11. Lurch 3 Linearadd conical 3-D motion of the diffuse cloud

The model and its justification
    Like Lurch 2 Linear, Lurch 3 Linear calculates the speed of the rearward lurch of the upper body from the bullet, the body, the cloud, and the large fragments. To these it adds conical three-dimensional motion of the diffuse cloud. This three-dimensionality decreases the intensity of the lurch dramatically, from -2.18 ft s-1 to -1.06 ft s-1 for default values of the variables. This result shows that the 3-D motion of the diffuse cloud is an extremely important property of this system.

Solving the simultaneous equations
    As before, the equations for conservation of X-momentum and total energy are solved simultaneously. To the version in Lurch 2 is added a term that express the three-dimensionality of the motion of the diffuse cloudfxcl. The fxcl corrects the total velocity of the conical cloud, assumed to be the same in all directions, to its average X-component. The value of fxcl is calculated from Qcl by the formula given below. Its default value is 0.59.
    The solutions for vbodyafter and vcloud, -1.06 ft s-1 and 388 ft s-1, respectively, are shown below the variables. The lurch is drastically reduced from Lurch 2 Linear, but the speed of the cloud remains the same.

Conservation of X-momentum

 

Conservation of total energy

 

Default values of the 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 = 300 ft s-1
Q = 12  Qfrags23 = 70  vfrags23 = 300 ft s-1
PE = 300 ft-lb Qcl = 70  fxcl = 0.59 (calc.)

Default solutions to the simultaneous equations

vbodyafter = -1.06 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 = -1.41   KEbodyafter = 0.4
  Pcloud = 2.15   KEcloud = 705
  Pfrag1 = 0.32   KEfrag1 = 105
  Pfrags23 = 0.05   KEfrags23 = 39
      PE = 300

    As required by the drop in the lurch, the distribution of momentum differs significantly from those in Lurch 2 Linear. Interestingly, however, the distribution of KE remains nearly the same, except for a drop in the term for the body. This doesn't affect the other energies because the body gets such a small part of the total energy.

Sensitivity analysis
    

Sensitivity analysis, Lurch 3 Linear
(Standard conditions in boldface)

mbullet

vlurch

mbody

vlurch

mcloud

vlurch

PE

vlurch

vbullet

vlurch

 

 

 

 

 

 

0

-1.378

 

 

156

-1.047

 

 

 

 

50

-1.329

1750

-1.014

157

-1.051

65

-1.391

 

 

100

-1.279

1760

-1.024

158

-1.054

70

-1.292

 

 

150

-1.227

1770

-1.034

159

-1.057

75

-1.205

0.1

-0.375

200

-1.174

1780

-1.043

160

-1.060

80

-1.129

0.2

-0.764

250

-1.119

1790

-1.053

161

-1.063

85

-1.063

0.3

-1.063

300

-1.063

1800

-1.063

162

-1.066

90

-1.004

0.4

-1.316

350

-1.004

1810

-1.072

163

-1.069

95

-0.951

0.5

-1.539

400

-0.943

1820

-1.082

164

-1.071

100

-0.903

0.6

-1.741

450

-0.880

1830

-1.091

165

-1.074

105

-0.860

0.7

-1.928

500

-0.813

1840

-1.101

166

-1.077

 

 

0.8

-2.102

550

-0.743

1850

-1.110

 

 

 

 

 

 

600

-0.670

 

 

Sensitivity = 
[(-1.069 + 1.057)/4]/
[1.063/161] = -0.45

Sensitivity = 
[(-1.004 + 1.129)/10]/
[1.063/85] = 1.00

Sensitivity = 
[(-1.316 + 0.764)/0.2]/
[1.063/0.3] = -0.78

Sensitivity = 
[(-1.004 + 1.119)/100]/
[1.063/300] = 0.32

Sensitivity = 
[(-1.072 + 1.053)/20]/
[1.063/1800] = -1.61

Range = 0.03

Range = 0.53

Range = 1.73

Range = 0.71

Range = 0.10

vbulletafter

vlurch

Q

vlurch

Qcl

vlurch

mfrag1

vlurch

mfrags23

vlurch

 

 

7

-1.049

20

-2.074

0.022

-1.040

0.005

-1.065

0

-0.971

8

-1.051

30

-1.942

0.023

-1.044

0.006

-1.064

50

-0.997

9

-1.053

40

-1.766

0.024

-1.049

0.007

-1.064

100

-1.021

10

-1.056

50

-1.554

0.025

-1.054

0.008

-1.064

150

-1.043

11

-1.059

60

-1.316

0.026

-1.058

0.009

-1.063

200

-1.063

12

-1.063

70

-1.063

0.027

-1.063

0.010

-1.063

250

-1.081

13

-1.066

80

-0.804

0.028

-1.067

0.011

-1.062

300

-1.096

14

-1.070

90

-0.551

0.029

-1.072

0.012

-1.062

350

-1.110

15

-1.075

100

-0.312

0.030

-1.076

0.013

-1.061

400

-1.121

16

-1.079

110

-0.095

0.031

-1.081

0.015

-1.061

 

 

17

-1.084

120

+0.093

0.032

-1.085

0.015

-1.060

Sensitivity = 
[(-1.081 + 1.043)/100]/
[1.063/200] = -0.07

Sensitivity = 
[(-1.070 + 1.056)/4]/
[1.063/12] = -0.04

Sensitivity = 
[(-0.804 + 1.316)/20]/
[1.063/70] = 1.69

Sensitivity = 
[-1.067 + 1.058)/0.002]/
[1.063/0.027] = -0.11

Sensitivity = 
[(-1.061 + 1.064)/0.006]/
[1.063/0.010] = 0.00

Range = 0.15

Range = 0.04

Range = 2.17

Range = 0.04

Range = 0.00

vfrag1

vlurch

vfrags23

vlurch

Qfrag1

vlurch

Qfrags23

vlurch

250

-1.029

250

-1.076

 

 

 

 

300

-1.041

300

-1.075

20

-1.118

50

-1.098

350

-1.050

350

-1.073

25

-1.107

55

-1.090

400

-1.057

400

-1.071

30

-1.095

60

-1.081

450

-1.061

450

-1.067

35

-1.080

65

-1.072

500

-1.063

500

-1.063

40

-1.063

70

-1.063

550

-1.061

550

-1.057

45

-1.044

75

-1.053

600

-1.057

600

-1.051

50

-1.024

80

-1.043

650

-1.050

650

-1.044

55

-1.001

85

-1.033

700

-1.039

700

-1.035

60

-0.978

90

-1.022

700

-1.025

750

-1.026

 

 

 

 

Sensitivity = 
[(-1.057 + 1.057)/200]/
[1.063/500] = 0.00

Sensitivity = 
[(-1.057 + 1.067)/100]/
[1.063/500] = 0.05

Sensitivity = 
[(-1.044 + 1.080)/10]/
[1.063/40] = 0.14

Sensitivity = 
[(-1.053 + 1.072)/10]/
[1.063/70] = 0.12

Range = 0.00

Range = 0.05

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 0.00 0.00 Small
vfrag1 0.00 0.00 Small
vfrags23 0.05 0.05 Small
Qfrags23 0.12 0.08 Small
Qfrag1  0.14 0.14 Small
mbody 1.00 0.53 Medium
PE 0.32 0.71 Medium
Qcl 1.69 2.17 Large
Negative effect on lurch (increases rearward velocity)
mbullet -0.45 0.03 Small
mfrag1 -0.11 0.04 Small
Q  -0.04 0.04 Small
vbullet -1.61 0.10 Small
vbulletafter -0.07 0.15 Small
mcloud -0.78 1.73 Large

    Making the cloud three-dimensional changed the basic solution by generally decreasing the positive effects and increasing the negative effects. All the sensitivities and effects retained their signs, however. The half-angle of the cloud became the biggest positive effect and the biggest overall effect, reducing the three other important variables by up to half. The properties of the diffuse cloud (Qcl and mcloud, the two "large" variables) thus became all-important to calculating the lurch.

Summary
    Adding three-dimensionality to the diffuse cloud decreased the magnitude of the lurch by a factor of two, from -2.2 ft s-1 to -1.1 ft s-1. It also made the mass and 3-D shape of the cloud the most important variables (greatest range of lurch). As much as anything, this result expressed the difficulty is estimating the true attributes of the cloud.

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