19. Lurch 4 angular

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 1.44 ft s^-1 to 1.38 ft s^-1. This change amounts to only 4%, and means that the 3-D motion of the large fragments is an unimportant part of the solution. A similar result was found for 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 = 3.80	Ω_bulletafter = 0.43	KE_bullet = 1164	KE_bulletafter = 14
	Ω_bodyafter = -4.05		KE_bodyafter = 1
	Ω_cloud = 6.28		KE_cloud = 668
	Ω_frag1 = 0.97		KE_frag1 = 132
	Ω_frags23 = 0.16		KE_frags23 = 49
			PE = 300

m_bullet	v_lurch	m_body	v_lurch	m_cloud	v_lurch	PE	v_lurch	v_bullet	v_lurch
						0	-1.813
156	-1.356					50	-1.746	1750	-1.310
157	-1.361	65	-1.804			100	-1.677	1760	-1.324
158	-1.365	70	-1.675			150	-1.606	1770	-1.337
159	-1.370	75	-1.563	0.1	-0.474	200	-1.532	1780	-1.351
160	-1.374	80	-1.465	0.2	-0.985	250	-1.457	1790	-1.365
161	-1.378	85	-1.378	0.3	-1.378	300	-1.378	1800	-1.378
162	-1.383	90	-1.301	0.4	-1.710	350	-1.297	1810	-1.392
163	-1.387	95	-1.233	0.5	-2.003	400	-1.212	1820	-1.405
164	-1.391	100	-1.171	0.6	-2.269	450	-1.124	1830	-1.418
165	-1.395	105	-1.115	0.7	-2.513	500	-1.031	1840	-1.432
166	-1.399			0.8	-2.742	550	-0.933	1850	-1.445
						600	-0.828
Sensitivity = [(-1.383 + 1.374)/2]/ [1.378/161] = -0.53		Sensitivity = [(-1.301 + 1.465)/10]/ [1.378/85] = 1.01		Sensitivity = [(-1.710 + 0.985)/0.2]/ [1.378/0.3] = -0.79		Sensitivity = [(-1.297 + 1.457)/100]/ [1.378/300] = 0.35		Sensitivity = [(-1.392 + 1.365)/20]/ [1.378/1800] = -1.76
Range = 0.04		Range = 0.69		Range = 2.27		Range = 0.98		Range = 0.14

v_bulletafter	v_lurch	Q	v_lurch	Q_cl	v_lurch	m_frag1	v_lurch	m_frags23	v_lurch
		7	-1.359	20	-2.704	0.022	-1.356	0.005	-1.390
0	-1.255	8	-1.362	30	-2.531	0.023	-1.360	0.006	-1.387
50	-1.290	9	-1.366	40	-2.301	0.024	-1.365	0.007	-1.385
100	-1.322	10	-1.369	50	-2.023	0.025	-1.369	0.008	-1.383
150	-1.352	11	-1.374	60	-1.701	0.026	-1.374	0.009	-1.381
200	-1.378	12	-1.378	70	-1.378	0.027	-1.378	0.010	-1.378
250	-1.402	13	-1.383	80	-1.038	0.028	-1.383	0.011	-1.376
300	-1.423	14	-1.389	90	-0.705	0.029	-1.387	0.012	-1.374
350	-1.440	15	-1.394	100	-0.391	0.030	-1.391	0.013	-1.371
400	-1.455	16	-1.401	110	-0.106	0.031	-1.396	0.015	-1.369
		17	-1.407	120	+0.142	0.032	-1.400	0.015	-1.366
Sensitivity = [(-1.402 + 1.352)/100]/ [1.378/200] = -0.07		Sensitivity = [(-1.389 + 1.369)/4]/ [1.378/12] = -0.04		Sensitivity = [(-1.038 + 1.711)/20]/ [1.378/70] = 1.71		Sensitivity = [-1.383 + 1.374)/0.002]/ [1.378/0.027] = -0.09		Sensitivity = [(-1.374 + 1.383)/0.004]/ [1.378/0.010] = 0.02
Range = 0.20		Range = 0.05		Range = 2.85		Range = 0.04		Range = 0.02

v_frag1	v_lurch	v_frags23	v_lurch	Q_frag1	v_lurch	Q_frags23	v_lurch	f_kefrag1	v_lurch
250	-1.366	250	-1.409					1.00	-1.420
300	-1.377	300	-1.406	20	-1.453	50	-1.426	1.05	-1.412
350	-1.384	350	-1.401	25	-1.439	55	-1.415	1.10	-1.403
400	-1.387	400	-1.395	30	-1.421	60	-1.403	1.15	-1.395
450	-1.385	450	-1.388	35	-1.401	65	-1.391	1.20	-1.387
500	-1.378	500	-1.378	40	-1.378	70	-1.387	1.25	-1.378
550	-1.367	550	-1.367	45	-1.353	75	-1.365	1.30	-1.370
600	-1.349	600	-1.355	50	-1.325	80	-1.351	1.35	-1.361
650	-1.327	650	-1.340	55	-1.295	85	-1.338	1.40	-1.353
700	-1.297	700	-1.324	60	-1.264	90	-1.324	1.45	-1.344
700	-1.261	750	-1.306					1.50	-1.336
Sensitivity = [(-1.367 + 1.385)/100]/ [1.378/500] = 0.06		Sensitivity = [(-1.367 + 1.388)/100]/ [1.378/500] = 0.08		Sensitivity = [(-1.353 + 1.401)/10]/ [1.378/40] = 0.14		Sensitivity = [(-1.365 + 1.391)/10]/ [1.378/70] = 0.13		Sensitivity = [(-1.370 + 1.387)/0.10]/ [1.378/1.25] = 0.15
Range = 0.10		Range = 0.10		Range = 0.19		Range = 0.10		Range = 0.08

f_kefrags23	v_lurch	f_I	v_lurch
1.00	-1.394	1.06	-1.443
1.05	-1.391	1.07	-1.430
1.10	-1.388	1.08	-1.417
1.15	-1.384	1.09	-1.404
1.20	-1.381	1.10	-1.391
1.25	-1.378	1.11	-1.378
1.30	-1.375	1.12	-1.366
1.35	-1.372	1.13	-1.354
1.40	-1.369	1.14	-1.342
1.45	-1.366	1.15	-1.330
1.50	-1.363	1.16	-1.319
Sensitivity = [(-1.375 + 1.381)/0.10]/ [1.378/1.25] = 0.05		Sensitivity = [(-1.366 + 1.391)/0.02]/ [1.378/1.11] = 1.01
Range = 0.03		Range = 0.12

As expected from the fact that the addition of three-dimensionality to the large fragments didn't change the lurch perceptibly, the basic pattern of sensitivities remained nearly unchanged, although the minor terms m_frags23 and v_frag1 moved from negative to positive. The two new parameters, f_kefrag1 and f_kefrags23, also had extremely small sensitivities. The two dominant sensitivities remained Q_cl and m_cloud, both properties of the cloud.

Summary
The addition of three-dimensionality to the motion of the large fragments changed the solution by only a few percent, and left the basic pattern of sensitivities unchanged. The variables of the cloud remained by far the most important.

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 = 500 ft s^-1
Q = 17°	Q_frags23 = 70°	v_frags23 = 500 ft s^-1
PE = 200 ft-lb	Q_cl = 70°	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)
m_frags23	0.02	0.02	Small
f_kefrags23	0.05	0.03	Small
f_kefrag1	0.15	0.08	Small
v_frags23	0.08	0.10	Small
v_frag1	0.06	0.10	Small
Q_frags23	0.13	0.10	Small
f_I	1.01	0.12	Small
Q_frag1	0.14	0.19	Small
m_body	1.01	0.69	Medium
PE	0.35	0.98	Medium
Q_cl	1.71	2.85	Large
Negative effect on lurch (increases rearward velocity)
m_bullet	-0.53	0.04	Small
m_frag1	-0.09	0.04	Small
Q	-0.04	0.05	Small
v_bullet	-1.76	0.14	Small
v_bulletafter	-0.07	0.20	Small
m_cloud	-0.79	2.27	Large