Simulation-technical evaluation of ancient catapult systems

If Hellenistic catapults and their Roman reproductions to center of the 1st century a. C. are shown a short a distance of the torsion bars as possible out, here seems in the 2nd half of the 1st century a. C. a radical change of the Roman torsion artillery to have come. This manifests itself by the first time with the representations of arrow catapults on the Trajans Column in Rome.

If the clamping frames of the traditional catapults were implemented in wood post and beam construction way with iron sheet metal reinforcing, now a type of bolt- firers with complete metal frames dips itself in addition on which is distinguished by a clearly increased distance of the torsion systems.




Illustration 1

Legionary fire also an arrow catapult (ballista) out a
fastened position. Trajans Column Rome
© German Archaeological Institut Rome

Michael Lewis dares it in a publication:

Current World of Archeology

to speak  of a revolution in the Roman catapult nature.

Trajan's atlilery.The Archeology of a novel Technological revolution

Lewis places there a reconstructed manuballista (torsion arm chest) also after interiorswivelling sheet arms forwards and this design is als postulates for on the Trajans Column represented types.
Aitor Iriarte even sets up the thesis that starting from the 1st century a. C. all Roman catapult types were equipped with internal catapult systems.

Duncan B. Campell
graphically reconstructed  the torsion catapult of Hatra as type swivelling inward. This sholuld be examined:

a. The effect of an increase of the distance of the torsion bars on the physical efficiency of a catapult..

b. The effects of a conversion of conventional external, outward swivelling to an internally working, inward oscillating catapult arm system.

Manfred Böhnisch developed for this purpose a computational program, which permits the simulation of different catapult variants.


Input table computational program (program provides by M. Böhnisch with Microsoft Exel)

Simulation of conventional designs (external catapult systems, external swivelling)

Computationally different variants were simulated. The dimensions - in the context of usual catapult sizes -  were selected randomly and do not correspond to documented finds.

In each case only the socket distance was changed. Clamping angles, the length of the bow- string and the pre-loading of the torsion bars remained the same in all three cases.

By continuous length of the bow- string and clamping angle the length of the catapult arms changed in each of the three cases by the enlargement of the socket distance.

Clamping angle: in each case 50°

Pre-loading torsion: in each case 15 Nm

Variant 1               Variant 2              Variant 3

Distance of  torsion sockets:          300 mm                 600 mm                 1200 mm
Length of the catapult arms :          700 mm                 560 mm                 215 mm
Arrow speed :                                  63.8 m/s                47.2 m/s                 21.4 m/s
Range :                                              415 m                  227 m                      45 m

 Table 1

It is clearly recognizable that by increasing the distance of the
torsion bars no improvement but rather a deterioration in performance is achieved.






 Pattern 1 to table 1

Variant 1 (Hellenistic type also closely standing torsion systems)


Pattern 2 to table 1


 Variant 2 (with increased bush distance)








Pattern 3 to table 1

Variant 3 wit a distance of 1200mm)




Two ballistics were build by us according to motives of the Trajans Column. between 1998 and 2002.

Catapult 1 (built 1998) had a distance of 450 between the torsion bars, with an arm length of 520mm and a length of the bow- string
of 1400mm.




Illustration 2

Catapult 2 built. 2002

(Rekonstruction und picture, author)

With catapult 2 (built 2002, illustration 2)  the distance of the torsion sockets were increased to 600mm. The catapult arms were 400mm and the chord 1200mm long.

This change was made, because a assumption existed that the shortened arms should positively be affected by their smaller weight the efficiency of the catapult. The dimensions of the two torsion systems remained unchanged during the renovation.

With practical attempts it then turned out that the firing ranges of the predecessor model were not reached any longer.

The geometrical data of the two models were entered into the simulaton program.



model from 1998      

model from 2002




Distance of torsion sockets:  :

450 mm

600 mm

Lenght of catapult arms       :

520 mm

400 mm

Arrow speed :

  52,97 m/s  

44,7 m/s

Range                            :

286,3 m        

204,6 m


Table 2


The actual ranges were the reality  approx. 25-30% under the computational values, because the effect of air resistance with the projectile pins and friction losses were neglected with the catapult mechanics.

The conditions of the parameters are dtermining to each other.

The recent model shows here, although it is from the concept actually more modern, rather disadvantages instead of an improvement.


Simulation of new designs (internal catapult system, interior swivelling)

The idea of a interiorswivelling catapult arm system is not new.

In the 2nd half 19. century the Frenchman Victor Prou allready reconstructed a stone thrower (palintonon) with internal catapult system. Michael Lewis, Duncan B. Campell and Aitor Iriarte continued to work on this t
up-to-date topic . (See menu option refernces)

Recalling the negative effects the one enlargement of the distances of the torsion sockets on the efficiency of such torsion catapults not only purely computationally but also practically has,  we had tried to graphically grasps the conditions with interiorswwivelling catapults..


The basis for this were the catapult findings of Orsova (Romania)

Both variants were examined once conventionally and once with an internal catapult system.

Here only graphically the possible length of the catapult arms swivelling inward was determined. Here restrictions are available, since the arms with the shot and the chord in the strained condition not allowed only such an angle may not to collide to insert the projectile pin into the arrow gutter which is still permitted. The lever arms were specified on 450mm and maintained also for the conventional interpretation.





Pattern 4


Even this simple graphic reconstruction shows on arrangement from advantage resulting that  a clamping angle of over 100° is possible.
This is thereby twice as large as during a conventional arrangement.
Thus the spacer enlargement would itself simply explain thereby that interior swivelling catapult systems were used.

The simulator program was now extended for an interior swivelling system.
As distance of the torsion systems 1,25m was selected.











Distance  torsions sockests :

1250 mm


Length catapult arms    :

  450 mm


Arrow speed           :

   45 m/s  


Range    :

   245 m         

490 m


Table 3

Simulation catapult conventionally and interior swivelling.

Also under the premise that this are theoretical values, the difference is so significant that one would like to exclude that the Orsava catapult was ever equipped with externalswivelling catapult arms.

Clearly advantages regarding range and speed for the projectile pins showed up from this system.
Table 3

Michael Lewis describes in his publication

Trajan’s artillery

The Archeology of a Roman Technological Revolution


The advantages of an internal catapult system in a way that nothing more is to add:


“The greatly increased arm-swing of the new machines compared with the old – it was at least doubled – had two advantages: more torsion energy was stored in the springs as the bowstring was withdrawn; and that energy had longer to act on the missile as it sped down the stock when the trigger was released.”


Illustration 3

 Ballista with interior catapult arm system.
Poor one and sees in starting position.
(Reconstruction and picture, author)


With ours - in the meantime ready for operation - reconstruction (illustration 4) with interiorswivelling arms a further advantage resulted. The rope bundles need only to be linked up only relatively weakly. In principle after drawing in the catapult arms only a 180° turn of the clamping chucks.

The energy is only introduced by the large clamping angle by stretching before the shot into the torsion bundles. In the conventional execution of 360° and more were necessary to achive sufficient shot performance.

These were very rope-careful systems because the demand in a state of rest was only weak and only with the real application to come to carrying.




Illustration 4

Ballista strained.

(Reconstruktion and picture, author)


Statement by Manfred Böhnisch

Creator of the catapult simulation.

It was our intention to create a tool that allowed us to compare different designs of torsion catapults. Especially conventional catapult systems with outside swinging arms should be compared with systems equipped with inside swinging arms.

It seemed to make sense at first to concentrate on the conventional design.

The existing catapult has already once been modified. By this modification the Distance of the catapult arms fulcrums had been increased and their length was reduced with bad effect to the performance of the catapult.

We thought about a simulation tool that allowed to compare different geometrical design in order to reach maximum performance.

It seemed to be helpful to divide the firing procedure into small steps and get by this a rough simulation of the entire process.

The simulation should be realized using Microsoft Excel and its Visual Basic functions.

Soon the concept for the simulation was defined.

It should work in the following way:

The power of the system is stored in tension of the torsion ropes.

For the simulation torsion ropes with linear spring characteristics seems to be acceptable. The spring characteristics will be taken from the existing catapult.

Illustration 5
Measurement on catapult arm. 
Interior catapult system
(Picture, author)

Different spring characteristics and pre-loadings shall be adjustable as parameters for the calculation.

The calculation should be done for a half system (one torsion arm and half arrow and chord/bowstring mass).

Torsion arm, arrow and cord should be considered as mass to be accelerated .

The torsion arm will be viewed as a bar with constant moment of inertia over its length. Since motion of torsion arm, arrow and cord is different during the process, they must be joined to a common moment of inertia. I decided to do this by transforming the moment of inertia of the arrow to a mass located at the end of the catapult arm. Half of the cord mass will be considered as being located at the end of the arm, the rest as pert of the arrows mass. That should give a sufficient approximation in in view of the small chord mass.

The speed ratio between the rotation of the torsion arms and the linear movement of the arrow constantly changes during the firing process. To handle this, the simulation will be accomplished in small angular steps. For each step the actual transmission ratio will be used to calculate the systems common moment of inertia. The converted energy per step results from the angle step the torsion ropes are relaxing.

At the end of the firing procedure, when the cord becomes straight, the speed ratio theoretically nears infinite and the cords modulus of elasticity wins of importance. I decided to fade out this not easy to handle condition by ending calculation short before this point. At this moment, arrow and catapult arm have geometrical determined speeds and by this also defined kinetic energy.

Comparing the kinetic energy of the arrow with the kinetic energy supplied by the torsion ropes, the effectiveness of the system can be calculated.

The cord must take the remaining energy of the catapult arms by acting as a spring. To be able to calculate its stress caused by this we decided to measure the spring rate of an original cord from the existing catapult. Knowing the spring rate allows to calculate the cords stress pending on the energy to be absorbed.

I did not know if this stress at the end of the firing procedure or the stress at the beginning, when the system is tensioned would be higher. So both values will be calculated and the bigger one compared with an estimated tensile strength of the cord.

The simulation program should be able to calculate with different initial values and list them together with th results of the calculation in a spread sheet.

Pattern 5
The shooting process is performed in steps to approximate simulation decomposed.

In order to get confirmed (or refuted) our ideas about the details of the firing process born in discussions, the process should be visualized graphically.

I decided to plot following values per angle step in two diagrams:

  • The development of peripheral speed if the torsion arms

  • The development of the arrows speed

  • The development of the speed ratio between arrow and torsion arm

  • The development of the arrows movement

As the simulation should only compare different geometrical constellations, I first wanted to ignore friction and air drag. Later I added a factor, that takes energy from the system pending on square of catapult arms peripheral speed.

Following substantial results of the simulation are being displayed or logged.

  • The arrows initial velocity

  • The theoretical range of fire (pending on initial firing angel and ignoring air drag)

  • The force needed to bend the system

  • The stress of the cord (absolute and compared with its firmness)

  • The efficiency of the system (part of the available Energy stored in the torsion ropes that became kinetic energy of th arrow)

The following values can be set as initial variables (pending on the design variant some values are dependent on others and not selectable)

  • The distance of fulcrums of the catapult arms.

  • The length of the torsion arms

  • The weight per meter of the torsion arms

  • The length of the cord/bowstring

  • The angle of rotation of the arms in tensed condition (starting angle)

  • The angle of rotation of the arms in relaxed condition (final angle)

  • The cords weight per meter

  • The arrows weight

  • The spring rate of the Tension ropes

  • The pre-loading of the tension ropes

  • The tensile strength of the cord

  • The spring characteristics of the cord

  • The firing angle (to calculate the range of fire)

  • The angle step the simulation should be carried out (radian measure, 0.005 – 0.01 proved to be useful)

I realized the simulation step by step. First the existing catapult with arms swiveling outwards should be viewed. This provided the opportunity to compare simulation with reality. As the simulation seemed to work for this I took the real bulks from the catapult such as arrow weight, chord weight, spring rates and also meter weight of catapult arms. For this we used a scale, a spring balance, a folding ruler and a protractor.

With these values we could now calculate and we where very pleased, that the simulation results showed, that the catapult in its original configuration in deed a must have had a better performance. Also the calculated amount of performance loss corresponded to reality.

Short time later the module for simulation of the version with inside swiveling arms where finished. The simulation results suggested a significant better performance of this design.

Pattern 6
External swivelling catapult system.



Diagram 1                                                                        Diagram 2                                                                                                                                                                                                                                                                                                                                                                                                                                   

A substantial factor for this are more favorable geometrical conditions at the end of the firing process. The speed ratio between arrow movement and peripheral speed of the torsion arm is determined at two points. On the one hand by the aspect ratio of the chord. This partial factor with both variants will ever more favorably, the more the string becomes straight. The second point is the angle between chord and torsion arm. And here the variant swiveling inward is clear in the advantage. With the conventional design this partial factor becomes more and more unfavorable at the end of firing.


 Pattern 7
Interior swivelling
catapult system.

Diagram 3                                                                                              Diagram 4

A comparison of the diagrams shows that during the firing procedure in both cases (with the interior swiveling variant even more) most energy is taken up by the torsion arms. At the end of the procedure the biggest part of this energy is being transferred to th arrow. How effective this can happen is pending on progress of the transmission ratio of their motions.

When we did run the simulation with different catapult arm masses, the performance of the System with inside swiveling arms was not much affected by heavy arms in contrast ti the conventional design.

Due to the much higher angle of rotation of the interior swiveling system the torsion rope should have a softer spring characteristics. In principle rather flat characteristics with high pre-loading are favorable for high shot performance.

Unfortunately the high stress even in relaxed condition will be a problem then.

When the catapult was reconstructed inti the interior swiveling design, the torsion rope design remained unchanged. This led to higher stress to the larger angle of rotation. So the pre-loading of the torsion ropes had to be reduced substantially. Their load in relaxed condition was the comfortable low. However, better performance could be reached by using longer and by this softer torsion ropes and higher pre-loading.

Sometime when discussing we spoke about the possibility to use a kind of hoisting drums instead of catapult arms on which the cord is rolled up when firing. I added another module to the simulation program to analyze this design.

The performance to be expected from this design is not very promising and so we didn't continue discussing about it (see also menu option “Differents”)


Designs and pictures: Author

Computational program provides by M. Böhnisch

Source references References to used publications and Internet sites are under the menu option references.