22--L7A

22. Lurch 7 Angular—add four time intervals

The model and its justification

The rotational analogue of Lurch 7 Linear. Procedure for calculating speed of rearward lurch of upper body from bullet, body, conical cloud, and 3-D motions of large fragments and body: (1) Use v_snap to calculate t_snap. (2) Combine with t_delay and t_transit to calculate t_lurch.(3) Use t_lurch and d_frags to calculate v_frags. (4) Set v_cloud = v_frags and solve for m_cloud and v_bodyafter.

Solving the simultaneous equations

Conservation of angular X-momentum

Conservation of total energy

Default values of 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_head = 7 lb	Q_frag1 = 40°	f_kefrag1 = 1.25
Q = 17°	Q_frags23 = 70°	f_kebody = 1.2
PE = 200 ft-lb	Q_cl = 45°	f_I = 1.11
d_transit = 4 in	t_delay = 0.002 s	d_frag1 = 6 ft
d_snap = 2.2 in	R_bullet = 5.75 in	R_head = 4.5 in

Solutions to simultaneous equations

v_bodyafter = -0.33 ft s^-1 m_cloud = 0.083 lb

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 = -0.97		KE_bodyafter = 0.1
	Ω_cloud = 3.13		KE_cloud = 772
	Ω_frag1 = 0.96		KE_frag1 = 130
	Ω_frags23 = 0.16		KE_frags23 = 48
			PE = 200

Sensitivity analysis

Sensitivity analysis, Lurch 7 angular (Conical)
(Standard conditions in boldface)

m_bullet	v_lurch	m_body	v_lurch	m_head	v_lurch	Q	v_lurch	PE	v_lurch
156	-0.198					12	-0.401
157	-0.224	65	-0.432	5	-1.510	13	-0.389	0	-0.606
158	-0.250	70	-0.401	5.5	-1.208	14	-0.376	50	-0.537
159	-0.276	75	-0.374	6	-0.909	15	-0.362	100	-0.468
160	-0.303	80	-0.351	6.5	-0.615	16	-0.347	150	-0.399
161	-0.330	85	-0.330	7	-0.330	17	-0.330	200	-0.330
162	-0.358	90	-0.312	7.5	-0.062	18	-0.313	250	-0.261
163	-0.385	95	-0.295	8	+0.169	19	-0.294	300	-0.193
164	-0.413	100	-0.281	8.5	+0.266	20	-0.275	350	-0.124
165	-0.441	105	-0.267	9		21	-0.254	400	-0.055
166	-0.469					22	-0.232
Sensitivity = [(-0.358 + 0.303)/2]/ [0.330/161] = -13.42		Sensitivity = [(-0.312 + 0.351)/10]/\| [0.330/85] = 1.005		Sensitivity = [(-0.062 + 0.615)/1]/ [0.330/7] = 11.73		Sensitivity = [(-0.313 + 0.317)/2]/ [0.330/17] = 0.88		Sensitivity = [(-0.261 + 0.399)/100]/ [0.330/200] = 0.84
Range = 0.27		Range = 0.165		Range = 1.78		Range = 0.157		Range = 0.55

v_bullet	v_lurch	v_bulletafter	v_lurch	m_frag1	v_lurch	m_frags23	v_lurch	Q_frag1	v_lurch
1750	-0.161			0.022	-0.303	0.005	-0.336
1760	-0.194	0	-0.676	0.023	-0.308	0.006	-0.335	20	-0.404
1770	-0.227	50	-0.601	0.024	-0.314	0.007	-0.334	25	-0.390
1780	-0.261	100	-0.519	0.025	-0.319	0.008	-0.333	30	-0.373
1790	-0.295	150	-0.428	0.026	-0.325	0.009	-0.331	35	-0.353
1800	-0.330	200	-0.330	0.027	-0.330	0.01	-0.330	40	-0.330
1810	-0.366	250	-0.227	0.028	-0.336	0.011	-0.329	45	-0.305
1820	-0.403	300	-0.121	0.029	-0.341	0.012	-0.328	50	-0.278
1830	-0.440	350	-0.017	0.03	-0.347	0.013	-0.327	55	-0.248
1840	-0.477	400	+0.072	0.031	-0.352	0.014	-0.325	60	-0.217
1850	-0.516			0.032	-0.358	0.015	-0.324
Sensitivity = [(-0.366 + 0.295)/20]/ [0.330/1800] = -19.36		Sensitivity = [(-0.227 + 0.428)/100]/ [0.330/200] = 1.22		Sensitivity = [(-0.336 + 0.325)/0.002]/ [0.330/0.027] = -0.450		Sensitivity = [(-0.328 + 0.333)/0.004]/ [0.330/0.010] = 0.038		Sensitivity = [(-0.305 + 0.353)/10]/ [0.330/40] = 0.58
Range = 0.36		Range = 0.75		Range = 0.055		Range = 0.012		Range = 0.187

Q_frags23	v_lurch	Q_cl	v_lurch	f_kefrag1	v_lurch	f_kebody	v_lurch	f_I	v_lurch
								1.06	-0.346
50	-0.378	25	-0.877					1.07	-0.343
55	-0.367	30	-0.711					1.08	-0.339
60	-0.355	35	-0.567	1.0	-0.379	1.0	-0.330	1.09	-0.336
65	-0.343	40	-0.442	1.1	-0.360	1.1	-0.330	1.10	-0.333
70	-0.330	45	-0.330	1.25	-0.330	1.2	-0.330	1.11	-0.330
75	-0.317	50	-0.231	1.4	-0.301	1.3	-0.330	1.12	-0.327
80	-0.304	55	-0.141	1.5	-0.281	1.4	-0.330	1.13	-0.324
85	-0.290	60	-0.060					1.14	-0.322
90	-0.276	65	+0.014					1.15	-0.319
								1.16	-0316
Sensitivity = [(-0.317 + 0.343)/10]/ [0.330/70] = 0.55		Sensitivity = [(-0.231 + 0.442)/10]/ [0.330/45] = 2.88		Sensitivity = [(-0.301 + 0.360)/0.3]/ [0.330/1.25] = 0.74		Sensitivity = [(-0.330 + 0.330)/0.2]/ [0.330/1.2] = 0.000		Sensitivity = [(-0.316 + 0.346)/0.1]/ [0.330/1.11] = 1.01
Range = 0.102		Range = 0.89		Range = 0.098		Range = 0.000		Range = 0.03

d_transit	v_lurch	d_snap	v_lurch	d_frag1	v_lurch	t_delay	v_lurch
0	-0.363	1.4	-1.842	2	-2.865
1	-0.355	1.6	-1.454	3	-1.573
2	-0.346	1.8	-1.071	4	-0.940	0.000	-0.528
3	-0.338	2	-0.695	5	-0.570	0.001	-0.428
4	-0.330	2.2	-0.330	6	-0.330	0.002	-0.330
5	-0.322	2.4	+0.006	7	-0.166	0.003	-0.234
6	-0.314	2.6	+0.254	8	-0.048	0.004	-0.141
7	-0.306	2.8	-0.466	9	+0.038	0.005	-0.050
8	-0.298	3.0	+2.033	10	+0.103	0.006	+0.036
9	-0.290
Sensitivity = [(-0.322 + 0.338)/2]/ [0.330/4] = 0.097		Sensitivity = [(0.006 + 0.695)/0.4]/ [0.330/2.2] = 11.68		Sensitivity = [(-0.166 + 0.570)/2]/ [0.330/6] = 3.67		Sensitivity = [(-0.234 + 0.428)/0.002]/ [0.330/0.002] = 0.59
Range = 0.073		Range = 3.88		Range = 2.97		Range = 0.56

R_bullet	v_lurch	R_head	v_lurch
4.75	+0.269	3.50	-1.241
5.00	+0.188	3.75	-1.008
5.25	+0.021	4.00	-0.778
5.5	-0.157	4.25	-0.551
5.75	-0.330	4.50	-0.330
6.00	-0.495	4.75	-0.119
6.25	-0.650	5.00	+0.074
6.50	-0.795	5.25	+0.226
6.75	-0.931	5.50	+0.253

Sensitivity = [(-0.495 + 0.157)/0.5]/ [0.330/5.75] = -11.78		Sensitivity = [(-0.119 + 0.551)/0.5]/ [0.330/4.5] = 11.78
Range = 1.20		Range = 1.49

Ordered summary of sensitivities

Variable	Sensitivity of v_lurch	*Range of v_lurch, ft s^-1*	Magnitude
Positive effect on lurch (reduces rearward velocity)
f_kebody	0.000	0.00	Small
m_frags23	0.038	0.01	Small
f_I	1.01	0.03	Small
d_transit	0.097	0.07	Small
Q_frags23	0.55	0.10	Small
f_kefrag1	0.78	0.10	Small
Q_frag1	0.58	0.16	Small
Q	0.88	0.16	Small
m_body	1.00	0.16	Small
t_delay	0.59	0.56	Medium
PE	0.84	0.55	Medium
v_bulletafter	0.1.22	0.75	Medium
R_head	11.78	1.49	Large
m_head	11.73	1.78	Large
Q_cl	2.88	1.89	Large
d_frag1	3.67	2.97	Large
d_snap	11.68	3.88	Large
Negative effect on lurch (increases rearward velocity)
m_frag1	-0.45	0.06	Small
m_bullet	-13.42	0.27	Small
v_bullet	-19.36	0.36	Small
R_bullet	-11.78	1.20	Large

Summary

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