If on completion of a group tuning exercise at 70 meters some bare shaft arrows are shot then their relative hit positions with respect to the fletched arrows are
typically those as illustrated in the attached photograph. The bare shaft arrows hit above the fletched arrows and laterally displaced often to the left
as illustrated i.e. stiff for a right hand archer.
The reasons for the bare shaft hitting above the fletched arrows are fairly well understood, a combination of the bareshaft arrow getting more aerodynamic lift and being lighter, the bare shaft arrow having a slightly higher launch speed. The bare shaft relative height variation with distance sometimes becomes incorporated into basic bare shaft tuning methods as in the bare shaft at 30 meters hitting "slightly below" the fletched arrows
No reason has been put forward as to why the bare shaft arrows should be laterally displaced (left, right or center) with respect to the fletched arrows but the phenomenon has again been incorporated into some basic tuning suggestions as in "bare shaft arrow at 30 meters hits slightly low and left". It is also fairly common for archers to say that they get the best results from a bow set up on the "stiff side".
It is sometimes suggested that bare shaft arrow should be back weighted so that the arrow has the same all up mass (and sometime the same FOC) as the fletched arrow.
The reasons given
are:
a) That with the same mass the bare shaft will have the same launch speed as the fletched arrow and so consequent differences in hit height will be removed.
b) That the additional weight to the rear of the fletched arrow (fletching mass and aerodynamic "added mass" ) will result in the fletched arrow shaft acting "stiffer"
on the bow in a tuning sense. e.g. for a right handed archer the effect would be the bare shaft hitting to the right of the fletched arrow (weaker). This is
assumed to distort the result of a bare shaft tuning process. Backweighting the bare shaft gives it the same on-bow "stiffness" behaviour as the fletched arrow.
The overall suggestion to back weight bare shafts and the associated explanations for doing so are not valid. In terms of the relative vertical hit positions this is mainly due to the aerodynamic differences not the mass differences so backweighting the bare shaft has the result of moving the hit point of the bare shaft from one arbitrary position to another arbitrary position - pointless. As regards the rear weight "stiffness" argument you have exactly the same problem in that all you do is move the hit point of the bare shaft from one arbitrary position to another arbitrary position - again pointless. Where the bare shaft hits horizontally relative to the fletched arrow is the overall net result of many interrelated factors. Trying to make a minor contribution to this topic is the point of this web page.
If you look at the above photo for a group tuned system you can see that the unweighted bare shaft is acting "stiffer" then the fletched shaft, the complete opposite of the bare shaft backweighting argument. All that backweighting the bare shaft would presumably do is shift the hit point even further to the left.
One speculative factor affecting where the bare shaft hits laterally with respect to the fletched arrow is called the Magnus Effect. Flow over a spinning ball or cylinder
generates a lift force on the object resulting from the consequent non symmetric flow separation. The underlying principle (the Coanda effect) is similar to the
destabilising torque on an arrow discussed on the page on Vortex Shedding
.
The image is a "real flow" picture of assymetric separation over a spinning ball.
For the bare and fletched shafts to hit at different points laterally must be down to some difference between them and one obvious difference is that the fletched arrow is spun up by the fletchings and the bare shaft isn't.
A summary of a "back of the envelope" calculation on the effect of arrow spin on the relative hit positions of fletched and bare shafts is made here to assess if the Magnus effect is a candidate for explaining the observed arrow behaviour.
The arrow used for the estimate calculation is an (assumed non vibrating) 29.5" ACE 370 with 120 grain point launched at 204 fps. The fletchings are assumed to be a
typical spin wing size and weight.
Not having a spin wing fletching spin model available the spin calculator located on this site was used on the basis of a flat fletching with a 2 degree offset. This produced the attached graph of arrow spin RPM with distance.
The spin rate acceleration decreases over time towards the normal "terminal velocity" situation with the bulk of the RPM increase being generated in the first half of the flight. The spin rate and spin direction are two of the factors determining the direction and magnitude of the Magnus force. A right handed archer with the arrow rotating clockwise from the archer's viewpoint is assumed.
The arrow flight properties in the vertical plane were generated using the "Flight" simulator (see downloads page). This produced the arrow speed and airflow
attack angle variations over the flight assuming that the arrow is not spinning and there is no rotation of the arrow in the horizontal plane.
The conseqence of the vertical arrow acceleration under gravity is that that arrow porpoises. The airflow is intially on the underside of the arrow and switches to the arrow topside and so on. The airflow speed and angle of attack are both parameters affecting the Magnus force,
Combining the wind tunnel spin data and the vertical plane flight properties data allows an approximate estimate to be made of the lateral Magnus force variation over the flight at 5 meter
intervals. The arrow is assumed to be restrained so that it cannot rotate about a vertical axis but is allowed to rotate about a horizontal axis. The circulation is calculated
at a specific "flight" horizontal distance from the shaft radius and the RPM at that distance. The arrow spin direction is assumed to be in a clockwise direction
when viewed from the nock end. The fluid flow rate normal to the arrow shaft at any distance is taken from the output of the Flight" model which provides the total
velocity and angle of attack. Adding in the assumed air density of 1.2 gms/litre and the 29.5" length of the shaft allows calculation of the total Magnus force on
the shaft at 5 meter intervals. This data is given in the attached graph.
All arrows will porpoise to some extent in flight because of the action of gravity acceleration generated torque on the arrow. Intially the air flow acts on the bottom of the arrow shaft. In this case at around 34 meters the air flow will switch to acting on the top surface of the shaft. The direction of the Magnus force (acting horizontally) will change direction through 180 degrees when the flow switches from bottom to top of the shaft. In the attached chart you get a positive force with the flow on the bottom surface of the shaft and a negative force with the flow on the upper surface of the shaft. The points where the Magnus force is zero are where the flow switches between top and bottom surfaces. Asssuming a clockwise spin viewed from the nock end then a positive force is in the direction to move the arrow to the left (stiff) and vice versa for a negative force.
Keeping our wind tunnel model and not allowing the arrow to rotate around a vertical axis we can convert the arrow linear acceleration under the Magnus force into
a nominal horizontal arrow flight path. In this graph a positive value is a displacement to the stiff side (to the left for a right handed archer) and a negative
value is a displacement to the weak side.
The arrow initially curves to the left under the action of the Magnus force. When the airflow on the arrow reverses from bottom to top the direction of the Magnus force switches from right to left to left to right decelerating the arrow leftward movement and ultimately moving the arrow back to the right. The calculation predicts that the fletched arrow will hit around 6-7 centimetres to the left of the bareshaft arrow (which is assumed to fly straight).
Like most wind tunnel type models the above calculation bears little relation to what actually happens to an arrow in free flight. The arrow behavour is actually going to resemble the effect of asymmetric flow separation fron an attack angle discussed on the vortex shedding page. Flow separation acts as a control mechanism. The center of pressure of the force created by flow separation is located at the rear of the arrow generating a torque on the arrow which acts to increase the airflow angle of attack and hence the lateral drag force on the shaft resulting in a curved arrow flight path in the opposite direction to the flow separation force itself. In a similar way the centre of pressure of the Magnus force will act to the rear of the arrow centre of mass so again a torque is generated acting to rotate the arrow generating lateral drag in the opposite direction to that of the Magnus force itself. As any horizontal rotation of the arrow is going to affect the magnitude and direction of the Magnus force there is a bit of a feeback loop operating so as ever the actual arrow behaviour is going to be highly complex. Let's assume, from the above graph, an average Magnus force right to left of 0.01 Newtons acting from 0 to 35 metres and an average Magnus force left to right of 0.02 Newtons acting on the fletched arrow from 35 to 70 metres.
A Magnus effect force of 0.01 Newtons (stiff) is equivalent to a cross wind on the arrow of around 1.2 metres/second, A Magnus effect force of 0.02 Newtons (weak) is equivalent to a cross wind on the arrow of around 1.8 meters/second.
The Magnus force is going to cause the arrow to fishtail and the lateral drag forces on the arrow as a consequence of the fishtailing are going to be in the opposite direction to the Magnus force itself.
The effect of a 1.2m/s cross wind (right to left) from release until the vertical attack angle equals zero (35 meters) is estimated (using the Drift simulator) as resulting in a leftward drift of the fletched arrow of around 3.8cm. This is a lot less than the estimated 7.3cm leftward drift of the Magnus force only in a wind tunnel as per the graph above.
The values of the 35m flight direction (0.06 degrees), arrow speed (197fps), horizontal angle (3 degrees) and arrow angular velocity ((.05 rads.sec) at 35 meters distance from the Drift program are used along with the 1.8m/s (Magnus force equivalent) cross wind as the inputs to calculate the arrow horizontal behaviour over the 35 to 70 meter flight distance.
The result of the simulation is that the fletched arrow drifts to the right by 2.61cm. Subtract the inital 3.8cm drift left and overall the fletched arrow will hit 1.19cm to the left (stiff side) of the bareshaft arrow. So the result overall is that the Magnus force in principle can move the fletched arrow left or right a distance measured in centimetres. The calculation is not accurate enough to determine whether the overall direction is to the stiff or weak side.Possibly depending the conditions both stiff and weak results are feasible. A simplification in the above calculation is the assumption that the arrow has zero angular velocity and zero angle of attack at launch. As suggested by Zanevsky this cannot by the case, these values at launch will be determined by the group tuning and will have a significant effect on how the Magnus force affects the arrow flight. The discussion on tuning suggests that for an optimum tuning the arrow will be launched rotating in the vertical plane in the sense nose down, nock up. As far as the Magnus force goes this will have the effect of moving the fletched arrow more in the weak direction (to the right for a RH archer) than obtained from the above calculation.
The above estimates are extremely crude but they do not exclude the Magnus force from being a contributor to the bare shafts hitting with some lateral offset to the fletched arrow group of tuned fletched arrows at 70m. (Note it is the Fletched arrows being moved laterally and the bare shafts assumed to fly straight.) The obvious test is look at the relative hit positions of bare shaft and fletched arrows at 70 meters and other distances where the fletched arrows are spun in opposite directions or with different spin rates. It goes without saying that as the Magnus effect has an impact on arrow rotation it is a "tuning parameter" and will be catered for in any group tuning methodology.
A practical demonstration (fletched = spin, bare shaft = no spin)
Last Revision 13 February 2015