Siege Engines

A bit of History

Siege engines are machines, more or less complex, used mainly to attack or defend fortifications, like castles and fortresses.
There are a number of varieties of siege engines, ranging from simple ladders and mantlets to battering rams, siege towers and trebuchets.

Catapults, ballistas, trebuchets and its variations are the basic siege artillery: they are machines built for throwing projectiles, either very far away and/or very fast.
Due to its technology they are sometimes referred to as neuroballistic and baroballistic artillery (from the greek words for ‘sinew’ and ‘weight’).
These machines ruled the siege fields for millennia, until the advent of the so-called piroballistic artillery (from the greek word for ‘fire’): in the second half of the XIV th century the gunpowder cannon started making it’s practical appearance.


Four Types of Technology


Tension (Spring) Engines

Spring engines get their power from the tension of a bow spring (wood, composite or metal). The origin of all these machines is the simple structure known as a… bow. What a bow does is "transforming" the strength (force) into elastic energy and finally in kinetic energy. Being complex in its physics, it's quite simple to make.
The very first spring engines were just oversized bows that no human arms could possibly draw. So that required an arming structure and release mechanism. From a simple bow to a tension catapult it was a very fast, but important step.
These engines were the first ones and were present from the early greek beginning, throughout the Middle Ages, until being finally ousted by the cannon.



Counterweight and Traction Engines

This kind of engines take their energy from the weight and/or muscle power. Their work principle is simple: a shorter arm is pulled to provide a longer sling arm with more momentum for throwing the projective as far as possible.
In traction engines, the short arm requires someone to pull it down, with their muscles and/or weight. They are simply oversized slings, and were used all through history due to their simplicity, not effectiveness. They could achieve a high rate of fire, but had little accuracy.



In counterweight engines, as the name implies, the shorter arm has a heavy weight on it and gravity pulls it down, its swing providing the momentum for the longer sling arm to throw its massive load against an unsuspecting wall. Although it uses the same principle as traction engines, the fact that it always applying the exactly same amount of force, means they are able to achieve some measure of precision.

Trebuchets are counterweight engines, the pinnacle of baroballistic artillery. Trebuchets were the most powerful and effective artillery built before gunpowder finally took over the battlefield. These engines could hurl massive stones against walls hundreds of meters away, and crumble the defences of a castle much faster than anything else. However, they were complex, hard to build (between weeks to a few months), cumbersome and dangerous to operate. And heavy: for throwing a 100 kg (220 lbs) projectile some mere 200 meters (220 yards) it requires something like a 7 ton (15500 lbs) counterweigh, which is not exactly manoeuvrable.
These sizable machines would take the resources only available to a few.
Counterweight engines were not used in the classical roman and greek era. Its origin lays somewhere on the East (probably even China) and it started to be used in Western Europe only in the Middle Ages, on the wake of the first crusades.


Torsion (rope) Engines

Their energy comes from the twisting of rope bundles, made from animal hair or sinew, since both have excellent elastic properties. More accurately, these engines use the elastic properties of a spring coil to provide torque to one or more swing arms that in turn will propel the projectiles. Their working principle is simple: force is stored in an elastic material, in slow steady steps, and upon arming the weapon and releasing a catch or trigger, all that force will be suddenly applied to a projectile, making it achieve an enormous speed in a short span.
Balistas, Catapults, Onagers and Espringals are all torsion engines. Although artillery using this technology survived well until the cannon era, it’s golden age were the greek and roman classical era. 



Its development can be traced to IV century B.C
Light arrow-shooting balistae (scorpios) were very effective as battlefield anti-personnel weapons. With a high exit velocity the bolt’s "strait" trajectory allows it to be easily aimed, much like a crossbow or bow. Ancient sources credit small Roman scorpios with the kind of precision shooting we associate with modern "sniper" weapons, being able to take out key elements of the opposing armies.
Other heavier
arrow shooting and stone-shooter like catapults use the same torque principle  but achieve their working range after a typical parabolic curved trajectory; they were mainly used against walls and fortifications.

Cerco 21’s creations are all torsion engines: so let’s find more about them:

Ballistas - Scorpio

The word “ballista”, comes from the greek ‘ballistra’ (“thrower”), which was the original two-arm stone-shooter torsion engine. That configuration as a stone throwing device (literally, a ‘lithobolos’) disappeared by the end of the Roman Empire to be replaced by the much simpler one-arm Onager. As an arrow shooter, it lingered on. Today, in the common usage, we associate the term catapult with these two-arm engines. On the other hand, the word “catapult” actually comes from the greek ‘katapeltes’ (“shield-piercer”) and meant a two-arm arrow-shooter based on the ballista design. Latter any arrow-shooter came to be called a balista (along with and a string of other names during the Middle Ages).

So, what today we call a ballista is a two-arm arrow-shooter, its working principle being similar to a huge bow. A Scorpio was just a roman denomination for a light (small, portable) arrow-shooter.
The two-springs / two-arm torsion technology is complex, but with the right tweaking and tunning is also a reliable
and accurate one.

Catapult - Onager

This exclusively stone-throwing device appeared towards the end of the Roman Empire. Some claim that the simple technology hallmarks the decline of the Roman Empire, but perhaps some Generals appreciated the usefulness of something that is robust, cheaper to produce and simpler to operate.
For the Romans it was known as onager (from wild-ass because of its kick or squeaking sound; in another bit of name-mixing it was probably sometimes also referred as Scorpio). This is the engine that one more easily identifies with the word catapult.



Technically is like half a ballista, with one rope-coil and one-arm that strikes against a buffer in a cross-beam. There are two possible configurations of the throwing arm: with a sling, which is much more efficient and allows for low-trajectory shooting and it was used in classical times; or with a “spoon” or “basket” that only allows high-trajectory over-the-wall shooting and it was the mostly used in medieval times.


The Math behind all this

The beauty of the torsion engines is that, besides being a full lesson on history (extremely well documented archaeological, epigraphically and bibliographical evidence), they are also a belly-full of engineering, physics, mathematic and geometry. And yes, we like that.

The classical level of engineering is astonishing. After its early development, it took only some years of trial and error for the greek-speaking to determine empirically the precise optimal proportions all these torsion engines should obey to be at its top efficiency.
They reached such a good set of proportions, that their Romans successors never really changed them, doing only slight improvements.

These proportions where based on the “D”, that is diameter of the "spring" (the cylinder of coil of sinew rope), which was a function of the length of the bolt or the weight of the rock one wished to shoot. “D” would condition the size of every single piece on the machine.
For arrow-shooting engines the relation was relatively simple: a direct multiplication of the desired length of the projectile (L) would give you the diameter (D):


For instance, hole-carriers frames should be 6,5 x D, and the limbs or arms would be 7 D.
Cerco 21’s Scorpio, for instance has a D of 90 mm, so its designed to shoot 810 mm arrows and it’s frames are 585 mm long.

For rock throwing engines the formula was far more complicated: using “M” as desired weight (ancient unit ‘minas’, aprx. 436 g), D is obtained (in the ancient length unit of ‘dactyls’, aprx. 19 mm) by applying a cubic root:

It is clear that the inventors recognized that the power of the machine is a function of the volume of the sinew-rope bundle, but took no short-cuts or approximations, although it was not easy to calculate such a cubic root using classical-era mathematics. Remember, Algebra hadn’t been invented yet! So all computations were made using geometry in methods developed.


Modern Newtonian physics can also give a hand. As show by research done in the last decades, we can approximate a bundle of rope to a cylinder of the same material and the elastic properties will be roughly the same. Our experience showed us that indeed the torsion spring is only dependent on the outside diameter of the rope coil.

In a closed system and assuming near 0 friction (not true, but easier on the math) we have that potential energy can be transformed into kinetic energy and vice-versa without loss. We are disregarding all non-conservative forces in our approximation

. - Gravitic Potential Energy

 - Elastic Potential Energy

 - Kinetic Energy

 - Elastic Force (from Hooke’s law)

F = system Force (i.e. a 300kg Balista needs around 2940N to bring the string to the cocked position)

x = distance traveled by either the string from the cocked to the released position, or the counterweight from maximum height to minimum height.

k = Elastic Constant - Material Characteristic

M = counterweight mass

m = projectile mass

g = 9,8m/s2 - gravitic aceleration

v = exit velocity of the projectile

e = eficiency quoeficient (empiric)


So for a Torsion Engine we'll have:

-  This will be the bolt’s exit velocity dependent on all construction variables. Torsion Force (F), string travel distance (x) and weight of the projectile (m).

While for a Counterweight Engine we'll have:


- This would be the "stone" exit velocity also dependent on all construction variables, but this velocity would be increased by the length coefficient between the two arms. I.E. if the counterweight arm is 5 times shorter than the projectile arm the exit velocity will increase 5 times. Being the same throwing rigid arm pivoting around the same spot, angular speed must be the same. So:

So what we are doing when arming the engines is giving the system potential energy that it will be transformed into kinetic energy.

For a traction or counter weight engine you'll have Force -> Gravitic Potential Energy -> Kinetic Energy.

You have a mass M at a Height x, that when droped will cause the binary around a the pivoting spot, hurling the projectile m at a speed v. You transform traction, or pull force into gravitic potential energy and then into kinetic energy.

For a torsion engine you'll have Force -> Elastic Potential Energy -> Kinetic Energy.

Transforming Force into elastic potential energy is a question of "twist" (arming the weapon provides a torsion of the bundle) The force released is the same as the force inputed, however the diference is the time you take to twist the torsion bundle, and the time it takes to un-twist it.

We then end up with final velocity of V=ev (our e is determined for each siege engine and is someting that takes into account all the little variables like friction, and other non conservative forces).

From the exit velocity, angle and mass of the projectile we can then move towards the Balistics equations.


Balistic Equations (or "How far does it go and how hard does it hit?")

From the equations above we can compute our exit velocity based solely on our construction variables (overall size and spring size). Using our Medieval Balista we've computed and exit velocity of 132m/s however field testing and high speed photographs showed an exit velocity of around 120 m/s.

Now how does this all comes together?

Balistic equations tell us how far a projectile will travel based on exit velocity and angle. In very simple terms we can decompose a balistic motion in two components, it's horizontal and vertical motions. (we are disregarding shear wind that would bring the third axis).

So will have both equations in function of time:



To find drag acceleration we take into account the drag force on our bolts:

From which we compute our drag acceleration:

Both gravitic and drag acceleration are opposed to the motion and as such they are negative terms.

We then used excel to write a very simple discrete system. And we come up with these mind boggling inovative graphics (not really but maybe the google text crawler will believe it):

Our Construction Variables:

- Drag Coefficient (0,35 for bolt and 0,47 for spheres).

- Bolt Cross Section Area.

- Air Density

At 5º Angle:

From motion equation we can also find the impact energy of a projectile. We compute the velocity before impact and with the kinetic energy formula above we can determine the impact energy in joules from one of our bolts.

One of our bolts has a impact energy of 1213 J at 60 m.

As a comparison a Magnum .357 has a impact energy of 782 J at 60 m.

To reach a similar impact energy our bolts need to travel 250m!!!!