Topics on Archery Mechanics Joe Tapley Topics on Archery Mechanics Introduction The basic physics of archery has in principle been understood for around 80 years. The last topic to be theoretically described was vortex shedding (aerodynamics) in the 1920's related to developments in the aircraft industry. While the principles of archery are understood, in practice the behaviour of the bow/arrow/archer system (termed 'interior ballistics') and the arrow in flight (termed 'exterior ballistics') are somewhat complicated. In order to understand the mechanics of archery computer models are required. Models related to interior ballistics have been developed over the years becoming more realistic (and complex). A few related papers are listed below: Kooi, B.W. 1994. The Design of the Bow. Proc.Kon.Ned.Akad. v. Wetensch , 97(3), 283-309 The design and construction of various bow types is investigated and a mathematical model is used to assess the resulting effects on the (point mass) shot arrow. Kooi, B.W. & Bergman, C.A. 1997. An approach to the study of Ancient Archery using Mathematical Modelling. Antiquity, 71:124-134. Interesting comparison between the characteristics and performance of various historical and current bow designs. Kooi, B.W. & Sparenberg, J. A. 1997. On the Mechanics of the Arrow: Archer's Paradox. Journal of Engineering Mathematics 31(4):285-306 A mathematical model of the behaviour of the arrow when being shot from a bow including the effects of the pressure button and bow torsional rigidity. The string forces applied to the arrow are derived from the bow model referenced above. You can download most of the archery papers by Bob Kooi et al (in PDF/Postscript format) from the following link Archery Mechanics Papers. While inevitably many of the papers have a highly technical content there are non-mathematical versions which cover the general concepts and conclusions. Realistic mathematical models for the exterior ballistics of arrow flight, while simpler than the interior ballistics, are as far as I know non-existant. You can at least download a couple of (incomplete) models from this site. I have attempted here to put together a basic (laymans) guide to the bow/arrow system and arrow flight. The guide is not complete and comprises bits and pieces added when and if I have the time and inclination. Joe Tapley June 2000 Last Revision 1 July 2009 BOW MECHANICS Bow Efficiency And The Concept Of Virtual Mass Defining Virtual Mass Starting with a stationary bow at the brace height position, as you draw it back you store energy in the bow. The area under the force draw curve equals the total energy stored (Et). When the arrow is released the bow ends up stationary again at the bracing height. The total energy stored has gone somewhere. Most of this energy ends up where you want it as arrow linear kinetic energy (Ea) but the rest (Ew)is wasted. The energy equation for the bow is thus: Et = Ea + Ew The energy efficiency of the bow (F) is defined as the ratio of the arrow energy to the total stored energy. i.e. F = Ea/Et Supposing the arrow leaves the bow with a particular speed (S) then you can write the total stored energy in the bow as being equal to some imaginary mass (M) travelling at the same speed as the arrow. i.e. 2 Et = MS /2 The value "M" includes the mass of the arrow (m) with the remaining mass (v) called the Virtual Mass. i.e. M = m + v 2 2 2 2 This gives you that Et = MS /2 = (m + v)S /2 = mS /2 + vS /2 2 2 mS /2 is just the kinetic energy of the arrow and vS /2 is the wasted energy Ew in tems of the kinetic energy of an imaginary Virtual Mass. The bow energy efficiency F = Ea/Et thus becomes: F = (mS2/2)/((m + v)S2/2) = m/(m + v) The bow energy efficiency can be defined in terms of the arrow mass and the value of the virtual mass. What makes this useful is that it is found by experiment that for a given bow the value of the virtual mass is a constant over a sensible range of arrow mass. Note that this expression for bow energy efficiency and the value of v being constant indicates that the heavier the arrow then the more energy efficient the bow becomes. We'll look at the practical effects of this later on. Finding the Value of the Virtual Mass In order to see how varying arrow weight effects bow efficiency or arrow speed we need to know value of the (constant) virtual mass for the bow. There are a number of ways you can do this. The first method (maybe not recommended) is just to make a guess. Let's say a typical recurve bow has an energy efficiency between 70% and 80% and a typical arrow weight is 300 grains. Using the energy efficiency equation we have now got we get: with 70% efficiency 0.7 = 300/(300+v) which gives a value of v = 128 grains With 80% efficiency 0.8 = 300/(300+v) which gives a value of v = 75 grains So lets take a typical value of virtual mass at somewhere around 100 grains Probably the easiest method to measure the virtual mass reasonably accurately is shoot arrows of different weights and use a chrono to estimate the respective arrow speeds. Suppose we have two arrows of weight m1 and m2 and measure their respective speeds out of the bow at S1 and S2. The using the equation for total bow energy above we get: (m1 + v)S12/2 = (m2 + v)S22/2 or tidying things up (m1+v)/(m2+v) = S22/S12 From which you can calculate the value of v The third method that can be used, though a bit cumbersome, is to measure the speed out of the bow of an arrow of known weight and estimate the total energy stored in the bow from the force draw curve. You can then use the above equation to get the value of v: 2 Et = (m + v)S /2 To get the value for the total energy you can either plot a draw weight - draw length curve on graph paper and measure the area under it (A), from which Et = gA or assume the draw force curve is a straight line which gives Et = gLW/2. Where "L" is the draw length, "W" the draw weight and "g" the gravitational acceleration. Don't forget to include "g". In archery, energy is often quoted in foot-lbs which is not a unit of energy. To convert you need to multiply by "g". Example of Estimating Arrow Speed Suppose you have a 300 grain arrow (m1) and a measured speed of 210 feet per second (S1). What speed (S2) would you get with an increased pile weight say giving an overall 330 grain arrow (m2). Let's say the virtual mass value for the bow is 100 grains If you ignore the effect of arrow mass on bow efficiency then the two arrows would have the same kinetic energy i.e. S2 = S1*Squareroot(m1/m2) = 210*Sqrt(300/330) = 200 fps With the 300 grain arrow the bow energy efficiency = 300/(300+100) = 0.75 (75%) With a 330 grain arrow the bow energy efficiency = 330/(330+100) = 0.77 (77%) so the extra arrow weight increases the bow efficiency by around 2% The speed of the heavier arrow would be 210*Sqrt((300+100)/(330+100)) = 203 fps So the speed effect of the heavier arrow on bow efficiency is in this case around 3 feet per second How Well Does The Virtual Mass Approach Work To test whether the virtual mass approach works in predicting speeds for arrows of different masses clearly we need to use the method to predict arrow speeds and compare the prediction with actually measured speeds. Bertil Olssen on his excellent site has produced numbers we can be confident about so we can make the comparison. The following table lists measured different arrow speeds for different mass arrows and compares the results with speeds calculated using the above virtual mass method.The first two arrows in the table are used for the virtual mass calculation. Arrow Mass Measured Speed Calculated Speed Calculated-Measured Speed grams m/s m/s Difference 16.5 60.80 60.80 0.00 19.4 57.21 57.21 0.00 22.6 53.81 53.90 0.09 27.8 49.41 49.57 0.16 30.6 47.41 47.63 0.22 33.5 45.61 45.85 0.24 42.1 41.31 41.54 0.23 As can be seen in this case when more than doubling the arrow mass the estimated speed is withing 0.25 m/s BOW SIGHTS Bow sights are used to aid the archer to align the vertical and horizontal position of the bow so that the arrow hits the centre of the target. The reference (sighting) line runs from the archer's eye through the sight pin to the target centre. By moving the sight pin up/down, side to side or backwards and forwards the orientation of the bow is changed consistently with respect to this reference line. The main elements of a bow sight are represented in the attached diagram. What recurve bow sights do not do is assist in keeping the axis of the bow in the vertical plane (or canted at some angle if required).