ABSTRACT
Earlier in the text, we stated that projectiles rarely leave the tube with their velocity vectors aligned with the geometric axis of the gun tube. Chapters 11 and 12 describe this behavior. The result of this behavior is weapon inaccuracy and it must be well understood by the practicing ballistician because, although it is not practical to completely eliminate the behavior, we would like to reduce it to acceptable levels. The first component of this behavior is known as lateral throwoff. It is a dynamic response of the projectile to either a static or a dynamic imbalance and will now be described in detail. If we imagine a projectile with a mass asymmetry as depicted in Figure 10.3, we can
imagine the spinning motion as viewed from the rear. If we ignore the axial velocity by simply spinning the projectile at a high rate, say, between two flexible supports on a test stand, we would see a wobble develop as a result of the centrifugal action on the center of mass. All the time the projectile is being spun up in the gun, the tube walls and stiffness of the supporting members prevent this wobble (to the extent the clearances allow) from developing. At the instant, the projectile is free from the constraints of the tube we expect it to become affected by this centrifugal loading. This is lateral throwoff because the effect is to fling the projectile in a direction off the tube centerline. We can use the analogy of a vacuum trajectory to examine the lateral throwoff effect
generated by either a static or a dynamic imbalance. Consider the projectile asymmetry from Figure 10.3. If we examine the projectile over a short period of flight, ignoring gravity as well as assuming no drag because of the vacuum assumption, we would see the dynamic forces acting on the projectile as depicted in Figure 11.1. In this figure, the only force acting is the centrifugal force due to spin. This dynamic action will result in the force vector changing direction, though since there is no angular acceleration or deceleration it maintains a constant magnitude. It is worth noting that we have resorted to our complex plane in this example as it is convenient to use in our development. At the instant, in time depicted here, we can break the force into a component in the y-direction and one in the iz-direction. We are not necessarily concerned with the force acting on the CG per se. We want to see
where the projectile moves because of this force. To accomplish this, we need to use Newton’s second law. We know that
Fr ¼ mar (11:1)
This is the centripetal force. The centrifugal force would be equal but opposite in sign. From dynamics [1], we recall that
ar ¼ rp2 (11:2)
In the case we are considering here, we see that
r ¼ « and F ¼ pt (11:3)
With this, we can write the magnitude of the force as
Fr ¼ m« p2 (11:4)
and the centripetal acceleration in the complex plane as
a ¼ Fr m [ cos ( pt)þ i sin ( pt)] ¼ « p2[cos ( pt)þ i sin ( pt)] (11:5)
The complex velocity can therefore be expressed as
V ¼ « p2 ðt 0
[ cos ( pt)þ i sin ( pt)]dt (11:6)
Evaluating the integral and assuming that as the projectile leaves the muzzle we have an initial orientation of the mass asymmetry of F¼F0 yields
V ¼ «p[ sin ( ptþF0) i cos ( ptþF0)] ¼ «p[ sin ( ptþF0)þ i cos ( ptþF0)] (11:7)
To see how much lateral movement has developed, we can integrate again
r ¼ «p ðt 0
[sin (ptþF0)þ i cos (ptþF0)]dt (11:8)
The evaluation of which yields
r ¼ «[ cos (ptþF0)þ i sin (ptþF0)] (11:9)
ε
Theory and Design of Guns and
As an example, if we were only concerned with motion in the crossrange direction, we could state
z ¼ Im{«[ cos (ptþF0)þ i sin (ptþF0)]} ¼ « sin (ptþF0) (11:10)
To apply numbers to this example, let us consider a projectile that weighs 100 lbm and is spinning at a rate of 270 Hz. We shall assume the projectile has a CG offset of 0.25 in. If this were the case, the velocity in the z-direction as well as the motion for the first 4 s of flight can be seen in Figures 11.2 and 11.3. Here we have assumed that the CG offset has emerged from the weapon at the twelve o’clock position. The most interesting observation between the figures is that for this arbitrary emergence
of the CG offset, we see that the projectile would like tomove laterally to the right for a righthand spin. This is commonly known as drift. Just to put things into perspective, the muzzle velocity consistent with the 270-Hz spin rate is about 2,750 ft=s so the projectile would only have gone about 0.4 ft to the right after it traversed 11,000 ft downrange. We must always bear in mind that this example was an idealized situation. In the case of
a real projectile, there are other forces acting which complicate the motion; however, it is instructive to look at simplifications such as this to see the phenomenon at work. We will now move on to examine the dynamic behavior in terms of the equations of motion of a projectile from statically imbalanced and dynamically imbalanced projectiles. We shall see how this affects lateral throwoff.
