Lateral deflection of the arrow: Zanevskyy 2001

Twenty pages of mechanics, no arrow shot. Ihor Zanevskyy, a biomechanist in Lviv, builds a model of the bow and arrow taken together that holds, in one piece, the shaft's deflection in the lateral plane, the concentrated masses at its two ends, the lateral elasticity of string and limbs and the bow's virtual mass. Then he uses it to answer the question an archer asks at the bench: which shaft goes with this bow.

The answer is a condition on timing. The arrow has to have completed nearly a whole cycle of oscillation by the moment it leaves the string. Too stiff and it oscillates too fast to get round the handle; too soft and the tail flexes outward and strikes it.

The worked example is a 202-newton bow with two 660-millimetre shafts. The first, less stiff, stays in common motion with the string for 15.5 milliseconds against a period of 18.9: it leaves at three quarters of a cycle and the tail brushes the handle. The second, stiffer, stays 16.2 milliseconds against a period of 15.6: it has closed the cycle and starts again in the first quarter of the next, so the tail goes round the handle. The second is the right match, and the article notes it agrees with what a 1992 high-speed film showed and with Kooi's conclusions.

Here comes the part worth the price of admission. The static buckling test, with the rule that the bow's weight should sit two and a half times above the shaft's critical force, picks the other one: the first arrow gives 2.5 times 84.3, that is 211 newtons, almost exactly the bow's weight, while the second gives 290. The most used bench method and the dynamic model point to different shafts, and the article says why: in the static test spine does not depend on mass, in the shot it does.

The arithmetic recomputes. From the dimensionless frequencies, the stiffness and the shaft's mass the periods come out at 18.9 and 15.6 milliseconds, the printed values, and the ratio of the free periods squared, 1.23, is exactly the product of the mass ratio and the stiffness ratio.

The arrow is a vibrating beam, handled with the technical theory of beam vibration with concentrated masses at the ends: the point on one side, the nock plus the string's virtual mass on the other. Attached to the nock is an oscillator representing string, limbs and grip through concentrated elastic and inertial elements. The problem is a fourth-order differential equation with boundary conditions, solved as a polynomial series by successive approximations, and the natural frequencies and modes come out as an eigenvalue problem.

One choice underlies everything: the longitudinal and lateral motions are separated, in the sense that the lateral deflection is determined by the longitudinal motion but does not influence it. The article states that the error this separation introduces does not exceed one per cent of the system's energy.

The model is first explored in the abstract. The first four natural frequencies and their shapes are computed, and the modes that are periodic oscillations are distinguished from those in which the bend grows monotonically, that is those where the arrow loses stability. Raising the bow's force is found to lower the arrow's natural frequencies, and when the force reaches a critical value the frequency goes to zero: at that point the periodic oscillation turns into a deviation that keeps growing in the same direction. For a bow and arrow together the critical dimensionless force is 14.79, while in the static buckling test it is 9.87, a third lower.

Then come the real numbers. For the international federation's target bows: force at full draw from 121 to 244 newtons, efficiency from 70 to 80 per cent, arrow length from 622 to 826 millimetres, total mass from 16.6 to 23.8 grams, shaft mass from 13 to 17, distributed flexural stiffness from 2.73 to 5.96 newtons per square metre. With the average values, 180 newtons, 0.75 efficiency, 700 millimetres, 20 grams, the arrow's mean acceleration comes out at 6,750 metres per second squared and the dimensionless force at 12.3, that is inside the interval between the two critical forces.

The closing example is Table 2, with a 202-newton bow, a draw of 186 millimetres beyond the braced position, a brace height of 474, 76 per cent efficiency, a 6-gram string, a lateral virtual mass of 35.3 grams and a lateral stiffness of string and limbs of 278 newtons per metre. The two arrows are 660 millimetres long, with shafts of 14.1 and 15.8 grams, points of 3.5 and 3.9, tails of 2.8 and 2.9, buckling critical forces of 84.3 and 116.0 newtons, bend-test deflections of 14.3 and 10.0 millimetres and stiffnesses of 3.72 and 5.12. Out of that come exit speeds of 62.6 and 59.5 metres per second and the timings that settle the match. The two shafts are identified neither by make nor by spine number.

There is no measurement. Not one in the whole article: the work is theoretical and the comparison with reality sits in a single sentence, the one by which the computed parameters show a good correspondence with known experimental results obtained by video analysis. The reference for those results is a high-speed film from 1992, cited as a film and not as an article, with no numbers, no pages, no data. The second support is Kooi's theoretical work, which in its turn compared itself with a single shot read off a figure. So the only empirical anchor of this construction cannot be checked by a reader.

The example is one combination: one bow and two shafts. There is no series of spines, no series of bows, no replication, and the two shafts have no names: there is no knowing what make they are, what spine number they carry, whether they are aluminium or carbon. An archer therefore cannot connect the result to anything on the shelf, and a reader cannot verify the starting values of stiffness and critical force, which are where everything else comes from.

Some parameters are exactly the hard ones to measure, and their provenance is not stated: the 76 per cent efficiency, the 35.3-gram lateral virtual mass, the 278 newtons per metre of lateral stiffness of string and limbs. Those are three numbers that move the result and of which one does not read how they were obtained for this bow. The same holds for the energy breakdown given in the text, eleven per cent in the limbs, nine in the string, under two in the air and under three in hysteresis: the four values close with the declared efficiency, but they have no source.

The most interesting conclusion is also the least verified. The article closes by writing that spine ought to be directly proportional to the arrow's mass. It is a consequence of the algebra, because the period of oscillation depends on the ratio of mass to stiffness, and in these data it shows clearly: the ratio of the free periods squared, 1.23, is exactly the product of the mass ratio, 0.892, and the stiffness ratio, 1.376. But it is also a statement the manufacturers' charts do not follow, since they index on draw weight and arrow length rather than on mass, and the article does not test it against any real set of matched shafts. It remains a prediction to be checked, not a result.

The estimate of the separation error too, that splitting the longitudinal problem from the lateral one costs less than one per cent of the energy, appears in the conclusions and not in the results: the calculation supporting it is not to be found in the body of the article.

Accuracy is absent. That a tail brushing the handle causes an error in the result is written as a premise, not measured: no score, no dispersion, no comparison between shots. The model says where the tail passes, and the step from the tail to the target remains an assumption.

A note on the notation, because it misleads a reader. In the list of symbols `ma` is the arrow's total mass, while in the formula for the dimensionless force `ma` is the product of the shaft's mass and the acceleration: using the total mass that figure comes out at 16.5 rather than the printed 12.3, using the shaft's mass it comes out at 12.4. It is not an arithmetic error, it is the same symbol for two things.

This is the article to read if one wants to understand why a spine test done at the bench is not enough, and the reason is a single one: the bench measures a stiffness, the shot brings a stiffness and a mass into play together.

The strongest practical result is the disagreement between the two methods on the same bow. With a 202-newton bow, the classic buckling rule, by which the bow's weight should sit two and a half times above the shaft's critical force, picks the softer arrow, because 2.5 times 84.3 makes 211 newtons against the bow's 202, while the stiffer one would give 290. The dynamic model picks the stiffer, because that is the one closing its cycle of oscillation before it leaves the string. Both cannot be right, and the article puts the timings in a table: 16.2 milliseconds of common motion against a 15.6-millisecond period for the stiff arrow, 15.5 against 18.9 for the other.

The criterion to take away is this one, and it is easy to keep in mind: the arrow has to close one turn of oscillation while it is still on the string. Not half, not three quarters. One leaving at three quarters has its tail in the worst place as it passes the handle.

The second useful thing is knowing what weighs and what does not. Among the bow's parameters, the lateral stiffness of string and limbs counts more than the bow's mass for the shaft's deflection: so changing string or limbs shifts the spine match, while adding or removing weight on the stabilisers shifts it far less. And raising the draw weight lowers the arrow's natural frequency, which is another way of saying it pushes towards a stiffer shaft.

The third, to be treated as a hypothesis and not as a rule, is that spine ought to grow in proportion to the arrow's mass. If that holds, two shafts of the same declared spine but different weight are not interchangeable, and the heavier behaves as though it were softer. In the manufacturers' charts mass does not enter, so it is something to check on one's own bow with the bare shaft, not to take on trust.

What is not here: a number to take to the bench. No shaft is identified by make or spine, the shafts computed are two, the bow one, and no arrow was shot.