Moment of inertia is a measurement of the 'rotatibility' of an object.
If you apply a force to an object it accelerates. The relationship between
the force and resulting acceleration is the Newtonian definition of 'inertial
mass'. i.e. acceleration = force/mass. If you apply a force to an object
which rotates it (a torque) then the relationship between the angular acceleration
and the torque is the moment of inertia i.e. angular acceleration = torque/moment
of inertia. If you apply a torque to an object and want to know how much
it will rotate in a given time then you need to know the object's moment
of inertia i.e.
rotation angle = torque x time squared/ twice the moment of inertia.
The value of the moment of inertia depends on the axis around which the
object rotates. If you hold a bow normally at the grip and twist it about
the feel is very different to holding it at the end of the long rod and twisting
it about. The respective moments of inertia are very different because the
rotation axes are different. The moment of inertia of mass around an axis
of rotation (assuming an infinitely stiff connection) is the mass multiplied
by the square of the distance to the rotation axis. (In practice it will
always be effectively less than this as the connection between the mass and
the rotation axis, e.g. a long rod, will bend). The total moment of inertia
of a complicated shape, e.g. a bow, is the momentum of inertia of all the
individual small masses at different distances from the rotation axis added
together.
The two main areas in which moment of inertia is important in archery
is in how the bow rotates when you are shooting an arrow and how the arrow
rotates when it's flying through the air. Both these rotations affect where
the arrow eventually hits the target. The affect of arrow moment of inertia
is covered in various other sections so I'll stick to some aspects relating
to the bow. During the power stroke of the bow accelerating the arrow,
the axis of rotation of the bow is the centre of pressure of the bow hand
- bow grip contact. The main forces generating torque in the bow around this
point of rotation result from the accelerations of the bow limbs, arrow and
string.
Always what you are looking for in archery is consistency. If the bow
hand position shifts about from shot to shot (or even during a single shot)
then the axis of rotation shifts - the moment of inertia changes as well
as the torques being applied to the bow and consequently the rotation behaviour
of the bow changes. Result - the arrow hits at a different place. If the
draw length or release varies then the forces generating bow torque will
vary and again the bow rotation characteristics change. Result - the arrow
hits at a different place. To cater for the natural variations of the archer
the bow design is aimed at minimizing the effects of the archer variations.
The first point is the vertical position of the arrow - pressure button
contact with respect to the bow axis of rotation.
If the pressure button -arrow contact point is vertically above the throat of the grip then it lies on the vertical rotation axis and any horizontal rotation of the bow will have less effect on the arrow-pressure button interaction. While the bow rotates,
the arrow (inertia again) tends to stay where it is so any movement of the
contact point with respect to the arrow will effect the action of the
pressure button and hence where the arrow eventually hits. While the arrow
has two contact points with the bow, nock and rest/pressure button then any
rotation of the bow can affect the direction the arrow is shot as well as
the rotation the arrow has when it leaves the bow. Once the arrow has only
one contact point, the nock, then any bow rotation will affect the direction
of the string force acting on the arrow as well as nock movement rotating
the arrow in the horizontal and vertical planes.
The primary use of the bow stabiliser system is to position the bow centre of gravity to minimise the generation of torques acting to rotate the bow. While You can't completely eliminate these torques you can can reduce their effect by increasing the bow moment of inertia. The higher the moment of inertia the lower the angular acceleration for a given torque and hence the less the bow will physically rotate during the shot. Some moment of inertia is built into the bow design but the expectation is that stabilisers will be
added to increase its value. The additional moment of inertia generated by a stabilizer depends on three factors: mass, distance of mass to the bow rotation axis and the stiffness of the stabilizer rod. Don't forget the last item. You won't get much benefit from a tonne weight on the end of 50 metres of string. This makes assessing stabilizers tricky. If you use the same end weight on a longer rod you could end up with less effective moment of inertia then with the shorter rod because of the increased flexibility
of the longer rod. When shooting an arrow you have high torques over a short
time time period, 10-15 milliseconds say, so waving the bow around
in the air to assess its moment of inertia is not much help. Like a lot of
things in archery it's play about and try to evaluate the effects of changes.
There is a simple stabiliser moment of inertia calculator on the "Downloads" page.
Measuring the Moment of Inertia of a Bow (MoI)
It is fairly straightforward to measure the 'static' bow MoI. By 'static'
I mean that the bow is in the bracing height position and the accelerations
of the bow are small so that components of the bow don't deform in any way.
This 'static' measurement relates to the dynamic MoI so you can use it to
look at the effects of e.g. different stabiliser configurations on how it
affects the bow MoI.
The main requirement to measure bow MoI is having a solid pivot point from
which you can hang the bow with the bow being free to swing (e.g. a nail
in a doorframe. Tying a string loop on the bow and hanging the bow from a
hook is a simple option). The following is a suggested procedure for measuring
the bow MoI.
step 1 - Find the overall bow weight
Weigh the bow including all attachments. Call the mass of the bow M
(in grams).
step 2 - Find the bow centre of gravity
Hang the bow at some point on the pivot and hang a plumb line from the pivot.
With say masking tape, mark on the bow at two points the alignment of the
plumb line with respect to the bow.
Hang the bow from the pivot at a different point and hang a plumb line from
the pivot. Place a straight edge along the masking tape alignment marks.
The point where the straight edge crosses the plumb line is the location
of the bow centre of gravity. (you could say mark the plumb line at this
point).
Measure the distances (in centimetres) from the bow centre of gravity to
the throat of the bow grip (b) and to the pivot point (h).
step 3 - Measure the bow swing time
There are 3 possible axes of bow rotation and associated MoI values and the
axes are assumed to run through the throat of the bow grip (approx. the hand
- bow pressure point). Refer to the 'stabiliser' link on the main page for
some nice drawings of these 3 axes if needed.
In order to measure the MoI about a particular axis the bow has to be swung
(like a pendulum) around the pivot point (for which h was measured)
parallel to this axis. The time (T secs) for the bow to swing forward
and back to the same point has to be measured. The maximum amount the bow
swings should be as small as possible (less than 10 degrees from 'vertical')
and its more accurate to time a number of swings (say ten) and divide the
total time by the number of swings to get the average swing time.
If you need to change the pivot point to get the bow to swing parallel to
a given axis then the value of h needs to be measured for the
new pivot point.
The bow MoI for a particular axis running through the throat of the bow grip
Ibx is then given by
Ibx = M(24.85hT2-h2+b2)
If you add/remove mass from the bow then you have to start from step 1
again. If you just rearrange the stabilisers then you have to restart from
step 2.
If the pressure button -arrow contact point is vertically above the throat of the grip then it lies on the vertical rotation axis and any horizontal rotation of the bow will have less effect on the arrow-pressure button interaction. While the bow rotates,
the arrow (inertia again) tends to stay where it is so any movement of the
contact point with respect to the arrow will effect the action of the
pressure button and hence where the arrow eventually hits. While the arrow
has two contact points with the bow, nock and rest/pressure button then any
rotation of the bow can affect the direction the arrow is shot as well as
the rotation the arrow has when it leaves the bow. Once the arrow has only
one contact point, the nock, then any bow rotation will affect the direction
of the string force acting on the arrow as well as nock movement rotating
the arrow in the horizontal and vertical planes.