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RE: Wheels go round and round

To: mgs@autox.team.net
Subject: RE: Wheels go round and round
From: "Gosling, Richard B" <Richard.Gosling@atkinsglobal.com>
Date: Fri, 3 Jun 2005 15:56:25 +0100
I've just looked at what I've written below, and decided that this message
should be prefaced by a WARNING - VERY LONG RAMBLING MESSAGE FROM MILDY
BOOZED-UP ENGINEER.  Still with me?  Don't say I didn't warn you...


I'm kind of with Robert Houston on this.  Even if a tyre is running quite
flat, the distance around the outside of the tyre, at the thread, won't
change (or at least it will change very little).  For one rotation of the
tyre, unless the tyre is slipping on the road, the tyre will move forwards
by the distance arounds its tread.  Unless the tyre is slipping on the
wheel, therefore, for one wheel rotation the car will move forward by the
distance around its tread.

Now, what is the distance around the tread?  Well, imagine the shape it
makes - mostly a circle, but with a short straight bit at the bottom.
Calculate based on the distance to the tyre at the top, you'll get a value
that's a bit high, but calculate on the distance to the flattest point on
the bottom, you'll get a value that's too small.  The true circumference
will be somwhere between the two, but a lot closer to the value based on the
radius of the "free" (uncompressed) tyre.

Imagine a car running on square, wooden, wheels.  Turn those wheels round
one revolution, the car undoubtably moves forward by 4x the length of the
side of that square - i.e. the distance around the edge.  Does this make the
theory any easier to visualise?

Of course, the situation gets even more complex when the car is accelerating
or braking hard, as there will be increased circumferential tension
immediately behind or in front of the contact point respectively, but let's
not go there...

Actually why not get really deep into it?  It's Friday afternoon, I've had
two pints of Stella and a shot of Stolichnaya with my lunch so consequently
don't feel like doing ANY work, and I'm an engineer so I enjoy this sort of
thing.  PLEASE, if engineering talk hurts your brain, read no further!!

How can my theory above fit with Paul H.'s assertion about "effective"
radius?  We need to understand how the tyre deforms as it goes past the
contact patch.  Lets follow a specific point on the tyre tread.  We can
imagine we've painted a little mark on it.  At the very top of its travel,
it will be directly above it's corresponding point on the wheel - we can
paint a mark on the wheel there too.

Now here's my theory.  The wheel turns until our marked point is just
touching the tarmac.  At this point the wheel hub has turned, say, 170
degrees.  However, the tread has not moved round quite so far, say only 167
degrees, relative to the axle centre.  This is achieved by a slight shear
deformation in the sidewall.  As the wheel continues to rotate, the distance
between the axle centre and our marked point on the tyre (the tread radius)
will decrease, as the tyre moves through the flat point.  This enables our
marked point to "catch up", so that when the wheel has rotated to 180
degrees, our point on the tyre has also rotated 180 degrees.  For the rest
of the time the tyre is in contact with the road, it is still rotating at
more degrees per second than the wheel, so that by the time it loses contact
it is, rotationally, slightly ahead - say 193 degrees to the wheels 190
degrees.  As it makes its way back up to the top, that lead will be lost
again, until we have completed one revolution and are back where we started.

Therefore, while the radius of the tyre may be less at the centre of the
contact patch, the rotational speed in degrees per second is higher
(remember, 26 degrees to get through the contact patch, in the same time the
wheel has rotated 20 degrees, in my example), so the reduced radius and
increased rotational speed largely cancel each other out.

Well, that's my theory.  Feel free to pick flaws, demolish, or come up with
counter-arguments (preferably more detailed than "that's a load of B.S.!"),
if you can make any sense of it to start with (and you've actually managed
to read this far!!!).  If it makes no sense, blame those Stellas and the
vodka!

Richard Gosling
Senior Engineer
Operations - Engineering
Atkins
6 Golden Square, Aberdeen AB10 1RD
Tel: 01224 619487
Fax: 01224 647652
E-mail: richard.gosling@atkinsglobal.com
www.atkinsglobal.com




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