17. Lurch 2 Angular—add large fragments

The model and its justification
    Lurch 2 Angular is the rotational analog of Lurch 2 Linear. It calculates the speed of the rearward lurch from the bullet, the body, the diffuse cloud of fragments, and the large fragments. Its result for the lurch, -2.94 ft s-1, is within 1% of Lurch 1 Angular (-2.92 ft s-1) and 35% greater than for Lurch 2 Linear (-2.18 ft s-1).

Solving the simultaneous equations
    As with Lurch 2 Linear, the large fragments are represented by six terms, two for masses, two for exit speeds, and two for exit angles. 

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

fI = 1.11

 

Solutions to simultaneous equations

vbodyafter = -2.94 ft s-1  vcloud = 387 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 = -8.64

 

KEbodyafter = 4

 

Ωcloud = 10.88

 

KEcloud = 701

 

Ωfrag1 = 0.97

 

KEfrag1 = 105

 

Ωfrags23 = 0.16

 

KEfrags23 = 39

 

 

 

PE = 300

    The angular momentum and energy are distributed similarly to the linear momentum and energy. The momentum is partitioned roughly equally between the forward movement of the cloud and the rearward movement of the body. By contrast, nearly all the kinetic energy goes to the cloud, except for the 300 units consumed in the potential energy of breaking the skull and scalp.

Sensitivity analysis
    The sensitivity analysis for Lurch 2 Angular is very similar to that for Lurch 2 Linear. Only three terms matter much, the mass of the body, the mass of the cloud, and the potential energy. The first and last of these are negatively related to the strength of the lurch, whereas the second is positively related. The latter (the mass of the cloud), is also the strongest.

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

mbullet

vlurch

mbody

vlurch

mcloud

vlurch

PE

vlurch

vbullet

vlurch

 

 

 

 

 

 

0

-3.660

 

 

156

-2.883

 

 

 

 

50

-3.549

1750

-2.807

157

-2.896

65

-3.849

 

 

100

-3.435

1760

-2.835

158

-2.908

70

-3.574

 

 

150

-3.317

1770

-2.862

159

-2.919

75

-3.335

0.1

-1.379

100

-3.196

1780

-2.889

160

-2.931

80

-3.127

0.2

-2.264

250

-3.072

1790

-2.916

161

-2.943

85

-2.943

0.3

-2.943

300

-2.943

1800

-2.943

162

-2.955

90

-2.779

0.4

-3.515

350

-2.809

1810

-2.970

163

-2.966

95

-2.633

0.5

-4.018

400

-2.671

1820

-2.996

164

-2.978

100

-2.501

0.6

-4.472

450

-2.526

1830

-3.023

165

-2.989

105

-2.382

0.7

-4.890

500

-2.375

1840

-3.049

166

-3.000

 

 

0.8

-5.278

550

-2.216

1850

-3.075

 

 

 

 

 

 

600

-2.048

 

 

Sensitivity = 
[(-2.955 + 2.931)/2]/
[2.943/161] = -0.66

Sensitivity = 
[(-2.779 + 3.127)/10]/
[2.943/85] = 1.00

Sensitivity = 
[(-3.515 + 2.264)/0.2]/
[2.943/0.3] = -0.64

Sensitivity = 
[(-2.809 + 3.072)/100]/
[2.943/300] = 0.27

Sensitivity = 
[(-2.970 + 2.916)/20]/
[2.943/1800] = -1.74

Range = 0.12

Range = 1.47

Range = 3.90

Range = 1.61

Range = 0.27

vbulletafter

vlurch

Q

vlurch

mfrag1

vlurch

mfrags23

vlurch

vfrag1

vlurch

 

 

7

-2.924

0.022

-2.933

0.005

-2.967

250

-2.981

0

-2.835

8

-2.927

0.023

-2.935

0.006

-2.962

300

-2.985

50

-2.869

9

-2.930

0.024

-2.937

0.007

-2.957

350

-2.983

100

-2.898

10

-2.934

0.025

-2.939

0.008

-2.952

400

-2.976

150

-2.923

11

-2.938

0.026

-2.941

0.009

-2.948

450

-2.962

200

-2.943

12

-2.943

0.027

-2.943

0.010

-2.943

500

-2.943

250

-2.958

13

-2.948

0.028

-2.945

0.011

-2.938

550

-2.917

300

-2.968

14

-2.953

0.029

-2.947

0.012

-2.933

600

-2.885

350

-2.974

15

-2.959

0.030

-2.949

0.013

-2.928

650

-2.845

400

-2.974

16

-2.965

0.031

-2.950

0.014

-2.924

700

-2.798

 

 

17

-2.972

0.032

-2.952

0.015

-2.919

750

-2.743

Sensitivity = 
[(-2.958 + 2.923)/100]/
[2.943/200] = -0.02

Sensitivity = 
[(-2.953 + 2.934)/4]/
[2.943/12] = -0.02

Sensitivity = 
[(-2.945 + 2.941)/0.002]/
[2.943/0.027] = -0.02

Sensitivity = 
[(-2.938 + 2.948)/0.002]/
[2.943/0.010] = 0.02

Sensitivity = 
[(-2.917 + 2.962)/100]/
[2.943/500] = 0.08

Range = 0.14

Range = 0.05

Range = 0.02

Range = 0.05

Range = 0.24

vfrags23

vlurch

Qfrag1

vlurch

Qfrags23

vlurch

fI

vlurch

250

-2.992

 

 

 

 

1.06

-3.081

300

-2.986

20

-3.017

50

-2.991

1.07

-3.053

350

-2.979

25

-3.003

55

-2.980

1.08

-3.024

400

-2.969

30

-2.986

60

-2.968

1.09

-2.997

450

-2.957

35

-2.966

65

-2.956

1.10

-2.970

500

-2.943

40

-2.943

70

-2.943

1.11

-2.943

550

-2.927

45

-2.918

75

-2.930

1.12

-2.917

600

-2.908

50

-2.890

80

-2.916

1.13

-2.891

650

-2.888

55

-2.861

85

-2.903

1.14

-2.866

700

-2.865

60

-2.829

90

-2.889

1.15

-2.841

750

-2.840

 

 

 

 

1.16

-2.817

Sensitivity = 
[(-2.927 + 2.957)/100]/
[2.943/500] = 0.05

Sensitivity = 
[(-2.918 + 2.966)/10]/
[2.943/40] = 0.06

Sensitivity = 
[(-2.930 + 2.956)/10]/
[2.943/70] = 0.06

Sensitivity = 
[(-2.917 + 2.970)/0.02]/
[2.943/1.11] = 1.00

Range = 0.15

Range = 0.19

Range = 0.10

Range = 0.26

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 0.05 Small
vfrags23 0.05 0.15 Small
Qfrags23 0.06 0.10 Small
Qfrag1  0.06 0.19 Small
vfrag1 0.08 0.24 Small
fI 1.00 0.26 Small
mbody 1.00 1.47 Large
PE 0.27 1.61 Large
Negative effect on lurch (increases rearward velocity)
mfrag1 -0.02 0.02 Small
Q  -0.02 0.05 Small
mbullet -0.66 0.12 Small
vbulletafter -0.02 0.14 Small
vbullet -1.74 0.27 Small
mcloud -0.64 3.90 Large

    Adding the large fragments and their six variables does not affect the sensitivity analysis materially, for all six terms have small sensitivities (<0.1) and small effects (<0.25 ft s-1). All the other terms remain as before, with the body, the cloud, and the potential energy still being the only large terms.

Summary
    Adding the large fragments hardly changes the lurch or any of the basic sensitivities. The six new variables all have small sensitivities and effects.

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