> I never claimed that there isn't a POINT of instantaneous zero velocity,
> just that the entire contact patch does not have zero velocity relative to
> the pavement.
>
Nope, just the great majority of it, where the tire is flattened to the
pavement. There is an "entry region" and "exit region" where their will be
some slippage between the two surfaces, but this is very small in comparison
to the area of the contact patch. Consider this extreme example of a
similar system; a tank's tracks. All along the bottom of the track where it
is flat their is no relative velocity between the surfaces. Also, bear in
mind that as an element of the tire (a differential mass if you will) enters
the contact patch, it is not only translating, but rotating, therefore moves
up or down rather rapidly, so the amount of time that it slips is very short
in comparison to the amount of time that element spends in the contact
patch.
Yes, the slippage that occurs is a wear mechanism, but I don't think that
you have described why tires cup. This wear mechanism you describe would
provide even wear on the tire, not uneven wear.
I consider the dynamic system like this; With vulcanized rubber and
pressurized air in the tire, there is a coefficient of restitution, or
spring constant for the system. There is also a damping constant, (which I
have no clue as to what it is, but it must exist and offhand I'd say the
system is underdamped.) The normal force between the tire and the surface,
and the force due to static friction contribute to the forcing function.
(Alternating stresses on each side of the contact patch, compression and
tension.) Therefore, this is an oscillatory system. I don't know how many
degrees of freedom. I do know that the natural frequencies of the system
are influenced by the rotational velocity of the tire, and the tire tread
pattern. Perhaps it is then a "beating phenomenon" that creates the tire
cupping, or simple resonance. This I don't know. As Tom pointed out, this
is a very complicated dynamic system, so I'm sure I don't have much hope
creating a mathematical model, even with finite element analysis. So you
see, it is because of the zero velocity point that tire cupping occurs, as
this is the point at which the forcing function occurs, and my original
explanation is valid, with refinement.
Mr. Morehead, I might respectfully suggest that you check the validity of
your own arguments before you make disparaging allusions to the intelligence
of others. I am not infallible, and I do appreciate corrections to any
theory I might propose, as this is a very important part of the learning
process and I still have much to learn. However you came across as rather
unprofessional. If you have any doubt as to the professional conduct and
ethics of an engineer, then I refer you to the National Society of
Professional Engineers Code of Ethics:
http://www.nspe.org/ethics/eh1-code.asp
More specifically:
Engineers shall:
6. Conduct themselves honorably, responsibly, ethically, and lawfully so as
to enhance the honor, reputation, and usefulness of the profession.
and
7. Engineers shall not attempt to injure, maliciously or falsely, directly
or indirectly, the professional reputation, prospects, practice, or
employment of other engineers.
Respectfully,
Jeff Walker