18. Lurch 3 angular

The model and its justification
Lurch 3 Angular calculates the speed of the rearward lurch of the upper body from the bullet, the body, the cloud, the large fragments, and conical 3-D motion of the diffuse cloud. As with the linear analogs, the three-dimensionality of the cone decreases the intensity of the lurch by about half, from the 2.94 ft s^-1 of Lurch 2 Angular to 1.44 ft s^-1 here. This shows how important the three-dimensionality of the cloud is to the final solution.

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 = -4.22		KE_bodyafter = 1
	Ω_cloud = 6.45		KE_cloud = 704
	Ω_frag1 = 0.97		KE_frag1 = 105
	Ω_frags23 = 0.16		KE_frags23 = 39
			PE = 300

m_bullet	v_lurch	m_body	v_lurch	m_cloud	v_lurch	PE	v_lurch	v_bullet	v_lurch
						0	-1.861
156	-1.414					50	-1.795	1750	-1.369
157	-1.419	65	-1.878			100	-1.727	1760	-1.383
158	-1.423	70	-1.744			150	-1.657	1770	-1.396
159	-1.427	75	-1.627	0.1	-0.507	200	-1.586	1780	-1.409
160	-1.431	80	-1.525	0.2	-1.032	250	-1.512	1790	-1.422
161	-1.435	85	-1.435	0.3	-1.435	300	-1.435	1800	-1.435
162	-1.439	90	-1.355	0.4	-1.776	350	-1.356	1810	-1.448
163	-1.443	95	-1.284	0.5	-2.077	400	-1.274	1820	-1.461
164	-1.447	100	-1.219	0.6	-2.350	450	-1.188	1830	-1.474
165	-1.451	105	-1.161	0.7	-2.601	500	-1.098	1840	-1.486
166	-1.455			0.8	-2.835	550	-1.004	1850	-1.499
						600	-0.904
Sensitivity = [(-1.439 + 1.431)/2]/ [1.435/161] = -0.45		Sensitivity = [(-1.355 + 1.525)/10]/ [1.435/85] = 1.01		Sensitivity = [(-1.776 + 1.032)/0.2]/ [1.435/0.3] = -0.78		Sensitivity = [(-1.356 + 1.512)/100]/ [1.435/300] = 0.33		Sensitivity = [(-1.448 + 1.422)/20]/ [1.435/1800] = -1.63
Range = 0.04		Range = 0.72		Range = 2.33		Range = 0.96		Range = 0.13

v_bulletafter	v_lurch	Q	v_lurch	Q_cl	v_lurch	m_frag1	v_lurch	m_frags23	v_lurch
		7	-1.416	20	-2.796	0.022	-1.404	0.005	-1.438
0	-1.311	8	-1.419	30	-2.619	0.023	-1.411	0.006	-1.438
50	-1.346	9	-1.423	40	-2.382	0.024	-1.417	0.007	-1.437
100	-1.379	10	-1.426	50	-2.097	0.025	-1.423	0.008	-1.436
150	-1.408	11	-1.431	60	-1.777	0.026	-1.429	0.009	-1.436
200	-1.435	12	-1.435	70	-1.435	0.027	-1.435	0.010	-1.435
250	-1.459	13	-1.440	80	-1.086	0.028	-1.441	0.011	-1.435
300	-1.480	14	-1.446	90	-0.744	0.029	-1.447	0.012	-1.434
350	-1.498	15	-1.451	100	-0.422	0.030	-1.454	0.013	-1.433
400	-1.514	16	-1.457	110	-0.129	0.031	-1.460	0.015	-1.433
		17	-1.464	120	+0.126	0.032	-1.466	0.015	-1.432
Sensitivity = [(-1.459 + 1.408)/100]/ [1.435/200] = -0.07		Sensitivity = [(-1.446 + 1.426)/4]/ [1.435/12] = -0.04		Sensitivity = [(-1.086 + 1.777)/20]/ [1.435/70] = 1.68		Sensitivity = [-1.441 + 1.429)/0.002]/ [1.435/0.027] = -0.11		Sensitivity = [(-1.432 + 1.438)/0.010]/ [1.435/0.010] = 0.00
Range = 0.20		Range = 0.05		Range = 2.92		Range = 0.06		Range = 0.01

v_frag1	v_lurch	v_frags23	v_lurch	Q_frag1	v_lurch	Q_frags23	v_lurch	f_I	v_lurch
250	-1.390	250	-1.453					1.06	-1.503
300	-1.406	300	-1.452	20	-1.510	50	-1.483	1.07	-1.48
350	-1.419	350	-1.450	25	-1.496	55	-1.472	1.08	-1.475
400	-1.428	400	-1.446	30	-1.478	60	-1.460	1.09	-1.462
450	-1.433	450	-1.441	35	-1.458	65	-1.448	1.10	-1.448
500	-1.435	500	-1.435	40	-1.435	70	-1.435	1.11	-1.435
550	-1.433	550	-1.428	45	-1.410	75	-1.422	1.12	-1.422
600	-1.428	600	-1.419	50	-1.382	80	-1.408	1.13	-1.410
650	-1.418	650	-1.409	55	-1.352	85	-1.395	1.14	-1.397
700	-1.403	700	-1.398	60	-1.321	90	-1.381	1.15	-1.385
700	-1.384	750	-1.385					1.16	-1.373
Sensitivity = [(-1.428 + 1.428)/200]/ [1.435/500] = 0.00		Sensitivity = [(-1.428 + 1.441)/100]/ [1.435/500] = 0.04		Sensitivity = [(-1.410 + 1.458)/10]/ [1.435/40] = 0.13		Sensitivity = [(-1.422 + 1.448)/10]/ [1.435/70] = 0.13		Sensitivity = [(-1.422 + 1.448)/0.02]/ [1.435/1.11] = 1.01
Range = 0.01		Range = 0.07		Range = 0.19		Range = 0.10		Range = 0.13

This ordering of sensitivities and effects is virtually identical to that of Lurch 3 Linear, except with greater effects. It is also very similar to the previous one, except for the new large effect for Q_cl. The other large effect, for m_cloud, has also been seen before. That both these large effects belong to properties of the diffuse cloud is an important feature of all subsequent simulations.

Summary
Making the diffuse cloud into a conical shape reduces the strength of the rearward lurch by a factor of two (2.9 ft s^-1 to 1.4 ft s^-1) and makes the half-angle of the cone one of the two most-sensitive variables, the mass of the cloud being the other one. This both brings the calculated lurch nearly into line with the observed lurch and emphasizes the all-importance of the properties of the cloud. Unfortunately for this work, the properties of the cloud are very difficult to estimate, and probably form the limiting factor for the accuracy of this series of simulations.

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 = 12°	Q_frags23 = 70°	v_frags23 = 500 ft s^-1
PE = 300 ft-lb	Q_cl = 70°	f_I = 1.11

Variable	Sensitivity of v_lurch	*Range of v_lurch, ft s^-1*	Magnitude
Positive effect on lurch (reduces rearward velocity)
m_frags23	0.00	0.01	Small
v_frag1	0.00	0.01	Small
v_frags23	0.04	0.07	Small
Q_frags23	0.13	0.10	Small
f_I	1.01	0.13	Small
Q_frag1	0.13	0.19	Small
m_body	1.01	0.72	Medium
PE	0.33	0.96	Medium
Q_cl	1.68	2.92	Large
Negative effect on lurch (increases rearward velocity)
m_bullet	-0.45	0.04	Small
Q	-0.04	0.05	Small
m_frag1	-0.11	0.06	Small
v_bullet	-1.63	0.13	Small
v_bulletafter	-0.07	0.20	Small
m_cloud	-0.78	2.33	Large