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3 Results and discussion

The bows we used are specified in Table I. The specifications have the following meaning. For example the OK Bow 68" 38.5# has a top to bottom length of 1.73 m (68") and needs, according to the manufacturers specifications, a force (called weight by archers) of 171 N (38.5 lbs) for complete draw. We call this the nominal weight. The code CV means that the limbs have been reinforced with carbon fibres. The arrows we used were all aluminum Easton arrows, with the appropriate length for each bow.

Table I. Bows used.
SpecificationsuperflexsuperflexsuperflexTD 350
Type68" 38.5#70" 34#66" 40#68" 30#
Eff. draw [cm]43.050.944.042.9
Nom. weight [N]171151178133
Eff. Weight [N]170.5170178130
Bracing ht [cm]21.8-2221.9*)21.4-22
W [J]42.650.345.633.2
Hysteresis [%]0.2*)*)0.9
*) has not been measured.

For bows OK1 and GH we determined the force-draw curves with increasing and decreasing load; for the other bows with increasing load only. The hysteresis was less then 1 %, see Table I. Maybe these results are not completely reliable, because of the process of putting masses on the scale and taking them off again to exchange them with heavier ones. The relaxation of the material of the limbs may have been enhanced by this process. The calculation of the energy, being the area below the force-draw curve, was done with the trapezoid formula for numerical integration. The form of the curves is not very different from that of the curves published formerly [1-4].

Since the method for the measurement of the velocity appeared to be very accurate (most series of shots gave a standard deviation of about 0.2 % or less in the measured velocities), it was possible to detect small effects. On the other hand it was difficult to control all design parameters of the bow-arrow combination, such as the bracing height (the distance between the undrawn string and the grip of the bow, |OH| in Figure 3), the length of draw and the way of releasing.

Fig. 3, different situations of the working recurve bow
Fig.3: Different situations of the working-recurve bow; the upper half of the bow is shown. The X-axis coincides with the line of aim. O: unbraced (squares are stations for measurement of the shape); H: braced and D: fully drawn.*) has not been measured.

The difference in shooting from hand and from the rack was surprisingly large: the velocity of arrows shot from the rack was 3 % higher. Possibly during loosing by hand the length of draw is not precisely defined: the string is sliding along the finger-tips, so the energy transferred to the arrow will be less than when released in a mechanical way. Another cause may be that the bow hand is yielding a little during the shot. Further the arrows loose energy during their flight because of the friction with the air. For different arrows the loss of velocity in the first 5 m appeared to vary from 0.25 % per metre to 0.5 % per metre. Since the distance to the first coil was 0.5 to 1 m for shooting from the rack and about 3 m for shooting from hand, this explains a part of the difference. This also means that the real initial velocity, defined as the velocity of the arrow at the moment it leaves the string, was a little higher: about 1 to 2 % for shooting from hand, and 0.5 to 1 % for shooting from the rack. These small corrections have not been applied to the data in the tables and figures.

Table II. Velocity as a function of the number of strands. Fast Flight string on GH.
Number of strandsMassVelocity
166.4551.88 (6)
145.8251.43 (6)
125.2851.62 (10)
104.6451.76 (7)

One of the small effects we could detect is the influence of the stabilizers. Modern competition bows are equipped with several stabilizing elements to reduce rotations and other movements of the bow. This is shown in Figure 4. With complete stabilization the velocity is highest. Since the inertial mass of the stabilizers reduces the movement of the bow hand this is in agreement with the above mentioned yielding of the bow hand. Perhaps more important for target shooters is the result that the standard deviation is smallest with complete stabilization.

Fig. 4, Stabilizing elements of the bow
Fig.4: Stabilizing elements of the bow: S is string, T is top stabilizer, F is front stabilizer and V is V-bar. T and F are in the vertical plane of the bow, F pointing horizontally forward and T slantwise forward. V is in an almost horizontal plane, with both legs pointing a little backwards and a little downwards.

The effect of different string masses on the velocity has already been investigated by Hickman et al [3,pp.45-47]. His conclusion was that the effect depends on the specifications of the bow, and that 'the velocity of an arrow is reduced about the same amount as if the arrow were increased in weight by one-third the increase in weight of the string.' This can easily be explained if it is assumed that at arrow exit the ends of the string are in rest, the centre of the string has the same velocity as the arrow and the parts between have a velocity that linearly depends on the position along the string.

Yet the string can be subject of several investigations. In [5] it was pointed out that there are two counteracting effects: on one hand the velocity is higher when the string is lighter; on the other hand the thicker the string, the stiffer it is, and the higher the velocity. We prepared a Fast Flight string of 16 strands and removed consecutively two, four and six of its strands. In Table II it is shown that the differences are small. To exclude the effect of the mass of the string, we prepared strings of different materials, but with almost the same mass. In Table III the results indicate that the Dacron strings give a distinctly lower velocity. This is in agreement with its lower modulus of elasticity [5, p.119]. For Dacron B50 and Kevlar the force to elongate a string was determined: for Dacron the force is about 22 N per % of elongation per strand; for Kevlar 70 N per % per strand. Twaron too is known for its high stiffness. Apparently the Fast Flight strings are best; the material is Dyneema SK60.

Table III. The velocity, kinetic energy Ta and the efficiency for different strings: a) from hand with OK3 and complete stabilization, arrow mass 20.1 g; b) from rack with GH, no stabilizers, arrow mass 18.1 g.
StringMass [g]Velocity [m/s]Ta [J]Eff. [%]
Dacron6.1055.44 (12)30.868.0
Kevlar5.9558.07 (5)33.874.0
Twaron5.5557.77 (10)33.573.0
FastFlight6.0258.38 (12)34.275.0
Dacron6.5549.73 (4)22.570.0
Kevlar6.4851.25 (10)23.973.8
Twaron6.5351.68 (7)24.375.0
FastFlight6.4552.09 (9)24.776.2

An important quantity for the archer is the bracing height. This distance determines the amount of energy to be stored in the bow during drawing the arrow until the full draw. Therefore archers always measure this bracing height very precisely to secure reproducibility of the shots. Since it depends rather strongly on the length of the string, and strings may tend to yield a little when they are strung, sometimes the archer has to correct the length of the string by twisting it. By doing so the elastic properties are changed a bit, but probably this is a minor effect. To investigate the effect of bracing height, we followed the same procedure. We used one string and twisted it a number of times. Strangely enough the force at full draw does not change very much, though the string is shorter. This fact was discovered by Hickman [3, pp.18-21]. We illustrate it in Figure 5(a) with the force-draw curve of the Greenhorn bow for different bracing heights. This means that the amount of elastic energy W stored in the limbs is roughly a linear function of the bracing height b: the larger b, the smaller W, because the effective draw length decreases. If now the efficiency (the ratio of the kinetic energy of the arrow at exit and the elastic energy stored in the bow before release of the arrow) does not depend on the bracing height, the kinetic energy of the arrow will also be a linear function of the bracing height. In Figure 5(b) we have plotted the kinetic energy of the arrow as a function of the bracing height for the OK1 bow. The slope of the line is -95(4) J/(m^-1), which means that the dependency of the apparent efficiency on variations in bracing height is about 2 % per cm for this bow.

Fig. 5, Force-draw curve
Fig.5: a) The force-draw curve for four different bracing heights of the GH bow with Dacron string. b) The kinetic energy Ta as a function of the bracing height for the OK1 bow

Klopsteg [2] improved Hickman's rule by stating that a certain part r of W, the deformation energy of the bow (and string), is converted to kinetic energy of the arrow, the string and the bow limbs:

formula 1

where r is a number < 1, because there is some energy loss by hysteresis, m is the arrow mass and K represents one third of the string mass plus an unknown added mass, accounting for the kinetic energy of the limbs. In fact K also accounts for the excess of elastic energy in the limbs and the string at arrow exit compared with the undrawn, braced situation. In principle some added mass of air, that is dragged with the arrow, string and limbs will be included in K also. K is called virtual mass. Experimentally Klopsteg showed that K is a constant for a specific bow. In fact K is a phenomenological quantity that cannot simply be identified with physical properties of the bow. K can be determined by measuring the velocity as a function of arrow mass.

Table IV. Arrow mass dependence of velocity. Arrows shot from the rack. Linear least squares fit for 2W/(v^2) = (m + K)/r.
W [J]50.333.2
K [g]5.8 (5)9.2 (5)
r1.02 (2)1.04 (2)
Note: The OK bow was used with higher effective weight than nominal; the Greenhorn bow slightly lower

We measured the arrow mass dependence of the velocity for two bows. Some arrows were prepared with higher masses by moulding polymer material into the points. The results are summarized in Table IV. To find r and K we applied a linear least squares fit of 2W/(v^2) as a function of m, see Figure 6. The values of r are striking, they mean that a very small amount of energy is lost in hysteresis. This agrees well with the low values for the hysteresis we found with the force-draw curves.

Klopsteg's rule
Fig.6:Klopsteg's rule: the mass dependence of the kinetic energy for the OK2 bow (squares) and the GH bow (O). Linear least squares fit for 2W/(v^2)=(m+K)/r.

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