the excess energy in the other recoiling parts. Ideally, it would be desirable for a gas-operated gun not to recoil at all or at least to recoil with a low velocity in order to minimize the problem of energy dissipation. In such a case, entire dependence for the production of bolt velocity could be placed on the piston and blowback. Unfortunately, the requirements of modern gun design as stated above, make a high recoil energy unavoidable, and the designer of the gas-operated gun is required to make the best of the situation. (From the standpoint of efficient design, it is somewhat paradoxical that, when the gas system is employed, a considerable amount of energy is accumulated, this energy is then discarded without being used, and finally energy for operating the gun mechanism is obtained from another source.)
After unlocking occurs and just before the piston strikes the bolt, the gun itself, the gas piston, and the bolt are all moving to the rear. The piston is moving at much higher velocity than the gun because of the action of the gas pressure on the relatively light piston mass. The bolt is also moving at a higher velocity than the gun because of the effect of blowback. When the piston strikes the bolt, the impact causes the piston velocity to decrease. The factors involved in this action can be seen clearly by considering the fundamental mathematical relationships affecting the conditions of impact. These relationships are expressed as follows:
Equation 3-3 is based on the assumption that the spring forces, friction, or other forces acting during the impact are negligible when compared to the impact forces. Therefore the net impulse received during the period of impact is zero and the total momentum of the system remains the same. Mi
and M2 are the masses of the piston and bolt, V\ and V2 their velocities before impact, and V'x and V'2 their velocities after impact. Equation 3-4 states that the relative velocity of separation of two bodies after impact is directly proportional to their relative velocity of approach before impact. The constant of proportionality is the coefficient of restitution, e, and the value of this constant depends on the materials (approximately 0.55 for steel).
Solving equation 3-4 for Vi gives:
V'i=V'2—e( V,—V2) Substituting the result in equation 3-3 gives:
Solving for V2:
This equation may be rewritten in the following form:
Now if V- is zero or is very small when compared with Vi, this equation rcduces to:
Examination of equation 3-7 alone might lead to the erroneous conclusion that the piston mass Mi should be heavier than the bolt mass M2 in order to produce a high bolt velocity V'2, or even to the conclusion that the final bolt velocity depends only on the ratio between the masses. However, it must be remembered that the piston velocity V, imparted by the gas pressure will be inversely proportional to the piston mass. This factor can be taken into account by setting Vi equal to I/Mi, where I is the impulse of the powder gases on the piston. It is best to make this substitution in the complete equation 3 6 giv-ing:
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