine:
https://www.wolfgang-menn.de/sosenka.htm

Speedo

Well, I have a write up of the equations. The basic differential equation is based on solving an equation of the power sources and the power sinks for the bike acceleration. The bike acceleration can then be integrated in whatever tool you have available to get speed vs time.

The model includes a pedal power model, rotating and non-rotating inertia, a hill (if desired!), and aerodynamic drag. The model doesn't include rolling resistance and bearing friction. I found references for those terms, but thought that going without them would be better for a start. I'm not sure there is a closed form solution to the differential equation. It's complicated enought that I'm just going to go after it numerically.

What I have is a Word document that has been converted to pdf. I used Microsoft equations; this won't look very good in plain text. I'm not sure if I can paste either of those here. I think I can covert the individual pages to an image. If I can't turn it into jpegs, I'll just try sending it to interested parties...

Speedo

jur

I am off on my 3 week cycle tour of Tassie tomorrow, so won't be able to look at this until end March. I also came up with a block diagram (scribbled on paper), so we can compare notes when I get back.