Step

2. The loss in velocity due to the initial compression of the springs is equal to:

Determine the velocity loss for various values of t, substract each from the corresponding ordinate of the free recoil velocity curve and draw a curve through the resulting points. If the effect of the spring constant proves to be negligible, this curve is the retarded velocity curve.

3. Integrate under the curve drawn in step 2 to obtain the displacement curve.

4. Assume that the curve drawn in step 3 represents the actual time-travel curve and use this curve to determine the retardation due to the spring constant. (Use the combined spring constant for the barrel spring and bolt spring, K1 + K2.) Ordinarily, it will be found that this retardation is so small that it will not have any effect worthy of consideration.

5. In the event that the retardation determined in step 4 is sufficient to affect the velocity, use it to modify the curve drawn in step 2 and then integrate under the new curve to obtain a corrected displacement curve.

6. Steps \ and 5 can be repeated as often as is necessary until no significant change occurs in the displacement curve. Actually, this process of successive approximation should never be necessary and satisfactory results should be achieved in the first three steps or at least in the first five steps.

Fig. 2-10 shows the curves obtained for the gun of the example. The total loss in velocity due to the combined effect of the initial compressions of the springs during the first 0.010 second is:

The loss due to the combined effect of the spring constants as determined by the method of step 4 is only about 0.345 foot per second. The final curves shown in fig. 2-10 are the result of performing step 5. Since the velocity loss due to the cffcct of the spring constant is so small, step 6 need not be taken.

The remainder of the displacement curve for the recoil stroke can now be determined analytically by using equation 2-10:

. KD + F, . T / K • _, F„ 1 Fo d- K *in LVKTr KD + F0K

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