18. Lurch 3 Angular—add 3-D conical diffuse cloud

The model and its justification
Lurch 3 Angular calculates the speed of the rearward lurch of the upper body from the bullet, the body, the cloud, the large fragments, and conical 3-D motion of the diffuse cloud. As with the linear analogs, the three-dimensionality of the cone decreases the intensity of the lurch by about half, from the 2.94 ft s-1 of Lurch 2 Angular to 1.44 ft s-1 here. This shows how important the three-dimensionality of the cloud is to the final solution.

Solving the simultaneous equations

##### 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° vfrag1 = 500 ft s-1 Q = 12° Qfrags23 = 70° vfrags23 = 500 ft s-1 PE = 300 ft-lb Qcl = 70° fI = 1.11

### Solutions to simultaneous equations

vbodyafter = -1.44 ft s-1  vcloud = 388 ft s-1

## 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 = -4.22 KEbodyafter = 1 Ωcloud = 6.45 KEcloud = 704 Ωfrag1 = 0.97 KEfrag1 = 105 Ωfrags23 = 0.16 KEfrags23 = 39 PE = 300

Sensitivity analysis

Sensitivity analysis, Lurch 3 Angular
(Standard conditions in boldface)

 mbullet vlurch mbody vlurch mcloud vlurch PE vlurch vbullet vlurch 0 -1.861 156 -1.414 50 -1.795 1750 -1.369 157 -1.419 65 -1.878 100 -1.727 1760 -1.383 158 -1.423 70 -1.744 150 -1.657 1770 -1.396 159 -1.427 75 -1.627 0.1 -0.507 200 -1.586 1780 -1.409 160 -1.431 80 -1.525 0.2 -1.032 250 -1.512 1790 -1.422 161 -1.435 85 -1.435 0.3 -1.435 300 -1.435 1800 -1.435 162 -1.439 90 -1.355 0.4 -1.776 350 -1.356 1810 -1.448 163 -1.443 95 -1.284 0.5 -2.077 400 -1.274 1820 -1.461 164 -1.447 100 -1.219 0.6 -2.350 450 -1.188 1830 -1.474 165 -1.451 105 -1.161 0.7 -2.601 500 -1.098 1840 -1.486 166 -1.455 0.8 -2.835 550 -1.004 1850 -1.499 600 -0.904 Sensitivity =  [(-1.439 + 1.431)/2]/ [1.435/161] = -0.45 Sensitivity =  [(-1.355 + 1.525)/10]/ [1.435/85] = 1.01 Sensitivity =  [(-1.776 + 1.032)/0.2]/ [1.435/0.3] = -0.78 Sensitivity =  [(-1.356 + 1.512)/100]/ [1.435/300] = 0.33 Sensitivity =  [(-1.448 + 1.422)/20]/ [1.435/1800] = -1.63 Range = 0.04 Range = 0.72 Range = 2.33 Range = 0.96 Range = 0.13

 vbulletafter vlurch Q vlurch Qcl vlurch mfrag1 vlurch mfrags23 vlurch 7 -1.416 20 -2.796 0.022 -1.404 0.005 -1.438 0 -1.311 8 -1.419 30 -2.619 0.023 -1.411 0.006 -1.438 50 -1.346 9 -1.423 40 -2.382 0.024 -1.417 0.007 -1.437 100 -1.379 10 -1.426 50 -2.097 0.025 -1.423 0.008 -1.436 150 -1.408 11 -1.431 60 -1.777 0.026 -1.429 0.009 -1.436 200 -1.435 12 -1.435 70 -1.435 0.027 -1.435 0.010 -1.435 250 -1.459 13 -1.440 80 -1.086 0.028 -1.441 0.011 -1.435 300 -1.480 14 -1.446 90 -0.744 0.029 -1.447 0.012 -1.434 350 -1.498 15 -1.451 100 -0.422 0.030 -1.454 0.013 -1.433 400 -1.514 16 -1.457 110 -0.129 0.031 -1.460 0.015 -1.433 17 -1.464 120 +0.126 0.032 -1.466 0.015 -1.432 Sensitivity =  [(-1.459 + 1.408)/100]/ [1.435/200] = -0.07 Sensitivity =  [(-1.446 + 1.426)/4]/ [1.435/12] = -0.04 Sensitivity =  [(-1.086 + 1.777)/20]/ [1.435/70] = 1.68 Sensitivity =  [-1.441 + 1.429)/0.002]/ [1.435/0.027] = -0.11 Sensitivity =  [(-1.432 + 1.438)/0.010]/ [1.435/0.010] = 0.00 Range = 0.20 Range = 0.05 Range = 2.92 Range = 0.06 Range = 0.01

 vfrag1 vlurch vfrags23 vlurch Qfrag1 vlurch Qfrags23 vlurch fI vlurch 250 -1.390 250 -1.453 1.06 -1.503 300 -1.406 300 -1.452 20 -1.510 50 -1.483 1.07 -1.48 350 -1.419 350 -1.450 25 -1.496 55 -1.472 1.08 -1.475 400 -1.428 400 -1.446 30 -1.478 60 -1.460 1.09 -1.462 450 -1.433 450 -1.441 35 -1.458 65 -1.448 1.10 -1.448 500 -1.435 500 -1.435 40 -1.435 70 -1.435 1.11 -1.435 550 -1.433 550 -1.428 45 -1.410 75 -1.422 1.12 -1.422 600 -1.428 600 -1.419 50 -1.382 80 -1.408 1.13 -1.410 650 -1.418 650 -1.409 55 -1.352 85 -1.395 1.14 -1.397 700 -1.403 700 -1.398 60 -1.321 90 -1.381 1.15 -1.385 700 -1.384 750 -1.385 1.16 -1.373 Sensitivity =  [(-1.428 + 1.428)/200]/ [1.435/500] = 0.00 Sensitivity =  [(-1.428 + 1.441)/100]/ [1.435/500] = 0.04 Sensitivity =  [(-1.410 + 1.458)/10]/ [1.435/40] = 0.13 Sensitivity =  [(-1.422 + 1.448)/10]/ [1.435/70] = 0.13 Sensitivity =  [(-1.422 + 1.448)/0.02]/ [1.435/1.11] = 1.01 Range = 0.01 Range = 0.07 Range = 0.19 Range = 0.10 Range = 0.13

Ordered summary of sensitivities

 Variable Sensitivity of vlurch Range of vlurch, ft s-1 Magnitude Positive effect on lurch (reduces rearward velocity) mfrags23 0.00 0.01 Small vfrag1 0.00 0.01 Small vfrags23 0.04 0.07 Small Qfrags23 0.13 0.10 Small fI 1.01 0.13 Small Qfrag1 0.13 0.19 Small mbody 1.01 0.72 Medium PE 0.33 0.96 Medium Qcl 1.68 2.92 Large Negative effect on lurch (increases rearward velocity) mbullet -0.45 0.04 Small Q -0.04 0.05 Small mfrag1 -0.11 0.06 Small vbullet -1.63 0.13 Small vbulletafter -0.07 0.20 Small mcloud -0.78 2.33 Large

This ordering of sensitivities and effects is virtually identical to that of Lurch 3 Linear, except with greater effects. It is also very similar to the previous one, except for the new large effect for Qcl. The other large effect, for mcloud, has also been seen before. That both these large effects belong to properties of the diffuse cloud is an important feature of all subsequent simulations.

Summary
Making the diffuse cloud into a conical shape reduces the strength of the rearward lurch by a factor of two (2.9 ft s-1 to 1.4 ft s-1) and makes the half-angle of the cone one of the two most-sensitive variables, the mass of the cloud being the other one. This both brings the calculated lurch nearly into line with the observed lurch and emphasizes the all-importance of the properties of the cloud. Unfortunately for this work, the properties of the cloud are very difficult to estimate, and probably form the limiting factor for the accuracy of this series of simulations.