Bob Schadewald, who isn't on this list, has written about trebuchets,
analyzed them and built them. He gave me permission to include some bits
from recent e-mail exchanges. They support my hunch that the trebuchet is
capable of efficiences nearly equal to one (in the absence of friction)
while the older rigid-arm catapult cannot come near this, even in the
absence of friction. The "secret" is in the simple linkage transfering
energy to the sling, making the counterweight slow to nearly a dead stop
at the instant the projectile is released. So there's no left-over kinetic
energy wasted. This relatively simple physics observation does not seem to
be found in the literature of the history of mechanisms, or even in the
modern literature of trebuchet hobbyists. Surely someone will correct me,
with a reference, if I'm wrong, for I've only read a small fraction of the
literature on this subject.
Ludwik was right, I think, in expecting an upper limit of
efficiency near 1 for the trebuchet. But for the rigid-arm
catapult, what is the upper limit? Someone want to do the
derivation? It's much easier than the trebuchet mathematics. It's
almost "conceptual physics".
This is the efficiency which matters, for it means that the projectile
gains higher speed, and therefore greater range. But a high price is paid,
from a practical point of view. The release mechanism (a curved hook) is
not consistent in its operation, so the firing angle and range may vary a
lot in successive shots. The changes required to reset the machine for a
desired range and projectile weight are interactive in a complicated way.
Certainly a set of tables would have to have been devised for the military
personnel. The re-adjustments would be time-consuming, and in the heat of
battle time delays are important.
The paper refered to below is one Bob wrote back in 1980 for *The Vector*
a publication I edited. One of these days soon I'll put up a version on my
web page. My web page doesn't have enough diversity, you see. Later Bob
will supply some nice pictures to go with it.
[Begin Bob Schadewald's comments]
The thing [trebuchet] still is a minor passion of mine, but I haven't done
anything about it for years. Either while I still was in high school or
right after I got out, a friend and I built a model with an arm about a
foot long. It would throw a marble 10 feet or so. A few years after I
got out of the Navy (1969), I built a larger model with about an eight
foot arm and a hundred pound concrete counterweight. This was
disappointing, because the whole thing was too flimsy for the stresses
involved, but it would throw a couple-pound rock about 40 feet. It was
very inefficient, though.
...
Back when I wrote the piece you published in *The Vector*, I was convinced
that 100% efficiency should be achievable in theory (less friction losses),
and I still am. The whole secret of the trebuchet is the crack-the-whip
motion of the projectile in the sling. Remember, the projectile starts out
heading north and ends up flying south. When the contraption is triggered,
the arm begins rotating, and its angular velocity increases rapidly.
Meanwhile, the projectile is first dragged (north), then lifted, and then
swings outward. When the crack-the-whip motion of the projectile begins,
and it goes from heading north to straight up to up-and-south, the angular
velocity of the arm/counterweight system slows dramatically. Essentially,
the sling steals kinetic energy from the arm/counterweight system and
transfers it to the projectile. What is required for 100% efficiency is
that the arm/counterweight system be dragged to a halt just as it hits dead
vertical, at which time the sling also releases. To get optimal range from
the projectile, the sling also must have achieved the optimal angle with
the arm at release. For a large trebuchet, the projectile might be 80 feet
in the air when it is released, so the optimal angle is much flatter than
45 degrees. Soooooo, all you need to do is simultaneously optimize the
relationships between counterweight, arm length, pivot position, arm moment
of inertia, sling length, and the angle at which the sling is released. I
have long thought this should be achievable with computer modeling.
...
The release on a trebuchet is simply a fairly flat hook, and a loop on one
end of the sling hooks onto it. When the sling gets to a certain angle,
the loop slips off the hook and the projectile is free. It would be
relatively trivial to make an adjustable hook that could be set for any
desired angle. The problem is figuring would what the angle and sling
length should be.
Because of the complex relationships between the arm, sling, and
projectile, the equations describing the system get pretty ugly...
...
The U. S. Military Academy trebuchet program is great, but I wish it gave
more information and provided tools for optimization. Whatever, here is a
set of empirically determined parameters that achieves nearly 100%
efficiency.
When you watch the animation, you will see that the arm barely oscillates
after the projectile is released. In my little paper way back when, I
derived a trivial formula for calculating the minimum energy required to
throw a projectile some given distance (based on the optimal parabolic
path). The Trebuchet program gives the distance traveled, so actual
efficiency for any throw could be calculated directly from distance and the
other parameters.
After I thought a little, it only took a few minutes to figure out the
efficiency of a trebuchet. For any set of parameters for which the release
angle of the sling is adjusted to maximize range, the projectile must be
released at a point on a parabolic path that impacts the ground at 45
degrees. The total energy of the projectile is the same at any point on
the path. Ergo, some simple manipulations yield
Efficiency = 2wh/WH
where w = projectile weight, h = maximum height achieved by the projectile,
W = weight of the counterweight, and H = distance the counterweight drops
to dead center.
efficiency is about 98.5%. But the range is far too short for military
purposes. BTW, the optimal angle is closer to 144 degrees.
... from a practical point of view, the medieval military engineer was not
really concerned with high efficiency. To achieve long ranges -- and wise
engineers built their engines beyond crossbow range -- you had to use
relatively light projectiles, resulting in relatively low efficiencies.
High efficiencies mean heavy projectiles and shorter ranges. As a result,
I suspect real siege trebuchets rarely if ever achieved high efficiencies.
[End Bob Schadewald's comments.]
-- Donald
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Donald E. Simanek
dsimanek@eagle.lhup.edu http://www.lhup.edu/~dsimanek
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