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.