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

f_I	v_lurch	d_transit	v_lurch	d_snap	v_lurch	d_frag1	v_lurch	t_delay	v_lurch
1.06	-1.457
1.07	-1.444	0	-1.394			2	-1.359
1.08	-1.430	1	-1.394	1.6	-1.393	3	-1.387
1.09	-1.417	2	-1.393	1.8	-1.401	4	-1.402	0.000	-1.403
1.10	-1.404	3	-1.392	2.0	-1.405	5	-1.404	0.001	-1.399
1.11	-1.392	4	-1.392	2.2	-1.392	6	-1.392	0.002	-1.392
1.12	-1.379	5	-1.391	2.4	-1.301	7	-1.363	0.003	-1.379
1.13	-1.367	6	-1.390	2.6	-0.536	8	-1.316	0.004	-1.360
1.14	-1.355	7	-1.389	2.8	—	9	-1.248	0.005	-1.328
1.15	-1.343	8	-1.388			10	-1.153	0.006	-1.275
1.16	-1.332							0.007	-1.184
Sensitivity = [(-1.379 + 1.404)/0.02]/ [1.392/1.11] = 1.00		Sensitivity = [(-1.388 + 1.394)/8]/ [1.392/4] = 0.00		Sensitivity = [(-1.301 + 1.405)/0.4]/ [1.392/2.2] = 0.41		Sensitivity = [(-1.363 + 1.404)/2]/ [1.392/6] = 0.09		Sensitivity = [(-1.379 + 1.399)/0.002]/ [1.392/0.002] = 0.01
Range = 0.12		Range = 0.01		Range = 0.86		Range = 0.21		Range = 0.22

R_bullet	v_lurch	R_head	v_lurch
4.75	—	3.50	-1.398
5.00	-1.042	3.75	-1.402
5.25	-1.292	4.00	-1.405
5.50	-1.365	4.25	-1.403
5.75	-1.392	4.50	-1.392
6.00	-1.402	4.75	-1.355
6.25	-1.405	5.00	-1.249
6.50	-1.405	5.25	-0.856
6.75	-1.404	5.50	—
Sensitivity = [(-1.402 + 1.365)/0.50]/ [1.392/5.75] = -0.31		Sensitivity = [(-1.355 + 1.403)/0.50]/ [1.392/4.50] = 0.31
Range = 0.45		Range = 0.54

Just as in the linear case, the sensitivities of Lurch 6 Angular are very similar to those of Lurch 5 Angular in spite of the differences in calculation for Lurch 6. The previous variables maintained very similar sensitivities and ranges, with Q_cl and m_cloud continuing to dominate. Six of the eight new variables related to the timing and the snap had positive sensitivities, that is, they worked to increase the velocity of the lurch (to reduce its rearward velocity). Five of the eight had small sensitivities, with only R_head, R_bullet, and d_snap having medium sensitivities. The lurch hardly changed, and if anything, increased in rearward speed (from -1.38 ft s^-1 to -1.39 ft s^-1). This tiny effect is probably because the sensitivities and effects of the new variables were so small. Thus, adding the four time intervals did not change the basic result.

Summary
The basic solution remained virtually unchanged when the four time intervals were added: The lurch of -1.38 ft s^-1 in Lurch 5 Angular increased (in magnitude) only to -1.39 ft s^-1 in Snap/Lurch 6 Angular. This important result was probably a consequence of the small sensitivities and effects of the eight new variables.

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

Variable	Sensitivity of v_lurch	*Range of v_lurch, ft s^-1*	Magnitude
Positive effect on lurch (reduces rearward velocity)
f_kebody	0.00	0.00	Small
m_frags23	0.01	0.01	Small
d_transit	0.00	0.01	Small
Q_frags23	0.12	0.09	Small
f_kefrag1	0.17	0.10	Small
f_I	1.00	0.12	Small
Q_frag1	0.13	0.17	Small
d_frag1	0.09	0.21	Small
t_delay	0.01	0.22	Small
m_head	0.35	0.28	Small
R_head	0.31	0.54	Medium
m_body	1.01	0.70	Medium
d_snap	0.41	0.86	Medium
PE	0.34	0.96	Medium
Q_cl	1.72	2.90	Large
Negative effect on lurch (increases rearward velocity)
Q	-0.03	0.03	Small
m_frag1	-0.09	0.05	Small
m_bullet	-0.75	0.07	Small
v_bullet	-2.00	0.16	Small
v_bulletafter	-0.04	0.16	Small
R_bullet	-0.31	0.54	Medium
m_cloud	-0.80	2.31	Large