The trebuchet is one of the largest and most destructive siege weapons to come out of the
middle ages. These massive war machines were commonly used
to break down enemy fortifications by launching heavy payloads, typically rocks and boulders
weighing as much as 180 kg, or 400 lbs. Projectiles could also include flammable material
intended to cause fires, and things like sewage and animal carcasses would sometimes be thrown
over castle walls in an attempt to spread disease.
Range and accuracy were far superior to other weaponry at the time as well, with some trebuchets
being able to exceed a throwing distance of 400 m, or about a quarter mile, depending
on the weight of the object. Although they have not been used for warfare
in more than 500 years, trebuchets have still remained quite popular throughout much of
modern history, and they are often constructed today for educational purposes and recreation;
Whether it’s for an engineering challenge, yeeting pumpkins in a fruit-throwing competition,
or taking out a castle with one hit in Age of Empires.
These impressive machines have some serious power despite their simple design, however
the underlying physics is actually fairly complicated, and we are going to find out
why by taking a closer look at how the mechanism works.
A typical trebuchet consists of a sturdy base and a large frame, and this supports a long
beam that is mounted on an axle. Construction is primarily done with heavy
timber, however the components may also be reinforced with leather, rope, and metal,
among other materials. The beam is positioned off-center so that
one side is approximately 4 to 6 times longer than the other, and a counterweight is suspended
from the shorter end, usually with a hinged connection so it can swing freely.
This is basically just a large wooden box filled with a heavy substance like sand, rocks,
or lead, and it can weigh as much as 20 metric tonnes depending on the size of the payload
being thrown. The short end of the beam is often thicker
or reinforced in order to carry the load from the counterweight, which also has the advantage
of lowering its moment of inertia by moving the center of mass towards the pivot point.
At the opposite end, the projectile is carried by a sling that is attached to the trebuchet
with 2 ropes; One that is securely fixed to the beam, and
another that slides freely over a finger so the payload can be released.
This finger can be any kind of projection, such as a peg or a hook, and its angle can
be adjusted in order to set the release point. To launch the projectile, a trebuchet employs
the law of conservation of energy by storing potential energy in the counterweight and
then converting it to kinetic energy. The counterweight is first lifted to a certain
height using a winch or treadwheel that is connected to the throwing end of the beam,
where its potential energy can be calculated by multiplying its mass by acceleration due
to gravity and its height above the bottom of the swing.
Since the beam acts as a lever and the lifting force is applied at the longer end, the force
that is required to lift the counterweight is actually a lot less than the counterweight
itself. If the throwing end is 4 times longer than
the short end, then the lifting force only needs to be ¼ of the counterweight, however
the displacement also needs to be 4 times greater.
Once the beam is in the correct orientation, it is tied down to a release mechanism and
detached from the winch, and the sling is positioned on a guide chute along the base
of the trebuchet where the projectile is loaded. When the beam is released, the falling counterweight
causes rotational acceleration because the applied torque is significantly greater than
the resistance from the payload, and the projectile is launched into the air as the free end of
the sling slides off the peg. The torque from the counterweight is not constant
through the entire motion, however, because the distance between the line of action and
the pivot point changes, and the acceleration of the beam will reach a maximum when it is
perfectly horizontal. The speed of the projectile will also be many
times faster than the speed of the counterweight since the sling follows a much wider arc,
which again comes back to the ratio between the long and short end of the beam.
If the ratio is still 4:1 like before, then the tip of the long end will travel 4 times
faster than the tip of the short end, however centripetal acceleration also causes the sling
to pivot around the end of the beam at an even faster rate, which increases the radius
and further amplifies the linear velocity. This double pendulum action is the thing that
gives trebuchets their awesome power and range, but it also makes their behaviour difficult
to analyze and predict. If all of the potential energy in the counterweight
were converted to kinetic energy in the projectile, then we could simply apply the law of conservation
of energy to compute the theoretical velocity and throwing distance, but in reality, these
machines are not 100% efficient. There will always be some amount of energy
remaining in the system after the payload is released because the beam and counterweight
will still be in motion, and so the maximum energy that can be transferred to the projectile
is therefore between 70% and 80% before accounting for losses due to friction and air resistance.
If we want to study the physics and model the behaviour accurately, then it becomes
necessary to break the trebuchet down into its primary components so that the equations
of motion can be derived for each part individually. This is done by drawing free-body diagrams
and summing the forces and torques in each direction, which can be a little tedious because
of the complex geometry, but the end result is a system of differential equations that
describes the movement of the components in relation to one another.
Now, unfortunately these equations cannot be solved analytically to obtain a closed-form
solution, however the system can still be solved iteratively by using a numerical integration
method. This type of problem needs to be divided into
a number of small time-increments and solved sequentially, as opposed to finding a single
governing equation, which means that the entire motion must be simulated from start to finish
in order to obtain a full a solution. Something that makes the problem slightly
more complicated is the fact that the constraints are not constant, and so the analysis also
needs to be broken up into three different stages.
When the beam and counterweight are first released, the sling is initially in contact
with the guide chute and it is only able to travel in one direction parallel to the surface,
but at the moment the sling is lifted away from the chute, its movement becomes unconstrained
and the equations of motion change. The equations change again just a few moments
later when the sling is released from the beam, at which point energy is removed from
the system, and the projectile continues off on its own following a simple parabolic trajectory.
Because the constraints on the projectile are slightly different at each stage, a separate
analysis needs to be conducted for each part of the sequence, where the end point of the
previous stage is taken as the initial condition for the next.
Once this has been completed and the differential equations are solved, the final result is
a complete simulation of the trebuchet, which can be used to track the path of the projectile
during the throw, along with its velocity and acceleration at every point.
It is particularly useful to find the speed and direction at the release point since this
determines where the projectile will strike, and it can also be used as a design tool to
help optimize a trebuchet by finding a set of parameters that will maximize the throwing
distance. For example, it can be shown that the range
will be greatest when the sling and the throwing end of the beam are approximately equal in
length, when the long end of the beam is roughly 4 times greater than the short end, and when
the beam is initially set at a 45-degree angle. The counterweight should also be about 100
times heavier than the payload in order to get as close to a free-fall as possible, and
the projectile should be released from the sling at 45-degrees from horizontal, which
happens to be the optimal angle for a standard projectile motion problem.
Together, these parameters will produce a design that is very close to optimal, and
although they have been derived using modern physics principles, it’s interesting to
see that many trebuchets from the middle ages were actually constructed with similar proportions.
The trebuchet was invented more than 500 years before Isaac Newton was around to establish
the basic laws of motion, and I think this says a great deal about the ingenuity of the
medieval engineers who built these incredible machines.
Castle defenses were simply no match for one of the most powerful siege weapons of all
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