22. Snap/Lurch 7 Angularsolve for lurch and PE

The model and its justification

This simulation differs from Snap/Lurch 6 Angular only in that it solves 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, however. It gives -0.91 ft s-1 for the lurch (a 35% contraction) and 576 ft-lb for the PE (double the previous default value). The sensitivities and effects also change considerably. As with Snap/Lurch 7 Linear, it appears that the default values of the 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 = -0.91 ft s-1  PE = 576 ft-lb
vfrag1
= vfrags23 = 458 ft s-1
vsnap
= 4.7 ft s-1

ttransit
= 0.33 ms
tsnap
= 39.2 ms
tlurch
= 13.1 ms

## Distributions of momentum and energy

 Angular momentum Energy, ft-lb Before After Before After Ωbullet = 3.80 Ωbulletafter = 0.43 KEbullet = 1164 KEbulletafter = 14 Ωbodyafter = -2.66 KEbodyafter = 0.5 Ωcloud = 4.99 KEcloud = 422 Ωfrag1 = 0.89 KEfrag1 = 111 Ωfrags23 = 0.15 KEfrags23 = 41 PE = 300

Sensitivity analysis

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

 mbullet vlurch mbody vlurch mcloud vlurch mhead vlurch vbullet vlurch 4.0 -0.708 156 -0.979 4.5 -0.724 1750 -0.980 157 -0.964 65 -1.187 5.0 -0.744 1760 -0.965 158 -0.949 70 -1.102 5.5 -0.769 1770 -0.949 159 -0.934 75 -1.028 0.1 +0.226 6.0 -0.801 1780 -0.935 160 -0.920 80 -0.963 0.2 -0.339 6.5 -0.844 1790 -0.920 161 -0.906 85 -0.906 0.3 -0.906 7.0 -0.906 1800 -0.906 162 -0.893 90 -0.856 0.4 -1.475 7.5 -1.002 1810 -0.893 163 -0.880 95 -0.811 0.5 -2.044 8.0 -1.169 1820 -0.879 164 -0.867 100 -0.770 0.6 -2.615 8.5 -1.536 1830 -0.866 165 -0.854 105 -0.733 0.7 -3.188 9.0 -2.980 1840 -0.853 166 -0.842 0.8 -3.761 9.5 +4.604 1850 -0.841 10.0 +0.696 Sensitivity =  [(-0.893 + 0.920)/2]/ [0.906/161] = 0.00 Sensitivity =  [(-0.856 + 0.963)/10]/ [0.906/85] = 1.00 Sensitivity =  [(-1.475 + 0.339)/0.2]/ [0.906/0.3] = -1.88 Sensitivity =  [(-1.002 + 0.844)/1.0]/ [0.906/7.0] = -1.22 Sensitivity =  [(-0.893 + 0.920)/20]/ [0.906/1800] = 2.68 Range = 0.14 Range = 0.45 Range = 3.99 Range = 1.40 Range = 0.14

 vbulletafter vlurch vcloud vlurch Q vlurch Qcl vlurch mfrag1 vlurch 0 +0.793 50 +0.510 7 -0.871 20 -1.965 0.022 -0.850 0 -0.671 100 +0.227 8 -0.876 30 -1.827 0.023 -0.861 50 -0.723 150 -0.057 9 -0.883 40 -1.642 0.024 -0.873 100 -0.779 200 -0.340 10 -0.890 50 -1.421 0.025 -0.884 150 -0.840 250 -0.623 11 -0.898 60 -1.172 0.026 -0.895 200 -0.906 300 -0.906 12 -0.906 70 -0.906 0.027 -0.906 250 -0.982 350 -1.190 13 -0.916 80 -0.636 0.028 -0.918 300 -1.070 400 -1.473 14 -0.927 90 -0.371 0.029 -0.929 350 -1.180 450 -1.756 15 -0.938 100 -0.121 0.030 -0.940 400 -1.327 500 -2.039 16 -0.951 110 +0.105 0.031 -0.951 550 -2.323 17 -0.965 120 +0.302 0.032 -0.962 600 -2.606 Sensitivity =  [(-0.982 + 0.840)/100]/ [0.906/200] = -0.31 Sensitivity =  [(-1.190 + 0.623)/100]/ [0.906/300] = -1.88 Sensitivity =  [(-0.927 + 0.890)/4]/ [0.906/12] = -0.12 Sensitivity =  [(-0.636 + 1.172)/20]/ [0.906/70] = 2.07 Sensitivity =  [-0.918 + 0.895)/0.002]/ [0.906/0.027] = -0.34 Range = 0.66 Range = 3.40 Range = 0.09 Range = 2.27 Range = 0.11

 mfrags23 vlurch Qfrag1 vlurch Qfrags23 vlurch fkefrags vlurch fkebody vlurch 0.005 -0.881 1.00 -0.906 0.006 -0.886 20 -0.975 50 -0.950 1.05 -0.906 1.00 -0.906 0.007 -0.891 25 -0.962 55 -0.940 1.10 -0.906 1.05 -0.906 0.008 -0.896 30 -0.946 60 -0.929 1.15 -0.906 1.10 -0.906 0.009 -0.901 35 -0.927 65 -0.918 1.20 -0.906 1.15 -0.906 0.010 -0.906 40 -0.906 70 -0.906 1.25 -0.906 1.20 -0.906 0.011 -0.911 45 -0.883 75 -0.894 1.30 -0.906 1.25 -0.906 0.012 -0.916 50 -0.858 80 -0.882 1.35 -0.906 1.30 -0.906 0.013 -0.921 55 -0.830 85 -0.869 1.40 -0.906 1.35 -0.906 0.015 -0.926 60 -0.801 90 -0.856 1.45 -0.906 1.40 -0.906 0.015 -0.931 1.50 -0.906 Sensitivity =  [(-0.911 + 0.901)/0.002]/ [0.906/0.010] = -0.06 Sensitivity =  [(-0.883 + 0.927)/10]/ [0.906/40] = 0.19 Sensitivity =  [(-0.894 + 0.918)/10]/ [0.906/70] = 0.18 Sensitivity =  [(-0.906 + 0.906)/0.50]/ |[0.906/1.25] = 0.00 Sensitivity =  [(-0.906 + 0.906)/0.40]/ [0.906/1.20] = 0.00 Range = 0.05 Range = 0.17 Range = 0.09 Range = 0.00 Range = 0.00

 fI vlurch dtransit vlurch dsnap vlurch dfrag1 vlurch tdelay vlurch 1.06 -0.949 1.07 -0.940 0 -0.898 2 -0.671 1.08 -0.932 1 -0.900 1.6 -0.748 3 -0.730 1.09 -0.923 2 -0.902 1.8 -0.782 4 -0.789 0.000 -0.860 1.10 -0.915 3 -0.904 2.0 -0.831 5 -0.848 0.001 -0.881 1.11 -0.906 4 -0.906 2.2 -0.906 6 -0.906 0.002 -0.906 1.12 -0.898 5 -0.909 2.4 -1.038 7 -0.965 0.003 -0.936 1.13 -0.890 6 -0.911 2.6 -1.327 8 -1.024 0.004 -0.970 1.14 -0.883 7 -0.913 2.8 -2.468 9 -1.083 0.005 -1.011 1.15 -0.875 8 -0.916 10 -1.141 0.006 -1.061 1.16 -0.867 0.007 -1.124 Sensitivity =  [(-0.898 + 0.915)/0.02]/ [0.906/1.11] = 1.04 Sensitivity =  [(-0.916 + 0.898)/8]/ [0.906/4] = -0.01 Sensitivity =  [(-1.038 + 0.831)/0.4]/ [0.906/2.2] = -1.26 Sensitivity =  [(-0.965 + 0.848)/2]/ [0.906/6] = -0.39 Sensitivity =  [(-0.936 + 0.881)/0.002]/ [0.906/0.002] = -0.06 Range = 0.08 Range = 0.02 Range = 3.22 Range = 0.47 Range = 0.26

 Rbullet vlurch Rhead vlurch 4.75 -1.506 3.50 -0.766 5.00 -1.193 3.75 -0.789 5.25 -1.047 4.00 -0.818 5.50 -0.962 4.25 -0.856 5.75 -0.906 4.50 -0.906 6.00 -0.867 4.75 -0.977 6.25 -0.838 5.00 -1.082 6.50 -0.816 5.25 -1.257 6.75 -0.798 5.50 -1.605 Sensitivity =  [(-0.867 + 0.962)/0.50]/ [0.906/5.75] = 1.21 Sensitivity =  [(-0.977 + 0.856)/0.50]/ [0.906/4.50] = -1.20 Range = 0.71 Range = 0.84

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 fkefrag1 0.00 0.00 Small fI 1.04 0.08 Small Qfrags23 0.18 0.09 Small mbullet 0.00 0.14 Small vbullet 2.16 0.14 Small Qfrag1 0.19 0.17 Small mbody 1.00 0.45 Small Rbullet 1.21 0.71 Medium Qcl 2.07 2.27 Large Negative effect on lurch (increases rearward velocity) dtransit -0.01 0.02 Small mfrags23 -0.06 0.05 Small Q -0.12 0.09 Small mfrag1 -0.34 0.11 Small tdelay -0.06 0.26 Small dfrag1 -0.39 0.47 Small vbulletafter -0.31 0.66 Medium Rhead -1.20 0.84 Medium mhead -1.22 1.40 Large dsnap -1.26 3.22 Large vcloud -1.88 3.40 Large mcloud -1.88 3.99 Large

These sensitivities and effects are very similar to those of the linear analog Snap/Lurch 7 Linear. They differ from Snap/Lurch 6 in that they are systematically shifted toward negative effects. As opposed to Snap/Lurch 6, which had 15 positive effects and 7 negative sensitivities, Snap/Lurch 7 has 10 positive and 12 negative effects, for a net shift of five effects from positive to negative. One might then expect the rearward lurch to become stronger, but the opposite happenedit decreased in intensity from -1.39 ft s-1 to -0.91 ft s-1, a loss of 35%. As before, however, there was a great dichotomy between the few strong terms (five, with effects between 1.40 ft s-1 and 3.99 ft s-1) and the many small and medium terms (17, with effects between 0.00 ft s-1 and 0.84 ft s-1). Of the five strong terms, three had to do with the cloud (mcloud, vcloud, and Qcl) and two with the snap (mhead and dsnap). This reinforces the importance of the cloud to the final solution.

Summary
Solving for PE instead of vcloud created five more negative effects but reduced the strength of the negative lurch from -1.39 ft s-1 to -0.91 ft s-1. Of the five large effects, three were properties of the cloud, two of the head and snap.