10. Lurch 2 linearadd large fragments

The model and its justification
    The next step in making the simulation more realistic is to add the two large fragments to it. These are the while lines seen in Z313 and to a small extent in Z314. Their masses can be estimated from the pictures in Lattimer's book and the density of bone. They come out to be 0.027 gr for fragment 1 (with the upward angle of 40 and 0.01 gr for the one with an angle of 70 (here called frags23 because it may include a smaller third fragment as well). We use a default velocity for each fragment of 500 ft s-1, as measured on Z313 (keeping in mind that the path represents less than a full frame's exposure). These velocities turn out to be unimportant.

Solving the simultaneous equations
    The equations to be solved simultaneously are shown below. Note that the mass of the lurching body has been reduced by the masses of the fragments. The starting values of the variables are shown below the equations.
    The solutions for vbodyafter and vcloud, -2.18 ft s-1 and 388 ft s-1, respectively, are shown below the variables. Both are very similar to the values from the first linear simulation, -2.17 and 425 ft s-1.

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  vfrag1 = 500 ft s-1
Q = 12    Qfrags23 = 70   vfrags23 = 500 ft s-1
PE = 300 ft-lb     

Default solutions to simultaneous equations

vbodyafter = -2.18 ft s-1  vcloud = 388 ft s-1

Distributions of momentum and energy

Momentum

Energy, ft-lb

Before

After Before After
Pbullet = 1.26 Pbulletafter = 0.14  KEbullet = 1164  KEbulletafter = 14
  Pbodyafter = -2.89    KEbodyafter = 2
  Pcloud = 3.63   KEcloud = 704
  Pfrag1 = 0.32   KEfrag1 = 105
  Pfrags23 = 0.05    KEfrags23 = 39
      PE = 300

    The table of distributions is very similar to the one without the large fragments, particularly for the momentum. The fragments removed 144 ft-lb of KE from the cloud, however (17% of its value). But the lurch changed by only 1%. Thus adding the fragments didn't change the basic solution very much.

Sensitivity analysis
    Although the sensitivity analysis looks different from the previous one, its essentials are the same. None of the six new variables (two masses, two speeds, two angles) have important sensitivities or ranges.

Sensitivity analysis, Lurch 2 linear
(Standard conditions in boldface)

mbullet

vlurch

mbody

vlurch

mcloud

vlurch

PE

vlurch

vbullet

vlurch

 

 

 

 

 

 

0

-2.715

 

 

156

-2.139

 

 

 

 

50

-2.632

1750

-2.082

157

-2.148

65

-2.857

 

 

100

-2.548

1760

-2.103

158

-2.157

70

-2.652

 

 

150

-2.461

1770

-2.123

159

-2.166

75

-2.475

0.1

-1.021

100

-2.371

1780

-2.143

160

-2.174

80

-2.320

0.2

-1.678

250

-2.279

1790

-2.163

161

-2.183

85

-2.183

0.3

-2.183

300

-2.183

1800

-2.183

162

-2.192

90

-2.061

0.4

-2.609

350

-2.084

1810

-2.203

163

-2.200

95

-1.953

0.5

-2.986

400

-1.981

1820

-2.223

164

-2.209

100

-1.855

0.6

-3.327

450

-1.874

1830

-2.242

165

-2.217

105

-1.766

0.7

-3.641

500

-1.762

1840

-2.262

166

-2.225

 

 

0.8

-3.933

550

-1.644

1850

-2.281

 

 

 

 

 

 

600

-1.519

 

 

Sensitivity = 
[(-2.192 + 2.174)/2]/
[2.183/161] = -0.66

Sensitivity = 
[(-2.061 + 2.320)/10]/
[2.183/85] = 1.01

Sensitivity = 
[(-2.609 + 1.678)/0.2]/
[2.183/0.3] = -0.64

Sensitivity = 
[(-2.084 + 2.279)/100]/
[2.183/300] = 0.27

Sensitivity = 
[(-2.203 + 2.163)/20]/
[2.183/1800] = -1.65

Range = 0.09

Range = 1.09

Range = 2.91

Range = 1.20

Range = 0.20

vbulletafter

vlurch

Q

vlurch

mfrag1

vlurch

mfrags23

vlurch

vfrag1

vlurch

 

 

7

-2.169

0.022

-2.175

0.005

-2.200

250

-2.211

0

-2.102

8

-2.171

0.023

-2.177

0.006

-2.197

300

-2.214

50

-2.128

9

-2.174

0.024

-2.179

0.007

-2.194

350

-2.213

100

-2.150

10

-2.176

0.025

-2.180

0.008

-2.190

400

-2.207

150

-2.168

11

-2.180

0.026

-2.182

0.009

-2.187

450

-2.197

200

-2.183

12

-2.183

0.027

-2.183

0.010

-2.183

500

-2.183

250

-2.194

13

-2.187

0.028

-2.184

0.011

-2.179

550

-2.164

300

-2.202

14

-2.191

0.029

-2.186

0.012

-2.176

600

-2.140

350

-2.206

15

-2.195

0.030

-2.187

0.013

-2.172

650

-2.111

400

-2.206

16

-2.199

0.031

-2.189

0.014

-2.169

700

-2.076

 

 

17

-2.204

0.032

-2.190

0.015

-2.165

750

-2.035

Sensitivity =
[(-2.194 + 2.168)/100]/
[2.183/200] = -0.02

Sensitivity =
[(-2.191 + 2.176)/4]/
[2.183/12] = -0.02

Sensitivity =
[(-2.186 + 2.180)/0.004]/
[2.183/0.027] = -0.02

Sensitivity =
[(-2.179 + 2.187)/0.002]/
[2.183/0.01] = 0.02

Sensitivity =
[(-2.164 + 2.197)/100]/
[2.183/500] = 0.08

Range = 0.10

Range = 0.04

Range = 0.02

Range = 0.04

Range = 0.18

vfrags23

vlurch

Qfrag1

vlurch

Qfrags23

vlurch

 

250

-2.219

 

 

 

 

 

300

-2.215

20

-2.238

50

-2.218

 

350

-2.209

25

-2.228

55

-2.210

 

400

-2.202

30

-2.215

60

-2.202

 

450

-2.193

35

-2.200

65

-2.192

 

500

-2.183

40

-2.183

70

-2.183

 

550

-2.171

45

-2.164

75

-2.173

 

600

-2.157

50

-2.144

80

-2.163

 

650

-2.142

55

-2.122

85

-2.153

 

700

-2.125

60

-2.098

90

-2.143

 

750

-2.106

 

 

 

 

 

Sensitivity = 
[(-2.171 + 2.193)/100]/
[2.183/500] = 0.05

Sensitivity = 
[(-2.164 + 2.200)/10]/
[2.183/40] = 0.07

Sensitivity = 
[(-2.173 + 2.192)/10]/
[2.183/70] = 0.06

Range = 0.11

Range = 0.14

Range = 0.08

 

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 0.04 Small
 Qfrags23 0.06 0.08 Small
vfrags23 0.05 0.11 Small
Qfrag1  0.07 0.14 Small
vfrag1 0.08 0.18 Small
mbody 1.01 1.09 Medium
PE 0.27 1.20 Medium
Negative effect on lurch (increases rearward velocity)
mfrag1 -0.02 0.02 Small
Q  -0.02 0.04 Small
mbullet -0.66 0.09 Small
vbulletafter -0.02 0.10 Small
vbullet -1.65 0.20 Small
mcloud -0.64 2.91 Large

    None of the six new variables (associated with the mass, velocity, and angle of inclination of the two large fragments) affect the speed of the lurch very much, as seen above. They all have low sensitivities and small ranges of effects. Five of them have positive effects, one negative. The effects of the previous variables remain much the same. In other words, the large fragments are not important to the solution, at least for these early calculations.

Summary
    The answer with the two large fragments is fundamentally the same as without them. They are thus minor variables. This is confirmed by the six sensitivities and effects associated with them, which are all very small. The next step is to add three-dimensional motions, beginning with the diffuse cloud of fragments.

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