On October 29th 1999, I received the following e-mail from Joe Tapley:

Most impressed by your web page. Has a useful content. Some while ago for my own benefit I wrote an arrow flight simulator. It has a number of simplifications to a complete mathematical model but is useful for understanding arrow flight. A copy is appended. Please copy/distribute as you see fit.

As you can see, I decided to publish it on the sagittarius site, which has had too little updates anyway and this seemed a good addition.

Obviously, I cannot give any guarantees about the quality and validity of the model used in this software. So you can just decide for yourself how you want to use it. Don't blame me (or Joe Tapley) for any results that do not satisfy you, the consequences of your actions are your own!

The program simulates arrow flight for bare shaft, walk-back and distance shooting by calculating the effect of drag forces on arrow speed, orientation and rotation.

The basic arrow and bow data inputs are entered under Arrow Setup and Bow Setup. Most of the numbers can be got with a tape measure and kitchen scales. The initial arrow speed for recurves would typically be 50-70 M/s. Initial arrow speed is calculated using a Bow Speed Factor which is for a known combination of arrow speed (M/s) and arrow mass (grammes) the mass*velocity squared. E.g. if the Speed is 60 m/s and the arrow weighs 15 grams then the bow speed factor is 15x60x60 = 54000. The bow speed factor allows the model to adjust the initial arrow speed as the arrow weight changes. The fletching area can be measured by tracing round the fletchings on graph paper. Note that the area excludes the shaft area. The arrow weight and balance point can be calculated by entering the pile and shaft weights & pressing calc. If calculated the apropriate items are given a red background.

Archer Group Setup allows definition of the range of arrow angular offsets in the vertical and horizontal plane due to variations in the archer's shot.

Walkback Setup and Flight Setup define the appropriate distances required. Wind direction is assumed to be from left to right (right to left if negative). The wind angle can vary from 0 (headwind) to 180 (tailwind). Checking Wind Gusts doubles the wind speed for 1/2 second at the half distance point. Aim-off angles are that resulting from horizontal/vertical pin movements to compensate for bow tuning and wind. Aim-off should be used, in Flight Calc. to bring the arrow hit point back to the Target Centre.

The Exclude Gravity option allows flight patterns to be created without the vertical gravity effect.

Note that Aim-off has a different effect on arrow dynamics than offset angle. Try adjusting Aim-off and offset angle to compensate for a wind to see the difference.

Constants Setup define the air density and assumed overall drag coefficients for the arrow.

Tuning Setup specifies how well the bow is tuned in terms of the horizontal and vertical angle to the direction of flight when the arrow leaves the bow. The initial arrow rotation is calculated from the initial angle. This is only an approximation and the Rotation Coefficient can be used to scale the calculated value between 0 (no rotation) and 1 (calculated value). Best guess is between 0.5 and 1.

The Mho effect torque on the arrow is not calculated with the current simulator and is just scaled with respect to the bare shaft fletching equivalent. i.e. 0 = no Mho torque, 1 = Mho torque equals shaft torque, greater than 1 then the Mho torque greater than the shaft torque. The value of the coefficient may be greater or less than 1 depending on the arrow.

A practical way to estimate the Mho coefficient is to do a bare shaft arrow walkback over say 5 to 15 metres with a slightly off-tune bow. If the walkback pattern is concave (bends towards the centreline) then the coefficient is greater than 1. If a convex pattern only can be obtained than the coefficient is less than 1. As a default use a value of 1.

File - Save Configuration will save all the current arrow etc. data to a file.

File - Load Configuration will retrieve the saved configuration.

The convention on directions is as follows:

The direction of flight and direction to the east of flight direction are positive.

Angles are positive in a clockwise direction to the direction of flight.

The Walk-Back Calc. button calculates the arrow flight with the defined bow/arrow and tuning over the defined walk-back distance. The boss arrow pattern is shown on a nominal 1 meter by 2 meter target. The arrow hits on the target are numbered 1,2...

The Flight Calc. button calculates the arrow flight with the defined bow/arrow and tuning over the defined distance. The horizontal/vertical scale is set by the first analysis and will remain until the clear button is clicked or a walk-back analysis run. To simulate actual flight the Aim-off should be adjusted to bring the arrow position at the flight distance back to the target centre. Flight shows the through air arrow path, Target shows the arrow hits on the target.

The Groups button calculates the arrow hits obtained from the archers variation from the standard bow tuning to represent the typical archers group size. Note that vertical variation of arrow angle has little effect, vertical variation is mainly due to arrow speed variation.

In all cases the box on the right contains arrow position, arrow speed and arrow angle at one metre intervals.

The result of an analysis can be saved using the File - Save Analysis option for subsequent import in a spreadsheet or whatever.

Any analysis file can be printed using e.g. Notepad.

Flight and walk-back plots can be superimposed as required. The colour of the centre line matches the colour of the last plot. The Clear button removes the current arrow plots.

Clicking and dragging on the flight path will create straight lines for marking eye-pin line, distances etc.

The simulator calculates form drag force using the standard equation:

drag force = drag coefficient*air density*velocity squared*arrow area/2

Shear drag effects are not calculated in the current model. Arrow rotational effects (lift & gyro effects) are not considered.

In the model the drag force is calculated by applying the composite velocity (arrow speed + wind) at right angles to the arrow shaft and pile areas & resolving the drag force in the forward & lateral directions.

The usual drag coefficient used for total drag on a cylinder is 1.2. This may be upped for the (flat) fletchings and reduced for the (parabolic) pile.

Drag force torque is calculated in the same way by applying the drag
force to the total fletching area which is assumed to act at 5 cm from the
nock. Torque is calculated with this force acting at the associated point of
percussion.

As the arrow centre of mass is towards the pile end there is
a length of shaft at the back of the arrow which has no counterbalancing
drag torque at the front. This area is added to the entered fletching area
to give the total area.

Arrow moment of inertia is calculated about an
axis through the point of rotation.

The effect of arrow rotation is not included in the shaft drag force calculation. A correction is made to fletching torque for arrow rotation.

Calculation of the arrow flight path is calculated in an incremental way. The rotational and linear accelerations are calculated at a specific point and then assumed to be constant over a small time interval (the model uses 1/1000th of a second). The consequent arrow positions & velocities calculated provide the input to the next time interval. No provision for added (virtual) mass is made.

It seems logical that if an arrow comes off the bow at angle to the bow line then it will have rotation resulting from string torque on the arrow. The model estimates the initial arrow angular velocity using the energy equation:

moment of intertia*angular velocity squared/2 = integral over the angle of torque at angle

There are a number of assumptions made in this approximation.

The rotation coefficient allows the calculated rotation to be adjusted to assess sensitivity.

Air vortices axially shed from the arrow will produce a torque (the Mho/Munk effect) in the direction to increase offset angle. For the current model the Mho torque is entered as a multiplier of the bare shaft torque. If the Mho coefficient is greater that 1 the arrow will therefore fly in a curve without fletchings as the arrow will rotate away from the direction of flight.

This arrow flight simulator is based on a simplified model where these coupling effects are neglected. This is done by assuming that the drag force is uniform along the arrow shaft. This is regarded as acceptable as the arrow velocity is much higher than the arrow yaw and pitch angular velocities.

The only area where this assumption seriously falls down is in the flight of bare shaft arrows. In the model the relative strength of the bare shaft fletching torque to the torque from vortex shedding is assumed to be a constant ratio, the 'Mho coefficient'. In practice this ratio is dependant on the arrow rotational characteristics at any specific moment and as the ratio for 'real' arrows is very close to 1 it is not possible to predict up front whether a bare shaft arrow will fly straight or in a curve. It's 'pick a number' time or do a bare shaft walk back with your setup to see what happens.

With fletched arrows the simplification results in a smoothing out of arrow flight paths. Walk back patterns for example show smoother variations in flight then you will get with an actual walk back.

The effect of shear (frictional) drag on arrow flight is not calculated so predicted arrow speeds will be higher than actual.

The other major model approximation is with fletchings. To accurately define the effect of the fletchings on arrow flight you would need to define to the model the exact fletching position and profile. This is not practical so the fletchings are lumped as an area providing torque to a single fixed point on the shaft.

The aim of the model is not to exactly predict arrow flight (no can do!) but to have a good enough model to enable understanding of how an arrow behaves and to assess what effect changing the arrow setup or bow tuning will have.

Joe Tapley - Greenwood Osterley Archers (tapley@cwcom.net)