11. Lurch 3 linear

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 cloud—f_xcl. The f_xcl corrects the total velocity of the conical cloud, assumed to be the same in all directions, to its average X-component. The value of f_xcl is calculated from Q_cl by the formula given below. Its default value is 0.59.
The solutions for v_bodyafter and v_cloud, -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 solutions to the simultaneous equations

Distributions of momentum and energy

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.

m_bullet	v_lurch	m_body	v_lurch	m_cloud	v_lurch	PE	v_lurch	v_bullet	v_lurch
						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

v_bulletafter	v_lurch	Q	v_lurch	Q_cl	v_lurch	m_frag1	v_lurch	m_frags23	v_lurch
		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

v_frag1	v_lurch	v_frags23	v_lurch	Q_frag1	v_lurch	Q_frags23	v_lurch
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

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 (Q_cl and m_cloud, 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.

m_bullet = 161 gr	v_bullet = 1800 ft s^-1	m_frag1 = 0.027 gr
m_body = 85 lb	v_bulletafter = 200 ft s^-1	m_frags23 = 0.01 gr
m_cloud = 0.3 lb	Q_frag1 = 40°	v_frag1 = 300 ft s^-1
Q = 12°	Q_frags23 = 70°	v_frags23 = 300 ft s^-1
PE = 300 ft-lb	Q_cl = 70°	f_xcl = 0.59 (calc.)

Momentum		Energy, ft-lb
Before	After	Before	After
P_bullet = 1.26	P_bulletafter = 0.14	KE_bullet = 1164	KE_bulletafter = 14
	P_bodyafter = -1.41		KE_bodyafter = 0.4
	P_cloud = 2.15		KE_cloud = 705
	P_frag1 = 0.32		KE_frag1 = 105
	P_frags23 = 0.05		KE_frags23 = 39
			PE = 300

Variable	Sensitivity of v_lurch	*Range of v_lurch, ft s^-1*	Magnitude
Positive effect on lurch (reduces rearward velocity)
m_frags23	0.00	0.00	Small
v_frag1	0.00	0.00	Small
v_frags23	0.05	0.05	Small
Q_frags23	0.12	0.08	Small
Q_frag1	0.14	0.14	Small
m_body	1.00	0.53	Medium
PE	0.32	0.71	Medium
Q_cl	1.69	2.17	Large
Negative effect on lurch (increases rearward velocity)
m_bullet	-0.45	0.03	Small
m_frag1	-0.11	0.04	Small
Q	-0.04	0.04	Small
v_bullet	-1.61	0.10	Small
v_bulletafter	-0.07	0.15	Small
m_cloud	-0.78	1.73	Large