22. Snap/Lurch 7 Angularsolve for lurch and PE

The model and its justification
   
This simulation differs from Snap/Lurch 6 Angular only in that it solves 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, however. It gives -0.91 ft s-1 for the lurch (a 35% contraction) and 576 ft-lb for the PE (double the previous default value). The sensitivities and effects also change considerably. As with Snap/Lurch 7 Linear, it appears that the default values of the 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 = -0.91 ft s-1  PE = 576 ft-lb
vfrag1
= vfrags23 = 458 ft s-1
vsnap
= 4.7 ft s-1

ttransit
= 0.33 ms
tsnap
= 39.2 ms
tlurch
= 13.1 ms

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 = -2.66

 

KEbodyafter = 0.5

 

Ωcloud = 4.99

 

KEcloud = 422

 

Ωfrag1 = 0.89

 

KEfrag1 = 111

 

Ωfrags23 = 0.15

 

KEfrags23 = 41

 

 

 

PE = 300

 

Sensitivity analysis
    

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

mbullet

vlurch

mbody

vlurch

mcloud

vlurch

mhead

vlurch

vbullet

vlurch

 

 

 

 

 

 

4.0

-0.708

 

 

156

-0.979

 

 

 

 

4.5

-0.724

1750

-0.980

157

-0.964

65

-1.187

 

 

5.0

-0.744

1760

-0.965

158

-0.949

70

-1.102

 

 

5.5

-0.769

1770

-0.949

159

-0.934

75

-1.028

0.1

+0.226

6.0

-0.801

1780

-0.935

160

-0.920

80

-0.963

0.2

-0.339

6.5

-0.844

1790

-0.920

161

-0.906

85

-0.906

0.3

-0.906

7.0

-0.906

1800

-0.906

162

-0.893

90

-0.856

0.4

-1.475

7.5

-1.002

1810

-0.893

163

-0.880

95

-0.811

0.5

-2.044

8.0

-1.169

1820

-0.879

164

-0.867

100

-0.770

0.6

-2.615

8.5

-1.536

1830

-0.866

165

-0.854

105

-0.733

0.7

-3.188

9.0

-2.980

1840

-0.853

166

-0.842

 

 

0.8

-3.761

9.5

+4.604

1850

-0.841

 

 

 

 

 

 

10.0

+0.696

 

 

Sensitivity = 
[(-0.893 + 0.920)/2]/
[0.906/161] = 0.00

Sensitivity = 
[(-0.856 + 0.963)/10]/
[0.906/85] = 1.00

Sensitivity = 
[(-1.475 + 0.339)/0.2]/
[0.906/0.3] = -1.88

Sensitivity = 
[(-1.002 + 0.844)/1.0]/
[0.906/7.0] = -1.22

Sensitivity = 
[(-0.893 + 0.920)/20]/
[0.906/1800] = 2.68

Range = 0.14

Range = 0.45

Range = 3.99

Range = 1.40

Range = 0.14

 

vbulletafter

vlurch

vcloud

vlurch

Q

vlurch

Qcl

vlurch

mfrag1

vlurch

 

 

0

+0.793

 

 

 

 

 

 

 

 

50

+0.510

7

-0.871

20

-1.965

0.022

-0.850

0

-0.671

100

+0.227

8

-0.876

30

-1.827

0.023

-0.861

50

-0.723

150

-0.057

9

-0.883

40

-1.642

0.024

-0.873

100

-0.779

200

-0.340

10

-0.890

50

-1.421

0.025

-0.884

150

-0.840

250

-0.623

11

-0.898

60

-1.172

0.026

-0.895

200

-0.906

300

-0.906

12

-0.906

70

-0.906

0.027

-0.906

250

-0.982

350

-1.190

13

-0.916

80

-0.636

0.028

-0.918

300

-1.070

400

-1.473

14

-0.927

90

-0.371

0.029

-0.929

350

-1.180

450

-1.756

15

-0.938

100

-0.121

0.030

-0.940

400

-1.327

500

-2.039

16

-0.951

110

+0.105

0.031

-0.951

 

 

550

-2.323

17

-0.965

120

+0.302

0.032

-0.962

 

 

600

-2.606

 

 

 

 

 

 

Sensitivity = 
[(-0.982 + 0.840)/100]/
[0.906/200] = -0.31

Sensitivity = 
[(-1.190 + 0.623)/100]/
[0.906/300] = -1.88

Sensitivity = 
[(-0.927 + 0.890)/4]/
[0.906/12] = -0.12

Sensitivity = 
[(-0.636 + 1.172)/20]/
[0.906/70] = 2.07

Sensitivity = 
[-0.918 + 0.895)/0.002]/
[0.906/0.027] = -0.34

Range = 0.66

Range = 3.40

Range = 0.09

Range = 2.27

Range = 0.11

mfrags23

vlurch

Qfrag1

vlurch

Qfrags23

vlurch

fkefrags

vlurch

fkebody

vlurch

0.005

-0.881

 

 

 

 

1.00

-0.906

 

 

0.006

-0.886

20

-0.975

50

-0.950

1.05

-0.906

1.00

-0.906

0.007

-0.891

25

-0.962

55

-0.940

1.10

-0.906

1.05

-0.906

0.008

-0.896

30

-0.946

60

-0.929

1.15

-0.906

1.10

-0.906

0.009

-0.901

35

-0.927

65

-0.918

1.20

-0.906

1.15

-0.906

0.010

-0.906

40

-0.906

70

-0.906

1.25

-0.906

1.20

-0.906

0.011

-0.911

45

-0.883

75

-0.894

1.30

-0.906

1.25

-0.906

0.012

-0.916

50

-0.858

80

-0.882

1.35

-0.906

1.30

-0.906

0.013

-0.921

55

-0.830

85

-0.869

1.40

-0.906

1.35

-0.906

0.015

-0.926

60

-0.801

90

-0.856

1.45

-0.906

1.40

-0.906

0.015

-0.931

 

 

 

 

1.50

-0.906

 

 

Sensitivity = 
[(-0.911 + 0.901)/0.002]/
[0.906/0.010] = -0.06

Sensitivity = 
[(-0.883 + 0.927)/10]/
[0.906/40] = 0.19

Sensitivity = 
[(-0.894 + 0.918)/10]/
[0.906/70] = 0.18

Sensitivity = 
[(-0.906 + 0.906)/0.50]/
|[0.906/1.25] = 0.00

Sensitivity = 
[(-0.906 + 0.906)/0.40]/
[0.906/1.20] = 0.00

Range = 0.05

Range = 0.17

Range = 0.09

Range = 0.00

Range = 0.00

fI

vlurch

dtransit

vlurch

dsnap

vlurch

dfrag1

vlurch

tdelay

vlurch

1.06

-0.949

 

 

 

 

 

 

 

 

1.07

-0.940

0

-0.898

 

 

2

-0.671

 

 

1.08

-0.932

1

-0.900

1.6

-0.748

3

-0.730

 

 

1.09

-0.923

2

-0.902

1.8

-0.782

4

-0.789

0.000

-0.860

1.10

-0.915

3

-0.904

2.0

-0.831

5

-0.848

0.001

-0.881

1.11

-0.906

4

-0.906

2.2

-0.906

6

-0.906

0.002

-0.906

1.12

-0.898

5

-0.909

2.4

-1.038

7

-0.965

0.003

-0.936

1.13

-0.890

6

-0.911

2.6

-1.327

8

-1.024

0.004

-0.970

1.14

-0.883

7

-0.913

2.8

-2.468

9

-1.083

0.005

-1.011

1.15

-0.875

8

-0.916

 

 

10

-1.141

0.006

-1.061

1.16

-0.867

 

 

 

 

 

 

0.007

-1.124

Sensitivity = 
[(-0.898 + 0.915)/0.02]/
[0.906/1.11] = 1.04

Sensitivity = 
[(-0.916 + 0.898)/8]/
[0.906/4] = -0.01

Sensitivity = 
[(-1.038 + 0.831)/0.4]/
[0.906/2.2] = -1.26

Sensitivity = 
[(-0.965 + 0.848)/2]/
[0.906/6] = -0.39

Sensitivity = 
[(-0.936 + 0.881)/0.002]/
[0.906/0.002] = -0.06

Range = 0.08

Range = 0.02

Range = 3.22

Range = 0.47

Range = 0.26

Rbullet

vlurch

Rhead

vlurch

4.75

-1.506

3.50

-0.766

5.00

-1.193

3.75

-0.789

5.25

-1.047

4.00

-0.818

5.50

-0.962

4.25

-0.856

5.75

-0.906

4.50

-0.906

6.00

-0.867

4.75

-0.977

6.25

-0.838

5.00

-1.082

6.50

-0.816

5.25

-1.257

6.75

-0.798

5.50

-1.605

Sensitivity = 
[(-0.867 + 0.962)/0.50]/
[0.906/5.75] = 1.21

Sensitivity = 
[(-0.977 + 0.856)/0.50]/
[0.906/4.50] = -1.20

Range = 0.71

Range = 0.84

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
fkefrag1 0.00 0.00 Small
fI 1.04 0.08 Small
 Qfrags23 0.18 0.09 Small
mbullet 0.00 0.14 Small
vbullet 2.16 0.14 Small
Qfrag1  0.19 0.17 Small
mbody 1.00 0.45 Small
Rbullet 1.21 0.71 Medium
Qcl 2.07 2.27 Large
Negative effect on lurch (increases rearward velocity)
dtransit -0.01 0.02 Small
mfrags23 -0.06 0.05 Small
Q  -0.12 0.09 Small
mfrag1 -0.34 0.11 Small
tdelay -0.06 0.26 Small
dfrag1 -0.39 0.47 Small
vbulletafter -0.31 0.66 Medium
Rhead -1.20 0.84 Medium
mhead -1.22 1.40 Large
dsnap -1.26 3.22 Large
vcloud -1.88 3.40 Large
mcloud -1.88 3.99 Large

    These sensitivities and effects are very similar to those of the linear analog Snap/Lurch 7 Linear. They differ from Snap/Lurch 6 in that they are systematically shifted toward negative effects. As opposed to Snap/Lurch 6, which had 15 positive effects and 7 negative sensitivities, Snap/Lurch 7 has 10 positive and 12 negative effects, for a net shift of five effects from positive to negative. One might then expect the rearward lurch to become stronger, but the opposite happenedit decreased in intensity from -1.39 ft s-1 to -0.91 ft s-1, a loss of 35%. As before, however, there was a great dichotomy between the few strong terms (five, with effects between 1.40 ft s-1 and 3.99 ft s-1) and the many small and medium terms (17, with effects between 0.00 ft s-1 and 0.84 ft s-1). Of the five strong terms, three had to do with the cloud (mcloud, vcloud, and Qcl) and two with the snap (mhead and dsnap). This reinforces the importance of the cloud to the final solution.

Summary
    Solving for PE instead of vcloud created five more negative effects but reduced the strength of the negative lurch from -1.39 ft s-1 to -0.91 ft s-1. Of the five large effects, three were properties of the cloud, two of the head and snap.

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