21. Snap/Lurch 6 Angularadd four time intervals

The model and its justification
Snap/Lurch 6 is the angular equivalent of Snap/Lurch 6, which added four time intervals to the simulation. It calculates the speed of the rearward lurch from the bullet, the cloud, and the large fragments, with 3-D motions of the cloud, the large fragments, and the body: It begins by interposing 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 former procedure and (4) solves for vbodyafter and vcloud. To simplify matters a bit, it sets vfrags23 = vfrag1 and fkefrags23 = fkefrag1. These time intervals leave the default solution about the same, however: -1.39 and 386 ft s-1 for lurch and cloud, respectively, versus -1.38 and 378 ft s-1 in Lurch 5 Angular. The sensitivities and effects also remain similar.

(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 PE = 300 ft-lb tdelay = 0.002 s fxcl = 0.59 (calc.) dtransit = 4 in dsnap = 2.2 in dfrag1 = 6 ft

### Solutions to simultaneous equations

vbodyafter = -1.39 ft s-1  vcloud = 386 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 = -4.09 KEbodyafter = 1 Ωcloud = 6.42 KEcloud = 697 Ωfrag1 = 0.89 KEfrag1 = 111 Ωfrags23 = 0.15 KEfrags23 = 41 PE = 300

Sensitivity analysis

Sensitivity analysis, Lurch 6 angular
(Standard conditions in boldface)

 mbullet vlurch mbody vlurch mcloud vlurch mhead vlurch vbullet vlurch 4.0 -1.378 156 -1.353 4.5 -1.385 1750 -1.306 157 -1.362 65 -1.821 5.0 -1.392 1760 -1.325 158 -1.370 70 -1.691 5.5 -1.399 1770 -1.343 159 -1.378 75 -1.578 0.1 -0.468 6.0 -1.404 1780 -1.360 160 -1.385 80 -1.479 0.2 -0.990 6.5 -1.405 1790 -1.376 161 -1.392 85 -1.392 0.3 -1.392 7.0 -1.392 1800 -1.392 162 -1.398 90 -1.314 0.4 -1.731 7.5 -1.336 1810 -1.407 163 -1.403 95 -1.245 0.5 -2.030 8.0 -1.097 1820 -1.421 164 -1.409 100 -1.182 0.6 -2.300 8.5 — 1830 -1.435 165 -1.414 105 -1.126 0.7 -2.550 9.0 — 1840 -1.449 166 -1.419 0.8 -2.782 9.5 — 1850 -1.463 10.0 — Sensitivity =  [(-1.398 + 1.385)/2]/ [1.392/161] = -0.75 Sensitivity =  [(-1.314 + 1.479)/10]/ [1.392/85] = 1.01 Sensitivity =  [(-1.731 + 0.990)/0.2]/ [1.392/0.3] = -0.80 Sensitivity =  [(-1.336 + 1.405)/1.0]/ [1.392/7.0] = 0.35 Sensitivity =  [(-1.407 + 1.376)/20]/ [1.392/1800] = -2.00 Range = 0.07 Range = 0.70 Range = 2.31 Range = 0.28 Range = 0.16

 vbulletafter vlurch PE vlurch Q vlurch Qcl vlurch mfrag1 vlurch 0 -1.819 50 -1.753 7 -1.378 20 -2.744 0.022 -1.367 0 -1.280 100 -1.685 8 -1.380 30 -2.568 0.023 -1.372 50 -1.315 150 -1.615 9 -1.383 40 -2.333 0.024 -1.377 100 -1.347 200 -1.543 10 -1.385 50 -2.050 0.025 -1.382 150 -1.373 250 -1.468 11 -1.388 60 -1.731 0.026 -1.387 200 -1.392 300 -1.392 12 -1.392 70 -1.392 0.027 -1.392 250 -1.398 350 -1.312 13 -1.395 80 -1.045 0.028 -1.396 300 -1.383 400 -1.229 14 -1.398 90 -0.704 0.029 -1.401 350 -1.321 450 -1.143 15 -1.402 100 -0.383 0.030 -1.406 400 -1.124 500 -1.053 16 -1.405 110 -0.092 0.031 -1.411 550 -0.958 17 -1.408 120 +0.161 0.032 -1.415 600 -0.857 Sensitivity =  [(-1.398 + 1.373)/100]/ [1.392/200] = -0.04 Sensitivity =  [(-1.312 + 1.468)/100]/ [1.392/300] = 0.34 Sensitivity =  [(-1.398 + 1.385)/4]/ [1.392/12] = -0.03 Sensitivity =  [(-1.045 + 1.731)/20]/ [1.392/70] = 1.72 Sensitivity =  [-1.396 + 1.387)/0.002]/ [1.392/0.027] = -0.09 Range = 0.16 Range = 0.96 Range = 0.03 Range = 2.90 Range = 0.05

 mfrags23 vlurch Qfrag1 vlurch Qfrags23 vlurch fkefrags vlurch fkebody vlurch 0.005 -1.398 1.00 -1.439 0.006 -1.397 20 -1.460 50 -1.435 1.05 -1.429 1.00 -1.392 0.007 -1.396 25 -1.447 55 -1.425 1.10 -1.420 1.05 -1.392 0.008 -1.394 30 -1.431 60 -1.415 1.15 -1.410 1.10 -1.392 0.009 -1.393 35 -1.413 65 -1.403 1.20 -1.401 1.15 -1.392 0.010 -1.392 40 -1.392 70 -1.392 1.25 -1.392 1.20 -1.392 0.011 -1.390 45 -1.368 75 -1.379 1.30 -1.382 1.25 -1.392 0.012 -1.389 50 -1.343 80 -1.367 1.35 -1.373 1.30 -1.391 0.013 -1.387 55 -1.316 85 -1.354 1.40 -1.363 1.35 -1.391 0.015 -1.386 60 -1.287 90 -1.342 1.45 -1.353 1.40 -1.391 0.015 -1.384 1.50 -1.344 Sensitivity =  [(-1.389 + 1.394)/0.004]/ [1.392/0.010] = 0.01 Sensitivity =  [(-1.368 + 1.413)/10]/ [1.392/40] = 0.13 Sensitivity =  [(-1.379 + 1.403)/10]/ [1.392/70] = 0.12 Sensitivity =  [(-1.382 + 1.401)/0.10]/ [1.392/1.25] = 0.17 Sensitivity =  [(-1.391 + 1.392)/0.40]/ [1.392/1.20] = 0.00 Range = 0.01 Range = 0.17 Range = 0.09 Range = 0.10 Range = 0.00

 fI vlurch dtransit vlurch dsnap vlurch dfrag1 vlurch tdelay vlurch 1.06 -1.457 1.07 -1.444 0 -1.394 2 -1.359 1.08 -1.430 1 -1.394 1.6 -1.393 3 -1.387 1.09 -1.417 2 -1.393 1.8 -1.401 4 -1.402 0.000 -1.403 1.10 -1.404 3 -1.392 2.0 -1.405 5 -1.404 0.001 -1.399 1.11 -1.392 4 -1.392 2.2 -1.392 6 -1.392 0.002 -1.392 1.12 -1.379 5 -1.391 2.4 -1.301 7 -1.363 0.003 -1.379 1.13 -1.367 6 -1.390 2.6 -0.536 8 -1.316 0.004 -1.360 1.14 -1.355 7 -1.389 2.8 — 9 -1.248 0.005 -1.328 1.15 -1.343 8 -1.388 10 -1.153 0.006 -1.275 1.16 -1.332 0.007 -1.184 Sensitivity =  [(-1.379 + 1.404)/0.02]/ [1.392/1.11] = 1.00 Sensitivity =  [(-1.388 + 1.394)/8]/ [1.392/4] = 0.00 Sensitivity =  [(-1.301 + 1.405)/0.4]/ [1.392/2.2] = 0.41 Sensitivity =  [(-1.363 + 1.404)/2]/ [1.392/6] = 0.09 Sensitivity =  [(-1.379 + 1.399)/0.002]/ [1.392/0.002] = 0.01 Range = 0.12 Range = 0.01 Range = 0.86 Range = 0.21 Range = 0.22

 Rbullet vlurch Rhead vlurch 4.75 — 3.50 -1.398 5.00 -1.042 3.75 -1.402 5.25 -1.292 4.00 -1.405 5.50 -1.365 4.25 -1.403 5.75 -1.392 4.50 -1.392 6.00 -1.402 4.75 -1.355 6.25 -1.405 5.00 -1.249 6.50 -1.405 5.25 -0.856 6.75 -1.404 5.50 — Sensitivity =  [(-1.402 + 1.365)/0.50]/ [1.392/5.75] = -0.31 Sensitivity =  [(-1.355 + 1.403)/0.50]/ [1.392/4.50] = 0.31 Range = 0.45 Range = 0.54

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 0.01 0.01 Small dtransit 0.00 0.01 Small Qfrags23 0.12 0.09 Small fkefrag1 0.17 0.10 Small fI 1.00 0.12 Small Qfrag1 0.13 0.17 Small dfrag1 0.09 0.21 Small tdelay 0.01 0.22 Small mhead 0.35 0.28 Small Rhead 0.31 0.54 Medium mbody 1.01 0.70 Medium dsnap 0.41 0.86 Medium PE 0.34 0.96 Medium Qcl 1.72 2.90 Large Negative effect on lurch (increases rearward velocity) Q -0.03 0.03 Small mfrag1 -0.09 0.05 Small mbullet -0.75 0.07 Small vbullet -2.00 0.16 Small vbulletafter -0.04 0.16 Small Rbullet -0.31 0.54 Medium mcloud -0.80 2.31 Large

Just as in the linear case, the sensitivities of Lurch 6 Angular are very similar to those of Lurch 5 Angular in spite of the differences in calculation for Lurch 6. The previous variables maintained very similar sensitivities and ranges, with Qcl and mcloud continuing to dominate. Six of the eight new variables related to the timing and the snap had positive sensitivities, that is, they worked to increase the velocity of the lurch (to reduce its rearward velocity). Five of the eight had small sensitivities, with only Rhead, Rbullet, and dsnap having medium sensitivities. The lurch hardly changed, and if anything, increased in rearward speed (from -1.38 ft s-1 to -1.39 ft s-1). This tiny effect is probably because the sensitivities and effects of the new variables were so small. Thus, adding the four time intervals did not change the basic result.

Summary
The basic solution remained virtually unchanged when the four time intervals were added: The lurch of -1.38 ft s-1 in Lurch 5 Angular increased (in magnitude) only to -1.39 ft s-1 in Snap/Lurch 6 Angular. This important result was probably a consequence of the small sensitivities and effects of the eight new variables.