20. Lurch 5 Angularadd 3-D motion of body

The model and its justification
Lurch 5 Angular is the rotational analog of Lurch 5 Linear. Like Lurch 4 Angular, it calculates the speed of the rearward lurch from the bullet, the cloud, the large fragments, conical motion of cloud, and 3-D motion of the large fragments. To this it adds 3-D motion of the body. It turns out that the 3-D component of the body's KE does not change the answer perceptibly—the default velocity of -1.38 ft s-1 in Lurch 4 Angular remains -1.38 ft s-1 here. Thus the 3-D motion of the body is an insignificant term in the overall equation.

Solving the simultaneous equations
The three-dimensionality of the body's kinetic energy is expressed by the term fkebody, which is similar in form and function to fkefrag. We assume a range of 11.4 and a starting value of 1.2.

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° fkefrag1 = 1.25 fI = 1.11 fkebody = 1.2 fkefrags23 = 1.25

Solutions to simultaneous equations

vbodyafter = -1.38 ft s-1  vcloud = 378 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 = -4.05 KEbodyafter = 1 Ωcloud = 6.28 KEcloud = 668 Ωfrag1 = 0.97 KEfrag1 = 132 Ωfrags23 = 0.16 KEfrags23 = 49 PE = 300

Sensitivity analysis

Sensitivity analysis, Lurch 5 Angular
(Standard conditions in boldface)

 mbullet vlurch mbody vlurch mcloud vlurch PE vlurch vbullet vlurch 0 -1.813 156 -1.356 50 -1.746 1750 -1.309 157 -1.361 65 -1.803 100 -1.677 1760 -1.323 158 -1.365 70 -1.674 150 -1.606 1770 -1.337 159 -1.370 75 -1.562 0.1 -0.474 200 -1.532 1780 -1.351 160 -1.374 80 -1.464 0.2 -0.985 250 -1.457 1790 -1.365 161 -1.378 85 -1.378 0.3 -1.378 300 -1.378 1800 -1.378 162 -1.383 90 -1.301 0.4 -1.710 350 -1.297 1810 -1.392 163 -1.387 95 -1.233 0.5 -2.003 400 -1.212 1820 -1.405 164 -1.391 100 -1.171 0.6 -2.268 450 -1.124 1830 -1.418 165 -1.395 105 -1.115 0.7 -2.512 500 -1.031 1840 -1.431 166 -1.399 0.8 -2.740 550 -0.933 1850 -1.444 600 -0.828 Sensitivity =  [(-1.383 + 1.374)/2]/ [1.378/161] = -0.53 Sensitivity =  [(-1.301 + 1.464)/10]/ [1.378/85] = 1.00 Sensitivity =  [(-1.710 + 0.985)/0.2]/ [1.378/0.3] = -0.79 Sensitivity =  [(-1.297 + 1.457)/100]/ [1.378/300] = 0.35 Sensitivity =  [(-1.392 + 1.365)/20]/ [1.378/1800] = -1.76 Range = 0.04 Range = 0.69 Range = 2.23 Range = 0.98 Range = 0.14

 vbulletafter vlurch Q vlurch Qcl vlurch mfrag1 vlurch mfrags23 vlurch 7 -1.359 20 -2.702 0.022 -1.356 0.005 -1.389 0 -1.254 8 -1.362 30 -2.530 0.023 -1.360 0.006 -1.387 50 -1.290 9 -1.366 40 -2.300 0.024 -1.365 0.007 -1.385 100 -1.322 10 -1.369 50 -2.022 0.025 -1.369 0.008 -1.383 150 -1.351 11 -1.374 60 -1.711 0.026 -1.374 0.009 -1.381 200 -1.378 12 -1.378 70 -1.378 0.027 -1.378 0.010 -1.378 250 -1.402 13 -1.383 80 -1.038 0.028 -1.383 0.011 -1.376 300 -1.422 14 -1.389 90 -0.705 0.029 -1.387 0.012 -1.374 350 -1.440 15 -1.394 100 -0.391 0.030 -1.391 0.013 -1.371 400 -1.454 16 -1.401 110 -0.106 0.031 -1.396 0.015 -1.369 17 -1.407 120 +0.142 0.032 -1.400 0.015 -1.366 Sensitivity =  [(-1.402 + 1.351)/100]/ [1.378/200] = -0.07 Sensitivity =  [(-1.389 + 1.369)/4]/ [1.378/12] = -0.04 Sensitivity =  [(-1.038 + 1.711)/20]/ [1.378/70] = 1.71 Sensitivity =  [-1.387 + 1.369)/0.004]/ [1.378/0.027] = -0.09 Sensitivity =  [(-1.374 + 1.383)/0.004]/ [1.378/0.010] = 0.02 Range = 0.20 Range = 0.05 Range = 2.84 Range = 0.04 Range = 0.02

 vfrag1 vlurch vfrags23 vlurch Qfrag1 vlurch Qfrags23 vlurch fkefrag1 vlurch 250 -1.366 250 -1.408 1.00 -1.420 300 -1.377 300 -1.406 20 -1.453 50 -1.426 1.05 -1.412 350 -1.384 350 -1.401 25 -1.439 55 -1.415 1.10 -1.403 400 -1.387 400 -1.395 30 -1.421 60 -1.403 1.15 -1.395 450 -1.385 450 -1.388 35 -1.401 65 -1.391 1.20 -1.387 500 -1.378 500 -1.378 40 -1.378 70 -1.378 1.25 -1.378 550 -1.367 550 -1.367 45 -1.353 75 -1.365 1.30 -1.370 600 -1.349 600 -1.355 50 -1.325 80 -1.351 1.35 -1.361 650 -1.327 650 -1.340 55 -1.295 85 -1.338 1.40 -1.353 700 -1.297 700 -1.324 60 -1.264 90 -1.324 1.45 -1.344 700 -1.261 750 -1.305 1.50 -1.335 Sensitivity =  [(-1.367 + 1.385)/100]/ [1.378/500] = 0.06 Sensitivity =  [(-1.367 + 1.388)/100]/ [1.378/500] = 0.08 Sensitivity =  [(-1.353 + 1.401)/10]/ [1.378/40] = 0.14 Sensitivity =  [(-1.365 + 1.391)/10]/ [1.378/70] = 0.13 Sensitivity =  [(-1.370 + 1.387)/0.10]/ [1.378/1.25] = 0.15 Range = 0.10 Range = 0.10 Range = 0.19 Range = 0.10 Range = 0.08

 fkefrags23 vlurch fI vlurch fkebody vlurch 1.00 -1.393 1.06 -1.443 1.05 -1.390 1.07 -1.429 1.00 -1.378 1.10 -1.387 1.08 -1.416 1.05 -1.378 1.15 -1.384 1.09 -1.403 1.10 -1.378 1.20 -1.381 1.10 -1.390 1.15 -1.378 1.25 -1.378 1.11 -1.378 1.20 -1.378 1.30 -1.375 1.12 -1.366 1.25 -1.378 1.35 -1.372 1.13 -1.354 1.30 -1.378 1.40 -1.369 1.14 -1.342 1.35 -1.378 1.45 -1.365 1.15 -1.330 1.40 -1.378 1.50 -1.362 1.16 -1.319 Sensitivity =  [(-1.375 + 1.381)/0.10]/ [1.378/1.25] = 0.05 Sensitivity =  [(-1.366 + 1.390)/0.02]/ [1.378/1.11] = 0.97 Sensitivity =  [(-1.378 + 1.378)/0.40]/ [1.378/1.20] = 0.00 Range = 0.03 Range = 0.12 Range = 0.00

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 mfrags23 0.02 0.02 Small fkefrags23 0.05 0.03 Small fkefrag1 0.15 0.08 Small vfrags23 0.08 0.10 Small vfrag1 0.06 0.10 Small Qfrags23 0.13 0.10 Small fI 0.97 0.12 Small Qfrag1 0.14 0.19 Small mbody 1.00 0.69 Medium PE 0.35 0.98 Medium Qcl 1.71 2.84 Large Negative effect on lurch (increases rearward velocity) mbullet -0.53 0.04 Small mfrag1 -0.09 0.04 Small Q -0.04 0.05 Small vbullet -1.76 0.14 Small vbulletafter -0.07 0.20 Small mcloud -0.79 2.23 Large

This ordered summary of sensitivities is virtually identical to the one for Lurch 4 Angular. Only a few terms have changed, and then only slightly. This is a consequence of the fact that the addition of 3-D movement of the body didn't change the solution perceptibly. As expected from that result, the new term, fkebody, has an extremely small sensitivity and effect—both 0.00. As before, the two dominant sensitivities remained Qcl and mcloud, both properties of the cloud.

Summary
The addition of three-dimensionality to the motion of the body changed virtually nothing in the solution or the sensitivities. The variables of the cloud remained by far the most important. This means that the 3-D motion of the body is not a significant part of the solution.