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Re: trebuchet

On Fri, 12 Feb 1999, James McLean wrote:

"Donald E. Simanek" wrote:
A friend of mine built what he calls a 'small' trebuchet with a 100 pound
counterweight which droped 3.5 feet. The sling and throwing arm measured
11 feet. It threw a 5 pound rock 69 feet from the pivot, a mass ratio
of 20 and an efficiency of 59%. Not bad.

Efficiency would be calculated how, exactly? I'm guessing that it's
(dE_projectile)/(dPE_counterweight). But everyone is reporting
projectile weight and range, which isn't enough to calculate the
numerator (I think...).


I would say that the potential energy change is U = (WH-wh) where W is the
counterweight, w is the projectile weight, H is the drop of the
counterweight, h is the rise of the projectile before firing. Then the
efficiency is the kinetic energy of the projectile divided by U. H/h is
the ratio of the lever arms of the trebuchet, a fraction less than 1.

Of course if you compare not to U, but to the energy required to "cock"
the system, the efficiency is *much* lower. But we are talking here of the
efficiency of the trebuchet firing mechanism itself.

You are correct that reported figures often are incomplete.
They usually assume a firing angle for maximum trajectory, which is
somewhere near 45 degrees, but not exactly. Also, the firing angle isn't
that easy to measure, nor is it consistent from one firing to another, due
to the nature of the release hook. Anyway, according to the literature,
this works out to E = Wx/2 where E is the minimum energy of the
projectile, W is the counterweight, x is the range. But this isn't quite
correct, for it assumes projectile release and landing at the same height.
In a real trebuchet, the projectile is released high above the ground,
about 70 feet above the ground for the larger trebuchets. They do some
sneaky approximations here, calculating the adjusted range x' by
subtracting the release height from the range. (This approximation assumes
the trajectory path (at its far end) makes an angle of 45 degrees with the
ground.) The bottom line (from the literature of trebuchet enthusiasts is

x' = 2eh(W/w)

where e is the efficiency, W is the counter weight, w is the projectile
weight and h is the distance of fall of the counterweight. I had hoped
someone here would have derived the correct equation here on this list by
now.

Also, from my reading of this stuff, they are using the firing velocity,
not the landing velocity, in calculating the efficiency. Not quite fair,
if so. But they are interested in comparing the performances of the
mechanisms, of different machine designs, so I can understand this.

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

.....................................................................
Donald E. Simanek
dsimanek@eagle.lhup.edu http://www.lhup.edu/~dsimanek
.....................................................................