The topic of Centre of Pressure occasionally comes up in archery discussions so it may be worth reviewing what it is and and its relevence to arrow behaviour.
The total drag forces and Munk moment acting on an arrow at any specific point
in its flight have net result of moving the arrow as a whole and superimposed
on this movement there is some arrow rotation. The following diagram illustrates
these two effects.
The complex forces acting on the arrow pile,shaft and fletchings can be represented by 'hypothetical' force or forces acting at specific points to give the same overall arrow response behaviour. While you can describe the total drag force on the arrow in terms of a single drag force (and hence a single centre of pressure) it is convenient to describe the drag force in terms of three drag forces, and hence three centres of pressure. This is the approach taken here.
The first of these forces acts through the arrow centre of gravity and relates to moving the arrow as a whole, the second force acts at some point on the arrow shaft towards the nock end and provides a stabilising "fletching" torque rotating the arrow. The third force is the flow separation Munk moment which acts on the arrow shaft close to the nock and provides a destabilising "fletching" torque rotating the arrow.
Worth noting that the commonly expressed view that arrow "stabilisation requires the centre of pressure to be to the rear of the arrow centre of mass" is incorrect, the Munk moment being an example. Stability or not is defined by the force direction as well as the position of the centre of pressure.
As regards this section the first force, acting through the arrow centre of mass, which does not contribute any fletching action to the arrow is ignored. Only drag forces and the corresponding centre of pressure which contribute to arrow rotation are considered. In terms of arrow motion the linear acceleration of the arrow from the drag forces on arrow point, shaft, nock and fletchings are ignored. Only fletching action at a single instant in time (air flow velocity) is considered.
The Total Pressure acting at the Centre of Pressure have the
axis about which the arrow rotates and the arrow angular accleration
around this axis behaving exactly as under the actual drag etc. forces.
The Total Pressure is the sum of the (rotational) drag forces from the fletchings, shaft and Munk moment. The position of the centre of pressure is determined by requiring that the moment (force times distance) of the Total Pressure at the Centre of Pressure equals the sum of the moments of the actual rotational drag forces on the arrow.
The axis around which the arrow rotates is determined by the position of the Centre of Pressure. In the above diagram 'CG' is the position of the arrow centre of gravity, 'A' is the distance from the axis of rotation to the CG and 'B' is the distance from the CG to the Centre of Pressure. The relationship is that 'A' times 'B' is a constant value 'K'. The constant value is the square of a fixed property of the arrow known as its radius of gyration calculated at the CG.
For linear motion Newton's law is acceleration = force/mass.
The rotational equivalent of this is angular acceleration = torque/moment
of inertia.
The value of moment of inertia of the arrow depends on its axis of rotation. The position of the Centre of Pressure determines the axis of rotation and hence the moment of inertia.
The torque on the arrow is the Total Pressure multiplied by the length of the lever (A+B). The position of the Centre of Pressure determines the length of the lever. The arrow angular acceleration is therefore determined by the Total Pressure and the position of the Centre of Pressure.
The often quoted 'the further the centre of pressure is behind the centre of gravity then the more stable the arrow flight' is therefore not necessarily correct. Arrow flight stability is determined by the magnitude and direction of the angular acceleration. This depends on both the Total Pressure and the position of the Centre of Pressure. As an (extreme) example a bareshaft arrow with zero FOC has its centre of pressure at the nock, as far back as you can go. The arrow flight is completely unstable.
During its flight the drag etc. forces on the arrow are continuously changing and hence the position of the Centre of Pressure and Total Pressure continually change. While in theory it is possible to calculate the position of the Centre of Pressure at any moment during the arrow flight it is in fact impracticable. The major headache is the drag effect of the fletchings. Fletchings come in a variety of complicated shapes, they are attached to the shaft at a variety of angles and the drag effect varies as the arrow rotates around the shaft axis.
A common mistake is to assume that an arrow always rotates at its centre of gravity. I guess this is because the CG is the one identifiable point on the arrow shaft. In practice an arrow can never rotate at its CG under a single applied torque as the Centre of Pressure would have to be at an infinite distance (A=0 requires B to be infinity). If the Centre of Pressure is to the rear of the arrow centre of gravity then the point about which the arrow rotates has to be in front of the CG.
The overall arrow Centre of Pressure is likely to be somewhere where the fletchings are located and the rotation point of the shaft will be in the region 3-4 inches in front of the arrow centre of gravity. An example of calculating the position of the arrow centre of pressure is included. Centre of Pressure Calculation
Last Revision 1 July 2009