A transmission shaft, transmitting 8 kW of power at 400 rpm from a bevel gear G1 to a helical gear G2 and mounted on two taper roller bearings B1 and B2 is shown in Fig. 15.12(a). The gear tooth forces on the helical gear act at a pitch circle radius of 55 mm, while those on the bevel gear can be

Question

A transmission shaft, transmitting 8 kW of power at 400 rpm from a bevel gear [latex] G_{1} [/latex] to a helical gear [latex] G_{2} [/latex] and mounted on two taper roller bearings [latex] B_{1} ext { and } B_{2} [/latex] is shown in Fig. 15.12(a). The gear tooth forces on the helical gear act at a pitch circle radius of 55 mm, while those on the bevel gear can be assumed to act at the large end of the tooth at a radius of 50 mm. The diameter of the journal at the bearings [latex] B_{1} ext { and } B_{2} [/latex] is 40 mm. The load factor is 1.2 and the expected life for 90% of bearings is 10 000 h. Bearings [latex] B_{1} ext { and } B_{2} [/latex] are identical. The thrust force due to bevel and helical gears is taken by the bearing B2. Select suitable taper roller bearings for this application.

Accepted Answer

Given kW = 8 n = 400 rpm d = 40 mm
load factor = 1.2 [latex] L_{10 h }=10000 h [/latex].

Step I Radial and axial forces on bearings
The forces acting on the shaft in the vertical and horizontal planes are shown in Fig. 15.12(b).
Considering forces in the vertical plane and taking moments about the bearing [latex] B_{1} [/latex],

[latex] 3473(150)+439(100)-1319(50)-R_{V 2}(300)=0 [/latex].

[latex] herefore \quad R_{V 2}=1663 N [/latex].

Considering equilibrium of vertical forces,

[latex] 3473-R_{V 1}-1663-439=0 [/latex].

[latex] herefore \quad R_{V 1}=1371 N [/latex].

Considering forces in the horizontal plane and taking moments about the bearing [latex] B_{1} [/latex],

[latex] 3820(100)+1265(150)+1475(55)-R_{H 2}(300)=0 [/latex].

[latex] herefore \quad R_{H 2}=2176.25 N [/latex].

Considering equilibrium of horizontal forces,

[latex] 2176.25+3820-1265-R_{H 1}=0 [/latex].

[latex] herefore \quad R_{H 1}=4731.25 N [/latex].

The radial forces acting on the bearing are as follows:

[latex] F_{r 1}=\sqrt{\left(R_{V 1} ight)^{2}+\left(R_{H 1} ight)^{2}} [/latex].

[latex] =\sqrt{(1371)^{2}+(4731.25)^{2}}=4926 N [/latex].

[latex] F_{r 2}=\sqrt{\left(R_{V 2} ight)^{2}+\left(R_{H 2} ight)^{2}} [/latex].

[latex] =\sqrt{(1663)^{2}+(2176.25)^{2}}=2739 N [/latex].

[latex] ext { and } \quad K_{a}=1319+1475=2794 N [/latex].

Step II Tentative selection of bearing
From Eq. (15.9),

[latex] L_{10}=\frac{60 n L_{10 h }}{10^{6}} [/latex] (15.9)

[latex] L_{10}=\frac{60 n L_{10 h }}{10^{6}}=\frac{60(400)(10000)}{10^{6}} [/latex].

= 240 million rev.
For the purpose of referring to Figs 15.10 and 15.11, the bearing [latex] B_{2} [/latex] is called Bearing A, the bearing [latex] B_{1} [/latex] is called B, and face-to-face construction is selected. With these notations,

[latex] F_{r A}=2739 N [/latex].

[latex] F_{r B}=4926 N [/latex].

[latex] K_{a}=2794 N [/latex].

Trial 1 Tentatively, we select Bearing 30208.
Referring to Table 15.6

Y	e	Designation	C	B	D	d
1.6	0.37	32004 X	22 900	15	42	20
1.7	0.35	30204	26 000	15.25	47
2.0	0.30	30304	31 900	16.25	52
2.0	0.30	32304	41 300	22.25	52
1.4	0.43	32005 X	25 500	15	47	25
1.6	0.37	30205	29 200	16.25	52
1.05	0.57	32205 B	34 100	19.25	52
1.7	0.35	33205	44 000	22	52
2	0.30	30305	41 800	18.25	62
0.72	0.83	31305	35 800	18.25	62
2	0.30	32305	56 100	25.25	62
1.4	0.43	32006 X	33 600	17	55	30
1.6	0.37	30206	38 000	17.25	62
1.6	0.37	32206	47 300	21.25	62
1.05	0.57	32206 B	45 700	21.25	62
1.7	0.35	33206	60 500	25	62
1.9	0.31	30306	52 800	20.75	72
0.72	0.83	31306	44 600	20.75	72
1.9	0.31	32306	72 100	28.75	72
1.3	0.46	32007 X	40 200	18	62	35
1.6	0.37	30207	48 400	18.25	72
1.6	0.37	32207	61 600	24.25	72
1.05	0.57	32207 B	57 200	24.25	72
1.7	0.35	33207	79 200	28	72	35
1.9	0.31	30307	68 200	22.75	80
0.72	0.83	31307	57 200	22.75	80
1.9	0.31	32307	89 700	32.75	80
1.1	0.54	32307 B	88 000	32.75	80
1.6	0.37	32008 X	49 500	19	68	40
1.7	0.35	33108	74 800	26	75
1.6	0.37	30208	58 300	19.75	80
1.6	0.37	32208	70 400	24.75	80
1.7	0.35	33208	96 800	32	80
1.7	0.35	T2EE040	114 000	33	85
1.7	0.35	30308	80 900	25.25	90
0.72	0.83	31308	69 300	25.25	90
1.7	0.35	32308	110 000	35.25	90
1.5	0.40	32009 X	55 000	20	75	45
1.6	0.37	33109	79 200	26	80
1.5	0.40	30209	62 700	20.75	85
1.5	0.40	32209	74 800	24.75	85
1.5	0.40	33209	101 000	32	85
0.68	0.88	T7FC045	84 200	29	95
1.8	0.33	T2ED045	140 000	36	95
1.7	0.35	30309	101 000	27.25	100
0.72	0.83	31309	85 800	27.25	100
1.7	0.35	32309	132 000	38.25	100
1.1	0.54	32309 B	128 000	38.25	100
1.4	0.43	32010 X	57 200	20	80	50
1.9	0.31	33010	64 400	24	80
1.5	0.40	33110	80 900	26	85
1.4	0.43	30210	70 400	21.75	90
1.4	0.43	32210	76 500	24.75	90
1.5	0.40	33210	108 000	32	90
1.7	0.35	T2ED050	145 000	36	100
0.68	0.88	T7FC050	102 000	32	105
1.7	0.35	30310	117 000	29.25	110
0.72	0.83	31310	99 000	29.25	110
1.7	0.35	32310	161 000	42.25	110
1.1	0.54	32310 B	151 000	42.25	110
1.4	0.43	32012 X	76 500	23	95	60
1.8	0.33	33012	85 800	27	95
1.5	0.40	33112	110 000	30	100	60
1.5	0.40	30212	91 300	23.75	110
1.5	0.40	32212	119 000	29.75	110
1.5	0.40	33212	157 000	38	110
1.1	0.54	T5ED060	157 000	39	115
1.8	0.33	T2EE060	183 000	40	115
0.72	0.83	T7FC060	145 000	37	125	60
1.7	0.35	30312	161 000	33.5	130
0.72	0.83	31312	134 000	33.5	130
1.7	0.35	32312	216 000	48.5	130
1.1	0.54	32312 B	205 000	48.5	130
1.4	0.43	32014 X	95 200	25	110	70
2.1	0.28	33014	121 000	31	110
1.6	0.37	33114	161 000	37	120
1.4	0.43	30214	119 000	26.25	125
1.4	0.43	32214	147 000	33.25	125
1.5	0.40	33214	190 000	41	125
1.8	0.33	T2ED070	220 000	43	130
0.68	0.88	T7FC070	168 000	39	140
1.35	0.44	T4FE070	264 000	52	140
1.7	0.35	30314	209 000	38	150
0.72	0.83	31314	176 000	38	150
1.7	0.35	32314	275 000	54	150
1.1	0.54	32314 B	264 000	54	150
1.4	0.43	32016 X	128 000	29	125
2.1	0.28	33016	157 000	36	125
1.4	0.43	33116	168 000	37	130
1.4	0.43	30216	140 000	28.25	140
1.4	0.43	32216	176 000	35.25	140
1.4	0.43	33216	233 000	46	140
1.9	0.31	T2ED080	264 000	46	145
1.7	0.35	30316	255 000	42.5	170
0.72	0.83	31316	212 000	42.5	170
1.7	0.35	32316	358 000	61.5	170
1.1	0.54	32316 B	336 000	61.5	170
1.4	0.43	32018 X	157 000	32	140
2.2	0.27	33018	205 000	39	140
1.5	0.40	33118	238 000	45	150
1.8	0.33	T2ED090	270 000	46	155
1.4	0.43	30218	183 000	32.5	160	90
1.4	0.43	32218	238 000	42.5	160
1.7	0.35	30318	308 000	46.5	190
0.72	0.83	31318	251 000	46.5	190
1.7	0.35	32318	429 000	67.5	190
1.25	0.48	T4CB100	119 000	24	145	100
1.3	0.46	32020 X	161 000	32	150
2.1	0.28	33020	212 000	39	150
1.9	0.31	T2EE100	292 000	47	165
1.4	0.43	30220	233 000	37	180
1.4	0.43	32220	297 000	49	180
1.5	0.40	33220	402 000	63	180
1.7	0.35	30320	380 000	51.5	215
0.72	0.83	31320 X	352 000	56.5	215
1.7	0.35	32320	539 000	77.5	215
1.3	0.46	32030 X	347 000	48	225	150
1.4	0.43	30230	402 000	49	270
1.4	0.43	32230	682 000	77	270
1.7	0.35	30330	765 000	72	320
0.72	0.83	31330 X	837 000	82	320
1.5	0.40	32940	446 000	51	280	200
1.4	0.43	32040 X	704 000	70	310
1.4	0.43	30240	737 000	64	360
1.5	0.40	32240	1 140 000	104	360
1.5	0.40	32960	990000	76	420	300

e = 0.37 and Y = 1.6.

[latex] ext { Since } F_{r d}0 [/latex].

The load case is similar to Case 2(a) of Fig. 15.11.

[latex] F_{a B}=\frac{0.5 F_{r B}}{Y}=\frac{0.5(4926)}{1.6}=1539.38 N [/latex].

[latex] F_{a A}=F_{a B}+K_{a}=1539.38+2794=4333.38 N [/latex].

Step III Final check for selection
Bearing A

[latex] F_{r A}=2739 N ext { and } F_{a A}=4333.38 N [/latex].

[latex] F_{a A} / F_{r A}=(4333.38) /(2739)=1.58>e [/latex].

[latex] P=0.4 F_{r}+Y F_{a} [/latex].

= 0.4(2739) + 1.6 (4333.38) = 8029 N.

[latex] C=P\left(L_{10} ight)^{0.3} ext { (Load factor) } [/latex].

[latex] =8029(240)^{0.3}(1.2)=49877.66 N [/latex].

Bearing 30208 (C = 58 300 N) is suitable.
Bearing B

[latex] F_{r B}=4926 N ext { and } F_{a B}=1539.38 N [/latex].

[latex] F_{a B} / F_{r B}=(1539.38) /(4926)=0.31

[latex] P=F_{r}=4926 N [/latex].

[latex] C=P\left(L_{10} ight)^{0.3} ext { (Load factor) } [/latex].

[latex] =4926(240)^{0.3}(1.2)=30601.24 N [/latex].

Bearing 30208 (C = 58 300 N) is suitable.

Question 15.8: A transmission shaft, transmitting 8 kW of power at 400 rpm ...

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