The various mechanisms involved in arrow stabilisation after it has left the bow have already been covered on other pages
(notably the sections on fletched arrows and drag). It is however such a fundamental topic with
respect to understanding arrow design, arrow flight and bow tuning that it's probably worth pulling
all the relevant bits together into a single unit. I will not rehash material like arrow rotation
axis, centre of pressure etc. in detail as already covered on other pages. I'm assuming these have been read.
I am also going to ignore the effects of gravity and the Munk moment as these just complicate
things without adding anything to the general idea.
The words 'arrow stabilisation' are often come across but my definition is that arrow stabilisation
has occured when all the rotational kinetic energy that the arrow has at launch is reduced to zero,
or to put it another way all the arrow angular momentum at launch is reduced to zero.
The base issue is that if an arrow has some angular momentum at launch then during the stabilisation
period the arrow acquires via drag a changed velocity, both speed and direction being altered. If
the arrow was launched with the correct speed and direction to hit the target center then after
stabilisation the arrow speed and direction are no longer those to hit the target center. (For the
purposes of this description I'm assuming that what's wanted is a zero launch angular momentum, so
I am not considering tuning optimisation of the arrow launch angular momentum with respect to
distance or wind. These topics are covered separately under variable tuning wind/distance).
There are two aspects to arrow stabilisation, fletching action and drag effects. The two will be
considered separately and then various aspects combined.
Fletching Action
Suppose you have a weight sitting on a table. There is a downward constant force F on the weight,
the effect of gravity. To lift the weight you have to do work against this force so if you lift
the weight a distance h to have to expend an energy Fh. All this hard work you've done
doesn't disappear. This energy can be regarded as being stored in the weight as (gravitational)
Potential Energy. If you let the weight go it falls down and the potential energy is
converted to Kinetic Energy. Just as the weight reaches the table all the potential energy you put
in by lifting it has been converted to Kinetic Energy.
Suppose you have weight on string, a pendulum. If you displace the pendulum some amount you are
lifting the weight and storing potential energy in it. If you let the pendulum go it swings back. Same
as the weight above the potential energy is converted to kinetic energy. When the pendulum reaches the
lowest point all the potential energy (ignoring frictional etc. losses) has been converted to kinetic
energy. From its lowest point the pendulum keeps swinging (raising the weight) until all the kinetic
energy is converted back into potential energy. So a swinging pendulum is just an energy oscillation
between kinetic and potential energy and with no energy losses would go on indefinitely. The angle
through which the pendulum swings depends on how much total energy it has and time it takes to swing
depends on the gravity force on the mass. The length of the pendulum effects both the angle and the
time of swing.
What's all the above to do with fletchings?. Suppose you have an arrow with an overall fletching
surface area A with the arrow pointed directly into an overall air flow of speed S. The drag
force on the fletching surface area is
only frictional so it's negligible. If we push the arrow (at its centre of pressure) then the
arrow rotates about some axis (determined by the centre of pressure distance from the arrow centre of
gravity). If the arrow has been rotated by an angle a then the inertial drag force on the
fletching surface is proportional to ASin2(a)S2. If the distance from the
center of pressure to the arrow rotation axis is L and the arrow is pushed through a small
angle t then the distance through which the arrow has been pushed against the drag force
is Lt.
So the energy required (force times distance) to rotate an arrow from an angle a to an
angle a+t is proportional to [ASin2(a)S2][Lt]. If an arrow is
at an angle a to the airflow then it has (similar to the lifted weight) a (drag) potential
energy. The stored potential energy depends on the angle to the airflow, the overall fletching area, the
arrow speed and the distance L (which depends on the arrow length, diameter, FOC, centre of
pressure and moment of inertia). This potential energy at arrow stabilisation equals the rotational kinetic energy the arrow was
launched with as a result of the torque input to the arrow by the bow.
As the force against which the arrow is rotated is not constant but depends on the airflow angle
and the overall air flow speed (arrow linear speed and angular rotation speed) it's a more complicated case then lifting the weight. The more the arrow has
rotated the higher the force against which the arrow is moved.
The aim is to minimise the time it takes and minimise the angle through which the arrow rotates for
arrow stabilisation as this minimises the change in overall velocity of the arrow resulting from drag
(as we'll see in a minute).
Once the arrow has stabilised then you have an arrow with a changed velocity and with the arrow shaft
at some angle to the air flow. The behaviour of the arrow is then similar to the displaced pendulum
above. The arrow fishtails/porpoises back and forth with an energy oscillation between fletching potential
energy and and the arrow rotational kinetic energy. As discussed elsewhere fishtailing/porposing has a minimal effect on
the arrow overall flight direction as the drag effects are more or less symmetrical and cancel out. The launch alignment of
the arrow itself has no significant effect in itself in changing the overall arrow flight path.
The effect of arrow stabilisation on flight direction is illustrated in the following figures. In this case the comparison
is made between a fletched arrow (blue) and a bare shaft arrow (red). The fletched arrow stabilises faster and attains a
lower offset angle than the bare shaft because the increased fletching area means more potential energy per degree offset.
The result is the change in direction of travel is higher for the bare shaft than for the fletched shaft. This of course
is the basic principle behind the bare shaft tuning method.
The following graph illustrates the lateral drag acceleration on the same two arrows. The vertical black lines indicate
the respective stabilisation distances. The bare shaft experiences a higher lateral drag acceleration and over a longer
distance/time then the fletched shaft hence the bigger shift in travel direction.
Drag Effects
The drag effect were interested in is the change in acceleration of the arrow as a consequence
of its rotating to what the 'normal' arrow acceleration would have been with no rotation. To simplify
this am going to ignore the effect of gravity on the drag acceleration of the arrow and also ignore
the drag effects on the arrow pile.
If the arrow shaft is at an angle a to an overall airflow of speed S then there is a drag force
acting at 90 degrees to the shaft accelerating the arrow in some direction depending on the arrow's
orientation in space and its current velocity. If the arrow leaves the bow with some rotational kinetic energy the angle a
is dependent on the consequent angular velocity and will increase from about zero to some maximum angle
determined by the fletching action (as above). If the surface area of the shaft contributing to this
drag acceleration is W then the drag force on the shaft is proportional to
WSin2(a)S2 (with W being determined by the shaft length, diameter and the
arrow FOC). If dV is the change in arrow linear velocity as a result of the arrow rotational kinetic energy
over the stabilisation time t and the effective average acceleration of the arrow resulting from
the rotation is f then dV = ft. To minimise the change in velocity from the launch kinetic
energy what's wanted is to minimise the stabilisation time, the angle of rotation and the drag area
contributing to the arrow acceleration. The higher the arrow speed then the higher the average
acceleration also.
Combination of Fletching Effect and Drag
What constitues a 'good' arrow is one where its properties minimise the time and angle of rotation (fletching effect)
while at the same time minimising the consequent drag acceleration resulting from the launch angular momentum (drag effect).
We can look at the effects of some arrow properties to see how this works.
Arrow Diameter
Part of the shaft area contributes to the fletching area (FOC effect) so for the fletching action the larger the arrow
diameter the better. As regards the drag acceleration the larger the diameter the larger the drag area so in this case the
smaller the diameter the better. So whether a smaller or larger diameter is better as regards the change in velocity there
is no definite answer. It's going to depend on their relative values (FOC being the driving force here). However as we can
(in theory) always increase the fletching area by using larger fletchings the smaller arrow diameter is probably going
to be the preferred option.
Arrow Launch Speed
As both the fletching action and the drag acceleration both depend on the square of the arrow speed at first sight it
would appear that speed is fairly irrelevant.
However up till now we have ignored the pile drag. The ratio of shaft drag to
pile drag acting to drag accelerate the arrow is proportional to the square of the tangent of the airflow angle a.
So at launch (a =0) the pile drag is much greater than the shaft drag. As the arrow rotates the drag ratio rapidly
changes so when the angle gets to around a few degrees the shaft drag exceeds the pile drag. The lateral drag acceleration
from the pile is in the opposite direction to the lateral acceleration from the shaft so it offsets it. The above picture
illustrates how the lateral arrow drift changes direction as the shaft drag becomes more important than the pile drag.
The overall lateral arrow movement is still zero after about 9 metres. The result is that
the higher the arrow launch speed the lower the change in the arrow velocity at stabilisation. There is a clear benefit in
having a higher launch speed. (Note this means we should really revisit the Arrow Diameter issue as there is here a benefit
from a larger arrow diameter). The simple way to increase launch speed is to have a lower mass arrow,
FOC
FOC initially appears to be a no brainer. The higher the FOC the larger the fletching area and fletching drag torque and
so there's a clear benefit to the fletching aspect. The larger the FOC the shorter is the stabilisation time (higher
potential energy storage per degree) and the maximimum value of the angle a is reduced so overall the lateral
drag acceleration is reduced. However in practice to get a higher FOC you need a lighter shaft and/or
a heavier pile so there are knock on effects to arrow mass, launch speed, moment of intertia, diameter and Uncle Tom Cobbley.
Arrow Mass
Don't go there! Changing arrow mass has knock on effects to just about everything
What the above illustrates is that you can't isolate any individual arrow property and say that changing X in some way
is going 'to be better' in some way. Every arrow property is spaghetti entangled with every other property. It is only the
behaviour of the composite arrow that really means anything. Having said that an arrow with a heavy pile, light shaft,
small diameter, short arrow length and large fletchings is likely to acquire the minimum change in velocity through
having launch rotational kinetic energy.
Tuning
If an arrow has angular momentum at launch then during the arrow stabilisation period the initial launch velocity is
changed to a greater or lesser extent depending on the arrow properties. The obvious solution to this problem is
to minimise the launch angular momentum in the first place by the the setup of the bow/arrow/archer system. The less
angular momentum the arrow has at launch, for a given arrow, the less change in velocity we end up with at the end of the
stabilisation process. Minimising launch angular velocity is what we mean by 'basic bow tuning'. Optimising the launch
angular velocity to minimise groups i.e. 'group tuning' takes into account external sources of angular momentum into the arrow.
The base issue is that if an arrow has some angular momentum at launch then during the stabilisation
period the arrow acquires via drag a changed velocity, both speed and direction being altered. If
the arrow was launched with the correct speed and direction to hit the target center then after
stabilisation the arrow speed and direction are no longer those to hit the target center. (For the
purposes of this description I'm assuming that what's wanted is a zero launch angular momentum, so
I am not considering tuning optimisation of the arrow launch angular momentum with respect to
distance or wind. These topics are covered separately under variable tuning wind/distance).
Once the arrow has stabilised then you have an arrow with a changed velocity and with the arrow shaft
at some angle to the air flow. The behaviour of the arrow is then similar to the displaced pendulum
above. The arrow fishtails/porpoises back and forth with an energy oscillation between fletching potential
energy and and the arrow rotational kinetic energy. As discussed elsewhere fishtailing/porposing has a minimal effect on
the arrow overall flight direction as the drag effects are more or less symmetrical and cancel out. The launch alignment of
the arrow itself has no significant effect in itself in changing the overall arrow flight path.
The effect of arrow stabilisation on flight direction is illustrated in the following figures. In this case the comparison
is made between a fletched arrow (blue) and a bare shaft arrow (red). The fletched arrow stabilises faster and attains a
lower offset angle than the bare shaft because the increased fletching area means more potential energy per degree offset.
The result is the change in direction of travel is higher for the bare shaft than for the fletched shaft. This of course
is the basic principle behind the bare shaft tuning method.
The following graph illustrates the lateral drag acceleration on the same two arrows. The vertical black lines indicate
the respective stabilisation distances. The bare shaft experiences a higher lateral drag acceleration and over a longer
distance/time then the fletched shaft hence the bigger shift in travel direction.
However up till now we have ignored the pile drag. The ratio of shaft drag to
pile drag acting to drag accelerate the arrow is proportional to the square of the tangent of the airflow angle a.
So at launch (a =0) the pile drag is much greater than the shaft drag. As the arrow rotates the drag ratio rapidly
changes so when the angle gets to around a few degrees the shaft drag exceeds the pile drag. The lateral drag acceleration
from the pile is in the opposite direction to the lateral acceleration from the shaft so it offsets it. The above picture
illustrates how the lateral arrow drift changes direction as the shaft drag becomes more important than the pile drag.
The overall lateral arrow movement is still zero after about 9 metres. The result is that
the higher the arrow launch speed the lower the change in the arrow velocity at stabilisation. There is a clear benefit in
having a higher launch speed. (Note this means we should really revisit the Arrow Diameter issue as there is here a benefit
from a larger arrow diameter). The simple way to increase launch speed is to have a lower mass arrow,