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Background
In determining the feasibility of a real MAGLEV transportation system,
performance will be based upon speed, vehicle cost, and passenger capacity,
among other factors. Awards will be distributed based on these and other
performance categories. (Race cars carry one occupant; price is no object.
A bus is something else again):
A Real MAGLEV Must Meet Several Criteria at Once
- It should go as fast as possible. The faster you make trips between
any two points, the faster you collect fares - that is, make money.
- It should carry as many passengers as possible. Since each passenger
pays a fare, the more passengers you carry per trip, the more money
you make.
- It should be as cheap as possible. The cheaper the vehicle, the
fewer trips will be needed for you to pay back the vehicle's cost and
start making a profit. If the vehicle is too expensive, the number of
trips you would need to pay it off may even turn out to be greater than
the number of trips it can actually make before it wears out. So, you'd
never even get your money back, let alone make a profit.
- Other factors also enter, such as operating and maintenance costs,
how long the vehicle actually lasts, the route chosen, the number of
people who might want to travel this route, reliability, public confidence
and availability of financial support, etc.. However, to keep things
simple while illustrating basic engineering principles, in the MAGLEV
contest we confine ourselves to speed, number of passengers, and cost.
There Will Be Tradeoffs Between Design Requirements
All else being equal, adding passengers will tend to reduce a vehicle's
maximum speed. Passengers add weight. They also make the vehicle larger.
We will take pennies to represent passengers carried by a model vehicle.
All things being equal, increasing a vehicle's carrying capacity will
tend to increase cost. A heavy-duty MAGLEV suspension system to support
a larger car and the more people will probably cost more than a light-duty
system. We will take suspension costs to represented by the number of
magnets used in a model vehicle.
The Design Engineer Must Find the "BEST" Tradeoff
That is, design a vehicle which carries the most pennies the fastest
while using the least magnets. This is how real engineering problems
present themselves. The engineer who finds the best compromise is the
one who becomes rich and famous.
The engineer will need some way to actually judge which compromise
is best. Is a car which goes twice as fast, has half as many magnets
and half as many pennies as another car "better" or "worse"?
A Way is Needed to Compare These Tradeoffs
- Comparisons are often made in practice by defining a FIGURE OF
MERIT" (FOM) for the design problem
- The FOM is simply a mathematical way of assigning a single number
to each different compromise of the things you're looking at - in this
case, speed, number of passengers (pennies), and cost (no. of magnets.)
This is done in such a way that "better" compromises have
higher FOM's.
- Example: Suppose you're designing a bridge. You want it to be strong, but also
light. If don't care about weight, you can make a bridge as strong as
you want by adding beams to it. If you don't care about strength you
can make a bridge as light as you like by making beams thinner and thinner.
- But how to judge how well you did in keeping weight low and strength
high? You want a FOM that will increase when strength increases, and
decreases when weight increases. Here, a simple math expression that
has these properties is the strength-to-weight ratio:
FOM (bridge) = strength/weight
- In BNL's Model Bridge Contest, a bridge is weighed. Its strength is
determined by loading it until it breaks. By this FOM , a bridge that
holds 25Kg and weighs 10g (FOM of 2500) is a better design than one
which holds 50 Kg but weighs 50g (FOM of 1000).
Ratio performance measures of this sort are very common: price-to-earnings
ratio for stocks; miles-per-gallon for cars; price-per-pound for meat,
etc. By using them we can compare Apple with IBM, Volkswagens with Hummers,
and sirloin with hamburger. SIMILARLY, WE CAN CONSTRUCT AN FOM TO
COMPARE MAGLEV DESIGNS WITH DIFFERENT, SPEEDS, CAPACITY AND COST:
- Let S stand for measured speed. All else being equal, we want the FOM be greater for vehicles with greater speeds. So, we'll make the
FOM directly proportional to S.
- Let P stand for the number of pennies carried, representing passengers.
All else being equal, we want the FOM to be greater for more passengers
carried. So, we'll also make the FOM directly proportional to P
- Let N stand for the number of magnets in the vehicle's suspension,
each of which adds the same amount to vehicle's cost. All else being
equal we would like the FOM to be smaller for vehicles needing greater
the number of magnets. So, we'll make the FOM inversely proportional
to N.
- All three statements can be simultaneously expressed in a simple
algebraic relation:
This can be put in terms of what we will measure. Let the distance between
timing points on the track be X feet. Let the time to go between the
timing points be T seconds Remembering r that speed S = X/T gives you
(algebra):
For the MAGLEV contest, P, and/or N and/or T may be different
for each vehicle. X however will be the same for all vehicles, i.e.
the 12' between the timers. To get FOM, multiply the number of pennies
(P) by 12' (X) and divide the result by the number of magnets (N) and
again by the measured time (T, in seconds).
Guidelines on Vehicle Design
Performance data and FOM calculations at the contest
will be tabulated in standard form. An example of results for five hypothetical
students is given below. You may want to do your own calculations from
the data and make sure that you get the same FOMs as those in the table.
Table 1
If a vehicle carries no pennies, its FOM will be zero no matter how
fast to goes. On the other hand, for loads over some maximum number
of pennies, say Pmax, the vehicle won't move at all. Its FOM will also
be zero. It stands to reason that for some number of pennies (P) between
zero and Pmax, the vehicle's FOM will be as high as possible.
So, having first built a vehicle, you then find the number of pennies
which gives the highest FOM. You can do this by loading the vehicle
with different numbers of pennies, and determining the FOM for each
case by actual tests. You might get results like so:
Table 2
It may also be very helpful to make a graph of the data , showing
FOM vs. P
FOM vs. Number of Pennies
This curve shows the FOM is greatest for loading around 20 pennies
(mathematically, a maximum) So you'd do best with a loading of about
20 pennies. Since the graph is fairly flat near the maximum a few pennies
more or less won't make much difference to the FOM. (This would be good
- you don't want FOM to drop off sharply if a vehicle has a few passengers
more or less than the ideal number.)
Every design should have a curve generally like this. Adding more magnets
to your design might shift the position of the maximum FOM to more pennies,
since you could support more of them. However, the value of the FOM
could increase or decrease, depending on how well the vehicle drive
could handle the extra load, the fact that increasing number of magnets
increases cost, etc.
So one way to start is to build a car which at least goes fast empty,
and then determine the number of pennies to add (and the way to add
them) to get the greatest FOM. Then test to see if fiddling with the
number of magnets, car shape, etc. improve things.
Problem Solving Strategies
- To get near the number of pennies for the maximum FOM more rapidly.
You could begin by quickly finding the number of pennies PMAX, at which
the car stalls.
- You might then guess that the maximum FOM should be at about half
this value and determine it at the midpoint between zero and Pmax.
- You could then measure the FOM at loadings above and below the midpoint
, say near 1/4 Pmax, and 3/4 Pmax, If both lie lower that the FOM at
the midpoint you can be pretty certain that the maximum lies fairly
near the midpoint, and start homing in on it . For the example above
this would have gone like so:
Table 3
Using a search strategy to find the loading which give the greatest
FOM.
Data are the same as in Table 2
This sort of search is more than trial-and-error. It is based on a
mathematically optimal strategy for finding something located within
a certain interval in the fewest possible tries - from a maximum number
to a hostile submarine. Here, at least, the series of measurements converged
to the best FOM fairly quickly. However, one drawback of this method
is that you don't have the learning experience of seeing what happens
when you have to add pennies gradually (and decide where to put them).
If
you have questions about MAGLEV contact:
Bernadette Uzzi
Brookhaven National Laboratory
Building 400C, PO Box 5000
Upton, New York 11973-5000
(631) 344-2756 phone
(631) 344-5832 fax
Last Modified: January 31, 2008
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