MadSci Network: Physics Query:

### Re: Wheels for elastic band powered car

Date: Thu Feb 8 19:59:58 2007
Posted By: Joel Chapman, Undergraduate, Mechanical Engineering, NC State
Area of science: Physics
ID: 1170981348.Ph
Message:
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I think I can help you a bit on this one.  We are going to assume that for
each case, you store the same amount of potential energy in the elastic
band independent of all other design characteristics.  We are also going to
assume that friction in the drive axle is the same for each case.

The primary goal is speed, correct?  If it is some sort of a race, then you
will want to get from A to B in the shortest amount of time.  This means
that  you will want to maximize the amount of energy put into translational
motion (that is, the forward motion of this car).  There are three things
you'll need to look at, then:  traction, weight, and gear ratio.

Traction is of the utmost importance.  If the wheels do not slip, you are,
in the ideal case, not transmitting any energy into the ground as heat due
to friction.  Of course, you still will end up having some friction because
your wheels will be a bit toe-in or toe-out.  Your rear wheels should have
a soft, grippy surface such as rubber to eliminate slipping.  Your front
wheels should be made out of a slicker material, because if they are out of
alignment by a bit, you want as little transmission of friction to the
ground as possible.

Concerning weight, you want your wheels to be as light as possible.  The
front wheels should be small, thin, and as light as possible.  As long as
the front axles is well-greased, we can assume that there is no friction in
the axle, so rotational speed is not the big issue.  Rotational momentum,
however, IS an issue, since we want to translate energy into the
translation of the car, not into the spinning of the wheels.  Your rear
wheels should be as light as possible, also, but they may need some width
to help them hold traction off the starting line.  Do not go overboard on
the width, or your wheel weight will slow your car down.  Rotational
kinetic energy is 1/2 * I * w^2 (I is the moment of inertia and w^2 is the
rotational speed in radians).  If you increase your weight on a wheel that
is the same size, you will increase the moment of inertia, and thus
decrease your rotational speed.

Concerning size, the front wheels should have a pretty small radius for
weight and rotational mass purposes.  The rear wheels should primarily be
based on the gear ratio between the cylinder the elastic band is attached
to and the wheels themselves.  If the band is still unwinding after the
finish line, you did not get to convert that additional potential energy in
the elastic band into kinetic energy.  This would mean that your wheels
were too large.  Also, the larger the wheel, the lower the torque being
transmitted, so your acceleration will be terrible.

If your wheel is too small, you will transmit so much torque that the wheel
will spin.  Also, with a very small wheel, at the maximum rotational speed
achievable, you may not be going very fast.

Concerning the specific design process for the rear wheels:

The rim material should probably be a light plastic.  These wheels will not
be under much actual stress, so they do not need to be very strong, just
very very light.

Concerning gear ratio, try making the wheel about the right size so that
your elastic band just finishes unwinding at the finish line.  This should
allow you to reach a maximum speed, use all available energy stored in the
elastic band, and minimize slipping due to overtorque.  You might do some
tests with a wheel slightly smaller than this, also, but do not exceed this
size.

Once you've got a few potential wheels chosen, use the grippiest rubber you
can find for the contact surface.  Do not use a material that deforms very
much, though...too much wheel deformation causes rolling resistance that
will slow your car significantly.

Hope this helps you out,

Joel

Reference:

Hibbeler, RC.  "Engineering Mechanics:  Statics, Tenth Edition"

Friction:  pg 379-382
Rolling Resistance:  pg 426
Moments (Torque):  pg 121-147

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