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 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

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.

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Back to Lurch 5 Angular

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