     ### 3 Comparison of the test results with predictions

In Table 3 the experimental data is shown for the bows each shooting a number of arrows with different masses. The product q*h is calculated with experimental data only. The quantity q measures the amount of mechanical energy stored in the fully drawn bow divided by the product of the weight and the draw. This quality coefficient is large for static recurve bows because of the leverage action of the rigid ears. Observe that the amount of energy available for the acceleration of the arrow q*|OD| * F(|OD|) equals the amount of energy in the fully drawn bow Ab minus the amount of energy stored in the braced bow. The quantity h is the efficiency and this is defined as the part of the available energy transformed into kinetic energy of the arrow. Hence the product q*h equals the amount of kinetic energy of the arrow per weight and per draw of the bow.

Also the values for the modem working recurve bow are shown. On all mentioned bows excluding the replica of the Holmegaard bow Dacron bowstrings were used. The mass of these strings were 12, 10, 15 and 6 gram for the longbow Egyptian Tartar and modern bow respectively.

The results suggest that the initial velocity of the modern bow is large when the small weight of the bow is taken into account. The velocity of the 25 gram arrow shot with the Tartar composite bow is larger than when shot with the modern bow but the weight of the modern bow is much smaller and its efficiency is obviously much larger q*h=0.21 for the Tartar composite bow and q*h=0.28 for the modem working recurve bow. When the light 18 grams arrow is shot with the modern bow its velocity equals that of the 25 grams arrow shot with the Tartar composite bow and the efficiency is still better q*h=0.26. This shows the influence of the superior performance of the modem materials a good cast or fast arrow is combined with a high efficiency.

These measured values can be compared with calculated values for the different type of bows. The predictions are obtained with computer simulations by the use of the mathematical model described in Kooi 6,8. Note that the mentioned bows are not modelled based on the replica bows. This means that we did not use the dimensions of the replica bows. The comparison has therefore to be crude.

For the KL-bow AN-bow and TU-bow described in Table 1 in Kooi 5 the values with ma=0.077 mb. of the product q*h are: 0.32 0.28 and 0.30 respectively.

For the longbow (q*h=0.24) and Tartar bow (q*h=0.30) see Table 3 the results are not in contradiction with those for the KL-bow and the TU bow. The smaller experimental value for the longbow indicates that measured efficiency based on the calculated q=0.407 equals h=0.59 and this is smaller than the calculated value h=0.765. The static quality coefficient q of the TU-bow is q=0.491 and this implies that the measured and calculated efficiency equal h=0.62. The experimental data shows a relative bad performance of the Egyptian composite bow (about q*h=0.13). These results do not correlate with the results obtained with the mathematical model for the AN-bow. For the static quality coefficient q for the AN-bow we calculated q=0.395 and this implies that the efficiency of the replica bow would be only h=0.33 while the calculated value is h=0.72.

The results for the Holmegaard bow given in Table 3 show that the amount of energy of the arrow per weight per draw equals q*h=0.17 and this is a rather small value. For the HO-bow we calculated in Kooi 6 that q*h=0.31. The calculated static quality coefficient equals q=0.364 and this implies that the efficiency of the replica bow would be only h=0.47 while the calculated value for the HO-bow is h=0.78.

In the mathematical model losses due to neither damping nor hysteresis are taken into account. This implies that the calculated efficiency will generally be too high but this cannot account for the large differences found for replica bows of the Egyptian composite bow and the Holmegaard bow because we would expect about the same effects for the replica bows of the Tartar bow and the Medieval longbow.

In Tuijn 7 experiments showed a rather large difference in shooting from hand and from shooting machine. Possibly for loosing with the hand the length of the draw is not precisely defined This effect may be larger for the Egyptian composite bow with the large draw of 101.6 cm than for the older bows with a more conventional draw of for instance 81.3 cm. This remark could also be made with respect to the bad performance of the Holmegaard bow. Alrune writes: In one movement I draw lhe bow to my anchor and let go without any stop . If this technique was also used during the experiments this could imply a certain degree of uncertainty with respect to the draw which is on the other hand assumed to be only 26 inches. Alrune reports that the bow becomes at 24 inches very heavy to draw ('stacks'). Also in this case the actual available amount of energy in the fully drawn bow is perhaps smaller than the anticipated one. So when the actual draw al shooting was smaller than the mentioned draw this could explain part of the discrepancies.

On the other hand when a bow stacks the amount of energy stored in the fully drawn bow per weight per draw depends sensitively on the draw. This suggest that the calculated static quality coefficient for the HO-bow which equals q=0.364 is too small when the actual draw was smaller than 26 inches. Hence this counteracts somewhat the effect described above.

### 4 The kick

In Miller it is stated that: In shooting the reconstructed angular composites it was found that the central grip remains rigid throughout the draw, contributing to smooth action and greater accuracy and in McEwen 3: Releasing the bowstring produces no kick, which results in a smooth, accurate shot. The results given in Table 3, however, do not support for the angular Egyptian bow, the statement that: the composite bow is generally more efficient, so that no energy is dissipated in the kick and oscillation which characterize other bows, see reference 1. Alrune 4 states that for the Holmegaard bow: the replica is pleasant to shoot..

Hence, we conclude that for these two bows with a bad mechanical efficiency, shooters report a pleasant bow to shoot. This is in contradiction with Klopsteg's theory. Klopsteg in reference 9 page 170, writes: "The recoil, or kick, of a bow is found by experience to be small in bows of good cast, and large in sluggish, heavy bows. This is clearly a matter of efficiency. If the virtual mass is large in relation to arrow mass, the large amount of energy retained in the bow must somehow be dissipated, hence recoil becomes noticeable if not annoying". At page 101 in reference 9 Klopsteg states: "It can be said very definitely that smoothness of action and absence of kick in a bow, depend primarily on two factors. The first is dynamic balance of the limbs. The second condition is that the bow be highly efficient, a condition somewhat depending on the first factor of dynamic balance, on the quality of wood used, and on the design of the bow. When the efficiency of the bow is high, it means that a high percentage of the energy in the limbs is transferred to the arrow, leaving very little in the bow to produce unpleasant jar or kick. A bow of low efficiency, like some steel bows I have tested, is likely to kick severely".

On the other hand Hickman in reference 9 page 18, mentions that a bow which: "bends throughout its length in the arc of a circle (hence without rigid grip) as a rule is not a pleasant bow to shoot because it is likely to have a unpleasant kick. The 'dip' construction (which is credited to John Buchanan of England) decreases the kick and makes a sweeter bow to shoot".

In reference 6 we found that the string of a straight-end bow without a grip becomes slack after the arrow has left the bow. The large vibrating motions of the limbs cause the force in the string to become negative. When the string is suddenly stretched again it is possible that a kick is felt by the bowhand of the archer. It is tempting to claim that this explains the occurrence of a kick. If this is true then a larger internal or external damping of the material of the limbs causes a smaller efficiency but also a less severe vibrating motion of the bow and therefore probable, also after arrow exit, tension in the string. internal friction produces heat and is called damping because it decays free vibrations of the bow so that it returns to the braced situation. Also before the arrow has left the string damping is present and this causes loss of useful energy. External damping is the friction of the limbs and arrow in the air and depends on the velocity of the subjects and this produces heat too. Original energy is also dissipated partly by radiation of sound. The damping capacity of wood is higher than it is for most other structural materials. Steel is known to have a small damping capacity and thus Klopsteg's observation, that the steel bows he tested were likely to kick severely, supports our conjecture.

We conclude that the notion 'efficiency' has to be reconsidered with respect to the cause of a kick. There are two factors which contribute to the loss of energy, first the virtual mass of the limbs and secondly the internal and external damping. When the efficiency is high because of small virtual mass, this means that a relative high percentage of the energy in the limbs is transferred to the arrow, leaving very little in the bow to produce kick. When the efficiency is low because of a high damping so that a large part of the energy is dissipated as heat, a relative small amount of energy is left in the bow and this decreases the kick. This shows the influence of both factors clearly. The efficiency of the bow h defined above is the product of both factors.

In summary: a large damping for the replica of the Egyptian composite bow and of the Holmegaard s flat bow explains simultaneously, a low efficiency and a pleasant bow to shoot without a kick.

But why is the internal or external damping of the materials of these replica bows much larger than for instance the damping of the materials of the Tartar bow and, perhaps to a smaller extent, of the longbow? Otherwise the design of the angular bow and the flat bow could make them sweet bows to shoot. We. calculated in reference 6 that the force in the string of the AN-bow becomes negative after arrow exit. In Figure 1 the force in the string together with lhe acceleration force E and recoil force P as function of the time t arc shown.

The recoil force P is the force the bow exerts on the bowhand of the archer. For t=0 the force E and the recoil force P are just the weight of the bow E(|OD|). The results suggest that the force in the string has a rather large maximum being about 5 times as large as the weight. The recoil force shows an oscillatory behaviour after the arrow has left the string (fixed by the moment that the acceleration force becomes zero) and before the force in the string becomes zero.

### 5 Discussion

The cause of the bad performance of the replica bows of the Egyptian angular bow and the Holmegaard bow is still an open question. The results obtained with the mathematical model suggest that the efficiency should be in the same range as that of the Tartar composite bow and the longbow.

We encountered a number of contradicting statements about the kick. Further investigations should be done. Controlled experiments with a bow which is known to possess a kick could be performed. By introduction of artificial damping and measurement of the force in the string and the recoil force, our hypothesis, that the kick is caused by a slack string which becomes suddenly stretched, can be tested.

References     