15. Snap/Lurch 7 Linearsolve for lurch and PE

The model and its justification

This simulation is very similar to Snap/Lurch 6. It differs only in solving for PE rather than vcloud. It focuses on PE because it is so important in setting constraints for other variables.
As before, this procedure calculates the speed of the rearward lurch of JFK's upper body from the bullet, the body, the cloud, and the large fragments, and incorporates 3-D motions of the cloud, the large fragments, and the body: Before doing that, however, it interposes three steps having to do with the times. (1) It first calculates vsnap and uses it to calculate tsnap. (2) Then it calculates ttransit and combines it with tsnap with tdelay to calculate tlurch. (3) Then it uses tlurch and dfrags to calculate vfrags. At this point it returns to the procedure of Lurch 5 Linear and (4) solves for vbodyafter and vcloud. To simplify matters a bit, it sets vfrags23 = vfrag1 and fkefrags23 = fkefrag1.
This minor change in variables changes the default solution considerably. It gives -0.62 ft s-1 for the lurch (a 40% contraction) and 633 ft s-1 for the PE (double the previous default value). The sensitivities and effects also change considerably. It appears that the default variables are an improper combination, and consequently that this default solution should not be regarded seriously. That does not represent a problem, however, for the main purpose of Snap/Lurch 7 is for use in setting the constraints that are discussed in later sections.

(1) Equations for tsnap

(2) Equations for tlurch

(3) Equations for vfrags

(4) Simultaneous equations for vbodyafter and vcloud

fkefrags23 = fkefrag1

##### Conservation of total energy

Default values of variables

 mbullet = 161 gr vbullet = 1800 ft s-1 mfrag1 = 0.027 gr mbody = 85 lb vbulletafter = 200 ft s-1 mfrags23 = 0.01 gr mcloud = 0.3 lb Qfrag1 = 40° fkefrag1 = 1.25 mhead = 7 lb Qfrags23 = 70° fkefrags23 = fkefrag1 Q = 12° Qcl = 70° fkebody = 1.2 vcloud = 300 ft s-1 tdelay = 0.002 s fxcl = 0.59 (calc.) dtransit = 4 in dsnap = 2.2 in dfrag1 = 6 ft

### Solutions to simultaneous equations

vbodyafter = -0.62 ft s-1  PE = 633 ft-lb
vfrag1
= vfrags23 = 363 ft s-1
vsnap
= 5.1 ft s-1

ttransit
= 0.33 ms
tsnap
= 35.8 ms
tlurch
= 16.5 ms

## Distributions of momentum and energy

 Momentum Energy Before After Before After Pbullet = 1.26 Pbulletafter = 0.14 KEbullet = 1164 KEbulletafter = 14 Pbodyafter = -0.82 KEbodyafter = 0.2 Pcloud = 1.66 KEcloud = 422 Pfrag1 = 0.24 KEfrag1 = 70 Pfrags23 = 0.04 KEfrags23 = 26 PE = 633

Sensitivity analysis

Sensitivity analysis, Lurch 7 Linear
(Standard conditions in boldface)

 mbullet vlurch mbody vlurch mcloud vlurch mhead vlurch vbullet vlurch 4.0 -0.517 156 -0.658 4.5 -0.526 1750 -0.659 157 -0.650 65 -0.807 5.0 -0.538 1760 -0.650 158 -0.641 70 -0.749 5.5 -0.551 1770 -0.642 159 -0.633 75 -0.699 0.1 +0.221 6.0 -0.568 1780 -0.633 160 -0.625 80 -0.655 0.2 -0.197 6.5 -0.589 1790 -0.625 161 -0.617 85 -0.617 0.3 -0.617 7.0 -0.617 1800 -0.617 162 -0.609 90 -0.582 0.4 -1.037 7.5 -0.655 1810 -0.609 163 -0.601 95 -0.552 0.5 -1.459 8.0 -0.709 1820 -0.601 164 -0.593 100 -0.524 0.6 -1.881 8.5 -0.796 1830 -0.593 165 -0.585 105 -0.499 0.7 -2.305 9.0 -0.952 1840 -0.585 166 -0.578 0.8 -2.729 9.5 -1.322 1850 -0.577 10.0 -3.275 Sensitivity =  [(-0.609 + 0.625)/2]/ [0.617/161] = 2.09 Sensitivity =  [(-0.582 + 0.655)/10]/ [0.617/85] = 1.01 Sensitivity =  [(-1.037 + 0.197)/0.2]/ [0.617/0.3] = -2.04 Sensitivity =  [(-0.655 + 0.589)/1.0]/ [0.617/7.0] = -0.75 Sensitivity =  [(-0.609 + 0.625)/20]/ [0.617/1800] = 2.33 Range = 0.08 Range = 0.31 Range = 2.95 Range = 2.76 Range = 0.08

 vbulletafter vlurch vcloud vlurch Q vlurch Qcl vlurch mfrag1 vlurch 0 +0.641 50 +0.431 7 -0.596 20 -1.400 0.022 -0.584 0 -0.467 100 +0.222 8 -0.599 30 -1.298 0.023 -0.590 50 -0.502 150 +0.012 9 -0.603 40 -1.161 0.024 -0.597 100 -0.539 200 -0.197 10 -0.607 50 -0.997 0.025 -0.604 150 -0.577 250 -0.407 11 -0.612 60 -0.813 0.026 -0.610 200 -0.617 300 -0.617 12 -0.617 70 -0.617 0.027 -0.617 250 -0.660 350 -0.826 13 -0.622 80 -0.417 0.028 -0.623 300 -0.707 400 -1.036 14 -0.629 90 -0.221 0.029 -0.630 350 -0.760 450 -1.246 15 -0.635 100 -0.036 0.030 -0.636 400 -0.821 500 -1.455 16 -0.642 110 +0.132 0.031 -0.643 550 -1.665 17 -0.650 120 +0.278 0.032 -0.650 600 -1.874 Sensitivity =  [(-0.660 + 0.577)/100]/ [0.617/200] = -0.27 Sensitivity =  [(-0.826 + 0.407)/100]/ [0.617/300] = -2.04 Sensitivity =  [(-0.629 + 0.607)/4]/ [0.617/12] = -0.11 Sensitivity =  [(-0.417 + 0.813)/20]/ [0.617/70] = 2.25 Sensitivity =  [-0.623 + 0.610)/0.002]/ [0.617/0.027] = -0.28 Range = 0.35 Range = 2.52 Range = 0.05 Range = 1.68 Range = 0.07

 mfrags23 vlurch Qfrag1 vlurch Qfrags23 vlurch fkefrags vlurch fkebody vlurch 0.005 -0.602 1.00 -0.617 0.006 -0.605 20 -0.657 50 -0.642 1.05 -0.617 1.00 -0.617 0.007 -0.608 25 -0.649 55 -0.637 1.10 -0.617 1.05 -0.617 0.008 -0.611 30 -0.640 60 -0.630 1.15 -0.617 1.10 -0.617 0.009 -0.614 35 -0.629 65 -0.624 1.20 -0.617 1.15 -0.617 0.010 -0.617 40 -0.617 70 -0.617 1.25 -0.617 1.20 -0.617 0.011 -0.620 45 -0.603 75 -0.610 1.30 -0.617 1.25 -0.617 0.012 -0.623 50 -0.588 80 -0.602 1.35 -0.617 1.30 -0.617 0.013 -0.626 55 -0.572 85 -0.595 1.40 -0.617 1.35 -0.617 0.015 -0.628 60 -0.555 90 -0.587 1.45 -0.617 1.40 -0.617 0.015 -0.631 1.50 -0.617 Sensitivity =  [(-0.620 + 0.614)/0.002]/ [0.617/0.010] = -0.05 Sensitivity =  [(-0.603 + 0.629)/10]/ [0.617/40] = 0.17 Sensitivity =  [(-0.610 + 0.624)/10]/ [0.617/70] = 0.16 Sensitivity =  [(-0.617 + 0.617)/0.50]/ [0.617/1.25] = 0.00 Sensitivity =  [(-0.617 + 0.617)/0.40]/ [0.617/1.20] = 0.00 Range = 0.03 Range = 0.10 Range = 0.06 Range = 0.00 Range = 0.00

 dtransit vlurch dsnap vlurch dfrag1 vlurch tdelay vlurch 0 -0.613 2 -0.479 1 -0.614 1.6 -0.540 3 -0.513 2 -0.615 1.8 -0.558 4 -0.548 0.000 -0.594 3 -0.616 2.0 -0.583 5 -0.582 0.001 -0.605 4 -0.617 2.2 -0.617 6 -0.617 0.002 -0.617 5 -0.618 2.4 -0.667 7 -0.651 0.003 -0.630 6 -0.619 2.6 -0.751 8 -0.686 0.004 -0.645 7 -0.620 2.8 -0.915 9 -0.720 0.005 -0.663 8 -0.621 10 -0.755 0.006 -0.683 0.007 -0.707 Sensitivity =  [(-0.621 + 0.613)/8]/ [0.617/4] = -0.01 Sensitivity =  [(-0.667 + 0.583)/0.4]/ [0.617/2.2] = -0.75 Sensitivity =  [(-0.651 + 0.582)/2]/ [0.617/6] = -0.34 Sensitivity =  [(-0.630 + 0.605)/0.002]/ [0.617/0.002] = -0.04 Range = 0.01 Range = 0.38 Range = 0.28 Range = 0.11

Ordered summary of sensitivities

 Variable Sensitivity of vlurch Range of vlurch, ft s-1 Magnitude Positive effect on lurch (reduces rearward velocity) fkebody 0.00 0.00 Small fkefrags 0.00 0.00 Small Qfrags23 0.16 0.06 Small mbullet 2.09 0.08 Small vbullet 2.33 0.08 Small Qfrag1 0.17 0.10 Small mbody 1.01 0.31 Small Qcl 2.25 1.68 Large Negative effect on lurch (increases rearward velocity) dtransit -0.01 0.01 Small mfrags23 -0.05 0.03 Small Q -0.11 0.05 Small mfrag1 -0.28 0.07 Small tdelay -0.04 0.11 Small dfrag1 -0.34 0.28 Small vbulletafter -0.27 0.35 Small dsnap -0.75 0.38 Small mhead -0.75 2.76 Large vcloud -2.04 2.52 Large mcloud -2.04 2.95 Large

The biggest change in this ordered summary of sensitivities is the shift toward negative effects. As opposed to Snap/Lurch 6, which had 12 positive effects and 7 negative, Snap/Lurch 7 has 8 positive and 11 negative effects, for a net shift of four effects from positive to negative. One might then expect the rearward lurch to become stronger, but the opposite happenedit decreased in intensity from -1.04 ft s-1 to -0.62 ft s-1, a loss of 40%. As before, however, there was a great dichotomy between the few strong terms (four, with effects between 1.68 ft s-1 and 2.95 ft s-1) and the many small terms (14, with effects between 0.00 ft s-1 and 0.38 ft s-1). Of the four strong terms, three had to do with the cloud (mcloud, vcloud, and Qcl) and one with the head (mhead). This reinforces the all-importance of the cloud to the final solution.

Summary
Solving for PE instead of vcloud created four more negative effects but reduced the strength of the negative lurch from -1.04 ft s-1 to -0.62 ft s-1. Of the four large effects, three were properties of the cloud, one the mass of the head.