18--L3A

The model and its justification
Lurch 3 Angular is the rotational analog of Lurch 3 Linear. It is the procedure for calculating the speed of the rearward lurch of the upper body from the bullet, the body, cloud, the large fragments, and conical 3-D motion of the diffuse cloud. Actually, "half-conical" would be a better way to describe the shape of the could, since it is assumed to be conical in the XY plane but only half-conical (upward). Material of the cloud is assumed to be distributed evenly over the half-angle Q_cl. The two conservation equations remain the same because Q_cl is used to calculate the X-component of the velocity of the particles in the cloud, f_xcl, and the X-component of the kinetic energy of the cloud, f_kecl, and those terms are used in the equations. The two formulas are given in the next section.
The conical motion of the diffuse cloud greatly reduces the calculated strength of the rearward lurch, from the previous 3.29 ft s^-1 to 1.30 ft s^-1. This reduction by a factor of nearly three is very similar to the factor observed from Lurch 2 Linear to Lurch 3 Linear, 2.44 ft s^-1 to 0.88 ft s^-1.

Conservation of angular X-momentum

Conservation of total energy

Solutions to simultaneous equations

Distributions of momentum and energy

Angular momentum		Energy, ft-lb
Before	After	Before	After
Ω_bullet = 3.71	Ω_bulletafter = 0.43	KE_bullet = 1164	KE_bulletafter = 14
	Ω_bodyafter = -3.82		KE_bodyafter = 0.8
	Ω_cloud = 6.42		KE_cloud = 897
	Ω_frag1 = 0.58		KE_frag1 = 38
	Ω_frags23 = 0.10		KE_frags23 = 14
			PE = 200

m_bullet	v_lurch	m_body	v_lurch	m_cloud	v_lurch	PE	v_lurch	v_bullet	v_lurch
156	-1.290							1750	-1.256
157	-1.292	65	-1.701			0	-1.531	1760	-1.264
158	-1.294	70	-1.579			50	-1.475	1770	-1.273
159	-1.296	75	-1.473	0.1	-0.375	100	-1.418	1780	-1.282
160	-1.298	80	-1.381	0.2	-0.898	150	-1.359	1790	-1.291
161	-1.300	85	-1.300	0.3	-1.300	200	-1.300	1800	-1.300
162	-1.301	90	-1.227	0.4	-1.639	250	-1.238	1810	-1.308
163	-1.303	95	-1.162	0.5	-1.939	300	-1.174	1820	-1.317
164	-1.305	100	-1.104	0.6	-2.210	350	-1.109	1830	-1.326
165	-1.306	105	-1.051	0.7	-2.460	400	-1.041	1840	-1.334
166	-1.308			0.8	-2.694			1850	-1.343
Sensitivity = [(-1.303 + 1.296)/4]/ [1.300/161] = -0.22		Sensitivity = [(-1.227 + 1.381)/10]/ [1.300/85] = 1.01		Sensitivity = [(-1.639 + 0.898)/0.2]/ [1.300/0.3] = -0.86		Sensitivity = [(-1.238 + 1.359)/100]/ [1.300/200] = 0.186		Sensitivity = [(-1.317 + 1.282)/40]/ [1.300/1800] = -1.21
Range = 0.018		Range = 0.65		Range = 2.32		Range = 0.49		Range = 0.087

v_bulletafter	v_lurch	Q	v_lurch	m_frag1	v_lurch	m_frags23	v_lurch	v_frag1	v_lurch
		12	-1.271	0.022	-1.271	0.005	-1.292
0	-1.170	13	-1.276	0.023	-1.277	0.006	-1.293	100	-1.208
50	-1.206	14	-1.281	0.024	-1.283	0.007	-1.295	150	-1.235
100	-1.239	15	-1.287	0.025	-1.288	0.008	-1.296	200	-1.259
150	-1.270	16	-1.293	0.026	-1.294	0.009	-1.298	250	-1.281
200	-1.300	17	-1.300	0.027	-1.300	0.010	-1.300	300	-1.300
250	-1.326	18	-1.306	0.028	-1.305	0.011	-1.301	350	-1.316
300	-1.351	19	-1.314	0.029	-1.311	0.012	-1.303	400	-1.329
350	-1.373	20	-1.321	0.030	-1.316	0.013	-1.304	450	-1.340
400	-1.393	21	-1.329	0.031	-1.322	0.015	-1.306	500	-1.348
		22	-1.338	0.032	-1.328	0.015	-1.307
Sensitivity = [(-1.326 + 1.270)/100]/ [1.300/200] = -0.086		Sensitivity = [(-1.314 + 1.287)/4]/ [1.300/17] = -0.088		Sensitivity = [(-1.311 + 1.288)/0.004]/ [1.300/0.027] = -0.119		Sensitivity = [(-1.303 + 1.296)/0.004]/ [1.300/0.01] = -0.0135		Sensitivity = [(-1.316 + 1.281)/100]/ [1.300/300] = -0.081
Range = 0.22		Range = 0.067		Range = 0.057		Range = 0.015		Range = 0.14

v_frags23	v_lurch	Q_frag1	v_lurch	Q_frags23	v_lurch	Q_cl	v_lurch	f_I	v_lurch
								1.06	-1.361
100	-1.293	20	-1.344	50	-1.328	25	-2.005	1.07	-1.348
150	-1.296	25	-1.336	55	-1.322	30	-1.814	1.08	-1.336
200	-1.298	30	-1.325	60	-1.315	35	-1.634	1.09	-1.323
250	-1.299	35	-1.313	65	-1.307	40	-1.463	1.10	-1.311
300	-1.300	40	-1.300	70	-1.300	45	-1.300	1.11	-1.300
350	-1.299	45	-1.284	75	-1.292	50	-1.143	1.12	-1.288
400	-1.297	50	-1.268	80	-1.283	55	-0.993	1.13	-1.277
450	-1.294	55	-1.250	85	-1.275	60	-0.848	1.14	-1.265
500	-1.291	60	-1.231	90	-1.267	65	-0.709	1.15	-1.254
								1.16	-1.244
Sensitivity = [(-1.29 + 1.30)/200]/ [1.300/300] = 0.0115		Sensitivity = [(-1.268 + 1.325)/20]/ [1.300/40] = 0.088		Sensitivity = [(-1.283 + 1.315)/20]/ [1.300/70] = 0.086		Sensitivity = [(-1.143 + 1.463)/10]/ [1.300/45] = 1.11		Sensitivity = [(-1.244 + 1.361)/0.10]/ [1.300/1.11] = 0.99
Range = 0.009		Range = 0.113		Range = 0.061		Range = 1.30		Range = 0.12

This ordering of sensitivities and effects is very similar to the previous one, except for the new large effect for Q_cl. The other large effect, for m_cloud, was also seen before. That both these large effects belong to properties of the diffuse cloud is an important feature of all subsequent simulations.

Summary
Making the diffuse cloud into a conical shape reduces the strength of the rearward lurch by a factor of nearly three (3.3 ft s^-1 to 1.3 ft s^-1) and makes the half-angle of the cone into one of the two most-sensitive variables, the mass of the could being the other one. This both brings the lurch nearly into line with the observed velocity and emphasizes the all-importance of the properties of the cloud. Unfortunately for this work, the properties of the cloud are very difficult to estimate, and probably form the limiting factor for the accuracy of this series of simulations.

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 = 17°	Q_frags23 = 70°	v_frags23 = 300 ft s^-1
PE = 200 ft-lb	Q_cl = 45°	f_I = 1.11

Variable	Sensitivity of v_lurch	*Range of v_lurch, ft s^-1*	Magnitude
Positive effect on lurch (reduces rearward velocity)
v_frags23	0.012	0.01	Small
Q_frags23	0.086	0.06	Small
Q_frag1	0.088	0.11	Small
f_I	0.99	0.12	Small
PE	0.19	0.49	Small
m_body	1.01	0.65	Small
Q_cl	1.11	1.30	Large
Negative effect on lurch (increases rearward velocity)
m_frags23	-0.014	0.02	Small
m_bullet	-0.22	0.02	Small
m_frag1	-0.119	0.06	Small
Q	-0.088	0.07	Small
v_bullet	-1.21	0.09	Small
v_frag1	-0.081	0.14	Small
v_bulletafter	-0.086	0.22	Small
m_cloud	-0.82	2.32	Large