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The Track of a Bicycle Back Tire(28 posts)

The Track of a Bicycle Back TireET
Oct 16, 2001 5:20 AM
That is the title of an article in the latest (October) issue of the journal Mathematics Magazine. For an abstract of this article, see this link:

http://www.maa.org/pubs/mag_oct01_toc.html

Here is the end of the first paragraph of the article:

"Suppose we know the path of the front tire precisely; what is the path of the back tire? A related question is: If the front tire travels some distance, how far does the back tire travel? It seems obvious from experience that the back tire travels a shorter path than the front tire. Bicycle folklore says that after a long trip the back tire will show about 10% less wear than the front tire. Is it possible to verify the folklore?"

While some physics know-it-alls here may claim that this is all old hat and easy stuff, believe it or not, the results of the article apparently are new (the article points out that similar but different problems have been dealt with before, e.g. given the paths of both wheels, figure out which is the front and which is the back). The 15-page article was a collaborative effort of several physicists and mathematicians, and the journal, while not the absolute highest-level in terms of difficulty and sophistication of its contents (but often more interesting), has its articles very carefully reviewed, and in fact is the highest-circulated math journal in the US (nearly 100% of all universities, so if interested, you could gain access to the article at the research library of one of them).

The authors, upon assuming a few logical and fair simplifying assumptions, go on to give a method for solving the problem of finding the path of the rear tire given the front and the wheelbase. It involves nonlinear differential equations; several approaches are given. It is not at all a trivial paper. Unfortunately, the results may not be of much practical interest to most on the forum. For all but the trivial cases, nothing is easy or nice (because in general the differential equations have no nice closed-form solution, although numerical approximations can always be found). For a front-tire circular path, it is easy to see (even without the diff eqns) that the back-tire path will also be circular, and the distances between them easily compared. But even for the case of a sinusoidal front tire path, these methods can't precisely find the rear-tire path, which is not exactly a sine curve, but nearly so, with of course a phase shift and smaller amplitude. The authors end up by saying, "Can we verify the folklore that on a long trip the back-tire wear is less than that of the front tire? Probably not, even though the analysis in this article supports the folklore. Too many other variables intervene in the reality to be modelled so simply."
re: The Track of a Bicycle Back Tiremorey
Oct 16, 2001 5:46 AM
Both tires may not be equally round, or even the same size.
I believe this would make a difference?
re: The Track of a Bicycle Back TireChen2
Oct 16, 2001 5:59 AM
Interesting, but these mathematicians obviously aren't cyclists, and where did they get the folklore? The rear bicycle tire wears much faster than the front, not because of its relative path but because of the greater weight it carries.
-Al
Agreed -- never heard this folklorejtolleson
Oct 16, 2001 6:34 AM
Though it has almost a cute simplicity to it.

Even a non-cyclist with a rear wheel drive vehicle would understand that when a weight bearing set of wheels is also the source of the vehicle's forward drive, that rubber will wear (and a lot more than 10% faster in my experience!)

As for mental masturbation, don't be so hard on threads like this. This board has a fascinating cadre of engineer types and I'm always amazed at how they can reduce cycling to math.
Thanks butChen2
Oct 16, 2001 8:49 AM
I wasn't the one knocking the masturbation, after all it got me through adolescence mentally healthy.
-Al
sorry, you're wrongET
Oct 16, 2001 8:21 AM
All three authors (two physicists and one mathematician with a physics background) are cyclists. Two of them met on a 300-mile bike ride in Nebraska, where the third was a professor. This led to the collaboration.

Interesting everyone's making much of the one-line anecdote but showing no interest in the long article's results.
sorry, you're wrongChen2
Oct 16, 2001 8:43 AM
I'll bet these guys have a lot of flats.
-Al
somebody must've put LSD in their gatorade (nm)Rusty McNasty
Oct 16, 2001 10:13 AM
Uh, noKerry Irons
Oct 16, 2001 5:30 PM
Rear tires wear very slightly faster because of the weight they carry. They wear very much faster because they are transmitting power to the road. My wife weighs 1/3 less than me, but gets 2X the mileage on her rear tires. She can ride a front tire for 2 years (> 10K miles) and see minimal weight loss due to rubber wear. Meanwhile, her rear tire is losing 15% of its weight, due to rubber wear. My tires wear 2X faster, but I still see no weight loss on the front tire - only the rear. The wear is due to power being transmitted through the tire to the pavement.
mathematical masturbation -nmDA
Oct 16, 2001 6:02 AM
nm
what a load of cr@p!Rusty McNasty
Oct 16, 2001 6:34 AM
Anybody with 2 eyes in their head would see that the rear tire wears faster than the front on nearly every bike ever made! Where the hell does folklore say that the front wears faster??
This, of course, has nothing to do with distance travelled, but rather with the greater load, and therefore greater friction on the rear tire. Once again, mathematicians miss the utterly obvious, and obfuscate the situation with complex mathematics.
Don't these half-@$$ed mathematicians have anything better to do with their time-like calculate trajectories for cruise missiles?
easy boy... read the whole articleraboboy
Oct 16, 2001 6:54 AM
you are agreeing with the article writers. check the ending, especially the last line:

The authors end up by saying, "Can we verify the folklore that on a long trip the back-tire wear is less than that of the front tire? Probably not, even though the analysis in this article supports the folklore. Too many other variables intervene in the reality to be modelled so simply."
so what is the point?Rusty McNasty
Oct 16, 2001 8:00 AM
If the whole damn exercise was based on fallacy, why waste the time or server space? As I said before, just another case of a mathematician out of touch with reality.
It's what mathematician's do!mr_spin
Oct 16, 2001 8:27 AM
Maybe they are out of touch with reality, but so what? Modeling any real world activity mathmatically is a huge challenge that not many people can do.

They may have gotten the folklore wrong, but that only invalidates the reason why they went on the quest, not the results they found.

Why did we send men to the moon? Because it was a huge, impossible challenge. Why haven't we gone back? Because it's not a challenge anymore.
Heh, heh, heh....grzy
Oct 16, 2001 9:03 AM
They already have the cruise missle thing down pat - they're just looking for something to do. it may seem unimportant, but it probably opens the door to solving other types of complex problems.

Good point though - you gotta seperate the wear issue from the distance travelled. If bikes were like cars with more uniform weight distribution the front would wear faster.....but it's not and doesn't.
silly meET
Oct 16, 2001 9:49 AM
to waste time typing out such posts for you. We should limit this forum to "Should I get a Litespeed or an OCLV?" (As if that will stop the sniping. :-))
It depends... Campy or Shimano? LOL -NMTig
Oct 16, 2001 10:01 AM
hey, no LOL at your own posts :-). But LOL! (nm)ET
Oct 16, 2001 10:17 AM
.
tubular, not clinchers!!Rusty McNasty
Oct 16, 2001 10:19 AM
Really!
The main thing here is that these guys don't have even 1 foot in reality. Their brains must've been fried from the heat when they came up with this idea!
Of course, this is typical math/physics professor stuff: they've all been cycling for years, and haven't noticed which tire wears faster!! They probably don't know their wives birthdays, or that their socks don't match, either.
That's why the world needs engineers. To save us all from theoretical mathematicians and physicists.
Stop strokin your slide rule and go ride!ElvisVentoux
Oct 16, 2001 7:08 AM
"So far as the laws of mathematics refer to reality, they are not certain.
And so far as they are certain, they do not refer to reality."
Albert Einstein

"Some infinite sets are larger than other infinite sets"
"Mathmaticians usually die poor and insane."
Bertrand Russell

At least these guys knew their faults!

PS. Russell DID die penniless and in an asylum.
Stop strokin your slide rule and go ride!morey
Oct 16, 2001 7:20 AM
What is a "Slide Rule", boy, this really dates you!
slide ruleDebby
Oct 16, 2001 7:38 AM
Hey I remember them! We had to use them in electrical courses in 1979. Back in the stone age when the calculators were LED or whatever it was called.
Not Entirely Useless?Chen2
Oct 16, 2001 8:41 AM
There may be something to learn from this misguided mathematical exercise. If as postulated, the front wheel travels farther than the rear, this could be another logical reason to put our odometer/speedometer sensors and magnets on the rear wheel instead of the front. I suspect this doesn't make much difference though.
-Al
re: The Track of a Bicycle Back Tiregust-of-sun
Oct 16, 2001 9:48 AM
Well, i dropped out of advanced differential equations to pursue badminton instead, but the folk lore makes sense to me.
In the audio electronics realm it could be modelled as a transfer function. The front wheel traces a path like a waveform. The structure/mechanics/physics of the bike transfers that waveform to the back wheel which traces a related path. because of the reduced amplitude of the back tire's path, the front tire will always have a "curvier" path than the back tire; that is, at any given point it will be at a greater angle to the direction of travel than the rear tire as it passes through the same point. the greater sideways forces on the front tire with respect to the tire's rolling direction will wear it down faster than the rear tire which will see less sideways force. The difference in wear will be a function of just how "curvy" the front tire's path is. If the bike were travelling a perfectly straight line, neither tire would see sideways forces and they should wear equally. If the cyclist is always swerving to avoid broken glass, roadkill, bad drivers, and the like, then the front tire will trace a very curvy path and will wear faster than the rear.
It seems mountain bike tire manufacturers already know instictivly what the formulas show. Take a look at front and rear specific MTB tires: the fronts have side knobs for better cornering whereas the rear tires tend to have wider paddles for applying forward force. It would be interesting to know what type of path the mathematicians chose to analyze. Was it generally straight like a road bike, or really twisty like a MTB on singletrack.

Just my confused $.02
gust-of-sun
general pathET
Oct 16, 2001 10:14 AM
They derived the nonlinear differential equations for any path. But if you select anything other than a trivial one, a closed-form solution is intractible, but a very close numerical approximation can be graphed (and printed and visualized). "Trivial" front-tire paths discussed in the article (often to show that the solution to the diff eqns reduce to what one would expect) are a straight path, a large circular path, and a stunt circular turn with rear tire as pivot. A non-trivial example given is a sine path.
re: The Track of a Bicycle Back TireElvisVentoux
Oct 16, 2001 11:58 AM
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Ok, thesis is correct, application is wrongChrisZeller
Oct 16, 2001 12:17 PM
The applicatio of this thesis is what is not properly developed. Tire wear is a complex problem thattakes into account weight distribution, friction, etc as well as distance travelled and lateral forces (i.e. how curvy is the path). That's where they went wrong. This is the problem that often arrises when over-zelous researchers when they try to justify the importance of their work. Usually, these statements are not nearly as well thought out as the work itself, as they are an afterthought tat is in most cases best left out to begin with.

Not that this work is unimportant. It is an important application of mechanical theory that should apply to mechanisms in general. That shuld be sufficently important--certainly more important than which tire wears out faster.
Tandem Stokers don't ride as far.nee Spoke Wrench
Oct 17, 2001 5:43 AM
So I guess my wife gets less exercise benefit from each ride.