15. Snap/Lurch 7 Linearsolve for lurch and PE

The model and its justification
   
This simulation is very similar to Snap/Lurch 6. It differs only in solving for PE rather than vcloud. It focuses on PE because it is so important in setting constraints for other variables. 
    As before, this procedure calculates the speed of the rearward lurch of JFK's upper body from the bullet, the body, the cloud, and the large fragments, and incorporates 3-D motions of the cloud, the large fragments, and the body: Before doing that, however, it interposes 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 procedure of Lurch 5 Linear and (4) solves for vbodyafter and vcloud. To simplify matters a bit, it sets vfrags23 = vfrag1 and fkefrags23 = fkefrag1.
    This minor change in variables changes the default solution considerably. It gives -0.62 ft s-1 for the lurch (a 40% contraction) and 633 ft s-1 for the PE (double the previous default value). The sensitivities and effects also change considerably. It appears that the default variables are an improper combination, and consequently that this default solution should not be regarded seriously. That does not represent a problem, however, for the main purpose of Snap/Lurch 7 is for use in setting the constraints that are discussed in later sections.

(1) Equations for tsnap

 

 

(2) Equations for tlurch

 

 

(3) Equations for vfrags

 

 

(4) Simultaneous equations for vbodyafter and vcloud

fkefrags23 = fkefrag1

Conservation of 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
vcloud = 300 ft s-1 tdelay = 0.002 s fxcl = 0.59 (calc.)
dtransit = 4 in dsnap = 2.2 in dfrag1 = 6 ft

Solutions to simultaneous equations

vbodyafter = -0.62 ft s-1  PE = 633 ft-lb
vfrag1
= vfrags23 = 363 ft s-1
vsnap
= 5.1 ft s-1

ttransit
= 0.33 ms
tsnap
= 35.8 ms
tlurch
= 16.5 ms

Distributions of momentum and energy

Momentum Energy
Before After Before After
Pbullet = 1.26 Pbulletafter = 0.14 KEbullet = 1164 KEbulletafter = 14
  Pbodyafter = -0.82   KEbodyafter = 0.2
  Pcloud = 1.66   KEcloud = 422
  Pfrag1 = 0.24   KEfrag1 = 70
  Pfrags23 = 0.04   KEfrags23 = 26
      PE = 633

Sensitivity analysis
    

Sensitivity analysis, Lurch 7 Linear
(Standard conditions in boldface)

mbullet

vlurch

mbody

vlurch

mcloud

vlurch

mhead

vlurch

vbullet

vlurch

 

 

 

 

 

 

4.0

-0.517

 

 

156

-0.658

 

 

 

 

4.5

-0.526

1750

-0.659

157

-0.650

65

-0.807

 

 

5.0

-0.538

1760

-0.650

158

-0.641

70

-0.749

 

 

5.5

-0.551

1770

-0.642

159

-0.633

75

-0.699

0.1

+0.221

6.0

-0.568

1780

-0.633

160

-0.625

80

-0.655

0.2

-0.197

6.5

-0.589

1790

-0.625

161

-0.617

85

-0.617

0.3

-0.617

7.0

-0.617

1800

-0.617

162

-0.609

90

-0.582

0.4

-1.037

7.5

-0.655

1810

-0.609

163

-0.601

95

-0.552

0.5

-1.459

8.0

-0.709

1820

-0.601

164

-0.593

100

-0.524

0.6

-1.881

8.5

-0.796

1830

-0.593

165

-0.585

105

-0.499

0.7

-2.305

9.0

-0.952

1840

-0.585

166

-0.578

 

 

0.8

-2.729

9.5

-1.322

1850

-0.577

 

 

 

 

 

 

10.0

-3.275

 

 

Sensitivity = 
[(-0.609 + 0.625)/2]/
[0.617/161] = 2.09

Sensitivity = 
[(-0.582 + 0.655)/10]/
[0.617/85] = 1.01

Sensitivity = 
[(-1.037 + 0.197)/0.2]/
[0.617/0.3] = -2.04

Sensitivity = 
[(-0.655 + 0.589)/1.0]/
[0.617/7.0] = -0.75

Sensitivity = 
[(-0.609 + 0.625)/20]/
[0.617/1800] = 2.33

Range = 0.08

Range = 0.31

Range = 2.95

Range = 2.76

Range = 0.08

vbulletafter

vlurch

vcloud

vlurch

Q

vlurch

Qcl

vlurch

mfrag1

vlurch

 

 

0

+0.641

 

 

 

 

 

 

 

 

50

+0.431

7

-0.596

20

-1.400

0.022

-0.584

0

-0.467

100

+0.222

8

-0.599

30

-1.298

0.023

-0.590

50

-0.502

150

+0.012

9

-0.603

40

-1.161

0.024

-0.597

100

-0.539

200

-0.197

10

-0.607

50

-0.997

0.025

-0.604

150

-0.577

250

-0.407

11

-0.612

60

-0.813

0.026

-0.610

200

-0.617

300

-0.617

12

-0.617

70

-0.617

0.027

-0.617

250

-0.660

350

-0.826

13

-0.622

80

-0.417

0.028

-0.623

300

-0.707

400

-1.036

14

-0.629

90

-0.221

0.029

-0.630

350

-0.760

450

-1.246

15

-0.635

100

-0.036

0.030

-0.636

400

-0.821

500

-1.455

16

-0.642

110

+0.132

0.031

-0.643

 

 

550

-1.665

17

-0.650

120

+0.278

0.032

-0.650

 

 

600

-1.874

 

 

 

 

 

 

Sensitivity = 
[(-0.660 + 0.577)/100]/
[0.617/200] = -0.27

Sensitivity = 
[(-0.826 + 0.407)/100]/
[0.617/300] = -2.04

Sensitivity = 
[(-0.629 + 0.607)/4]/
[0.617/12] = -0.11

Sensitivity = 
[(-0.417 + 0.813)/20]/
[0.617/70] = 2.25

Sensitivity = 
[-0.623 + 0.610)/0.002]/
[0.617/0.027] = -0.28

Range = 0.35

Range = 2.52

Range = 0.05

Range = 1.68

Range = 0.07

mfrags23

vlurch

Qfrag1

vlurch

Qfrags23

vlurch

fkefrags

vlurch

fkebody

vlurch

0.005

-0.602

 

 

 

 

1.00

-0.617

 

 

0.006

-0.605

20

-0.657

50

-0.642

1.05

-0.617

1.00

-0.617

0.007

-0.608

25

-0.649

55

-0.637

1.10

-0.617

1.05

-0.617

0.008

-0.611

30

-0.640

60

-0.630

1.15

-0.617

1.10

-0.617

0.009

-0.614

35

-0.629

65

-0.624

1.20

-0.617

1.15

-0.617

0.010

-0.617

40

-0.617

70

-0.617

1.25

-0.617

1.20

-0.617

0.011

-0.620

45

-0.603

75

-0.610

1.30

-0.617

1.25

-0.617

0.012

-0.623

50

-0.588

80

-0.602

1.35

-0.617

1.30

-0.617

0.013

-0.626

55

-0.572

85

-0.595

1.40

-0.617

1.35

-0.617

0.015

-0.628

60

-0.555

90

-0.587

1.45

-0.617

1.40

-0.617

0.015

-0.631

 

 

 

 

1.50

-0.617

 

 

Sensitivity = 
[(-0.620 + 0.614)/0.002]/
[0.617/0.010] = -0.05

Sensitivity = 
[(-0.603 + 0.629)/10]/
[0.617/40] = 0.17

Sensitivity = 
[(-0.610 + 0.624)/10]/
[0.617/70] = 0.16

Sensitivity = 
[(-0.617 + 0.617)/0.50]/
[0.617/1.25] = 0.00

Sensitivity = 
[(-0.617 + 0.617)/0.40]/
[0.617/1.20] = 0.00

Range = 0.03

Range = 0.10

Range = 0.06

Range = 0.00

Range = 0.00

dtransit

vlurch

dsnap

vlurch

dfrag1

vlurch

tdelay

vlurch

0

-0.613

 

 

2

-0.479

 

 

1

-0.614

1.6

-0.540

3

-0.513

 

 

2

-0.615

1.8

-0.558

4

-0.548

0.000

-0.594

3

-0.616

2.0

-0.583

5

-0.582

0.001

-0.605

4

-0.617

2.2

-0.617

6

-0.617

0.002

-0.617

5

-0.618

2.4

-0.667

7

-0.651

0.003

-0.630

6

-0.619

2.6

-0.751

8

-0.686

0.004

-0.645

7

-0.620

2.8

-0.915

9

-0.720

0.005

-0.663

8

-0.621

 

 

10

-0.755

0.006

-0.683

 

 

 

 

 

 

0.007

-0.707

Sensitivity = 
[(-0.621 + 0.613)/8]/
[0.617/4] = -0.01

Sensitivity = 
[(-0.667 + 0.583)/0.4]/
[0.617/2.2] = -0.75

Sensitivity = 
[(-0.651 + 0.582)/2]/
[0.617/6] = -0.34

Sensitivity = 
[(-0.630 + 0.605)/0.002]/
[0.617/0.002] = -0.04

Range = 0.01

Range = 0.38

Range = 0.28

Range = 0.11

 

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
fkefrags 0.00 0.00 Small
Qfrags23 0.16 0.06 Small
mbullet 2.09 0.08 Small
vbullet 2.33 0.08 Small
Qfrag1  0.17 0.10 Small
mbody 1.01 0.31 Small
Qcl 2.25 1.68 Large
Negative effect on lurch (increases rearward velocity)
dtransit -0.01 0.01 Small
mfrags23 -0.05 0.03 Small
Q  -0.11 0.05 Small
mfrag1 -0.28 0.07 Small
tdelay -0.04 0.11 Small
dfrag1 -0.34 0.28 Small
vbulletafter -0.27 0.35 Small
dsnap -0.75 0.38 Small
mhead -0.75 2.76 Large
vcloud -2.04 2.52 Large
mcloud -2.04 2.95 Large

    The biggest change in this ordered summary of sensitivities is the shift toward negative effects. As opposed to Snap/Lurch 6, which had 12 positive effects and 7 negative, Snap/Lurch 7 has 8 positive and 11 negative effects, for a net shift of four effects from positive to negative. One might then expect the rearward lurch to become stronger, but the opposite happenedit decreased in intensity from -1.04 ft s-1 to -0.62 ft s-1, a loss of 40%. As before, however, there was a great dichotomy between the few strong terms (four, with effects between 1.68 ft s-1 and 2.95 ft s-1) and the many small terms (14, with effects between 0.00 ft s-1 and 0.38 ft s-1). Of the four strong terms, three had to do with the cloud (mcloud, vcloud, and Qcl) and one with the head (mhead). This reinforces the all-importance of the cloud to the final solution.

Summary
    Solving for PE instead of vcloud created four more negative effects but reduced the strength of the negative lurch from -1.04 ft s-1 to -0.62 ft s-1. Of the four large effects, three were properties of the cloud, one the mass of the head.

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