HOT: Active* forum members generally gain 5% discount at starbike.com store!
Weight Weenies
* FAQ    * Search    * Trending Topics
* Login   * Register
HOME Listings Blog NEW Articles FAQ Contact About




Post new topic Reply to topic  [ 54 posts ] 
Go to page Previous  1, 2, 3, 4
Author Message
PostPosted: Tue Jul 31, 2012 2:08 am 
Offline
User avatar

Joined: Tue Nov 20, 2007 7:18 pm
Posts: 589
Location: Lyon
[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?

_________________
BIKE DESIGN AND +

http://cds-0.blogspot.com/


Top
 Profile  
 
Posted: Tue Jul 31, 2012 2:08 am 


Top
  
 
PostPosted: Tue Jul 31, 2012 3:09 am 
Offline
User avatar

Joined: Wed Sep 26, 2007 1:57 pm
Posts: 7371
Location: San Francisco, CA
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

_________________
http://djconnel.blogspot.com/
Fuji SL/1
\


Top
 Profile  
 
PostPosted: Tue Jul 31, 2012 3:47 am 
Offline
User avatar

Joined: Thu May 04, 2006 4:43 pm
Posts: 5096
Location: Wherever there's a mountain beckoning to be climbed
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.


Top
 Profile  
 
PostPosted: Tue Jul 31, 2012 4:08 am 
Offline
User avatar

Joined: Wed Sep 26, 2007 1:57 pm
Posts: 7371
Location: San Francisco, CA
Unless there's wind on the course...

_________________
http://djconnel.blogspot.com/
Fuji SL/1
\


Top
 Profile  
 
PostPosted: Tue Jul 31, 2012 4:25 am 
Offline
User avatar

Joined: Thu May 04, 2006 4:43 pm
Posts: 5096
Location: Wherever there's a mountain beckoning to be climbed
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.


Top
 Profile  
 
PostPosted: Tue Jul 31, 2012 5:16 am 
Offline
User avatar

Joined: Tue Nov 20, 2007 7:18 pm
Posts: 589
Location: Lyon
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

_________________
BIKE DESIGN AND +

http://cds-0.blogspot.com/


Top
 Profile  
 
PostPosted: Tue Jul 31, 2012 12:32 pm 
Offline
User avatar

Joined: Thu May 04, 2006 4:43 pm
Posts: 5096
Location: Wherever there's a mountain beckoning to be climbed
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.


Top
 Profile  
 
PostPosted: Tue Jul 31, 2012 3:46 pm 
Offline
User avatar

Joined: Tue Nov 20, 2007 7:18 pm
Posts: 589
Location: Lyon
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

_________________
BIKE DESIGN AND +

http://cds-0.blogspot.com/


Top
 Profile  
 
PostPosted: Tue Jul 31, 2012 3:55 pm 
Offline
User avatar

Joined: Wed Oct 31, 2007 10:32 pm
Posts: 7412
Location: Los Angeles / Glendale, California
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).

_________________
Exp001 || TeamLACBC


Top
 Profile  
 
Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 54 posts ] 
Go to page Previous  1, 2, 3, 4


Who is online

Users browsing this forum: Google [Bot], Spinnekop and 43 guests


You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot post attachments in this forum

Search for:
Jump to:  

   Similar Topics   Author   Replies   Views   Last post 
There are no new unread posts for this topic. 2012 Ridley Helium - 2012 BH G5

in Road

1 Lite Ghisallo

4

677

Mon Aug 11, 2014 5:02 pm

willieboy View the latest post

There are no new unread posts for this topic. New aero test: 12 aero frames vs 12 "unaero" light frames

[ Go to page: 1 ... 17, 18, 19 ]

in Road

fa63

280

23576

Sun Oct 19, 2014 7:37 pm

dunbar42 View the latest post

There are no new unread posts for this topic. LiteSpeed 2012 C1R

in Road

Chubby

1

470

Sun May 25, 2014 9:44 pm

kode54 View the latest post

There are no new unread posts for this topic. 2012 Cannondale EVO Hi-Mod top tube

[ Go to page: 1, 2 ]

in Road

252002

22

2135

Tue Apr 29, 2014 5:16 pm

Qman View the latest post

There are no new unread posts for this topic. 2012 FSA Cranks w/Red BBright Logo

in Road

armcomdes

1

504

Sat Feb 01, 2014 6:12 am

dogg View the latest post


It is currently Mon Dec 22, 2014 1:11 pm

All times are UTC + 1 hour




Advertising   –  FAQ   –  Contact   –  Convert   –  About

© Weight Weenies 2000-2013
hosted by starbike.com


How to get rid of these ads? Just register!


Powered by phpBB