The center of gravity of an arrow is the point where the arrow will balance on a pivot. Many things can be used as a pivot, the important thing is that it leaves the arrow free to move, so it should not be more that say 1 mm wide at the contact point to achieve a 1 mm accuracy. The following diagram illustrates the pivot.
The front of center of an arrow is the tradition (although inadequate) way of expressing an arrow's flight stability. It is the distance between the center of the arrow and the CoG relative to the length of the arrow, expressed as a percentage. Use the formula:
FoC = (D/L - 0.5) . 100
Note that the arrow length is traditionally measured from the nock valley to the end of the shaft. The point is excluded.
The center of pressure of an arrow is the point along the shaft where the aerodynamic forces on an arrow fore and aft of this point are balanced. Because the aerodynamic centers of fletches and broadheads moves with angle of attack, the measurement is best done with the arrow near in aligned with the air stream.
The measurement requires a jig be made that can pivot the arrow about a vertical axis and be slid along the arrow shaft, eventually to the CoP point. A pivot should be low friction and vertical to avoid the deflections due to gravity.
In a strong wind (or a wind tunnel if you happen to have one), the aim is to find the CoP by moving the pivot to a point until there is no weather-cock effect. Ideally this would be done with a spinning arrow as the orientation of the fletches and broadhead have a significant effect (otherwise there would be no need to spin). To overcome this repeat the measurement with several different arrow orientation in the pivot jig and take an average.
Arrow Stability Margin
The stability margin is a measure of an arrow's aerodynamic stability. It is the distance between the CoG and CoP as a proportion of the arrow aerodynamic length. Use the equation:
SM = ( D - P ) / La
where the measurements are indicated in the above diagrams.
Higher values of SM imply higher stability, however can also lead to longer decay times.
In calculating an arrow's flight characteristics, it is important to know the mass and the center of gravity location for each component. Frequently the CoG can be estimated by sight, and that is usually sufficient. However, if you seek confirmation of the estimate, use the method as for the assembled arrow detailed above.
A bow's center of gravity determines how a bow moves during the release process (internal ballistics) and shortly afterwards. The CoG is relatively easy to measure on an un-drawn bow with the use of a plumb-bob.
With a plumb-bob cord tied to the string, and the bow hanging, mark the vertical line in the region of the handle. It may be necessary firmly attach a sheet of paper to the bow to allow an off bow CoG. Repeat this with plum-bob cord in a significantly different position. The two lines intersect at the CoG in the vertical plane. The process can be repeated to improve the accuracy - ±5 mm (1/4") is usually fine.
You may assume the lateral position of the CoG is on the center-line, but it is worth checking as it may point to a limb alignment issue. Using a short cord on the plumb-bob identify the lateral CoG point in a similar way to the other measurements. The distance from the center-line to this point is the Z dimension.
Now record the result in terms of the X, Y and Z dimensions above into FlyingSticks Stabilizer-Dimension. The reference point is the handle valley referred to as the bow's pivot point.
The CoG moves reward as a bow is drawn, but measuring this is more difficult and should not be attempted. Instead, FlyingSticks makes some assumptions about the CoG movement base on bow type.
Adding stabilizers tends to move the CoG forward, although side-rods as well will restrain that movement.
When a sighting system is dependent on the relative direction of gravity, then it will introduce an error by adding a azimuth component to the arrows launch direction when a bow is angled away from the vertical. If the sighting system maintained orientation to gravity and the arrow, canting would not be an issue. This is why instinctive bare-bow archers that use the arrow as their guide have no difficulty with canter and are able to exploit the advantages that it can provide at little cost to accuracy. Possible side effects of high canter can be changed draw length and rest position movement, both of which will impact accuracy.
However with conventional modern sights, there is a use for canter in compensating for wind drift. Interestingly both canter and wind-drift are effects that are exponentially proportional to range, leading to an alternate method for wind-drift compensation that is to some extent independent of range. By cantering into the crosswind (i.e. top bow limb to the direction form which the wind is coming). Typically, about 1.5° of canter per m/s of crosswind is required. FlyingSticks will work this out for your setup.
A smart phone can be a great measurement tool. Perhaps the most useful thing most can measure is sound frequency via the build in microphone.
The arrow's fundamental resonant frequency is closely related to the arrow's dynamic spine. It is typically in the 40 to 150 Hz range.
You can get a feel for the resonance by holding the arrow vertically between the thumb and forefinger at about a 25% down from the nock and flick the shaft at its midpoint. You will feel a vibration last for several seconds. By moving you hold point up or down the resonance may be more pronounced and last longer. Your hold point is then close to a null, where your fingers can have very little dampening effect. Invert the arrow and repeat - and you will find the second null closer to the point due to the anchoring effect of the point's mass. In both cases you will feel the same resonant frequency - it is the arrow's fundamental.
To measure this frequency and its dampening time-constant, download a free sound frequency or instrument tuning app. With the phone's microphone close to the edge of a table, support the arrow at one of the nulls with the fletches close to the microphone to ensure good low frequency sound coupling. Flick the arrow at its center or end to stimulate a resonant vibration and observe the waveform or frequency on the phone's screen.