Re: [Phys-l] Another tire question

[Brian Whatcott <betwys1@sbcglobal.net>]

At 05:07 PM 11/9/2007, you wrote:

> > > > > Sophisticated (in this context) means you can understand
> > > > > a pneumatic tire can be contrived with NO contribution
> > > > > from its upper half, and that this can be made to
> > > > > function on a spoked wheel assembly.
>
> > > > I guess I am not sophisticated because I cannot understand that
> > > >      unless the tire is internally sectioned.
> > > >
> > > > The spoked bicycle wheel/tire is no different than the auto
> > > > wheel/tire. The axle "hangs" from the top rim via the top spokes.
> > > > The rim "hangs"   from the top sidewall of the tire.
>
> > >No.  The bead hangs from the top sidewall.  The rim is supported from
> > >below by the bead.
>
>
> > Well, well, well, Jeffrey. You didn't start out as a physicist, did
> >  you?       :-)
> >
> > You might enjoy responding to the design issue of showing the
> > ice/snow loading capacity of a blown diaphragm roof on a circular
> > "pill-box" arena.
> > These tension roof skins can be held down by a rim, despite the air
> > pressure differential applied across them.  The side walls pull down
> > on the rim to keep the roof in place, unless the snow load gets too
> great!
> >
> > Wait: isn't that a bit like a tubeless tire? Oh yes!
> >
>
> There are similarities between the arena and the system we have been
> discussing but there are important differences.  The loading in the case
> of the arena is the snow pushing downward on the roof.  The question,
> analogous to the one being asked about tires is: What is the upward
> force on the roof that keeps the roof from accelerating downward once
> the load is applied?  The answer in the case of the roof is the pressure
> of the air upward on the roof integrated over the area of the roof.
> This force was always there but prior to the loading it was
> counterbalanced by the net downward component of the tension force
> exerted by the rim on the roof.  That tension force automatically
> adjusts to whatever it has to be to keep the roof from accelerating
> upward.  It decreases as you add snow to the roof.
>
> In the case of the car wheel, the loading is the axel of the car
> pressing downward on the central regions of the hub.  The (hub + bead)
> for this system, plays a role similar to the role played by the roof in
> the arena.  The question here is: "What keeps the (hub + bead) from
> accelerating downward once the load is applied?"  The answer in this
> case can't be the pressure of the air on the hub integrated over the
> surface of the hub in contact with the high pressure air.....


I will admit, I was hoping for more from you.

Here we go.    Tension structures have mechanical properties
that you might not expect from floppy rubber, fabric and
steel bands.
They can react tension, and they can react compression.
The tension structure we know and love as the pneumatic tire
 has two ways of reacting ground forces at the hub.

Where the hub makes a strong connection to the tire case,
the two dimensional stress field of the surface of the inflated
 tire has a particular pattern when reacting the ground force:
it is an enhanced tension region around roughly the top half
 of its area,  and a compression region in the lower half.

For a tire case not strongly connected to the wheel hub, the field
of compression is largely confined to the lower half of the case.
It is not overwhelmingly necessary that the tire's case deflect
in some particular way at the hub. The structure has stiffness
 due to inflation.

The tire's footprint area transmits internal air pressure,
as does the belt bending force and the sidewall bending force.
   A decent reading of the product of footprint area and
 internal air pressure   should provide a large fraction of the
total ground reaction force, in non-pathological cases.

Disappointing, but that's all there is.


Brian Whatcott    Altus OK    Eureka!