When an arrow is released using fingers/tab the nock end of the arrow
initially bends away from the bow. The nock end subsequently bends back towards
the bow and then away again. Archer's paradox is the term used to describe
this bending behaviour. (The original Archer's Paradox related to why a rigid arrow could be shot straight and not fly to the side because of the arrow to riser interaction (see Original Paradox Problem). The flexing of the arrow, the solution to the paradox issue, has these days become the general meaning of the term).
The aim is to have the nock (fletching) end of the
arrow bending away from the bow as it passes the body of the bow the purpose
being to avoid any collision between the rear of the arrow and the bow. This
section gives an overview of how this bending behaviour is generated. I will
assume a right handed archer when describing directions of travel etc.
Points to bear in mind are that the force exerted by the string on the
arrow varies as the arrow moves forward and also that the direction of the string force is always towards the plane of the bow. The static equilibrium point is
with the string back at the bracing height point.
The mechanism that initiates the Archer's Paradox behaviour is (Euler) buckling of the arrow shaft
Blythe example. If the axial stress
in the arrow shaft during launch, resulting from the string force acceleration of the point and shaft masses, exceeds the critical load (which by definition it
must do for a recurve arrow) then the arrow bends (buckles). There is some numerical data on this topic on the Forum
A well known static demonstration of this effect is that reported by Liston. You load the bow into the arrow and then instead of drawing the bow
you place the point of the arrow against a wall and push the riser slowly forwards thus applying the "draw" force to the arrow. The arrow shaft
fails (buckles) long before the full draw force is applied to the arrow. What is interesting about this demonstration is not so much that the arrow
shaft fails but that the failure is not catastrophic. The reason for this is that a consequence of the arrow failing and hence bending is that the
draw length is reduced lowering the loading on the arrow back down blelow the critical load. Pushing the riser further forward does not increase the
load on the arrow shaft, it only decreases the bend radius (and hence increases the stress in the shaft).
When an arrow is shot a similar safety relief mechanism exists and, as described below, as the arrow buckles the point of the arrow is
free to accelerate and relieve the bending stress in the shaft so no catastrophic failure occurs. There are of course limits to this failure
escape route. Too high an acceleration and, as reported by Hardy in the Blythe reference above, arrow failure will occur.
Strictly speaking then, any modelling of arrow Archers Paradox behaviour on the bow should be a model of dynamically unstable Euler buckling.
Unfortunately this topic is far too complicated mathematically, at least at the present time, to be doable. Instead the current standard arrow models
(those of Kooi and more recently Zanevskyy) are based on a perturbed simple beam vibration model. Technically iffy, but justified on the basis
that the results are in reasonable agreement with experiment
The Loose - Nock End
The pressure button is set up for a recurve bow so that the arrow at full
draw is pointing left away from the bow. The direction of the string force
at full draw points towards the bracing height position and is therefore
on the bow side of the arrow. If the loose was made using a mechanical release
then the nock end of the arrow would bend towards the bow and the the pile
end of the arrow would rotate away from the bow as illustrated. Ooops! The
arrow would end up flying uncontrollably off to the left.
What initialises the the Archer's Paradox effect is the action of the
fingers/tab on the bow string. At full draw the string force is balanced
by an equal and opposite force on the tab. At the loose the string force
causes the tab to rotate as the fingers are 'uncurled'
(ref).
At this point there are three forces acting, the string force towards
the bracing height position, the tab reaction force at a right angle to
the tab surface and a tab frictional force parallel to the tab surface.
These three forces add together
(ref) to produce a net force on the string forwards and to the left away from
the bow. The sideways acceleration of the string causes the arrow shaft
to bend away from the bow at the nock and as a consequence the string force
ends up running across the arrow shaft. At the same time the forwards acceleration
of the string transfers the load from the tab onto the nock end of the
arrow shaft. The purpose in having a 'slippy' tab is that the lower the
tab frictional force then the shorter is the time that the string is acting
on the tab during the loose i.e. there is less time to mess it up by moving
the string hand. The other effect the tab has in principle is on the effective
arrow dynamic spine. The 'slippier' the tab then the higher the nock acceleration
will be and the more the arrow will bend. This is because the higher the tab frictional force
the closer the net resultant force on the nock will lie towards the axis of the shaft.
The result is that when the string force ends up acting on the arrow nock
the string force runs across the arrow shaft and the reaction at the nock
(which is at a right angle to its surface) is pointing backwards and away
from the bow to the left. These two forces add together and the resultant
force direction is forwards and to the left away from the bow i.e. the
nock end of the arrow continues to bend away from the bow. This process
is similar to the behaviour of a pole vaulter's pole after it is planted
in the box.
The Loose - Pile End
As the nock end of the arrow bends away from the bow a torque is generated
on the arrow in the direction to rotate the arrow pile towards the bow. Because of the shaft intertia there will a small
amount of shaft bending caused by the rotational acceleration.and the shaft flexes in towards the bow. The rotating shaft
then runs into the button/riser as appropriate. Because the arrow is pushing against the bow there is an opposite
reaction with the bow/button applying a sideways force on the arrow. This force exerts a counter torque to the arrow rotation
and a bending moment on the arrow shaft. This force causes the shaft section at the button to swivel around at the button.
The section of shaft in front of the button moves away from the bow and the section to the rear of the button moves into
the bow. Little seems to be known about the behaviour of the shaft in front of the button at this point but it's reasonable
to assume that the behaviour will depend on the length, mass and stiffness of this front section. The longer this shaft
section then the more inertia it will have and the lower stiffness it will have. It's possible that an excessively long
section will continue to rotate around the button with the overall shaft forming an "S" shape. This might be a contributer
to the grouping problems experienced if the arrow shaft section in front of the button is too long.
The Straighten
If the nock end of the arrow continued to bend away from the bow then
eventually the arrow would snap. What limits the amount of this arrow bending?
The best way to look at this is to regard the arrow as a spring with a weight
on the front end being pushed from the rear. When the arrow is released the
nock end of the arrow accelerates forward faster than the pile end which
is heavy and has high inertia. The gap between pile and nock reduces and
this reduction goes into spring compression of the arrow (it bends). The
acceleration of the pile comes from shaft forces on it which comprise the
string force through the shaft and also from this spring compression (bending)
of the shaft. As the arrow bends more the pile acceleration keeps increasing
until it exceeds the nock acceleration. At some point the pile forward speed
catches up with the nock forward speed at which point the arrow stops bending.
The pile is still accelerating faster than the nock (string force + spring
force) so the pile now travels faster than the nock and the spring compression
comes out of the arrow i.e. it straightens up. As the rear of the arrow straightens
the force with which the pile end of the arrow is pressing against bow/button
from torque reduces and so the bend in the front part of the shaft reduces.
As the arrow straightens the pile acceleration decreases. You end up with
a more or less straight arrow with the pile and nock travelling forwards
at the same speed.
The differences between this 'straight' arrow and the one at full draw
is that in addition to the arrow being further forward and so the string
force and direction to the brace height position being different the arrow
shaft has a lateral velocity profile. Not all the spring energy goes into
pile acceleration, much of it has gone into a lateral shaft acceleration.
The Second Bend
When the arrow has straightened up, because it is moving it keeps going and the shaft bends outwards away from the bow.
This outward bending triggers a nock end bending effect similar to the loose. The effect of the string force running
across the shaft and the reaction force of the nock produce a net force accelerating
the arrow nock end forwards and laterally towards the bow. The lateral movement is assisted by the spring torsion
effect of the limbs which have been pulled out of line by the initial arrow bend. The nock end of
the arrow bends towards the bow as the arrow travels forward. You have exactly
the same pile/nock forward speed shuffle as before for the arrow to reach
a maximum bend and then straighten up again. There is no bow/button to counteract
the torque from the rear end bending this time and the torque results partly
in shaft rotation and partly in making the front part of the shaft bend.
The swinging of the front section of the shaft provides a counterbalancing
torque to the rear end bending. (This was a trick used by dinosaurs who used
a long tail with a lump of bone on the end as a weapon. When the tail was
swung the long neck/head was swung in the opposite direction, counterbalancing
the torque so the dino was not spun off its feet).
The Third Bend
Following the arrow straightening from the second bend the whole process
repeats with the nock end of the arrow bending away from the bow on its
third bend and so on as long as the string is accelerating the arrow.
For an illustration of the overall behaviour of the arrow on the bow click here
During this process, at around the point the arrow completes its second
bend the arrow leaves the string.. The aim is to have a clean separation
of the string from the nock and to have the rear end of the arrow sufficiently
bent away from the bow to provide good clearance as it passes the riser.
Having the right timing to do this relates to all the factors which affect
how much and how rapidly the arrow bends and how fast the arrow accelerates
forward. i.e. arrow length, mass, shaft spine, pile weight, draw weight,
bracing height and the bow force draw curve. Fortunately all the archer needs
to do all this is to select the correct arrow from a selection chart based
on accumulated experience of what works and what does not.
The amount the arrow bends ("weak/stiff") depends on the relative accelerations
of the nock and pile ends. Increasing the pile weight for example reduces
the pile acceleration and also increases the axial stress in the shaft and hence the arrow bends more ("weaker") and vice
versa. You can similarly change the amount the arrow bends (its "stiffness")
by changing the weight (and hence acceleration) at the nock end of the arrow.
As the nock end of the arrow is light the arrow stiffness is fairly sensitive
to changes in nock weight. e.g. adding brass nocking points or "pin nocks"
reduces the nock acceleration and therefore stiffens the arrow. A special
case is the fletchings. These increase the nock weight but in addition the
drag on the fletching surface additionally reduces the nock acceleration.
Fitting larger area fletchings (with the same weight) will stiffen the arrow
due to the increased drag.
The pressure button is set up for a recurve bow so that the arrow at full
draw is pointing left away from the bow. The direction of the string force
at full draw points towards the bracing height position and is therefore
on the bow side of the arrow. If the loose was made using a mechanical release
then the nock end of the arrow would bend towards the bow and the the pile
end of the arrow would rotate away from the bow as illustrated. Ooops! The
arrow would end up flying uncontrollably off to the left.
What initialises the the Archer's Paradox effect is the action of the
fingers/tab on the bow string. At full draw the string force is balanced
by an equal and opposite force on the tab. At the loose the string force
causes the tab to rotate as the fingers are 'uncurled' 
As the nock end of the arrow bends away from the bow a torque is generated
on the arrow in the direction to rotate the arrow pile towards the bow. Because of the shaft intertia there will a small
amount of shaft bending caused by the rotational acceleration.and the shaft flexes in towards the bow. The rotating shaft
then runs into the button/riser as appropriate. Because the arrow is pushing against the bow there is an opposite
reaction with the bow/button applying a sideways force on the arrow. This force exerts a counter torque to the arrow rotation
and a bending moment on the arrow shaft. This force causes the shaft section at the button to swivel around at the button.
The section of shaft in front of the button moves away from the bow and the section to the rear of the button moves into
the bow. Little seems to be known about the behaviour of the shaft in front of the button at this point but it's reasonable
to assume that the behaviour will depend on the length, mass and stiffness of this front section. The longer this shaft
section then the more inertia it will have and the lower stiffness it will have. It's possible that an excessively long
section will continue to rotate around the button with the overall shaft forming an "S" shape. This might be a contributer
to the grouping problems experienced if the arrow shaft section in front of the button is too long.
If the nock end of the arrow continued to bend away from the bow then
eventually the arrow would snap. What limits the amount of this arrow bending?
The best way to look at this is to regard the arrow as a spring with a weight
on the front end being pushed from the rear. When the arrow is released the
nock end of the arrow accelerates forward faster than the pile end which
is heavy and has high inertia. The gap between pile and nock reduces and
this reduction goes into spring compression of the arrow (it bends). The
acceleration of the pile comes from shaft forces on it which comprise the
string force through the shaft and also from this spring compression (bending)
of the shaft. As the arrow bends more the pile acceleration keeps increasing
until it exceeds the nock acceleration. At some point the pile forward speed
catches up with the nock forward speed at which point the arrow stops bending.
The pile is still accelerating faster than the nock (string force + spring
force) so the pile now travels faster than the nock and the spring compression
comes out of the arrow i.e. it straightens up. As the rear of the arrow straightens
the force with which the pile end of the arrow is pressing against bow/button
from torque reduces and so the bend in the front part of the shaft reduces.
As the arrow straightens the pile acceleration decreases. You end up with
a more or less straight arrow with the pile and nock travelling forwards
at the same speed. 
The differences between this 'straight' arrow and the one at full draw
is that in addition to the arrow being further forward and so the string
force and direction to the brace height position being different the arrow
shaft has a lateral velocity profile. Not all the spring energy goes into
pile acceleration, much of it has gone into a lateral shaft acceleration.
When the arrow has straightened up, because it is moving it keeps going and the shaft bends outwards away from the bow.
This outward bending triggers a nock end bending effect similar to the loose. The effect of the string force running
across the shaft and the reaction force of the nock produce a net force accelerating
the arrow nock end forwards and laterally towards the bow. The lateral movement is assisted by the spring torsion
effect of the limbs which have been pulled out of line by the initial arrow bend. The nock end of
the arrow bends towards the bow as the arrow travels forward. You have exactly
the same pile/nock forward speed shuffle as before for the arrow to reach
a maximum bend and then straighten up again. There is no bow/button to counteract
the torque from the rear end bending this time and the torque results partly
in shaft rotation and partly in making the front part of the shaft bend.
The swinging of the front section of the shaft provides a counterbalancing
torque to the rear end bending. (This was a trick used by dinosaurs who used
a long tail with a lump of bone on the end as a weapon. When the tail was
swung the long neck/head was swung in the opposite direction, counterbalancing
the torque so the dino was not spun off its feet).