As you draw an arrow back on a recurve bow the force on the fingers steadily
increases. How the weight on the fingers varies with the amount the arrow
is pulled back is called the draw force curve. The draw force curve has
the following important characteristics. Firstly the draw force curve determines
what weight the archer has on the fingers at full draw. Secondly the draw
force curve establishes how much energy is stored in the bow at full
draw potentially available to go into the arrow speed off the bow. Thirdly
the shape of the draw force curve near the full draw position determines
the 'stacking' property of the bow i.e. at what rate the weight on the fingers
varies with draw length.
What draw weight the archer is holding at full draw is down to archer preference
and physique. This is basically determined by the riser/limb combination
with some variation available to the archer via limb pocket adjustment or
bracing height modification.
The following chart illustrates draw force curves for draw weights of 30
and 40 pounds at full draw.
The energy stored in the bow limbs potentially available to go into arrow
speed is represented by the area under the draw force line (brown for the
30#, blue for the 40# draw weights). The area under the 40# draw force curve
is larger than under the 30# draw force curve i.e. more energy is potentially
available for 40# weight bow. The blue area not covered by the brown area represents the difference
in total area i.e. the difference in stored energy between the two draw weights.
While a lot of this energy is wasted in limb
movement, arrow vibration etc. on the shot the arrow will leave the bow
faster with the 40# weight. The energy stored in the bow relates to the
shape of the draw force curve.
The representation of a compound
bow draw force curve shows why much more energy (the blue area) is stored
in the bow with a much lower weight at full draw.
The actual shape of the force draw curved is determined by the limb 'spring'
characteristics and by the bow geometry. The bow limb is a complicated spring.
With one end fixed as you draw the arrow and pull on the other end the limb
shape changes and the spring force exerted by the limb changes. As you draw
the arrow the angle the string makes with the direction of limb spring force
and the angle the string makes with the arrow shaft change. The following
diagram outlines what is going on at a specific point as the arrow is being
drawn.
In the diagram 'F' is the force exerted by the bow limb. The string
is at an angle 'A' to the direction of the spring force. The tension in
the bow string is FCos(A). Where Cos (cosine) is a mathematical function
whose value depends on the angle 'A'. (Its value varies between 0 at 90
degrees to 1 at 0 degrees). The string is at angle 'B' to the arrow shaft.
The weight on the fingers is 2FCos(A)Cos(B). (The factor 2 comes from
having two, assumed identical, bow limbs). As the arrow is drawn back the
values of F,A and B continually change as the draw length changes. e.g. angle
B starts at 90 degrees at the bracing height position and decreases steadily
as the arrow is drawn. The following charts illustrate how the string geometry
and limb spring force vary over the draw length to produce the draw force
curve.
The basic assumption in the attached diagrams is that the force draw curve
is a straight line and the draw weight is 40 pounds with the arrow drawn
25 inches. The initial few inches of draw are not shown as here the string
is running around the bow limb so the geometry is more complicated and you
also have string-limb friction operating. The recurve limb changes shape as
the arrow is drawn and this affects the string geometry (angles 'A' and 'B').
The effect of this is ignored.
The Geometrical Draw Force graph shows how the value of 2Cos(A)Cos(B) typically
varies over the draw length. The Limb-Spring Force graph shows how the value
of 'F' varies over the draw length. You multiply these two values together
to get the draw weight at any specific draw length. What is interesting about
the spring force curve is that the force drops as the arrow is drawn and
towards the latter part of the draw effectively becomes a constant tension
spring. All done by clever limb construction, geometry and the reduction
in material towards the limb tip.
An interesting approach to looking at bow draw force behaviour is presented
by Ugo Bardi at the following link:
Look at the draw force curve for any recurve bow and what you see is, more
or less, a straight line. The old rule of thumb of 2 pounds/inch for draw
weight is based on the draw force curve being a straight line. Why straight?
One could produce a bow limb having a draw force curve with a hump in it,
something similar to a compound bow. This would give you a higher speed
arrow with the same weight on the fingers. The way this can be done is in
principle the same as for a compound bow i.e. the string tension is run
through a lever which gets longer as you approach full draw. In the case
of a recurve limb the lever is created by having a stiff section of limb
towards the limb tip. As the limb bends and rotates and the angle between
the limb and the string increases then the lever arm effectively gets longer
and the required draw weight decreases as illustrated in the diagram.
The downside of this approach is that the limb section near the tip is
not acting to store draw energy and is effectively 'dead weight'. Its rather
like having a weight at each limb tip, it reduces the bow efficiency and
can increase the shock loading on the bow from the shot. Overall you can
still get a net gain. The traditional mongolian bow is an example of a bow
using this approach. The practical problem in applying this approach to a
modern recurve limb is torsional stability. Using a lever results in
concentrating the stress in the bow limb below it making the limb more susceptible
to twisting. The effect of any torque going into the limb from the string
will be amplified because it acting through the limb lever. To get a significant
weight let-off requires increasing the limb torsional rigidity by complex
limb construction or by locally increasing the limb width. It is not currently
economic to mass produce limbs having significant weight let-off.
Most archers prefer that the weight on the fingers increases smoothly up
to the full draw position. When the draw weight rapidly increases near/up
to the full draw position this is called 'stacking' and is viewed as a 'bad
thing' (unless you are a compound archer and use a mechanical stop to generate 'infinite stacking' behaviour). Stacking can result from the spring
characteristics of the limb, from the bow geometry or a combination of both.
Usually if the limb and riser are bought as a package then stacking should
not be a problem as the limb 'spring' and bow geometry are designed as a
unit. With the advent of universal limb fitting however a problem has risen
from one manufacturer's limbs being used with a different manufacturer's
riser. Where this is done there is no guarantee that the limb spring properties
and the overall bow geometry will produce a non stacking combination. If
going the pick and mix route then you need to test before you buy.
While the area under the draw force curve represents the energy stored
in the bow not all this energy will end up where we want it as arrow kinetic
energy. The ratio of arrow kinetic energy to stored energy, expressed as
a percentage, is what is called the bow efficiency. For recurve bows the
efficiency is somewhere around 80% i.e. around a fifth of the energy stored
in the bow is 'wasted'. You can have two bows with identical draw force curves
but for a given draw length and arrow the arrow speeds off the bow can be
different i.e. the two bows have different efficiencies. One reason for this
could be that the two bows have diffferent geometrical and spring force characteristics
contributing to the (overall same) draw force curve. For example suppose
the bows have different limb lengths so one bow ends up longer than the other.
The 'geometrical draw force' for the longer bow will be lower than for the
shorter bow at all points during the draw because of the difference
in bow geometry (the difference in the angles 'A' and 'B'). To have
the overall same draw force then for the longer bow the 'spring force' must
be correspondingly greater at all points during the draw. A typical way
to get a stiffer spring force is to have more material in the limb, which
makes it heavier. Accelerating a heavier limb with the same force results
in a lower acceleration so the limbs (and hence the arrow) end up with a
lower final speed. The longer bow is less efficient than the shorter bow
although they both have the same draw force curve.
The design of the bow limb, its spring force characterisics, how it bends
and the mass distribution in the limb will all effect the resulting efficiency
of the bow. How the limb bends will also effect the bow 'geometrical draw
force' characteristics. To pick a simple example the fastest moving part
of the limb is the limb tip. The more mass there is in the limb towards the
tip then the lower the resulting bow efficiency.
Another way the draw force curve influences bow efficiency is its shape/slope.
Because the draw force increases as the bow is drawn the energy input per
unit of draw length increases towards the end of the draw. The reverse is
also true, the force on the arrow is highest at the point of the release
and decreases as the arrow moves forward. One effect of a finger release
is the Archers Paradox effect, the arrow bends. The higher the force on the
arrow during the initial part of the shot then the more the arrow will bend
. The energy that goes into the bending of the arrow is 'wasted' energy,
it ends up as arrow vibration. The steeper the draw force curve or a draw
force curve that bends upwards will result in more arrow bending and hence
lower bower efficiency.
The other option is 'reverse' stacking i.e. the rate of increase of draw
weight decreases as you approach full draw. This may be more comfortable
for some archers. There is also the argument that with reverse stacking the
arrow speed will be less sensitive to draw length as the variation of the
area under the force draw curve with draw length will be reduced. True but
a very marginal effect. The only practical way to get reverse stacking is
to have the limb spring force reducing near to full draw. This can be done
be reducing the amount of limb material towards the limb tip but this is limited
by the strength, durability and stability requirements for the limb.
Dynamic Force Draw Curve
All the previous discussion relates to what is called the Static Draw Force
Curve. When the static draw force is measured at different draw lengths nothing
is moving. When the arrow is being shot everything is moving: bow, limbs,
string, arrow, archer. The variation of the force exerted by the string on
the nock of the arrow during the shot is called the Dynamic Force Draw Curve
(even though the arrow is being shot rather than drawn). The dynamic force
draw curve relates to the static force draw curve but is determined by the
masses and accelerations of all the moving bits: bow, limbs arrow etc. A
typical peak (static) draw weight for a recurve would be around 40 pounds.
If you placed an arrow vertically on the ground and put a 40 pound weight
on it then 'snap' the arrow would break. Over most of the shot the Dynamic
Draw Force is lower than the Static Draw Force at any given draw length and
follows an up-and-down variation tied in to the mechanical action of the
bow and the Archers Paradox flexing of the arrow. Towards the latter
end of the shot the Dynamic Draw Force is higher than the Static Draw Force
as the momentum in the limbs feeds back into arrow acceleration.
It's the performance of the bow when being shot that matters so the Static
Draw Force Curve should not be given to much weight when comparing one bow
with another. e.g. if you replaced the working limb spacing material with
mercury then although the Static Draw Force Curve would be much the same the
performance of the bow would be considerably poorer.
The energy stored in the bow limbs potentially available to go into arrow
speed is represented by the area under the draw force line (brown for the
30#, blue for the 40# draw weights). The area under the 40# draw force curve
is larger than under the 30# draw force curve i.e. more energy is potentially
available for 40# weight bow. The blue area not covered by the brown area represents the difference
in total area i.e. the difference in stored energy between the two draw weights.
While a lot of this energy is wasted in limb
movement, arrow vibration etc. on the shot the arrow will leave the bow
faster with the 40# weight. The energy stored in the bow relates to the
shape of the draw force curve.
The representation of a compound
bow draw force curve shows why much more energy (the blue area) is stored
in the bow with a much lower weight at full draw. 
In the diagram 'F' is the force exerted by the bow limb. The string
is at an angle 'A' to the direction of the spring force. The tension in
the bow string is FCos(A). Where Cos (cosine) is a mathematical function
whose value depends on the angle 'A'. (Its value varies between 0 at 90
degrees to 1 at 0 degrees). The string is at angle 'B' to the arrow shaft.
The weight on the fingers is 2FCos(A)Cos(B). (The factor 2 comes from
having two, assumed identical, bow limbs). As the arrow is drawn back the
values of F,A and B continually change as the draw length changes. e.g. angle
B starts at 90 degrees at the bracing height position and decreases steadily
as the arrow is drawn. The following charts illustrate how the string geometry
and limb spring force vary over the draw length to produce the draw force
curve.
Look at the draw force curve for any recurve bow and what you see is, more
or less, a straight line. The old rule of thumb of 2 pounds/inch for draw
weight is based on the draw force curve being a straight line. Why straight?
One could produce a bow limb having a draw force curve with a hump in it,
something similar to a compound bow. This would give you a higher speed
arrow with the same weight on the fingers. The way this can be done is in
principle the same as for a compound bow i.e. the string tension is run
through a lever which gets longer as you approach full draw. In the case
of a recurve limb the lever is created by having a stiff section of limb
towards the limb tip. As the limb bends and rotates and the angle between
the limb and the string increases then the lever arm effectively gets longer
and the required draw weight decreases as illustrated in the diagram.