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Frank Day said:
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1. It will take less energy to make them go around one revolution as the up and down excursion of the thigh is less by twice the crank length difference.
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Well, I am waiting for you guys to show me where that statement is wrong. You can show me the math or you can show me a study. But, last I looked F=ma. If at BDC or TDC each thigh is stopped and at 90º later it is moving up or down at maximum velocity the "force" to cause that "acceleration" had to come from somewhere. Show me how it wasn't the muscles and I will concede defeat on this point.
Ok, first ever post here (man, it's noisy).
It seems you don't have the whole picture clear regarding the mechanics of the pedaling.
First, there are the conscious/unconscious muscle contraction forces that produce the driving power.
Second, there are the mechanical forces from the dynamics/motions of the limbs and cranks/pedals.
Read this out loud: Those two groups of forces are independent and superimposed
Mechanical forces: Yes, there is a metabolic cost to have to apply the forces fast (and maybe precise in time, depending on technique) with a high cadence/pedal velocity.
No, there is (virtually) no metabolic cost to continually stop and accelerate the limbs in a pendulum or circular way.
That would be true, if the total kinetic and potential energy of the system remained constant. In this instance it does not. So, the movement of the thigh cannot be analyzed as a pendulum (where the total energy of the system is constant less friction losses) even though it might superficially resemble one. So, if the energy of the parts is not kept constant when energy is lost it must be either lost as work or lost as heat. While I will agree that some is converted into work (see below) I submit that most is lost as heat. Unless you can show that the loss of energy in the thighs is totally accounted for by transfer to other elements in the leg (or to the bicycle as work) your analysis is wrong.
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You focus on the fact that the thighs are stopped and accelerated. There's a force required to accelerate the thighs, yes. That same kind of force, but the other way, decelaration, is being put to good use by adding extra pedal force when the thigh is being stopped
by the crank. Frank, ponder on this...
Well, that would all be cool if only the forces decelerating the thigh were tangential to the circle (in the direction of motion). They are not, AFAIK, so all that effort cannot be recovered as work but is, rather, lost as heat. Unless, of course, you can show me some work to demonstrate this is not the case.
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On top of that, the cranks work as elastic KE converters. Part of the momentum of the thighs is converted to a similar (but lower) momentum of the shanks (as in lower legs).
Thighs stopped = fast shanks.
Thighs fast = shanks stopped.
Yes, but because
the masses are substantially different but the speeds are much the same it is not possible to convert all of the energy contained in the thighs at maximum speed to the lower leg (which is moving up at the same time at pretty much the same speed). Unless, of course, you can show me some math that demonstrates it is possible. I look forward to seeing that. The problem with your simplistic analysis is you are ignoring the different masses and different motions (the lower leg is much closer to a circle, like the foot so the total energy variation of the lower leg is fairly small both because of the smaller mass of this element and the elliptical nature of the motion) of the different elements.
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To complete the picture, the feet and shoes rotate with basically constant KE/momentum.
I would agree. The feet and shoes act pretty much as a spinning disk or rod.
I disagree, see above. Show me the math that proves your point.
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Muscle forces: With just a basic level of technique and a suitable cadence, one can fire the muscles in symphony with the given motion of the pedals. Given, in the sense that the KE of the "vehicle" is in the order of 10-100 times the energy of one "pedal push", meaning you can't change the motion of the pedal "within the push" much no matter what you do.
Yes, but you can change the direction of the push to be either more or less tangential to the pedal circle. You would agree, I presume, that forces directed non-tangentially are "wasted". So, while it doesn't take much more than a "basic level" of coordination to make the pedals go around (3 yo's do it quite nicely) it seems to me that there is lots of room for efficiency improvement.
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So, the motion of the pedals is given, and hence the mechanical forces from the motions of the limbs. Superimposed on that are the muscle forces that do the actual work. This superimposion works perfect and unnoticed for a wide variety of cadences and power levels since the muscle forces always "overshines" the mechanical forces and the system never "backlashes".
If the cranks are connected, one leg can help the other.
Yes, that is part of the problem. It is very difficult to correct the muscle force direction inefficiencies if these inefficiencies are masked by overwhelming forces coming from the other side.
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Very high cadence: The system can fall apart in the very high cadence/lower power scenario. As the cadence increases, so does the acceleration of the thighs, proportional to (crank length) x (cadence)^2. If the muscle forces that makes it through to the pedals are lower that the acceleration forces required, the superimposion becomes "negative" and there is a "discontinuity" in the motion, unless you are riding a fixed (which will rob "vehicle KE" to get it done).
If you would actually sit down and do the energy calculations you will see that the energy requirements to make the pedals go around one revolution with zero outside work being done vary with the square of the cadence, making the power losses vary with the cube of the cadence.
It is very clear to me, I have done all this work. You ought to try it rather than your thought experiment.
And, of course, you have the problem of explaining all those studies that show overall cycling efficiency varies with cadence, especially the part where efficiency starts to drop above a "most efficient" cadence.
And, then there is this other way of looking at the issue.
The efficiency of the contracting muscle is about 40%. The overall efficiency of most cyclists at the wheel is about 20%. You can account for 1-2% of that difference through drive train losses. Account for all of these losses. I look forward to seeing how you do it without invoking pedaling motion losses.
edit: however the problem is analyzed one should come up with the same asnwer if the analysis is done correctly.
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Frank,
An original Ironman and the Inventor of PowerCranks