pressure against the base when the front bearing is unsupported, from the jump of the gun and from the powder blast.
Furthermore, at this point the air resistance begins to act in retarding the projectile. The action line of the resultant air pressure on a yawing projectile intersects the axis at a point in front of the centre of gravity. The resultant air resistance, therefore, exerts a moment around an axis through the centre of gravity, in such a direction as to increase the yaw. We then have a motion similar to that of a spinning top or gyroscope when an angular motion is given to the axis of spin, except that we have in addition a rapid motion of the centre of gravity.
In other words, the projectile has a motion of translation accom- panied by precession and nutation. The motion of the point pro- jected on a plane through the centre of gravity and normal to the trajectory, describes a rosette, as shown in fig. 5.
FIG. 5.
Condition of Stability. If the spin is insufficient the air moment may cause the projectile to tumble. The condition of stability may be expressed by the following equation:
A'N*
"46
where
5, is the stability factor. Its value must be greater than l-o if the projectile is not to tumble, and not lower than 1-5 or 2-0 for modern projectiles, if excessive yaws are to be avoided,
A, the moment of inertia of the projectile about the axis of spin,
B, moment of inertia about an axis at right angles through the centre of gravity,
N, the velocity of rotation about the longer axis in radians per second,
17, sin 8, the moment of the air resistance around an axis through the centre of gravity at right angles to the longer axis when the yaw is 5,
The value of 6 depends upon the air resistance, but is nearly in- dependent of a for small yaws.
8, the angle of yaw.
By an analysis of the results obtained in British and American jump-card experiments, R. H. Kent has determined that the value of the first maximum yaw outside the gun may be computed in terms of the stability factor and the yaw inside the gun by the follow- ing equation:
al
01 = A
where a is the first maximum yaw
. the yaw in the gun.
Figure 6 shows the values of aj in terms of s for a value of .- =8 and
=0-2.
It appears from this relation that the maximum yaw in front of the gun is principally due to the yaw in the gun, and that it is very little affected by the pressure of the powder gas, during the time the projectile is emerging from the muzzle, by the jump of the gun, or by the blast in front of the muzzle.
Orientation of the Yaw. The plane of yaw contains the path ot the centre of gravity and the axis of the projectile. The orientation of the yaw is the angle between this plane and the vertical plane containing the path of the centre of gravity. It is determined by measuring the angle between the traces of these two planes on the jump card. The precessional motion consists of rotation of the plane
STABILITY FACTOR
FIG. 6.
of yaw around the path of the centre of gravity, while the motion in nutation consists of oscillations around an axis through the centre of gravity normal to the plane of yaw.
For a small yaw not accompanied by nutations the rate of change of orientation is,
,_AN
7T
The motion in nutation causes abrupt changes in this rate in the neighbourhood of the minimum yaws.
Damping of the Yaw. Reduction in the yaw of the projectile, as it proceeds down the range, is principally due to the following fac- tors: (a) The component of the air resistance normal to the direction of motion of the yawing projectile causes motion of the centre of gravity in the direction of the yaw. The effect is a virtual reduction in the yaw accompanied by a helical motion of the centre of gravity; (b) the resultant angular motion of the axis of the projectile due to precession and nutation sets up an air-resistance couple which opposes that motion, and which is quite distinct from the air-resist- ance moment which causes the main part of the initial maximum yaw. The effect of the couple is first to damp out the nutations and finally to reduce the yaw; (c) as the velocity of the projectile de- creases, the air resistance also decreases. The consequent reduction in the air-resistance moment on a yawing projectile causes a reduc- tion in the maximum yaws.
Effect of Yaw on Range and Accuracy. The resistance of a yawing projectile is very much greater than that of a projectile moving in the direction of its axis. Experiments made by G. F. Hull and L. J. Briggs in an air stream indicate that at a velocity of 200300 metres per second the head-on resistance of a projectile of modern form yawing 15 is two to two and one-half times that of the same pro- jectile moving in the direction of its axis. A considerable yaw in front of the gun will, therefore, cause a rapid reduction in the velocity and a reduction in range.
It is readily seen that a variation in initial yaw between rounds will cause bad range dispersion. The same is true of dispersion in deflection. It may be stated that irregularity in initial yaw, what- ever may be its cause, forms one of the principal factors in dispersion of fire.
Drift. As the projectile proceeds along the trajectory, its axis tends to remain parallel to its original direction at the gun. Since the effect of gravity causes the trajectory to curve toward the earth, there is a gradual increase in the angle between the axis of the projectile and the trajectory. This yaw, due to gravity, is quite distinct from the initial yaw described above and does not begin to have an important effect on the flight of the projectile until after the greater part of the initial yaw has been damped out.
A yaw having been developed by gravity, the air-resistance moment /* sin s, tending to rotate the spinning projectile in the plane of yaw, causes motion in a plane at right angles. The effect with right hand rotation of the projectile, is to cause the point of the projectile to move at first to the right of the plane of fire and at a later period downward 4 the projectile being bodily displaced by the component of air resistance acting normal to the direction of motion, on the side presented by the yaw. The initial instability and the drift are phenomena of like nature. While the initial instability is caused by a suddenly applied yaw, or a high rate of change of yaw, and is accompanied by a rapid motion in precession and nutation, the drift is caused by the gradual yaw due to the action of gravity on the projectile, and is accompanied by a very much slower motion in precession without nutation. (W. H. T.)
