Leg speed

gwhite said:

Regardless of what definition is used, efficiency approachs zero as
cadence approaches zero*, and also approachs zero as it increases
"without limit." So there has to be at least one "max" between these
("ultra-low and ultra-high"😉 although that says little regarding how
"broad" a peak(s) may be.

* Otherwise a discontinuity would have to exist between zero cadence --
which produces zero power -- and some arbitrarily low cadence. One
could say zero cadence is indeterminant, but I see that as pointless
quibble.

It's not a pointless quibble, and your reasoning is completely unsound.
Efficiency does not necessarily approach zero as cadence approaches
zero. We haven't established exactly what the relationship is between
efficiency and cadence, but if, for example, they vary inversely to one
another (efficiency = N/cadence), efficiency would most definitely NOT
approach zero as cadence did, and there would be no "local maximum",
either:

http://melusine.eu.org/syracuse/bbgraf/albums/courbes_03/hyperbole.jpg

Again, that might not be the relationship, but not knowing otherwise
you can't say efficiency approaches zero as cadence does. And thinking
about it intuitively, I highly doubt that it does.

-Smurf

Ewoud Dronkert said:

gwhite said:

Regardless of what definition is used, efficiency approachs zero as
cadence approaches zero* [...]

* Otherwise a discontinuity would have to exist between zero cadence --
which produces zero power -- and some arbitrarily low cadence. One
could say zero cadence is indeterminant, but I see that as pointless
quibble.

But that's just it. The fraction a/b where both a and b tend to zero, is
usually indeterminate. Zero over zero is not zero.

Yes mr smartypants, that's why I said someone could quibble.

L'Hôpital's will save me! LOL

There is the definitional problem again. If it is Pout/Pin, then it is
zero by definition at zero cadence. (Pin at torque stall is not zero,
Pout -- rubber meets the road -- is zero.) For the Pout/Cadence
"cadence efficiency definition," then it is indeed 0/0 at zero cadence,
but that doesn't mean it still isn't zero. Zero is a finite number and
some indeterminate problems have finite answers -- so the answer could
be zero.

I don't know if a discontinuity exists. But even if it did, it still
doesn't say that thermodynamic efficiency doesn't eventually start going
down at some point of decreasing cadence. If you get to a point where
thermodynamic efficiency is still increasing as cadence decreases, but
insufficient power is delivered to allow a bike to move forward
(overcome all losses), or balance, then perhaps that could be called a
discontinuity (but the limit there happens simultaneously: we torque
stall even if we don't lose balance). But at such low powers, we get to
the problem with the thermodynamic definition -- it lost its usefulness
and practicality to the bicyclist.

I can't think of where a bicyclist would care about such a limiting case
of high thermodynamic efficency but ultra-low cadence. This is why the
other definitions came into being. At the point we let go of the
thermodynamic definition, we can define it however we want, with the
only constraint being that it makes sense for the "task at hand."

gwhite said:

I can't think of where a bicyclist would care about such a limiting case
of high thermodynamic efficency but ultra-low cadence.

Right. That's why Rompelberg got towed to well over 100 kph before he
could turn the gear himself on his speed record attempts.

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Torched Smurf said:

gwhite said:

Regardless of what definition is used, efficiency approachs zero as
cadence approaches zero*, and also approachs zero as it increases
"without limit." So there has to be at least one "max" between these
("ultra-low and ultra-high"😉 although that says little regarding how
"broad" a peak(s) may be.

* Otherwise a discontinuity would have to exist between zero cadence --
which produces zero power -- and some arbitrarily low cadence. One
could say zero cadence is indeterminant, but I see that as pointless
quibble.

It's not a pointless quibble, and your reasoning is completely unsound.
Efficiency does not necessarily approach zero as cadence approaches
zero.

What is thermo-efficiency at torque stall, or even approaching it? What
is cadence-efficiency at torque stall, or even approaching it? That's a
problem, because torque stall implies some "substantial" power output is
required. What is the efficiency at a speed insufficient to balance the
bike?

The "problem" is multidimensional. For example, we care about things
like Pin, Pout, cadence, time, torque, and efficiency. We always care
about Pout, sometimes as a dependent and sometimes independent
variable. To talk about efficiency without caring about the absolute
level of Pout is folly.

Efficiency itself says nothing about power output. 100W/500W = 1W/5W,
these two ratios show the same efficiency: bike riders care about a
total of 100W of output, they don't care about a total of 1W of output.
Basal metabolic rate probably implies that efficiency recedes anyway at
increasingly (decreasingly?) lower power, regardless of cadence.

Quoted message said:

We haven't established exactly what the relationship is between
efficiency and cadence, but if, for example, they vary inversely to one
another (efficiency = N/cadence),...

WTH is N? We have established we don't care about vanishing levels of
output power when they are associated with high efficiency, regardless
of how efficiency has been defined. IOW, as a cyclist I could not care
less if I achieve 99% efficiency if it comes at a cost of outputting < 1
pW.

"Mr. Keynes was a dangerously unsound thinker." -- Benjamin M. Anderson

"Ernst Noch" <[email hidden]> wrote in message
news:[email hidden]...

Quoted message said:

Relating to the "Ullrich grabs a bite" thread, this might explain Jan's
low cadence. He's afraid of losing too much weight in the TdF and hurting
is winter shape.

Earnst - that's just mean.