The design of torsion catapults

Michael French

Department of Engineering, University of Lancaster, Lancaster LA 1 4 YR, UK

This paper briefly describes the Ancient Greek catapults and shows the design to have been sophisticated but poor. I t then goes on to show how any of three design principles, properly understood and applied at a high level of abstraction, will lead readily to a much superior design. It is hoped that this strange stematic illustration may help to promote interest in the systematic use of such abstractions and principles in

practical design.

Keywords: design principles, engineering design, catapults

The Greeks appear to have invented the torsion catapult in about 370 ~c. It soon displaced the earlier catapults (gastraphetes), which used large bows as the springs, and was brought to a high pitch of refinement in the follow- ing century I. The new type of catapult used as springs or energy stores, two bundles of cords of sinew or hair twisted by rigid arms thrust through them (Figure la). These bundles were threaded through holes in a frame and wound round iron bars spanning the holes (Figure lb), after the fashion of a Spanish windlass. To increase the twist, the ends of the bundles were 'pretwisted' in the opposite direction to the arms. The two arms and the frame then functioned like a bow, the bending in the arms of the bow being replaced by twisting in the bundles. Special washers were used between the iron bars and the frame to assist pretwisting, and these were prevented from turning by pins inserted in unequally spaced patterns of holes in the washer and frame, an early application of the vernier principle.

A modular system of design was adopted, all dimen- sions being given in terms of a module, which was the diameter D of the holes in the frame. Thus the height of the frame was 6D, the section of the arms lAD x 1/2D, that of the iron bars l/sD x %D, and so on. The dimension D itself was determined from the size of

projectile: for arrows, D was one-ninth of the arrow length, while for stones D in dactyls (about 19.3 mm) was 1.1 times the cube root of the weight in drachmae (about 4.37 g). These rules are theoretically sound, as we can show today by dimensional analysis. The Greeks ex- perimented with the proportions--for instance, the ratio of height to diameter of the bundles--to obtain the best results, and it seems probable that the modular dimen- sions finally settled on are close to the optimum, given the form of spring adopted.

However, I shall demonstrate that they could have done very much better. That they did not is instructive, in my view, because it helps to show wherein the intellectual difficulty of functional design lies and how design can perhaps be taught.

We know so much about Ancient Greek artillery because three artillery manuals have been handed down to us, those of Bito, Hero and Philo. Translations with full and excellent commentaries have been made by Marsden 1, who also made small and full-size models of some of the machines. Earlier, full-scale models were made by Payne-Gallway 2 and Schramm 3. All these mod- els gave results comparable with the recorded perform- ances. However, no one seems to have studied the fundamental engineering. When this is done an asto-

208 0142-694X/89/040208-06 $03.00 © 1989 Butterworth & Co (Publishers) Ltd

I

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Pretwist

Iron b a r /

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Figure 1. Greek torsion catapult: (a) view of action; (b) action of single loop of bundle

nishing result is found, which is that the Greeks were using far more sinew than necessary.

Applying the formula for the diameter D to the 10 minae (4.37 kg) stone-throwing catapult gives 11 dactyls (212 mm), corresponding to a volume of sinew of about 88 litres. Using some figures kindly provided by Profes- sor R McNeill Alexander and Dr M Bennett of Leeds University for wallaby tail tendon, and assuming a working stress of 60% of the ultimate, this volume of sinew would store 425 kJ. Taking an efficiency conver- sion to kinetic energy in the stone of 0.93, which is reasonable, this corresponds to a launch velocity of 425 m/s, or an 'in vacuo' range of 18 km. The actual velocity was probably about 65 m/s, corresponding to an 'in vacuo' range of 431 m, requiring in theory only about 2.1 litres of sinew. Using my own results for tennis racket string, which may be better than anything the Greeks could have prepared, the figure falls to 0.83 litres.

This suggests that the sinew was greatly under-used, by a factor of between 40 and 100, which at first seems improbable. However, if we study the strength of the frame, the iron bars and the rigid arms, this result is supported. These parts are only strong enough to take about one-eighth of the load the sinew would exert if fully used. Since the energy stored is proportional to the square of the stress, this corresponds to an under-use of about 64 times. Table 1 gives estimated values for the percentage utilization based on various modes of col- lapse. Note that these are upper bounds, and assume the lowest wallaby tendon strength: for tennis racket quality, multiply by 0.4.

Bito's manual on artillery ~ includes descriptions of gastraphetes, the early giant crossbows. In one case, a design by Zopyrus of Tarentum, there is sufficient detail for estimates to be made of the amount of sinew in the bow (about 2 litres for projectiles with an estimated mass of 14 kg). Even assuming a lower launch velocity, this shows how little sinew was really necessary.

The small range of values of the upper bounds on the

utilization estimated in this way shows how well- balanced the design had become.

THE GREEK DESIGN OF TORSION SPRING

What was wrong with the Greek design? We know from Philo's account that the sinew cord and the frame itself were both near the limit of what they could stand. How

Table 1. Estimated utilization factors for sinew in Greek and Roman catapults

% utilization

Based on: Shear failure in iron bars* < 2.2 Shear in wooden beamst < 1.9 Bending in wooden beamst < 3.9 Bending in arms < 2.3 Reported range < 2.3 Comparison with Zopyrus' gastraphetes < 1.3

Assumed properties

Sinew. The working stress was taken as 187 N/ram 2, 60% of the lowest failure stress of wallaby tendon found by McNeill Alexander and Bennett (see text). Tennis racket string shows much higher strength, which would reduce the utilization factor by a factor of about 0.4 if the Greeks had equal material: calculation on Zopyrus' gastraphetes suggests they may have done.

Iron. Failure in shear at 130 N/mm 2 (i.e., equivalent to about 260 N/mm 2 tension on Tresca criterion).

Wood (e.g., oak or ash). Failure in bending, 100 N/mm2; in shear, 15 N/mm 2.

* No estimate has been included based on bending in the iron bars, because it depends too critically on the uncertain load distribution along them. Bronze bars, which Philo men- tions, would be weaker.

t For the arrow-firer. The beams in the stone-thrower were relatively much wider.

Vol 10 No 4 October 1989 209

could the sinew at the same time be so grossly under- used?

The answer lies in the poor geometry of the Greek spring. Where the bundles were pierced by the arms and the iron bars at the ends, the geometry of almost every cord was different. Moreover, cords in the middle of the bundle were hardly stretched at all. In a perfect design every cord would have the same stretch in it.

But this is not the only weakness. The average strain in a bundle having the Greek proportions, if it were simply twisted by the arm, would be altogether too small.

The effective length of the bundle in the arrow-firer was about 7.8D (3.5D for the side columns, 1D each for the two transverse beams, 3/40 each for the two washers, and about 0.4D allowance each for the parts going round the iron bars at the ends). The arm turned through about 35 ° according to Marsden 1, or 0.6 radians say. Figure 2 represents an outer string being deflected through 0.6 radians or 0.3D at its centre. The strain produced in the strand is only 0.003, whereas for full use about 0.06 is required. Since the energy stored is proportional to the square of the strain, the utilization would only be about 1/4oo in the most highly strained cords. Moreover, outside cords not passing close to the arm on the loaded side would not be deflected as far as 0.3D, nor would cords in the interior of the bundle.

The Greeks did two things to increase the strain and so the utilization. They pretensioned the cord as they wound it on the frame, a very time-consuming process, and they pretwisted the bundle, as has been explained, by turning the washers and the iron bars in the opposite direction to the movement of the arms, and so introduced a second amount of pretension and also altered the fundamental geometry of Figure 2. With no pretension, twisting the washers through 0.6 radians would give an initial deflec- tion of 0.3D and an initial strain of 0.003 as before, which would rise to a deflection of 0.6D and a strain of 0.012

l Strain = 0.003

0.6 o.sD...

Twisted cord

', \ 'l i ~ ---__~Untwisted ~ , cord

Figure 2. Strains in a single cord

~5

when the string was drawn back, a useful improvement but still far short of what is needed. This, too, would be only in outer strands.

Unfortunately, we have no information on either the amount of pretwist or the amount of pretension used by the Greeks. Philo remarks on the damaging effects of too much pretwist: ' the engine loses its springiness because the strands are huddled up into a thick spiral and the spring, becoming askew, is robbed of its natural force and liveliness through the excessive extra-twisting. Such a spring is hard to move and stubborn in pulling-back, but is weak and powerless in shooting, as it would be since the excessive extra-twist reverts to a similar relaxed state' (translated by Marsdenl). This is an interesting remark, because with modern knowledge we can under- stand that a bundle so heavily pretwisted as to 'huddle up ' would develop the hysteresis which Philo describes so well. He also remarks on the damage done to the cord by pretensioning during winding, when clips were ham- mered on to hold it temporarily, and these and the windlasses resulted in the cord being ' thoroughly squashed and torn' . However, we know this pretension cannot have been very much or it would have caused the frame to collapse.

By using a combination of pretension and pretwist, the average strain energy in the sinew before firing was probably raised to something like one-sixtieth of the potential, that is, to about one-eighth of the usable stress, since energy stored is proportional to stress squared. This is about what the frame could stand and the arms were strong enough to produce. On the other hand, the worst-placed cords would be subjected to a stress several times higher, and this together with the damage noted by Philo and the small radius (0.1D) on the iron bars might mean this was as much as the sinew itself could bear. Thus the design would be well-balanced, but nonetheless fundamentally bad.

A BETTER D E S I G N OF T O R S I O N SPRING

Using a fundamental approach, an improved design of torsion spring was devised, which uses about one-fiftieth as much sinew for roughly the same launch velocity and has a correspondingly light structure. Figure 3a shows the construction, and Figure 3b the comparison in size with the Greek weapon, shown in broken lines. The new scheme exemplifies a number of basic design principles, which are discussed in the next section.

The new design uses shorter, thinner and fewer cords; say, 1 m instead of 1.6 m, six strands instead of 15 in each spring, and a cord of about 2/7 of the diameter.

Each cord is deflected by the same amount*, and all

*To achieve this precisely, the cords must be slightly irregular- ly spaced round the axis, because of the shear on the bundle. To obtain equal loads on each strand, the resultant of all the (equal) transverse loads must be equal and opposite to the load in the bow-string. This implies two conditions on the spacing which can be met in many ways, e.g., with six cords by an angular spacing (o~, e~, ~, 13, 13, 13) and the appropriate choice of azimuth and the ratio ~/~.

210 DESIGN STUDIES

a

Figure 3. The new design: (a) spring; (b) comparison with Greek

the material is working equally and fully. There is no friction between cords and there are no sharp bends in them. The large compressive load exerted by the spring on the frame is carried in the most direct and economical way, by a column through the centre of the bundle. No direct pretension is applied, and the sinew can be relaxed simply and quickly by removing the bowstring and letting the arms go forward of the frame; this is done using a second bowstring in conjunction with the wind- lass, a procedure which is reversed for stringing. A large pretwist is used, but this is achieved not by rotating the ends of the bundle, but by stringing with the arms forward of the frame. The top cross-member can be removed to slip the arms over the columns. The cord is made in single lengths, with wooden 'buttons' or beads glued to the frayed-out ends to form a fixing of high efficiency.

A model was made to test the principle. For reasons of cost, it was made small, and instead of separate cords with buttons, a continuous cord was used, knotted at the ends and rendered round semicircular sheaves between end holes. This introduces friction and reduces the strain and hence the performance. Nevertheless, a discharge velocity of 57 m/s was achieved at once, without any development. With proper end-fittings and a little de- velopment it should be possible to equal the performance of the Greek weapons at say, 65 m/s, using about one-fiftieth as much sinew. The frame is 114 mm high, and it is strung with about 1.6 m of tennis racket string: for such a tiny thing, its performance is impressive, with a range of over 200 m.

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D E S I G N PRINCIPLES L E A D I N G T O T H E N E W D E S I G N

There are a number of design principles, any one of

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which would lead to this design without great conceptual leaps 4.

(1) Clari~y of function. This important principle, which is that functions should be performed in clear, unambiguous, simple ways, would lead the designer schooled in it to reject the Greek bundle, and isolate the basic mode of operation of each cord (as in Figure 2). The next step is to arrange for the same geometry of support for each cord. Also, since equality of strain means a hollow bundle, there must be a hole down the centre, and so on.

(2) Matching. One of the problems with the Greek design is that the aspect ratio of the bundle, the ratio of the height to the diameter, should be much smaller in order to obtain enough strain in the cords. However, with a smaller aspect ratio the loads on the frame would be much too high for any wooden construction or iron bar to bear. One answer might be to seek a spring material with more suitable properties, i.e., one better matched to the structure.

A reduced working strain (i.e., a stiffer material) would work well in a thin bundle. However, as the energy stored per unit volume is proportional to strain times stress, the forces on the frame would be even greater. An increased working strain would need an even thicker bundle. The conclusion is that this order (6% say) of working strain is about right, a lower working strain favouring a bow, as in the case of glass fibre, and a much higher working strain favouring the form of toy catapults using rubber.

What is wanted is a lower working stress, so that a thick bundle does not overload the frame and is not so stiff that it cannot be twisted with a wooden arm. Such a material was not available to the Greeks. On the other hand, the same effect can be obtained by

Vo110No4October 1989 211

spacing out strands of sinew with air between; the working strain will be unaltered but the effective stress will be reduced. Spacing out the strands means that since they do not react transversely one against another each must be individually deflected. This leads at once to the idea of plates with sets of holes in them, and so on.

Another way of approaching the solution via matching, perhaps a simpler one, is to note that the thicker bundle needed to obtain a useful strain has too great a cross-sectional area. We need to decouple thickness and cross-sectional area, and this can be done by making the bundle hollow, and so on.

(3) The principle of short direct force paths. Another way of arriving at the new design would be to concentrate on improving the structure rather than the spring. In the Greek design, the very large compressive loads exerted on the frame by the springs were transferred sideways to the columns by the cross-members acting as beams. Now beams are very inefficient structures, unlike columns. It is a well-known design principle to make force paths as direct and short as possible, and preferably to use only efficient structu- ral elements, i.e., struts and ties. In the case of the catapult, the elegant but obvious way to do this is to 'short circuit' the force path by a simple column down the centre of the spring: the amount of 'side- ways' transfer of load by beam action is then very small. Once this has been decided, the rest of the design follows almost automatically.

It is a simple matter to arrive at the improved version by the application of any of these three design principles, and the designer might well apply all of them and find himself irresistably carried along by the logic of the situation.

T H E USE OF PRINCIPLES IN D E S I G N

The example of the redesign of the torsion catapult shows the kind of use that can be made of principles in functional design, and also of concepts like 'decoupling'. The redesign was first done by the author in about 1955, but at that time he did not know the details of the Greek machines and so was unaware of how poor the spring design was. After reading Marsden, he decided to carry out tests on a small model, which was done in 1985. The redesign was based on making full and similar use of each cord, which is in accord with the principle of clarity of function. It resulted in the construction which has been described, including the central column.

An important aspect of the use of design principles in this way is that often both abstraction and physical insight are needed if the application is to prove fruitful. For example, the use of matching requires a rather abstract view of the torsion spring and the approach via structure needs insight into the load paths.

Design principles do not replace the need for invention

or creativity in the designer, but they do help. In the case of the catapult, the application is particularly simple, but not purely automatic: the steps are small and logical, but they must be taken. In other cases, the application may be much more difficult and the steps less obvious. The author has provided other examples and some guidance elsewhere 4'5 and a list of principles 6.

The origin of those design principles which are well known is not always clear. Perhaps the most useful, the principle of least constraint, was known to Kelvin but did not originate with him. The principle of short direct force paths, not under that name, was well known among aeroengine designers in the 1940s but has certainly had currency among all kinds of engineers for much longer. On the other hand, the idea of matching as a general principle, may have been first recorded by the author 7 but had been in use in particular contexts for many decades (e.g., in the choice of screw propellers and in the separation of gasesS). The principle of clarity of function may have been first enunciated by Pahl and Beitz 9 but has clearly been followed by good designers for much longer. The role of scholars should be to detect such principles, to formulate them, to extend them to other fields, to generalize, strengthen and clarify them with examples, theories and corollaries and, above all, to develop them into useful design tools with, wherever possible, quantitative expression.

As an example, the principle of short direct load paths may be expanded by a rule-of-thumb as follows. 'When space is unrestricted and strength is the determinant, it requires about three times as much material to transmit a force sideways as along its line of action: if the determi- nant is stiffness, the factor is about nine.'

This simple rule-of-thumb needs qualification, but is useful nevertheless. Other principles, like that of least constraint, have attached to them a great body of exam- ples which give them added substance and value, provid- ing a supply of generalized expedients which can be adopted when the circumstances suggest that kinematic design is appropriate. The designer first recognizes this appropriateness, and then reviews the body of examples for suitable models. A monograph on the principle of least constraint could be both a respectable academic work and an everyday help to practical designers.

S P E C U L A T I O N S

It is clear that the Greeks could have made much smaller, lighter and cheaper catapults. The engineer Philo saw some of the weaknesses and designed his own 'wedge' engine to overcome them. Unfortunately, his design was not very good and appears not to have been adopted.

What makes the task relatively easy today is not just the possession of sound design principles, but also our understanding of mechanics, particularly of the concepts of force, work and energy and their relationship. It appears that Archimedes himself may have believed that it was possible to design catapults either for greater range or for greater striking 'force', the latter having shorter,

212 DESIGN STUDIES

thicker springs (Ref 1, p 160). If the ancients had hit on this superior design, would it

have affected history? It would have been much cheaper, much lighter and easier to use in the field. The cords could be relaxed or made tight again in a small fraction of the time. Catapults were used as 'ship-to-ship' weapons, so that the new version, at about one-twelfth the weight, might have revolutionized naval warfare, just as the cannon did later. Catapults were only used in the field when there was some natural protection for them (e.g., at river crossings) but with the reduced weight and the quick readying for action, the new design might have found wide application as field artillery.

REFERENCES

1 blarsden, E W Greek and Roman artillery: technical treatises, Clarendon, Oxford (1971)

2 Payne-Gallway, R W F The crossbow, mediaeval and modern, military anti sporting, 2nd edition, London (1958)

3 Schramm, E Die antiken Geschiitze der Saalburg, Berlin (1918)

4 French M J Engineering design: the conceptual stage, Heine- mann Educational Books, London (1971)

5 French, M J 'The use of abstract formulations in design' Proc. Int. Conf. on Engineering Design, 1981 WDK, Zurich

6 French, M J 'A collection of design principles' Int. Work- shop on Engineering Design and Manufacturing Management, University of Melbourne, 1988

7 French, M J 'Aids to engineering design' The Engineer (19 May and 1 June 1967)

8 Ruhemann, M The separation of gases, Clarendon, Oxford (1940)

9 Pahl, G and Beitz, W Engineering design, English edition (Ed. K M Wallace), Design Council Books, London (1984)

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