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PostPosted: Tue Jul 31, 2012 2:08 am 
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[quote="HammerTime2"][/quote]

You don't say anything about the first sentence of my post? Can you point me out from where you have inferred that I don't know what I'm talking about, professor?

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PostPosted: Tue Jul 31, 2012 3:09 am 
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All of this discussion of the benefits of constant speed implies you want as much rotational inertia as possible, assuming speed isn't safety-limited.

Image

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PostPosted: Tue Jul 31, 2012 3:47 am 
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Epic-o wrote:
You don't say anything about the first sentence of my post? Can you point me out from where you have inferred that I don't know what I'm talking about, professor?
And your first sentence was
Epic-o wrote:
If the second derivative of the speed is positive, the acceleration of the cyclist always increases. That's not true
Power is a strictly convex function of speed, and is the integrand, to wit, the function to which Jensen's inequality is applied to deduce a conclusion relative to energy expended (integral of power) as a function of speed as a function of time. The result of which, under the stated assumptions, is that constant speed uniquely minimizes energy expended for a given average speed. In fact, even though the quantitative impact of departure from constant speed would differ, the same argument would hold via Jensen's inequality if the aerodynamic resistance were quadratic or quartic rather than cubic, such is the beauty of the approach. In fact, any exponent greater than one (whether an integer or not) would "do the trick" here.

Note that if power were a linear function of speed, then it would still be convex (but would not be strictly convex), and Jensen's inequality would still hold, but without strict inequality, and therefore the constant speed solution would still minimize energy, but would not be the unique solution to do so.


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PostPosted: Tue Jul 31, 2012 4:08 am 
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Unless there's wind on the course...

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PostPosted: Tue Jul 31, 2012 4:25 am 
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HammerTime2 wrote:
The result of which, under the stated assumptions, ...

As for those stated assumptions,
HammerTime2 previously wrote:
Ignoring the transient effect at the beginning of a ride, and assuming constant wind, constant surface, and no hills ...
And to clarify, by "assuming" constant wind, I mean a constant wind velocity vector (speed and direction) relative to the rider, and so am ruling out, for instance, an out and back course with meteorologically constant wind, but for which the wind velocity vector relative to the rider would not be constant over the duration of the ride. Given a head wind in one (say, the out) direction and a tail wind in the other direction, then a constant speed would not minimize energy expended. For a similar reason, I assumed no hills.


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PostPosted: Tue Jul 31, 2012 5:16 am 
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HammerTime2 wrote:
Epic-o wrote:
If the second derivative of the speed is positive, the acceleration of the cyclist always increases. That's not true
Power is a strictly convex function of speed


Do you notice that there is some conflict here? v(t) isn't a convex function so Jensen's inequality can't be applied. If you play with some type of cycling physics simulator, you will see that the jerk/jolt isn't always positive (so v(t) isn't convex) and the second derivative of the power to overcome aerodynamic drag isn't either

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PostPosted: Tue Jul 31, 2012 12:32 pm 
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Power is a convex function of speed, and is the integrand, so Jensen's inequality can be applied.

Epic-o, Jensen's inequality is a powerful though relatively simple tool in mathematics if understood and applied properly. I understand it and know how to apply it, while you don't. Let's leave it at that.


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PostPosted: Tue Jul 31, 2012 3:46 pm 
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HammerTime2 wrote:
Power is a convex function of speed, and is the integrand, so Jensen's inequality can be applied.

Epic-o, Jensen's inequality is a powerful though relatively simple tool in mathematics if understood and applied properly. I understand it and know how to apply it, while you don't. Let's leave it at that.


Ok HammerTime2, you haven't given any correct argument about why the speed function is convex yet. I'm tired of your condescendency so I give it up

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PostPosted: Tue Jul 31, 2012 3:55 pm 
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Thanks Epic-O, DJ and Strobbekoen for the analysis! Super helpful stuff (initially it was over the top of my head, but translating it helped immensely).

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