19. Lurch 4 Angularadd 3-D motion of large fragments

The model and its justification
    Lurch 4 Angular is the rotational analog of Lurch 4 Linear. As with Lurch 3 Angular, it calculates the speed of the rearward lurch from the bullet, the body, the conical cloud, large fragments, and the large fragments. To this it adds 3-D motion of the large fragments. This addition barely changes the results, with the rearward lurch decreasing in strength from 1.44 ft s-1 to 1.38 ft s-1. This change amounts to only 4%, and means that the 3-D motion of the large fragments is an unimportant part of the solution. A similar result was found for the linear calculations.

Solving the simultaneous equations
    

 

Conservation of angular X-momentum

 

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 = 17

Qfrags23 = 70

vfrags23 = 500 ft s-1

PE = 200 ft-lb

Qcl = 70

fkefrag1 = 1.25

fI = 1.11

 

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 4 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

-1.310

157

-1.361

65

-1.804

 

 

100

-1.677

1760

-1.324

158

-1.365

70

-1.675

 

 

150

-1.606

1770

-1.337

159

-1.370

75

-1.563

0.1

-0.474

200

-1.532

1780

-1.351

160

-1.374

80

-1.465

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

-2.269

450

-1.124

1830

-1.418

165

-1.395

105

-1.115

0.7

-2.513

500

-1.031

1840

-1.432

166

-1.399

 

 

0.8

-2.742

550

-0.933

1850

-1.445

 

 

 

 

 

 

600

-0.828

 

 

Sensitivity = 
[(-1.383 + 1.374)/2]/
[1.378/161] = -0.53

Sensitivity = 
[(-1.301 + 1.465)/10]/
[1.378/85] = 1.01

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 = 2.27

Range = 0.98

Range = 0.14

vbulletafter

vlurch

Q

vlurch

Qcl

vlurch

mfrag1

vlurch

mfrags23

vlurch

 

 

7

-1.359

20

-2.704

0.022

-1.356

0.005

-1.390

0

-1.255

8

-1.362

30

-2.531

0.023

-1.360

0.006

-1.387

50

-1.290

9

-1.366

40

-2.301

0.024

-1.365

0.007

-1.385

100

-1.322

10

-1.369

50

-2.023

0.025

-1.369

0.008

-1.383

150

-1.352

11

-1.374

60

-1.701

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

-1.423

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

-1.455

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 + 1.352)/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 = 
[-1.383 + 1.374)/0.002]/
[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 = 2.85

Range = 0.04

Range = 0.02

vfrag1

vlurch

vfrags23

vlurch

Qfrag1

vlurch

Qfrags23

vlurch

fkefrag1

vlurch

250

-1.366

250

-1.409

 

 

 

 

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

-1.387

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

-1.306

 

 

 

 

1.50

-1.336

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

1.00

-1.394

1.06

-1.443

1.05

-1.391

1.07

-1.430

1.10

-1.388

1.08

-1.417

1.15

-1.384

1.09

-1.404

1.20

-1.381

1.10

-1.391

1.25

-1.378

1.11

-1.378

1.30

-1.375

1.12

-1.366

1.35

-1.372

1.13

-1.354

1.40

-1.369

1.14

-1.342

1.45

-1.366

1.15

-1.330

1.50

-1.363

1.16

-1.319

Sensitivity = 
[(-1.375 + 1.381)/0.10]/
[1.378/1.25] = 0.05

Sensitivity = 
[(-1.366 + 1.391)/0.02]/
[1.378/1.11] = 1.01

Range = 0.03

Range = 0.12

Ordered summary of sensitivities

Variable Sensitivity of vlurch Range of vlurch, ft s-1 Magnitude
Positive effect on lurch (reduces rearward velocity)
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 1.01 0.12 Small
Qfrag1  0.14 0.19 Small
mbody 1.01 0.69 Medium
PE 0.35 0.98 Medium
Qcl  1.71 2.85 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 2.27 Large

    As expected from the fact that the addition of three-dimensionality to the large fragments didn't change the lurch perceptibly, the basic pattern of sensitivities remained nearly unchanged, although the minor terms mfrags23 and vfrag1 moved from negative to positive. The two new parameters, fkefrag1 and fkefrags23, also had extremely small sensitivities. The two dominant sensitivities remained Qcl and mcloud, both properties of the cloud.

Summary
    The addition of three-dimensionality to the motion of the large fragments changed the solution by only a few percent, and left the basic pattern of sensitivities unchanged. The variables of the cloud remained by far the most important.

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