act on these parts as the result of the accelerations and shocks to which they are subjected in operation and which vet remain to be determined. How-

j ever, the estimated weights obtained from a carefully made preliminary design and layout should be accurate enough to serve as a starting point for the calculations nccessary to determine the operating forces. It is these calculations which are the main conccrn of the following analysis.

As the analysis progresses, its applications will be illustrated by means of sample calculations. Although these calculations and the related graphs arc for a specific 20-mm cartridge and barrel and arc based on certain assumed weights and other charac teristics, the general approach described is applicable to short recoil guns of any caliber. The calculations cover the following important points:

1. Determination of the conditions of free rccoil.

2. Determination of correct time for unlocking.

3. Computation of data required for design of accelerator.

4. Selection of characteristics of barrel return spring and bolt driving spring and determination of data for design of backplate buffer.

5. Development of graphs showing how the velocity and travel of the barrel and bolt varv with time.

In the course of describing these calculations, the following fundamental formula will be developed and explained:

a. Momentum and velocity relation for time projectile is in bore.

b. Formula for determining velocity of free rccoil.

c. Expression for duration of residual pressure.

d. Formulas for determining spring retardations. (Because of the fact that before unlocking occurs there is no difference hetween the analysis of a short recoil gun and a long recoil gun, some of the explanations and derivations arc identical for both systems of operation. In such cases, to avoid the inconvenience and confusion of attempting to refer back to the explanations given under long recoil, the material will be repeated here.)

Under the heading "Principles of Rccoil" it was pointed out that, if a gun is mounted so that it can move freely without friction or any other restraint, the impulse of the recoil force will impart to the gun a rearward momentum equal to the total forward momentum of the projectile and powder gases.

For the time the projectile is in the bore, this momentum relationship is expressed by the equation:

Sincc the powder gases will be thoroughly mixed by the turbulence created in the explosion, it is reasonable to assume that the center of mass of the gases moves forward at one-half the velocity of the projectile. Actually, this is not quite accurate because the presence of the enlargement at the chamber and the fact, that the rifling does not extend the full length of the space occupied by the gases creates a condition in which the volume of the space is not uniformly distributed along its length. Nevertheless. the assumption is close enough for present purposes. Therefore equation 2-17 may be rewritten as:

NOTE: It should be pointed out here that the momentum equality expressed by equation 2-18 is not afTectcd by the internal frictional forces opposing the motion of the projectile and powder gases or by the forces incident to engraving the rifling band and to imparting the rotational velocity of the projectile. Although all of these forces retard the forward motion of the projectile and powder gases, they produce equal and opposite reactions on the barrel which result in a corresponding retardation of the rearward movement of the gun. In other words, the internal resistances merely decrease the cffcctivc impulse producing motion but they do not cause any inequality in the forward and rearward momentums.

Solving equation 2-10 for v-f gives the velocity of free recoil for the time the projectile is in the bore as:

Vrf=

Equation 2- 19 can be used to plot a curve showing the free rccoil velocity versus time for the period before the projectile leaves the muzzle. The weights of the projectile and powder charge are both known and it is assumed that the weight of the recoiling parts has been estimated in accordance with a preliminary design plan. Also, the velocity of the projectile at any time is known from the

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