21. Lurch 6 angular

The model and its justification
Snap/Lurch 6 is the angular equivalent of Snap/Lurch 6, which added four time intervals to the simulation. It calculates the speed of the rearward lurch from the bullet, the cloud, and the large fragments, with 3-D motions of the cloud, the large fragments, and the body: It begins by interposing three steps having to do with the times. (1) It first calculates v_snap and uses it to calculate t_snap. (2) Then it calculates t_transit and combines it with t_snap with t_delay to calculate t_lurch. (3) Then it uses t_lurch and d_frags to calculate v_frags. At this point it returns to the former procedure and (4) solves for v_bodyafter and v_cloud. To simplify matters a bit, it sets v_frags23 = v_frag1 and f_kefrags23 = f_kefrag1. These time intervals leave the default solution about the same, however: -1.39 and 386 ft s^-1 for lurch and cloud, respectively, versus -1.38 and 378 ft s^-1 in Lurch 5 Angular. The sensitivities and effects also remain similar.

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 = -4.09		KE_bodyafter = 1
	Ω_cloud = 6.42		KE_cloud = 697
	Ω_frag1 = 0.89		KE_frag1 = 111
	Ω_frags23 = 0.15		KE_frags23 = 41
			PE = 300

m_bullet	v_lurch	m_body	v_lurch	m_cloud	v_lurch	m_head	v_lurch	v_bullet	v_lurch
						4.0	-1.378
156	-1.353					4.5	-1.385	1750	-1.306
157	-1.362	65	-1.821			5.0	-1.392	1760	-1.325
158	-1.370	70	-1.691			5.5	-1.399	1770	-1.343
159	-1.378	75	-1.578	0.1	-0.468	6.0	-1.404	1780	-1.360
160	-1.385	80	-1.479	0.2	-0.990	6.5	-1.405	1790	-1.376
161	-1.392	85	-1.392	0.3	-1.392	7.0	-1.392	1800	-1.392
162	-1.398	90	-1.314	0.4	-1.731	7.5	-1.336	1810	-1.407
163	-1.403	95	-1.245	0.5	-2.030	8.0	-1.097	1820	-1.421
164	-1.409	100	-1.182	0.6	-2.300	8.5	—	1830	-1.435
165	-1.414	105	-1.126	0.7	-2.550	9.0	—	1840	-1.449
166	-1.419			0.8	-2.782	9.5	—	1850	-1.463
						10.0	—
Sensitivity = [(-1.398 + 1.385)/2]/ [1.392/161] = -0.75		Sensitivity = [(-1.314 + 1.479)/10]/ [1.392/85] = 1.01		Sensitivity = [(-1.731 + 0.990)/0.2]/ [1.392/0.3] = -0.80		Sensitivity = [(-1.336 + 1.405)/1.0]/ [1.392/7.0] = 0.35		Sensitivity = [(-1.407 + 1.376)/20]/ [1.392/1800] = -2.00
Range = 0.07		Range = 0.70		Range = 2.31		Range = 0.28		Range = 0.16

v_bulletafter	v_lurch	PE	v_lurch	Q	v_lurch	Q_cl	v_lurch	m_frag1	v_lurch
		0	-1.819
		50	-1.753	7	-1.378	20	-2.744	0.022	-1.367
0	-1.280	100	-1.685	8	-1.380	30	-2.568	0.023	-1.372
50	-1.315	150	-1.615	9	-1.383	40	-2.333	0.024	-1.377
100	-1.347	200	-1.543	10	-1.385	50	-2.050	0.025	-1.382
150	-1.373	250	-1.468	11	-1.388	60	-1.731	0.026	-1.387
200	-1.392	300	-1.392	12	-1.392	70	-1.392	0.027	-1.392
250	-1.398	350	-1.312	13	-1.395	80	-1.045	0.028	-1.396
300	-1.383	400	-1.229	14	-1.398	90	-0.704	0.029	-1.401
350	-1.321	450	-1.143	15	-1.402	100	-0.383	0.030	-1.406
400	-1.124	500	-1.053	16	-1.405	110	-0.092	0.031	-1.411
		550	-0.958	17	-1.408	120	+0.161	0.032	-1.415
		600	-0.857
Sensitivity = [(-1.398 + 1.373)/100]/ [1.392/200] = -0.04		Sensitivity = [(-1.312 + 1.468)/100]/ [1.392/300] = 0.34		Sensitivity = [(-1.398 + 1.385)/4]/ [1.392/12] = -0.03		Sensitivity = [(-1.045 + 1.731)/20]/ [1.392/70] = 1.72		Sensitivity = [-1.396 + 1.387)/0.002]/ [1.392/0.027] = -0.09
Range = 0.16		Range = 0.96		Range = 0.03		Range = 2.90		Range = 0.05

m_frags23	v_lurch	Q_frag1	v_lurch	Q_frags23	v_lurch	f_kefrags	v_lurch	f_kebody	v_lurch
0.005	-1.398					1.00	-1.439
0.006	-1.397	20	-1.460	50	-1.435	1.05	-1.429	1.00	-1.392
0.007	-1.396	25	-1.447	55	-1.425	1.10	-1.420	1.05	-1.392
0.008	-1.394	30	-1.431	60	-1.415	1.15	-1.410	1.10	-1.392
0.009	-1.393	35	-1.413	65	-1.403	1.20	-1.401	1.15	-1.392
0.010	-1.392	40	-1.392	70	-1.392	1.25	-1.392	1.20	-1.392
0.011	-1.390	45	-1.368	75	-1.379	1.30	-1.382	1.25	-1.392
0.012	-1.389	50	-1.343	80	-1.367	1.35	-1.373	1.30	-1.391
0.013	-1.387	55	-1.316	85	-1.354	1.40	-1.363	1.35	-1.391
0.015	-1.386	60	-1.287	90	-1.342	1.45	-1.353	1.40	-1.391
0.015	-1.384					1.50	-1.344
Sensitivity = [(-1.389 + 1.394)/0.004]/ [1.392/0.010] = 0.01		Sensitivity = [(-1.368 + 1.413)/10]/ [1.392/40] = 0.13		Sensitivity = [(-1.379 + 1.403)/10]/ [1.392/70] = 0.12		Sensitivity = [(-1.382 + 1.401)/0.10]/ [1.392/1.25] = 0.17		Sensitivity = [(-1.391 + 1.392)/0.40]/ [1.392/1.20] = 0.00
Range = 0.01		Range = 0.17		Range = 0.09		Range = 0.10		Range = 0.00

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°	f_kefrag1 = 1.25
m_head = 7 lb	Q_frags23 = 70°	f_kefrags23 = f_kefrag1
Q = 12°	Q_cl = 70°	f_kebody = 1.2
PE = 300 ft-lb	t_delay = 0.002 s	f_xcl = 0.59 (calc.)
d_transit = 4 in	d_snap = 2.2 in	d_frag1 = 6 ft

f_I	v_lurch	d_transit	v_lurch	d_snap	v_lurch	d_frag1	v_lurch	t_delay	v_lurch
1.06	-1.457