2. The loss in velocity due to the initial compression of the driving spring is equal to F«t/Mr. Determine the velocity loss for various values of t, subtract each from the corresponding ordinate of the free bolt velocity curve and draw a curve through the resulting points. If the effect of the spring rate 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 lhe actual bolt displacement curve and use this curve to determine the retardation due to the spring rate K. Ordinarily, it will be found that this retardation is so small that it will not have any effect worthy of consideration.
5. 11 the retardation determined in step 4 is sufficient to affect the velocity, use it to modifv the
curve drawn in step 2, and then integrate tinder the new curve to obtain a corrected displacement curve.
fi. Steps 4 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. 1-1G shows the curves obtained for the gun of the example. In this particular design, the spring is so weak that its retardation effects are entirely negligible during the first 0.010 second. To illustrate this point, the loss in velocity due to Fu over this interval is
The loss due to K, as determined by the method of step 4 is only about 0.0004 feet per second. Thus, in this gun the retarded velocity curve for the first 0.010 second is practically identical with the free velocity curve.
The remainder of the bolt displacement curve can now be determined analytically by using equation 1-10:
. KD + F. . f" /K . F0 "1 F„ ^ K smLVMr,"'Sm KD + fJ-K
However, since some bolt travel (cY) occurred during the first 0.010 second, the values of Fo, D. and t must be changed to take this motion into account and d' must be added to the resulting values obtained for d. The changed values to be used in equation 110 are:
The modified form of equation 1 10 is notv: gijv—i__F.+Kd' -| ?0+Kd'
This equation is employed to complete the bolt displacement curve. The orclinates of the displacement curve are then multiplied by K and increased by F0 to obtain a curve showing the variation of the total spring force with time. Integrating under this curve and dividing by Mr in accordance with equation 1-15 will give a graph of the loss in velocity due to the spring force. Subtracting this curve from the free bolt velocity curve will give the complete graph for the retarded bolt velocity.
Fig. 1-17 shows the displacement and velocity curves obtained by this method for the gun of the example. After the necessary substitutions are made in equation 1-10, the final form of the equation to be used after the first 0.010 second is:
The spring force curve obtained from the displacement curve of fig. 1-17 is shown in fig. 1-18.
If so desired in the course of a design, the effects of friction and loads incident to operating the gun mechanism may be taken into account in plotting the displacement and velocity curves. These forces are handled in the same way as the spring force. For example, the friction force resisting bolt motion will be essentially constant and therefore can be
taken into account by increasing F* in equation 1-10. A constant or varying load which exists for only a small portion of the cycle (such as the force
Figure 1-18. Variation of Spring Force With Time.
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