Re: How do trebuchets work?

    You have put your finger upon the problem when it comes to detailed
information or equations dealing with trebuchets; either too much or not
enough.  If you haven’t done so already you might try to view the video
“Medieval Siege” produced by NOVA (Annenburg CPB) and filmed for the most
part in your country.  I believe the two machines featured may still be at
Loch Ness.  They show a fair amount of the physics and engineering behind
these machines as well as a lot of history.  The trebuchet simulator you
will find at www.trebuchet.com is good or you could use a general physics
simulator like “Interactive Physics”.  
    As I am sure you are by now well aware there are many factors that
affect the range.  If you write out a full set of dynamical equations of
motion there are no general closed form solutions for them.  Also, at least
from my view point,  a trebuchet being at a minimum three simultaneously
interacting masses (either projectile, arm and counterweight or projectile,
arm, and rolling base) is an example of a “many body problem” which
inherently has no general solution due to having more variables than
independent equations from which to find them.  Your best bet at a
preliminary understanding is probably to write out the energy balance
(conservation of energy equation) and consider the values and relationships
of its individual terms at various points during the firing cycle.  This
should reveal the most efficient transfer of energy to the shot from the
counterweight occurs if you can get the sling to release the shot 45deg
above horizontal when the counterweight is at its lowest point.  However…
for given ratios of shot weight to counterweight and sling lengths to
throwing arm lengths you may find a different release angle with a higher
velocity has more range than at 45deg with the counterweight not at its
lowest point.  The timing is critical and not easily calculated at all in
general. 
     You could build a trebuchet without a sling but it would be difficult
to make it release well or work efficiently.  The sling really improves the
performance and should be considered, in my humble opinion, as a necessary
part of the device.  The sling length and hook angle at the tip of the arm
also greatly affect the timing.  You have the right desire experimentally
to change only one thing at a time and none of the others but this is one
of those annoying cases where this is not generally possible.  Based on the
latest work by a pair of my students, they were investigating a few of the
primary dimensionless ratios (PI parameters) and their effects, you might
try the following two approaches.  First you can adjust sling length for
maximum range for the same mass shot with various counterweights or perhaps
since trebuchets used for their intended purpose had a more or less fixed
distance they needed to throw you could adjust sling length to throw the
most massive shot at given distance for various counterweight masses.  You
mention changing the counterweight alone with a fixed sling length.  This
can also be revealing and you should try it paying careful attention to its
effect on the release timing.  I hope this response is not also too much or
way too little.  Have some safe and happy hurling.