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Re: trebuchet



If I had time I would try to simulate the mechanism (see below)
with Interactive Physics. It is designed for this kind of simulations
and allows us to "measure" everything we want. But this would not be
a reliable way of getting the efficiency. My guess is that in the
absence of friction (at joints), without air resistance, and for the
throwing arm (and rope) of negligible mass, the simulated efficiency
would be 1. By the way, air resistance is not simulated realistically
in the program.

Ludwik Kowalski
................................................................
"Donald E. Simanek" wrote:

.....
Exercise for student. Derive the *correct* equation for the efficiency,
in the absence of friction. This gives an upper limit on attainable
efficiency.

Data for the trebuchet Zephyrus: It had a hollow wooden throwing arm,
stiffened by stays and spreaders, like a sailboat's mast. The
counterweight was 2000 pounds of concrete in two 55 gallon drums, and
its cg is about 14 feet from the pivot. The throwing arm was 33 feet long
from pivot to sling pin, and the sling was 31.5 feet. When cocked, the
counterweight was about 53 degrees above horizontal. Rotating from the
cocked position to where the arm is upright, the center of mass of the
counterweight drops through a vertical distance of slightly more than 25
feet, giving up over 50,000 foot-pounds of energy. With a 100 lb
projectile this machine achieved a range of x = 579 feet. The adjusted
range x' is then 643.5 feet. The mass ratio is 20. This data is from Bob
Schadewald, 1978. In 1978 Zephyrus held the record for modern trebuchets,
but I haven't kept up with this field since. Bob calculates 64% efficiency
from this data. Once you folks calculate the upper limit of efficiency for
such a mechanism we can see how close Zephyrus was to its limit. There's
an annual National Catapult Contest, started in 1972 by Bernard Barcio, an
Indianapolis high school Latin teacher, as an outgrowth of a class project
for a Latin-related activity! If a dead-language teacher can handle the
math of this, I imagine you physics teachers can. But I'll bet it will
require a physics teacher to calculate the *upper limit* of efficiency
(ignoring all friction).

Those of you interested in smaller models could scale this down to a
manageable size. On the other hand, if you college folks want to stage an
assault on the administration building....

There's enough info given so far to calculate the upper limit of
efficiency. But for those still having trouble visualizing this

Trebuchet, cocked, ready to fire.

|\ d
| \ W is counterweight
_|_ \ d is counterweight lever arm
| W ||\ w is projectile weight
|___|| \ D is length of throwing arm
| \ L is length of loose sling (initially lying on base)
| \ R is a curved hook, to release sling when D is
| \ D at approximately 45 degrees elevation.
| \
| \
| \
| \
| \
| w \
| _ L \
| (_)--------R
______|________________ Base

_
(_) w
.
.
.
. L
.
.
.
R (releasese here)
/
/
/
/
/ D
/
/
/ At firing angle.
/
/
/
/
/|
d / |
_/_ |
| W ||
|___||
|
|
| (supporting structure, actually *much* sturdier than
| this diagram suggests)
|
|
|
|
______|________________

I'll venture an opinion that the best designs are those in which L and D
are parallel at the release point. So this becomes an interesting problem
in angular motion, probably requiring calculus (since the radius of the
path of the projectile varies continuously during the firing), to ensure
that L swings through the correct angle during the firing--not too small
an angle.
.....................................................................
Donald E. Simanek
dsimanek@eagle.lhup.edu http://www.lhup.edu/~dsimanek
.....................................................................