update on Howard's Theorem

Dec 31, 18 16:13

Post #1 of 22 (2077 views)

Hi all,

Its been some years sense I caused a fuss with my wacky theories on cycling performance. In the meantime I have been diligently studying physics and am now a Phd. in computational condensed matter physics. I have been trying to formulate my theorem into a more simplified mathematical construct that will allow some type of proof but have been unsuccessful as of yet. I think some people were right that it was not an actual theorem but more of a conjecture. So I think something along the line of the "Ideal inertia conjecture" would be appropriate. Aside from that I have been racking up some other significant mathematical results that are in top physics journals. The first issue of physical review B. of the new year will have a very detailed article between my adviser and I with some important results related to thermodynamics and calculation of free energies. Aside from my current research in battery technologies I consider it a life long goal to continue pursuing a proof of the "ideal inertia conjecture" and will continue to so. I have gotten as far as formulating what I am calling a "power action" and then trying to find extremums of this functional. I tried ordinary calculus of variations but there is some problems with the acceleration term. I think the problem may be possible to solve using optimal control theory. If there are any mathematicians out there how would like to collaborate please let me know.

happy new year,
Dr. Howard

Re: update on Howard's Theorem [honestly] [ In reply to ]

Dec 31, 18 17:01

Post #2 of 22 (2042 views)

So are you still inflating your tires with water? ;)

Re: update on Howard's Theorem [jkhayc] [ In reply to ]

Dec 31, 18 17:04

Post #3 of 22 (2036 views)

No. I only ride for recreation now.. I did do a study using water inflation for my senior thesis as undergraduate. The results are posted in biketechreview.

Re: update on Howard's Theorem [honestly] [ In reply to ]

Dec 31, 18 17:50

Post #4 of 22 (1998 views)

Could you provide a summary of your conjecture?

-------------
Ed O'Malley
www.VeloVetta.com
Founder of VeloVetta Cycling Shoes
Instagram • Facebook

Re: update on Howard's Theorem [RowToTri] [ In reply to ]

Dec 31, 18 17:54

Post #5 of 22 (1990 views)

For every rider course and conditions there will be an ideal wheel inertia for each wheel that will allow for optimal performance and this inertia will not in general tend towards zero.

Re: update on Howard's Theorem [honestly] [ In reply to ]

Dec 31, 18 18:01

Post #6 of 22 (1981 views)

Well, I might add a stipluation that the course must not be flat. Just a simple Newtonian calculation of energy required to keep the wheel in motion, even considering the "micro-accellerations" of uneven power application around the crank circle shows no change with different moments of inertia for the wheels when speed averaged over time is steady....

I've not done the math for hilly courses...

-------------
Ed O'Malley
www.VeloVetta.com
Founder of VeloVetta Cycling Shoes
Instagram • Facebook

Re: update on Howard's Theorem [honestly] [ In reply to ]

Dec 31, 18 18:02

Post #7 of 22 (1975 views)

Your seat's too high..

"Good genes are not a requirement, just the obsession to beat ones brains out daily"...the Griz

Re: update on Howard's Theorem [RowToTri] [ In reply to ]

Dec 31, 18 19:32

Post #8 of 22 (1920 views)

Ideally the analysis should be fully 3-d incorporating the fully kinematic interactions of the bike rider system. Possibly in the future a virtual rider on a 3-d virtual course with full computational fluid dynamics solutions for air drag would help to attack the problem. With the advancements in AI it may eventually be possible to train "virtual" rider and then to the simulations.

Re: update on Howard's Theorem [honestly] [ In reply to ]

Dec 31, 18 20:03

Post #9 of 22 (1885 views)

Sheldon, is that you?

Pink? Maybe. Maybe not. You decide.

Re: update on Howard's Theorem [honestly] [ In reply to ]

Dec 31, 18 20:35

Post #10 of 22 (1857 views)

Well, when I do a delayed detached eddy cfd simulation of just a lower leg and a shoe over a time frame of 0.6 seconds using a dual socket 32 core Epyc processor with 256 GB of memory takes about 4 hours. So doing it over an entire bike plus rider over an entire race might be a little too "computationally expensive"!!!

-------------
Ed O'Malley
www.VeloVetta.com
Founder of VeloVetta Cycling Shoes
Instagram • Facebook

Re: update on Howard's Theorem [RowToTri] [ In reply to ]

Dec 31, 18 21:13

Post #11 of 22 (1830 views)

RowToTri wrote:

Well, when I do a delayed detached eddy cfd simulation of just a lower leg and a shoe over a time frame of 0.6 seconds using a dual socket 32 core Epyc processor with 256 GB of memory takes about 4 hours. So doing it over an entire bike plus rider over an entire race might be a little too "computationally expensive"!!!

Ed - What is the bottom line here??? What's the point??? Of what use would this extensive series of calculations be??? What is the "practical engineering application"???

"Anyone can be who they want to be IF they have the HUNGER and the DRIVE."

Re: update on Howard's Theorem [ericmulk] [ In reply to ]

Dec 31, 18 21:26

Post #12 of 22 (1821 views)

What's practicality?

I think this discussion is somewhat pie-in-the-sky but interesting for overly physics-obsesed triathletes to think about.

Or are you talking about shoe CFD analysis, which is extremely practical when done right?

-------------
Ed O'Malley
www.VeloVetta.com
Founder of VeloVetta Cycling Shoes
Instagram • Facebook

Re: update on Howard's Theorem [RowToTri] [ In reply to ]

Dec 31, 18 21:38

Post #13 of 22 (1807 views)

RowToTri wrote:

What's practicality?

I think this discussion is somewhat pie-in-the-sky but interesting for overly physics-obsesed triathletes to think about.

Or are you talking about shoe CFD analysis, which is extremely practical when done right?

I was asking about Howard's original thesis about "optimum inertia". I suppose that your "aero shoe analysis" has some value; "aero shoes" might save you 60 sec over 112 miles but really, all this aero sheet is way over sold. It is the ENGINE that matters most by far.

"Anyone can be who they want to be IF they have the HUNGER and the DRIVE."

Re: update on Howard's Theorem [ericmulk] [ In reply to ]

Dec 31, 18 21:48

Post #14 of 22 (1795 views)

Speed matters most and speed is increased by propulsive force and reduced by drag force. You can work on both increasing propulsive force AND reducing drag force. Those efforts are not mutually exclusive. And focusing on the aerodynamics of shoes yields more like 60 s (or more) per 40k, not 180k!

Tuning the moment of inertia of wheels in triathlon generally gets 0s per 40k, but I can imagine situations where it's not 0, and maybe even significant for Sagan in a bunch sprint where he may win or lose by 0.1 s.

-------------
Ed O'Malley
www.VeloVetta.com
Founder of VeloVetta Cycling Shoes
Instagram • Facebook

Last edited by: RowToTri: Dec 31, 18 21:49

Re: update on Howard's Theorem [RowToTri] [ In reply to ]

Jan 1, 19 2:17

Post #15 of 22 (1718 views)

RowToTri wrote:

Could you provide a summary of your conjecture?

It's all about the details...
https://forum.slowtwitch.com/...d's_Therom_P3482781/

Re: update on Howard's Theorem [ianm] [ In reply to ]

Jan 1, 19 5:20

Post #16 of 22 (1626 views)

In this day and age, a reliable crank-based power meter and a disc-cover wheel with some weights can be used to demonstrate whether one wheel inertia gives more speed for a constant power. To be really convincing, demonstrating control of external variables with something like Tom Arnold's styrofoam spheres Chung method test would be useful. I'm sure there would be people on here, possibly including me, that would be willing to repeat your experiment if you demonstrate a significant benefit.

I have preferred how heavier wheels feel when I'm riding. I feel like it gives me something to push against -- especially in situations where there's a lot of drag from road surface or wind. I think you could also look for a human factors benefit where a heavier wheel shifts the power vs rpe curve. This is the domain of double-blind randomized testing and you would want to show an improvement over the best asymmetric chainrings.

Anyway, I would suggest a well-controlled experiment before trying to model an effect. The experimental chops will be required to validate a detailed model anyway.

Re: update on Howard's Theorem [japarker24] [ In reply to ]

Jan 1, 19 8:37

Post #17 of 22 (1529 views)

japarker24 wrote:

Sheldon, is that you?

Damn. Post-of-the-Year on Jan. 1.

Re: update on Howard's Theorem [honestly] [ In reply to ]

Jan 1, 19 9:50

Post #18 of 22 (1478 views)

A few years ago I got interested in instantaneous variation in crank angular velocity within the pedal cycle. I modeled power input as a rectified sin function to account for power input variation. I could then vary the amplitude and the frequency to give me any desired overall power or pedaling rate that would give me any desired average power. I then used a forward dynamic simulation to quantify the resulting variability in crank angular velocity and thus bike speed. I evaluate a wide range of powers and kinetic energies by modeling uphill, downhill, and flat courses. I also varied pedaling rate over reasonable ranges. The results were surprising in that crank angular velocity / bike speed is so nearly constant; within a few % even in the most extreme conditions. This went against my intuition based on how pedaling feels when climbing a steep grade at a low pedaling rate. In such a case I perceived (without data) that the crank was slowing way down in the dead spots thus giving lots of variability. The model did not agree with my perceptions.
So, what I'm leading up to is this: Given that the huge range in kinetic energy values that I used had a minuscule effect on crank angular velocity, the ranges that one might impart with changes in wheel inertia will likely be trivial.
But of course its fun to think about.
Cheers,
Jim

honestly wrote:

For every rider course and conditions there will be an ideal wheel inertia for each wheel that will allow for optimal performance and this inertia will not in general tend towards zero.

Re: update on Howard's Theorem [RowToTri] [ In reply to ]

Jan 1, 19 10:13

Post #19 of 22 (1454 views)

Well for the purposes of qualitative testing of the conjecture the accuracy of the CFD would not need to be as high as your simulations. I have not done CFD my self so I am not sure how coarse you could take the grid before it gets completely unreasonable. So with a more coarse mesh, a couple more doublings of computer power, and some algorithmic advancement it may be possible in 10-20 years. Aside from the troubles of doing the CFD it would be exceptionally challenging to program a life like fully kinematic simulation of a 3-d rider. The amusing part would be to use a neural network to train a virtual rider. After that in the simulation one would have complete control of the interia related variables without effecting the aerodynamics.

Last edited by: honestly: Jan 1, 19 10:14

Re: update on Howard's Theorem [Bio_McGeek] [ In reply to ]

Jan 1, 19 10:52

Post #20 of 22 (1416 views)

Bio_McGeek wrote:

A few years ago I got interested in instantaneous variation in crank angular velocity within the pedal cycle. I modeled power input as a rectified sin function to account for power input variation. I could then vary the amplitude and the frequency to give me any desired overall power or pedaling rate that would give me any desired average power. I then used a forward dynamic simulation to quantify the resulting variability in crank angular velocity and thus bike speed. I evaluate a wide range of powers and kinetic energies by modeling uphill, downhill, and flat courses. I also varied pedaling rate over reasonable ranges. The results were surprising in that crank angular velocity / bike speed is so nearly constant; within a few % even in the most extreme conditions. This went against my intuition based on how pedaling feels when climbing a steep grade at a low pedaling rate. In such a case I perceived (without data) that the crank was slowing way down in the dead spots thus giving lots of variability. The model did not agree with my perceptions.
So, what I'm leading up to is this: Given that the huge range in kinetic energy values that I used had a minuscule effect on crank angular velocity, the ranges that one might impart with changes in wheel inertia will likely be trivial.
But of course its fun to think about.
Cheers,
Jim

honestly wrote:

For every rider course and conditions there will be an ideal wheel inertia for each wheel that will allow for optimal performance and this inertia will not in general tend towards zero.

Nice.

So looking at the other side of the issue - effect of wheel inertia during this "micro accelerations". I used a very large speed differential (especially given your results) over the pedal crank cycle to amplify the effect, and predictably got that wheel inertia makes no difference in average speed - only a small difference in the amplitude of the oscillations. I took a wheel the weight of the Dash Disc and another the weight of the HED Jet Plus Disc, and did a simplified moment of inertia calculation. Then I assumed that the Dash wheel oscillated speed by 1mph and calculated the work required to create both the decelleration and acceleration. Then applied that same amount of drag to the HED wheel to calculate it's deceleration and then calculated the work to re-accelerate and of course it turned out to be equal.

So unless you are going steeply uphill, where weight will matter (but not moment of inertia), or if you are doing a big sprint to the line where moment of inertia will matter a little bit, wheel inertia is not something we need to concern ourselves with. Maybe this will put this argument to bed??

Here are my calculations:
(edit: it just occurred to me that I would have better illustrated my point by calculating work to accelerate/decelerate around the mean speed rather than to/from the peak speed, but the results would be the same)

-------------
Ed O'Malley
www.VeloVetta.com
Founder of VeloVetta Cycling Shoes
Instagram • Facebook

Last edited by: RowToTri: Jan 1, 19 10:58

Re: update on Howard's Theorem [honestly] [ In reply to ]

Jan 1, 19 14:04

Post #21 of 22 (1311 views)

If you don't make your mesh fine enough to resolve the boundary layer, the results are nonsense. This means you need several elements within the boundary layer. You really cannot go any courser than I am going. Reynolds averaged Navier Stokes models can be used instead of DDES which that same computer can solve in maybe 20 or 30 minutes instead of 4 hours. A method to apply that to an entire ironman course might take.... about a year or so to solve depending on the resolution unless you rent out thousands of cores in a cluster. But that's a steady-state solution. To do what you are suggesting likely requires a model for transient solutions.

I think that the results from the infinitely simpler simulations like those used by Best Bike Split have proven themselves to be good enough, and an ironman course can be solved in seconds. Seems much more practical.

-------------
Ed O'Malley
www.VeloVetta.com
Founder of VeloVetta Cycling Shoes
Instagram • Facebook

Re: update on Howard's Theorem [honestly] [ In reply to ]

Jan 2, 19 7:49

Post #22 of 22 (1113 views)

Just buy TrainerRoad, man.

Interval Design Studio
YouTube | SoundCloud

Our Partners

transition-DOT-area-AT-slowtwitch-DOT-com

Advertise with us

©1999-2024 Slowtwitch, Inc., and Slowtwitch.com
Reproduction of material from any Slowtwitch.com page
without written permission is strictly prohibited.