used is explained in paragraph 4 in connection with the plotting of the theoretical time-travel and time-velocity curvcs before acceleration.

Since the only purpose of the barrel return spring and the bolt driving spring is to assist the barrel and bolt to return to battery and neither spring is required to absorb all of the recoil energy' of the parts, the values of the forccs exerted by the springs are not critical. For this reason, the characteristics of these springs may be selected more or less arbitrarily.

The barrel assembly is assumed to weigh 45 pounds and to return this relatively heavy mass quickly to battery a fairly strong spring is indicated. As will appear, such a spring will not have any great effect on the rccoil motion and therefore the resistance of the spring can be made quite high. On this basis the initial compression of the spring will be selected as 250 pounds and the spring constant as 300 pounds per inch.

The bolt spring will be made very light so that it will not offer a high retardation to the five-pound bolt. An initial compression of 25 pounds and a spring constant of 10 pounds per inch will provide adequate force for assisting the closing of the bolt.

4. Theoretical time-travel and time-velocity curves before acceleration

Because of the complexities resulting from the multiplicity of actions during the recoil and counter-recoil movements in a short rccoil gun, it is not practical to attempt to derive analytically expressions for the time to rccoil and time to counter-recoil. Also, such derivations would be extremely complicated unless it is assumed that the initial kinetic energy is transferred instantaneously to the recoiling parts, ignoring the detailed cffects which occur during the action of the powder gas pressures. However, in high-rate-of-fire guns employing the short recoil system, the time of action of the powder gases is extremely significant and must be given due consideration in plotting the bolt motion curves. A detailed analysis of this type is particularly important because most of the critical actions and high accelerations occur during the progress of the pro-pcllant explosion and it is therefore highly desirable to determine what motion characteristics mav be

expected in the initial portion of the recoil stroke.

Since the cffects of the powder gas pressure can not be expressed by simple equations, a special method is employed to account for these effects in plotting the bolt motion curvcs. The method consists essentially of first plotting a curve of free recoil velocity versus time and then subtracting from each ordinate of this curve the velocity loss resulting from the retarding effects of the springs.

The curve showing the velocity of free recoil versus time for the time interval before unlocking was developed previously and is shown in fig. 2-21. This curve will be used to illustrate the following description of the method.

To determine the retarding effects of the springs, use is made of the law expressed by the equation :

This law states that the change in the momentum of a mass Is equal to the applied impulse (the product of the force and the time for which it is applied). Solving for dv gives :

To obtain the variation of the change in velocity with rcspect to time, this expression is integrated.

In accordance with equation 2-23, the retarding effect of a force on a given mass can be determined as follows:

1. Plot a curvc showing the variation of the force with respect to time.

2. Measure the area under the curvc between t=0 and some lime ti.

3. Divide the measured area by the mass. This gives the ordinate of the retardation curve for the time ti.

4. Repeat steps 2 and 3 for other values of t ana plot the retardation curvc.

Applying this procedure using the mass of the recoiling parts and the combined resistance of the barrel return spring and the holt driving spring produces a curvc showing the loss in rccoil velocity resulting from the action of the springs up to the time of unlocking. Since the free rccoil velocity curve shows the gain in velocity resulting from the thrust of the powder gases, the difference between the curves will

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