by J E Gordon
Some approximate figures for the strain energy storage capabilities of various materials are given in Table 3. The relative efficiencies of natural materials and of metals may come as a surprise to engineers, and some light is thrown on the performance of skiers and other animals by the figures for tendon and steel. It will be seen that the strain energy storage per unit weight is about twenty times higher for tendon than it is for modern spring steels. Although, considered as devices for storing strain energy, skiers are more efficient than most machines, yet even a trained athlete cannot compete with a deer upon a hillside or a squirrel or a monkey in a tree. It might be interesting to know the percentage of the body weight given up to tendon in these animals, as compared with people.
Animals like kangaroos progress by bounding. At each landing, energy has to be stored in the creature’s tendons, and I have been told by an Australian correspondent that the strain energy characteristics of kangaroo tendon are exceptionally good; but unfortunately I cannot quote any accurate figures. It occurs to me, however, that, if anyone should wish to revive the pogo-stick in a more efficient form, there would be a good deal to be said for making the spring from kangaroo tendon, or indeed from any form of tendon. Light aircraft, which have to be designed for bad landings on rough ground, often have their undercarriages sprung by means of rubber cords which have a strain energy capacity much better than that of steel springs, and are also better than tendon, though they are less durable.
TABLE 3
Approximate strain energy storage capacities of various solids
Material Working strain Working stress Strain energy stored Density Energy stored
% p.s.i. MN/m2 Joules × 106 per cubic metre kilograms per cubic metre Joules per kilogram
Ancient iron 0·03 10,000 70 0·01 7,800 1·3
Modern spring steel 0·3 100,000 700 10 7,800 130
Bronze 0·3 60,000 400 0·6 8,700 70
Yew wood 0·9 18,000 120 0·5 600 900
Tendon 8·0 10,000 70 2·8 1,100 2,500
Horn 4·0 13,000 90 1·8 1,200 1,500
Rubber 300 1,000 7 10·0 1,200 8,000
Besides its role in the suspensions of cars and aeroplanes and animals, strain energy plays an even more important part in the strength and fracture of all kinds of structures. However, before we pass on to the subject of fracture mechanics it may be worth spending a little time in discussing yet another application of strain energy, that is in weapons such as bows and catapults.
Bows
I will bring you the great bow of the divine Odysseus, and whosoever shall most easily string the bow with his hands, and shoot through all the twelve axes, with him will I go and forsake this house, this house of my marriage, so beautiful and filled with fair things, which I think I shall yet remember, aye, in a dream.
Penelope, in Homer, Odyssey XXI
The bow is one of the most effective ways of storing the energy of human muscles and releasing it to propel a missile weapon. The English longbow, which did so much execution at Crecy (1346) and Agincourt (1415), was nearly always made from yew. Because yew timber has not much commercial value nowadays, little scientific work was done on it until recently. However, my colleague Dr Henry Blyth, who is doing research on ancient weapons, has ascertained that yew (Taxus baccata) has a fine-scale morphology which is rather different from other timbers and seems to be peculiarly adapted for storing strain energy. Thus yew probably really is better than other woods for making bows.
Contrary to popular belief, English longbows were not, as a rule, made from English yew-trees, whether grown in churchyards or elsewhere. Most English bows were made from Spanish yew and it was legally compulsory to import consignments of Spanish bow-staves with each shipment of Spanish wine. In fact the yew-tree grows well, not only in Spain, but all over the Mediterranean area. It is growing wild today among the ruins of Pompeii for instance. In spite of this, one seldom hears of the use of yew bows in Spain or in the Mediterranean countries, either during the Middle Ages or in antiquity. Their use was almost confined to England and France and, to some extent, Germany and the Low Countries. English depredations generally stopped somewhere around Burgundy and hardly ever spread south of the Alps or the Pyrenees.
Although these facts seem surprising at first sight, Henry Blyth points out that, because of its rather special constitution, the mechanical properties of yew deteriorate more rapidly with increasing temperature than do those of other timbers. A yew bow cannot be used reliably above 35° C. As a weapon it is therefore pretty well confined to cool climates and is unsuitable for use in the Mediterranean summer. Thus, although yew wood was used for arrows, it was seldom used for bows in Mediterranean countries.
For this reason what was called a ‘composite’ bow was developed in these countries. Such bows had a core of wood which, being near the middle of the thickness of the bow, was only lightly stressed. To this core was glued a tension surface made from dried tendon and a compression face made from horn. Both these materials are even better at storing energy than yew. Furthermore they retain their mechanical properties better than yew in hot weather. After all, an animal normally operates at about 37° C. In practice, tendon does not deteriorate appreciably below about 55° C. As against this, dried tendon slackens and behaves badly in damp weather.
Composite bows of this kind were used both in Turkey and elsewhere down to comparatively recent times. Lord Aberdeen (1784-1860), travelling to the Congress of Vienna in 1813, wrote of the use of Tartar troops, armed with what seem to have been composite bows, against the armies of Napoleon which were retreating through eastern Europe. There is a good deal of evidence that composite bows were better in many respects than the English longbow. However, whereas the longbow was essentially a cheap and simple weapon to manufacture, the composite bow was a much more sophisticated affair, and presumably expensive. Greek bows were composite bows, and the bow of Odysseus, like that of Philoctetes, seems to have been a pretty special job.
Which brings us back to the unfortunate Penelope and the task she set her suitors of stringing the bow of Odysseus. As we all know, this turned out to be beyond the strength of any of them, even the technically-minded Eurymachus: ‘And now Eurymachus was handling the bow, warming it on this side and on that before the heat of the fire; yet even so he could not string it, and in his great heart he groaned mightily.’ But after all, why bother? Why didn’t the suitors, or Odysseus, or anybody else, just use a longer string?
The answer to this is ‘for a very good scientific reason’ – which is as follows. The energy which a man can put into a bow is limited by the characteristics of the human body. In practice, one can draw an arrow back about 0-6 metres (24 inches), and even a strong man cannot pull on the string with a force of more than about 350 Newtons (80 lb.). It follows that the available muscular energy must be around 0-6 metre × 350 Newtons, or about 210 Joules. This is the most that is available, and we want to store as much of it as possible as strain energy in the bow.
If we suppose that the bow is initially virtually unstressed and that the string is almost slack to begin with, then the archer starts to draw his arrow with a pull which is initially nearly zero, and he only works up to his greatest pull when the string reaches its maximum extension. This is expressed diagrammatically in Figure 2. In such a case, the energy put into the bow is the area of the triangle ABC, which cannot be more than half of the available energy, i.e. 105 Joules.
In practice the measured energy which was stored in an English longbow was a little less than this figure. However, Homer specifically says that the bow of Odysseus waspalintonos, that is, ‘bent or stretched backwards’. In other words the bow was initially bent in the opposite or ‘wrong’ direction, so that considerable forced had to be applied to it before it could be strung.
Figure 2. Energy stored in bow = ½ × 0·6 × 350 = 105 Joules.*
When we bow is strung in this way the archer is no longer starting to draw the bow from an initial condition of zero stress and strai
n; and, by intelligent design, it is now possible to arrange for the force-extension diagram to look something like Figure 4.
Figure 3. Greek stringing bow (vase painting).
Figure 4. Why a bow is ‘stretched backwards’ orpalintonos. Energy stored in bow is now area A B C D ≈ 170 Joules.
The area A BCD under such a diagram is now a very much higher fraction of the total available -energy and might perhaps reach about 80 per cent of it. So it is possible that about 170 Joules of energy can now be stored in the bow, as against only about 105 Joules for the bow that is not palintonos. This is clearly a great improvement for the archer – quite apart from any advantage it might have had for Penelope.
In fact all bows are more or less pre-stressed, in the sense that some kind of effort is needed to string them. However, since the longbow is a ‘self-bow’, that is to say, it is made from a stave which has been split from a log of timber and is therefore initially nearly straight, the effect in this case was small. It is much easier to arrange for the best initial shape with a composite bow, and these had usually a very characteristic form, from which we get the shape of a ‘Cupid’s bow’ (Figure 5).
Because the strain energy storage of horn and tendon, as materials, is better than that of yew, a composite bow can be made shorter and lighter than a wooden one. This is why we talk of a wooden bow as a ‘long’ bow. The composite bow could be made small enough to be used on horseback, as was indeed done by the Parthians and the Tartars. The Parthian bow was handy enough for the cavalrymen to be able to shoot backwards, as they retreated, at their Roman pursuers; from this we get the phrase ‘a Parthian shot’.
Figure 5. Composite bow, unstrung and strung.
Catapults
The greatest period of classical Greece came to an end when Athens fell in 404 B.C., and during the fourth century the Greek democratic governments declined and were superseded by dictatorships or ‘tyrannies’, which may have been more effective militarily, politically and economically. Both ashore and afloat the technology of warfare was changing, and the new rulers considered that there was a need for more modern and more mechanized weapons. Moreover, as the absolute masters of increasingly prosperous states, the dictators could well afford to pay the bills.
Development began in Greek Sicily. Dionysius I was a remarkable man who had risen from being a petty clerk in a government office to become Tyrant of Syracuse. During most of his reign, which lasted from 405 to 367 B.C., he made his country the leading power in Europe. As a part of his military programme he founded what was probably the first government research laboratory for weaponry, and for this establishment he recruited the best mathematicians and the best craftsmen from all over the Greek world.
The natural starting point for Dionysius’s experts was the traditional composite hand-bow. If one mounts such a bow upon some kind of stock and arranges to draw the string by means of mechanical gearing or levers, then the bow itself can be made much stiffer and so be enabled to store and deliver several times as much energy. Thus we arrive at the cross-bow, whose missile can generally penetrate any practicable thickness of body-armour.* The cross-bow has remained in use, with only minor variations, down to the present time. It is said to be in use in Ulster today. However, it is curious that, as a weapon, it never seems to have played any really decisive military role.
Furthermore, the cross-bow is essentially an infantry or antipersonnel weapon and it never fulfilled the requirement for a weapon which could do worthwhile damage to the hulls of ships or to fixed fortifications. Although the Syracusans enlarged the cross-bow type of catapult and put it on a proper mounting, like a gun-mounting, there seem to be certain physical limitations to this line of development, and catapults of the bow type do not seem ever to have been powerful enough to breach the heavy masonry of fortresses.†
The next step was therefore to abandon the bow type of construction and to store the strain energy in twisted skeins of tendon,‡ much like the skeins of rubber cord which are used to drive model aeroplanes. In such a skein all the cords, that is, the whole of the tendon material, are being stretched in tension as the skein is twisted, so that as an energy storage device it is very effective indeed.
There are various ways in which skeins of tendon rope can be used in weaponry, but by far the best was the device known to the Greeks as the palintonon and to the Romans as the ballista. In this very lethal piece of artillery there were two vertical tendon springs, each of them twisted by means of a rigid arm or lever, something like a capstan bar (Figure 6). The ends of these two arms were joined by a heavy bowstring, and the whole device worked much after the fashion of a bow. Indeed it got its Greek name from the fact that, in their relaxed position, the two arms point forward, like the arms of a composite bow; and the catapult was strung (by means of a powerful winch) in much the same way as a bow. The missile, which was often a stone ball, was propelled down a track which also served to mount the windlass that was needed to operate the weapon, whose draw force might be as high as a hundred tons.
Figure 6. A sketch of what original Greek catapults may have looked like.
The Romans copied the Greek catapults and Vitruvius, who was an artillery officer under Julius Caesar, has left us a handbook on ballistae which makes interesting reading. These weapons were made in sizes which ranged from one throwing a 5 lb. (2 kg) missile to one throwing a 360 lb. (150 kg) one. The effective range of all sizes was about a quarter of a mile or 400 metres. The standard Roman siege ballista seems to have been one throwing a 90 lb. (40 kg) ball.
At the final, dramatic, siege of Carthage in 146 b.c. the Romans filled in part of the shallow lagoon which lies against the city wall and proceeded to breach the defences with catapults. Archaeologists have recovered no fewer than 6,000 stone balls, weighing 90 lb. each, from the site.
Although ship-mounted catapults were used by both Julius Caesar and Claudius to clear the beaches of ancient Britons during their assault-landings on this island, the catapult never became a really dangerous ship-to-ship weapon. It seems likely that a ballista big enough to sink a ship with a single shot would have had a rate of shooting too slow for it to have had much chance of hitting a moving vessel.
Catapults sometimes threw incendiary missiles, but fires could generally be put out quite easily in simple ships which were full of men. One ingenious admiral won a sea-battle in 184 B.C. by shooting at the enemy brittle pots filled with poisonous snakes, but this lead does not seem to have been followed up. On the whole, catapults were not a success at sea.
Nevertheless, the palintonon or ballista was a most effective device for land warfare, although its construction and maintenance were a very sophisticated business indeed, and the Roman artillery officers and N.C.O.s must have been highly competent people. With the passing of the Roman Empire and of Roman technology such weapons became impracticable and were forgotten.* Medieval siege-warfare was reduced to using the weight-catapult or ‘trebuchet’.
This was a pendulum-like device using the potential energy of a raised weight. Even a large trebuchet was unlikely to involve raising more than say a ton (10,000 Newtons) through about 10 feet (3 metres). Thus the greatest potential energy stored probably did not much exceed 30,000 Joules. The same amount of strain energy could be stored in ten or twelve kilograms of tendon. Thus even a big trebuchet probably disposed of only about a tenth of the energy of the palintonon. Furthermore the efficiency of energy conversion seems to have been much lower. At its best the trebuchet could probably only make a nuisance of itself by lobbing big stones over a fortress wall; any assault upon heavy masonry would have been ineffectual.
Figure 7. The trebuchet or medieval weight-catapult – a most inefficient contrivance.
Regarded as machines for the conversion of energy, the bow and the palintonon both work on similar principles; it is not generally realized just how efficient an energy exchange mechanism is involved. In crude machines like the trebuchet, most of the energy which was available when the weapon was di
scharged went into accelerating the heavy lever or throwing-arm of the device and was ultimately lost in the necessary stop or braking system.
With a bow or a palintonon, when the bowstring is first released, some of the stored strain energy is communicated directly to the missile as kinetic energy. More of the available energy, however, is being used to accelerate the arms of the bow or the catapult, where it is temporarily stored as kinetic energy, much as it is in the trebuchet. In this case, though, as the discharge mechanism proceeds, the moving arms are slowed down, not by a fixed stop, but by the bowstring itself as it straightens and tautens. This further increases the tension in the string, enabling it to push yet harder on the missile and so speed it on its way. Thus, much of the kinetic energy stored in the arms is recovered.
Figure 8. Diagram of the mechanism of the palintonon or ballista.
(a) Ready to shoot. All the energy is stored in the tendon springs.
(b) Early stage of shooting operation. Heavy arms are being accelerated and so pick up much of the energy from the springs.
(c) Late stage of shooting operation. Heavy arms are being decelerated by increased tension in the string, and so their kinetic energy is transferred to the missile.
(d) Missile on its way, containing virtually all the energy which was stored in the system.
The mathematics of bows and catapults is difficult and, even when one has written down the equations of motion, they cannot be solved analytically. Fortunately, however, another colleague of mine, Dr Tony Pretlove, has been sufficiently interested in the problem to put the whole thing on a computer. It transpires that, rather surprisingly, the energy transfer process is in theory virtually 100 per cent efficient. In other words, practically the whole of the strain energy which was stored in the device can be converted into the kinetic energy of the missile. Therefore little energy is wasted or left behind to provide a recoil or to damage the weapon. In this respect, at least, bows and catapults are a great improvement on guns.