## Recoil Operation

Symbols Used in Analysis

A Area of bore cross-section—in.2 B Ratio between average forces of Darrcl spring and bolt spring. G Arbitrary constant of integration (also ratio between masses of barrel and bolt).

D Total recoil travel of bolt—ft. d Recoil travel of bolt for time t—ft. d Distance recoiled in first 0.010 second—ft. Er Initial bolt energy—ft. lb. Fav Average combined force of barrel and bolt spring over distance D—lb. FaV| Average force of barrel spring over distance D—lb. FaV2 Average force of bolt spring over distance D—lb. F0 Combined initial compression of barrel spring and bolt spring—lb. F0, Initial compression of barrel spring— lb.

FOJ Initial compression of bolt spring—lb. g Acceleration of gravity—32.2 ft./sec.2 K Combined spring rate of barrel and bolt springs—lb./ft. K, Spring rate of barrel spring—lb./ft. K2 Spring rate, of bolt spring—lb./ft. Mc Mass of powder charge—lb. sec.2/ft. MP Mass of projectile—lb. sec.yft. Mr Total mass of recoiling parts lb. sec.Vft.

Mt Mass of barrel—lb. sec.yft. M2 Mass of bolt—lb. sec. -'/ft. P Muzzle pressure—lb./in.2 T Time to recoil—sec. T' Total cycle time—sec. Tl Time for counter-recoil of barrel—sec. T^ Time for return of bolt—sec. t Time—sec.

tr Approximate total return time for barrel and bolt—sec. TrejtTime of duration of residual pressure—sec.

Vp Muzzle velocity of projectile—ft./sec. vp Velocity of projcctile in bore at time t -ft./sec.

vr Velocity of retarded recoil at time t—ft./sec.

Vrf Maximum velocity of free recoil— ft./sec.

Wc Weight of powder charge—lb. Wp Weight of projectile—lb. Wr Weight of recoiling parts—lb.

5. Computation of the power absorbed by the recoiling parts.

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

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

b. Formula for determining momentum and velocity of free recoil.

c. Expression for duration of residual pressure.

d. Formula for determining initial energy of the rccoiling parts.

e. Formulas for determining spring retardations.

f. Energy equation for recoiling parts and springs.

g. Formula for determining time to recoil.

h. Expression for computing rate of fire.

### 1. Condition of free recoil

Under the heading "Principles of Recoil" 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:

Since the powder gases will be thoroughly mixed by the turbulence created in the explosion it is reasonable to assume that the ccnter 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 2-1 may be rewritten as:

NOTE: It should be pointed out here that the momentum equality expressed by equation 2-2 is not affected 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

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