19--L4A

The model and its justification
Lurch 4 Angular is the rotational analog of Lurch 4 Linear. As with Lurch 3 Angular, it calculates the speed of the rearward lurch from the bullet, the body, the conical cloud, large fragments, and the large fragments. To this it adds 3-D motion of the large fragments. This addition barely changes the results, with the rearward lurch decreasing in strength from 1.30 ft s^-1 to 1.28 ft s^-1. This change is negligible, and means that the 3-D motion of the large fragments is an unimportant part of the solution. The same result was found in the linear calculations.

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 = -3.77		KE_bodyafter = 0.8
	Ω_cloud = 6.37		KE_cloud = 884
	Ω_frag1 = 0.58		KE_frag1 = 47
	Ω_frags23 = 0.10		KE_frags23 = 18
			PE = 200

m_bullet	v_lurch	m_body	v_lurch	m_cloud	v_lurch	PE	v_lurch	v_bullet	v_lurch
156	-1.274							1750	-1.239
157	-1.276	65	-1.680			0	-1.516	1760	-1.248
158	-1.278	70	-1.560			50	-1.460	1770	-1.257
159	-1.280	75	-1.455	0.1	-0.366	100	-1.403	1780	-1.266
160	-1.282	80	-1.364	0.2	-0.885	150	-1.344	1790	-1.275
161	-1.284	85	-1.284	0.3	-1.284	200	-1.284	1800	-1.284
162	-1.285	90	-1.212	0.4	-1.621	250	-1.221	1810	-1.292
163	-1.287	95	-1.148	0.5	-1.918	300	-1.157	1820	-1.301
164	-1.289	100	-1.091	0.6	-2.187	350	-1.091	1830	-1.310
165	-1.291	105	-1.038	0.7	-2.436	400	-1.023	1840	-1.319
166	-1.292			0.8	-2.668			1850	-1.327
Sensitivity = [(-1.287 + 1.280)/4]/ [1.284/161] = -0.22		Sensitivity = [(-1.364 + 1.212)/10]/ [1.284/85] = 1.01		Sensitivity = [(-1.621 + 0.885)/0.2]/ [1.284/0.3] = -0.86		Sensitivity = [(-1.221 + 1.344)/100]/ [1.284/200] = 0.19		Sensitivity = [(-1.301 + 1.266)/40]/ [1.284/1800] = -1.23
Range = 0.018		Range = 0.64		Range = 2.30		Range = 0.49		Range = 0.088

v_bulletafter	v_lurch	Q	v_lurch	m_frag1	v_lurch	m_frags23	v_lurch	v_frag1	v_lurch
		12	-1.255	0.022	-1.258	0.005	-1.278
0	-1.155	13	-1.260	0.023	-1.263	0.006	-1.279	100	-1.203
50	-1.190	14	-1.265	0.024	-1.268	0.007	-1.280	150	-1.228
100	-1.223	15	-1.271	0.025	-1.273	0.008	-1.281	200	-1.250
150	-1.255	16	-1.277	0.026	-1.278	0.009	-1.282	250	-1.268
200	-1.284	17	-1.284	0.027	-1.284	0.010	-1.284	300	-1.284
250	-1.310	18	-1.291	0.028	-1.289	0.011	-1.285	350	-1.295
300	-1.335	19	-1.298	0.029	-1.294	0.012	-1.286	400	-1.304
350	-1.357	20	-1.306	0.03	-1.299	0.013	-1.287	450	-1.309
400	-1.377	21	-1.314	0.031	-1.304	0.014	-1.288	500	-1.310
		22	-1.322	0.032	-1.310	0.015	-1.289
Sensitivity = [(-1.310 + 1.255)/100]/ [1.284/200] = -0.086		Sensitivity = [(-1.298 + 1.271)/4]/ [1.284/17] = -0.089		Sensitivity = [(-1.294 + 1.273)/0.004]/ [1.284/0.027] = -0.11		Sensitivity = [(-1.286 + 1.281)/0.004]/ [1.284/0.01] = -0.0097		Sensitivity = [(-1.295 + 1.268)/100]/ [1.284/300] = -0.063
Range = 0.22		Range = 0.067		Range = 0.052		Range = 0.011		Range = 0.107

v_frags23	v_lurch	Q_frag1	v_lurch	Q_frags23	v_lurch	Q_cl	v_lurch
100	-1.281	20	-1.328	50	-1.312	25	-1.984
150	-1.283	25	-1.320	55	-1.306	30	-1.795
200	-1.285	30	-1.309	60	-1.299	35	-1.616
250	-1.285	35	-1.297	65	-1.291	40	-1.446
300	-1.284	40	-1.284	70	-1.284	45	-1.284
350	-1.281	45	-1.268	75	-1.276	50	-1.128
400	-1.278	50	-1.252	80	-1.268	55	-0.979
450	-1.273	55	-1.234	85	-1.259	60	-0.835
500	-1.267	60	-1.215	90	-1.251	65	-0.697
Sensitivity = [(-1.267 + 1.284)/200]/ [1.284/300] = 0.020		Sensitivity = [(-1.268 + 1.297)/10]/ [1.284/40] = 0.090		Sensitivity = [(-1.276 + 1.291)/10]/ [1.284/70] = 0.082		Sensitivity = [(-1.128 + 1.446)/10]/ [1.284/45] = 1.11
Range = 0.014		Range = 0.113		Range = 0.061		Range = 1.29

f_kefrag1	v_lurch	f_kefrags23	v_lurch	f_I	v_lurch
				1.06	-1.344
				1.07	-1.332
				1.08	-1.319
1	-1.295	1	-1.288	1.09	-1.307
1.1	-1.291	1.1	-1.286	1.10	-1.295
1.25	-1.284	1.25	-1.284	1.11	-1.284
1.4	-1.277	1.4	-1.281	1.12	-1.272
1.5	-1.272	1.5	-1.279	1.13	-1.261
				1.14	-1.250
				1.15	-1.239
				1.16	-1.228
Sensitivity = [(-1.277 + 1.291)/0.3]/ [1.284/1.25] = 0.045		Sensitivity = [(-1.279 + 1.288)/0.5]/ [1.284/1.25] = 0.018		Sensitivity = [(-1.228 + 1.344)/0.1]/ [1.284/1.11] = 1.003
Range = 0.023		Range = 0.009		Range = 0.116

As expected from the fact that the addition of three-dimensionality to the large fragments didn't change the lurch perceptibly, the sensitivities and their orders also remained nearly unchanged. The two new parameters, f_kefrag1 and f_kefrags23, also had extremely small sensitivities.

Summary
The addition of three-dimensionality to the motion of the large fragments changed the solution by only negligible amounts, and didn't change the sensitivities either.

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_kefrag1 = 1.25
f_I = 1.11		f_kefrags23 = 1.25

Variable	Sensitivity of v_lurch	*Range of v_lurch, ft s^-1*	Magnitude
Positive effect on lurch (reduces rearward velocity)
f_kefrags23	0.018	0.01	Small
v_frags23	0.020	0.01	Small
f_kefrag1	0.045	0.023	Small
Q_frags23	0.082	0.06	Small
Q_frag1	0.090	0.11	Small
f_I	0.99	0.12	Small
PE	0.19	0.49	Small
m_body	1.01	0.64	Small
Q_cl	1.11	1.29	Large
Negative effect on lurch (increases rearward velocity)
m_frags23	-0.010	0.01	Small
m_bullet	-0.22	0.02	Small
m_frag1	-0.11	0.05	Small
Q	-0.089	0.07	Small
v_bullet	-1.23	0.09	Small
v_frag1	-0.063	0.11	Small
v_bulletafter	-0.086	0.22	Small
m_cloud	-0.86	2.30	Large