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

A transmission shaft, transmitting 8 kW of power at 400 rpm from a bevel gear G_{1} to a helical gear G_{2} and mounted on two taper roller bearings B_{1} \text { and } B_{2} 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 B_{1} \text { and } B_{2} is 40 mm. The load factor is 1.2 and the expected life for 90% of bearings is 10 000 h. Bearings B_{1} \text { and } B_{2} 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.

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

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 B_{1} ,

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

\therefore \quad R_{V 2}=1663 N .

Considering equilibrium of vertical forces,

3473-R_{V 1}-1663-439=0 .

\therefore \quad R_{V 1}=1371 N .

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

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

\therefore \quad R_{H 2}=2176.25 N .

Considering equilibrium of horizontal forces,

2176.25+3820-1265-R_{H 1}=0 .

\therefore \quad R_{H 1}=4731.25 N .

The radial forces acting on the bearing are as follows:

F_{r 1}=\sqrt{\left(R_{V 1}\right)^{2}+\left(R_{H 1}\right)^{2}} .

=\sqrt{(1371)^{2}+(4731.25)^{2}}=4926 N .

F_{r 2}=\sqrt{\left(R_{V 2}\right)^{2}+\left(R_{H 2}\right)^{2}} .

=\sqrt{(1663)^{2}+(2176.25)^{2}}=2739 N .

\text { and } \quad K_{a}=1319+1475=2794 N .

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

L_{10}=\frac{60 n L_{10 h }}{10^{6}}               (15.9)

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

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

F_{r A}=2739 N .

F_{r B}=4926 N .

K_{a}=2794 N .

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.

\text { Since } F_{r d}<F_{r B} \text { and } K_{a}>0 .

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

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

F_{a A}=F_{a B}+K_{a}=1539.38+2794=4333.38 N .

Step III Final check for selection
Bearing A

F_{r A}=2739 N \text { and } F_{a A}=4333.38 N .

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

P=0.4 F_{r}+Y F_{a} .

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

C=P\left(L_{10}\right)^{0.3} \text { (Load factor) } .

=8029(240)^{0.3}(1.2)=49877.66 N .

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

F_{r B}=4926 N \text { and } F_{a B}=1539.38 N .

F_{a B} / F_{r B}=(1539.38) /(4926)=0.31<e .

P=F_{r}=4926 N .

C=P\left(L_{10}\right)^{0.3} \text { (Load factor) } .

=4926(240)^{0.3}(1.2)=30601.24 N .

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

15.10
15.11

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