FINAL REPORT FOR THE METHODOLOGY STUDY OF THE CONSUMER BRAKING INFORMATION INITIATIVE

APPENDIX K. GRADE EFFECTS ANALYSIS

Known: Vehicle velocity at 0.1 second intervals with no surface grade.

Note: Refer to the attached spreadsheets when reviewing the process below.

1. Calculate stopping distance with no grade.

a. Determine deceleration (a1)

a1= D v1/ D t

b. Determine stopping force with no grade

F1=m*a1

c. Calculate stopping distance with no grade

s1_n=s1 _n-1+v1_n-1* D t

2. Calculate effects on stopping distance due to grade.

a. Calculate change in stopping force due to grade

b. Calculate change in deceleration due to grade

a2=F2/m Downgrade deceleration

a3=F3/m Upgrade deceleration

c. Calculate change in velocity due to grade

v2=a2*Dt Downgrade velocity

v3=a3*Dt Upgrade velocity

d. Calculate stopping distance as effected by grade

s2_n=s2_n-1+v2_n-1*Dt Downgrade stopping distance

s3_n=s3_n-1+v3_n-1*Dt Upgrade stopping distance

3. Refine results by additional iteration. Replace v1 with v2 and v3 in Equation 1a above until stopping distance converges. These results converged after three iterations.

	Vehicle Mass, slug		167.10
	Surface grade, deg		0.60

Time	v1	v1	a1	F1	s1	F2	F3	a2	a3	v2	v3	s2	s3
(sec)	(mph)	(ft/s)	(ft/s^2)	(lbf)	(ft)	(lbf)	(lbf)	ft/s^2	ft/s^2	ft/s	ft/s	(ft)	(ft)
0.00	61.00	89.66	0.00	0.00	0.00	56.34	-56.34	0.00	0.00	89.66	89.66	0.00	0.00
0.10	60.88	89.49	-1.78	-296.73	8.97	-240.38	-353.07	-1.44	-2.11	89.52	89.45	8.97	8.97
0.20	59.79	87.89	-15.98	-2670.07	17.92	-2613.73	-2726.42	-15.64	-16.32	87.95	87.82	17.92	17.91
0.30	57.61	84.69	-31.96	-5340.64	26.70	-5284.29	-5396.98	-31.62	-32.30	84.79	84.59	26.71	26.69
0.40	55.44	81.50	-31.96	-5340.39	35.17	-5284.05	-5396.74	-31.62	-32.30	81.63	81.36	35.19	35.15
0.50	52.78	77.59	-39.06	-6527.31	43.32	-6470.97	-6583.66	-38.73	-39.40	77.75	77.42	43.35	43.29
0.60	50.49	74.22	-33.74	-5637.12	51.08	-5580.78	-5693.47	-33.40	-34.07	74.42	74.01	51.13	51.03
0.70	48.68	71.55	-26.63	-4450.21	58.50	-4393.86	-4506.55	-26.29	-26.97	71.79	71.31	58.57	58.43
0.80	46.86	68.89	-26.63	-4450.45	65.66	-4394.11	-4506.80	-26.30	-26.97	69.16	68.62	65.75	65.56
0.90	44.93	66.05	-28.41	-4747.18	72.55	-4690.84	-4803.53	-28.07	-28.75	66.35	65.74	72.67	72.42
1.00	42.64	62.68	-33.74	-5637.12	79.15	-5580.78	-5693.47	-33.40	-34.07	63.01	62.33	79.30	79.00
1.10	40.22	59.13	-35.51	-5933.85	85.42	-5877.51	-5990.20	-35.17	-35.85	59.49	58.75	85.60	85.23
1.20	38.29	56.28	-28.41	-4747.18	91.33	-4690.84	-4803.53	-28.07	-28.75	56.68	55.88	91.55	91.11
1.30	36.24	53.27	-30.18	-5043.66	96.96	-4987.32	-5100.01	-29.85	-30.52	53.70	52.82	97.22	96.69
1.40	33.82	49.72	-35.51	-5933.85	102.29	-5877.51	-5990.20	-35.17	-35.85	50.18	49.24	102.59	101.98
1.50	31.52	46.34	-33.74	-5637.12	107.26	-5580.78	-5693.47	-33.40	-34.07	46.84	45.83	107.61	106.90
1.60	29.35	43.15	-31.96	-5340.39	111.89	-5284.05	-5396.74	-31.62	-32.30	43.68	42.60	112.29	111.48
1.70	27.54	40.48	-26.63	-4450.45	116.21	-4394.11	-4506.80	-26.30	-26.97	41.05	39.90	116.66	115.74
1.80	25.12	36.93	-35.51	-5933.85	120.26	-5877.51	-5990.20	-35.17	-35.85	37.53	36.32	120.76	119.73
1.90	23.07	33.91	-30.18	-5043.91	123.95	-4987.57	-5100.25	-29.85	-30.52	34.55	33.27	124.52	123.36
2.00	20.78	30.54	-33.74	-5637.12	127.34	-5580.78	-5693.47	-33.40	-34.07	31.21	29.86	127.97	126.69
2.10	18.96	27.88	-26.63	-4450.45	130.40	-4394.11	-4506.80	-26.30	-26.97	28.58	27.16	131.09	129.68
2.20	16.79	24.68	-31.96	-5340.39	133.18	-5284.05	-5396.74	-31.62	-32.30	25.42	23.93	133.95	132.39
2.30	14.86	21.84	-28.41	-4746.94	135.65	-4690.59	-4803.28	-28.07	-28.74	22.61	21.06	136.49	134.79
2.40	12.80	18.82	-30.18	-5043.91	137.84	-4987.57	-5100.25	-29.85	-30.52	19.63	18.01	138.75	136.89
2.50	10.63	15.62	-31.96	-5340.39	139.72	-5284.05	-5396.74	-31.62	-32.30	16.46	14.78	140.72	138.69
2.60	8.33	12.25	-33.74	-5637.12	141.28	-5580.78	-5693.47	-33.40	-34.07	13.12	11.37	142.36	140.17
2.70	6.28	9.23	-30.18	-5043.91	142.50	-4987.57	-5100.25	-29.85	-30.52	10.14	8.32	143.68	141.31
2.80	4.11	6.04	-31.96	-5340.39	143.43	-5284.05	-5396.74	-31.62	-32.30	6.98	5.09	144.69	142.14
2.90	1.93	2.84	-31.96	-5340.39	144.03	-5284.05	-5396.74	-31.62	-32.30	3.81	1.86	145.39	142.65
3.00	-0.48	-0.71	-35.51	-5933.85	144.32	-5877.51	-5990.20	-35.17	-35.85	0.30	-1.73	145.77	142.83

												1.45	-1.48

2nd Iteration
Downgrade								Upgrade
v1	a1	F1	s1	F2	a2	v2	s2	v1	a1	F1	s1	F3	a3	v3	s3
(ft/s)	(ft/s^2)	(lbf)	(ft)	(lbf)	ft/s^2	ft/s	(ft)	(ft/s)	(ft/s^2)	(lbf)	(ft)	(lbf)	ft/s^2	ft/s	(ft)
89.66	0.00	0.00	0.00	56.34	0.00	89.66	0.00	89.66	0.00	0.00	0.00	-56.34	0.00	89.66	0.00
89.52	-1.44	-240.38	8.97	-184.04	-1.10	89.55	8.97	89.45	-2.11	-353.07	8.97	-409.42	-2.45	89.41	8.97
87.95	-15.64	-2613.73	17.92	-2557.38	-15.30	88.02	17.92	87.82	-16.32	-2726.42	17.91	-2782.76	-16.65	87.75	17.91
84.79	-31.62	-5284.29	26.71	-5227.95	-31.29	84.89	26.72	84.59	-32.30	-5396.98	26.69	-5453.33	-32.64	84.49	26.68
81.63	-31.62	-5284.05	35.19	-5227.70	-31.28	81.76	35.21	81.36	-32.30	-5396.74	35.15	-5453.08	-32.63	81.22	35.13
77.75	-38.73	-6470.97	43.35	-6414.62	-38.39	77.92	43.39	77.42	-39.40	-6583.66	43.29	-6640.00	-39.74	77.25	43.25
74.42	-33.40	-5580.78	51.13	-5524.43	-33.06	74.62	51.18	74.01	-34.07	-5693.47	51.03	-5749.81	-34.41	73.81	50.98
71.79	-26.29	-4393.86	58.57	-4337.52	-25.96	72.02	58.64	71.31	-26.97	-4506.55	58.43	-4562.89	-27.31	71.08	58.36
69.16	-26.30	-4394.11	65.75	-4337.76	-25.96	69.43	65.84	68.62	-26.97	-4506.80	65.56	-4563.14	-27.31	68.35	65.47
66.35	-28.07	-4690.84	72.67	-4634.49	-27.73	66.65	72.79	65.74	-28.75	-4803.53	72.42	-4859.87	-29.08	65.44	72.30
63.01	-33.40	-5580.78	79.30	-5524.43	-33.06	63.35	79.45	62.33	-34.07	-5693.47	79.00	-5749.81	-34.41	62.00	78.85
59.49	-35.17	-5877.51	85.60	-5821.16	-34.84	59.86	85.79	58.75	-35.85	-5990.20	85.23	-6046.54	-36.19	58.38	85.05
56.68	-28.07	-4690.84	91.55	-4634.49	-27.73	57.09	91.77	55.88	-28.75	-4803.53	91.11	-4859.87	-29.08	55.47	90.88
53.70	-29.85	-4987.32	97.22	-4930.98	-29.51	54.14	97.48	52.82	-30.52	-5100.01	96.69	-5156.35	-30.86	52.38	96.43
50.18	-35.17	-5877.51	102.59	-5821.16	-34.84	50.65	102.90	49.24	-35.85	-5990.20	101.98	-6046.54	-36.19	48.77	101.67
46.84	-33.40	-5580.78	107.61	-5524.43	-33.06	47.35	107.96	45.83	-34.07	-5693.47	106.90	-5749.81	-34.41	45.33	106.55
43.68	-31.62	-5284.05	112.29	-5227.70	-31.28	44.22	112.70	42.60	-32.30	-5396.74	111.48	-5453.08	-32.63	42.06	111.08
41.05	-26.30	-4394.11	116.66	-4337.76	-25.96	41.62	117.12	39.90	-26.97	-4506.80	115.74	-4563.14	-27.31	39.33	115.28
37.53	-35.17	-5877.51	120.76	-5821.16	-34.84	38.14	121.28	36.32	-35.85	-5990.20	119.73	-6046.54	-36.19	35.71	119.22
34.55	-29.85	-4987.57	124.52	-4931.22	-29.51	35.19	125.09	33.27	-30.52	-5100.25	123.36	-5156.60	-30.86	32.63	122.79
31.21	-33.40	-5580.78	127.97	-5524.43	-33.06	31.88	128.61	29.86	-34.07	-5693.47	126.69	-5749.81	-34.41	29.19	126.05
28.58	-26.30	-4394.11	131.09	-4337.76	-25.96	29.29	131.80	27.16	-26.97	-4506.80	129.68	-4563.14	-27.31	26.45	128.97
25.42	-31.62	-5284.05	133.95	-5227.70	-31.28	26.16	134.73	23.93	-32.30	-5396.74	132.39	-5453.08	-32.63	23.19	131.62
22.61	-28.07	-4690.59	136.49	-4634.25	-27.73	23.39	137.35	21.06	-28.74	-4803.28	134.79	-4859.62	-29.08	20.28	133.93
19.63	-29.85	-4987.57	138.75	-4931.22	-29.51	20.43	139.69	18.01	-30.52	-5100.25	136.89	-5156.60	-30.86	17.20	135.96
16.46	-31.62	-5284.05	140.72	-5227.70	-31.28	17.31	141.73	14.78	-32.30	-5396.74	138.69	-5453.08	-32.63	13.93	137.68
13.12	-33.40	-5580.78	142.36	-5524.43	-33.06	14.00	143.46	11.37	-34.07	-5693.47	140.17	-5749.81	-34.41	10.49	139.08
10.14	-29.85	-4987.57	143.68	-4931.22	-29.51	11.05	144.86	8.32	-30.52	-5100.25	141.31	-5156.60	-30.86	7.41	140.12
6.98	-31.62	-5284.05	144.69	-5227.70	-31.28	7.92	145.96	5.09	-32.30	-5396.74	142.14	-5453.08	-32.63	4.14	140.87
3.81	-31.62	-5284.05	145.39	-5227.70	-31.28	4.79	146.76	1.86	-32.30	-5396.74	142.65	-5453.08	-32.63	0.88	141.28
0.30	-35.17	-5877.51	145.77	-5821.16	-34.84	1.31	147.24	-1.73	-35.85	-5990.20	142.83	-6046.54	-36.19	-2.74	141.37

							1.47								-1.47

3rd Iteration
Downgrade								Upgrade
v1	a1	F1	s1	F2	a2	v2	s2	v1	a1	F1	s1	F3	a3	v3	s3
(ft/s)	(ft/s^2)	(lbf)	(ft)	(lbf)	ft/s^2	ft/s	(ft)	(ft/s)	(ft/s^2)	(lbf)	(ft)	(lbf)	ft/s^2	ft/s	(ft)
89.66	0.00	0.00	0.00	56.34	0.00	89.66	0.00	89.66	0.00	0.00	0.00	-56.34	0.00	89.66	0.00
89.48	-1.78	-296.73	8.97	-240.38	-1.44	89.52	8.97	89.48	-1.78	-296.73	8.97	-353.07	-2.11	89.45	8.97
87.88	-15.98	-2670.07	17.91	-2613.73	-15.64	87.95	17.92	87.88	-15.98	-2670.07	17.91	-2726.42	-16.32	87.82	17.91
84.69	-31.96	-5340.64	26.70	-5284.29	-31.62	84.79	26.71	84.69	-31.96	-5340.64	26.70	-5396.98	-32.30	84.59	26.69
81.49	-31.96	-5340.39	35.17	-5284.05	-31.62	81.63	35.19	81.49	-31.96	-5340.39	35.17	-5396.74	-32.30	81.36	35.15
77.59	-39.06	-6527.31	43.32	-6470.97	-38.73	77.75	43.35	77.59	-39.06	-6527.31	43.32	-6583.66	-39.40	77.42	43.29
74.21	-33.74	-5637.12	51.08	-5580.78	-33.40	74.42	51.13	74.21	-33.74	-5637.12	51.08	-5693.47	-34.07	74.01	51.03
71.55	-26.63	-4450.21	58.50	-4393.86	-26.29	71.79	58.57	71.55	-26.63	-4450.21	58.50	-4506.55	-26.97	71.31	58.43
68.89	-26.63	-4450.45	65.66	-4394.11	-26.30	69.16	65.75	68.89	-26.63	-4450.45	65.66	-4506.80	-26.97	68.62	65.56
66.05	-28.41	-4747.18	72.54	-4690.84	-28.07	66.35	72.67	66.05	-28.41	-4747.18	72.54	-4803.53	-28.75	65.74	72.42
62.67	-33.74	-5637.12	79.15	-5580.78	-33.40	63.01	79.30	62.67	-33.74	-5637.12	79.15	-5693.47	-34.07	62.33	79.00
59.12	-35.51	-5933.85	85.42	-5877.51	-35.17	59.49	85.60	59.12	-35.51	-5933.85	85.42	-5990.20	-35.85	58.75	85.23
56.28	-28.41	-4747.18	91.33	-4690.84	-28.07	56.68	91.55	56.28	-28.41	-4747.18	91.33	-4803.53	-28.75	55.88	91.11
53.26	-30.18	-5043.66	96.96	-4987.32	-29.85	53.70	97.22	53.26	-30.18	-5043.66	96.96	-5100.01	-30.52	52.82	96.69
49.71	-35.51	-5933.85	102.28	-5877.51	-35.17	50.18	102.59	49.71	-35.51	-5933.85	102.28	-5990.20	-35.85	49.24	101.98
46.34	-33.74	-5637.12	107.25	-5580.78	-33.40	46.84	107.61	46.34	-33.74	-5637.12	107.25	-5693.47	-34.07	45.83	106.90
43.14	-31.96	-5340.39	111.89	-5284.05	-31.62	43.68	112.29	43.14	-31.96	-5340.39	111.89	-5396.74	-32.30	42.60	111.48
40.48	-26.63	-4450.45	116.20	-4394.11	-26.30	41.05	116.66	40.48	-26.63	-4450.45	116.20	-4506.80	-26.97	39.90	115.74
36.93	-35.51	-5933.85	120.25	-5877.51	-35.17	37.53	120.76	36.93	-35.51	-5933.85	120.25	-5990.20	-35.85	36.32	119.73
33.91	-30.18	-5043.91	123.94	-4987.57	-29.85	34.55	124.52	33.91	-30.18	-5043.91	123.94	-5100.25	-30.52	33.27	123.36
30.53	-33.74	-5637.12	127.33	-5580.78	-33.40	31.21	127.97	30.53	-33.74	-5637.12	127.33	-5693.47	-34.07	29.86	126.69
27.87	-26.63	-4450.45	130.39	-4394.11	-26.30	28.58	131.09	27.87	-26.63	-4450.45	130.39	-4506.80	-26.97	27.16	129.68
24.68	-31.96	-5340.39	133.17	-5284.05	-31.62	25.42	133.95	24.68	-31.96	-5340.39	133.17	-5396.74	-32.30	23.93	132.39
21.83	-28.41	-4746.94	135.64	-4690.59	-28.07	22.61	136.49	21.83	-28.41	-4746.94	135.64	-4803.28	-28.74	21.06	134.79
18.82	-30.18	-5043.91	137.82	-4987.57	-29.85	19.63	138.75	18.82	-30.18	-5043.91	137.82	-5100.25	-30.52	18.01	136.89
15.62	-31.96	-5340.39	139.71	-5284.05	-31.62	16.46	140.72	15.62	-31.96	-5340.39	139.71	-5396.74	-32.30	14.78	138.69
12.25	-33.74	-5637.12	141.27	-5580.78	-33.40	13.12	142.36	12.25	-33.74	-5637.12	141.27	-5693.47	-34.07	11.37	140.17
9.23	-30.18	-5043.91	142.49	-4987.57	-29.85	10.14	143.68	9.23	-30.18	-5043.91	142.49	-5100.25	-30.52	8.32	141.31
6.03	-31.96	-5340.39	143.41	-5284.05	-31.62	6.98	144.69	6.03	-31.96	-5340.39	143.41	-5396.74	-32.30	5.09	142.14
2.84	-31.96	-5340.39	144.02	-5284.05	-31.62	3.81	145.39	2.84	-31.96	-5340.39	144.02	-5396.74	-32.30	1.86	142.65
-0.71	-35.51	-5933.85	144.30	-5877.51	-35.17	0.30	145.77	-0.71	-35.51	-5933.85	144.30	-5990.20	-35.85	-1.73	142.83

							1.47								-1.47