Appendix

Euler'e equation, and Aerodynamic drag


Euler's formula for the buckling of a thin strut under axial load can be found in an engineering handbook, such as R.J. Roark, Formulas for Stress and Strain (McGraw Hill, 1954), in a number of variations to allow for different conditions of loading. Since the arrow is not held stiffly at either end, the version we require is that for a strut 'pin jointed at each end and subjected to both a point load P at one end and a distributed load p along its length L' for which the critical load is

Pcrit=(((pi^2)EI)/(L^2))-0.5pL......................(1)

where EI is an expression for the stiffness of the shaft, E being Young's modulus of elasicity for the material, and I the second moment of area of are of its cross section. Since Archers prefer to measure stiffness in terms of deflection under a sideways load, EI can be replaced using the standard formula

y=((W*S^3)/(48EI))......................(2)

Where y is the deflection caused by a load W in a standard spine test, and S is the distance between the supports. The first equation is therefore rewritten as

Pcrit+0.5 pL = ((pi^2)C)/(yL^2)......................(3)

When the arrow is being launched, the loads P and pL are produced by the inertia of the head and the shaft respectively, and equal to the product of their mass multiplied by the acceleration, i.e.

P + pL=a (M + mL)......................(4)

Where M is the mass of the head, mL is the mass of the shaft and a is the acceleration.

But the acceleration will be

a = f / (M + mL)......................(5)

where F is the force from the string. Hence the equation (3) may be rewritten

Fcrit * ((M + 0.5mL) / (M + mL) = ((pi^2)C)/(yL^2))......................(6)

from which the arrangement in the text follows immediately.

In calculating the effect of aerodynamic drag on the kinetic energy of the arrow, it is handy to express the energy U' remaining after distance S, as a proportion of the initial kinetic energy U0. his follows the usual 'die away curve' which often governs losses of energy, so that

U'=U0 e - ((2KS)/C)

where e = 2.718, C is the ballistic coefficient, and K is a constant depending on temperature, air pressure, and velocity. At arrow velocities K has a value around 000005 where energy is expressed in ft. Lb. and distance in feet. In the S.I. units J & mk is approximately 0.0002.


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