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 time, so it’s a good thing they are no longer
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1:01 is the best part obviously
Have an award and enjoy this video:
https://www.youtube.com/watch?v=CVlEKgywzUw