A gear-reduction unit uses the countershaft depicted in the figure. Find the two bearing reactions. The bearings are to be angular-contact ball bearings, having a desired life of 50 kh when used at 300 rev/min. Use 1.2 for the application factor and a reliability goal for the bearing pair of 0.96.

Question

A gear-reduction unit uses the countershaft depicted in the figure. Find the two bearing reactions. The bearings are to be angular-contact ball bearings, having a desired life of 50 kh when used at 300 rev/min. Use 1.2 for the application factor and a reliability goal for the bearing pair of 0.96. Select the bearings from Table 11–2.

Table 11–2 Dimensions and Load Ratings for Single-Row 02-Series Deep-Groove and Angular-Contact Ball Bearings	Bore, mm	OD, mm	Width, m m	Fillet Radius, mm	shoulder Diameter, mm		Load Ratings, kN
					shoulder Diameter, mm		Deep Groove		Angular Contact
					[latex]d _{ s }[/latex]	[latex]d _{ H }[/latex]	[latex]C_{10}[/latex]	[latex]C_{0}[/latex]	[latex]C_{10}[/latex]	[latex]C_{0}[/latex]
	10	30	9	0.6	12.5	27	5.07	2.24	4.94	2.12
	12	32	10	0.6	14.5	28	6.89	3.10	7.02	3.05
	15	35	11	0.6	17.5	31	7.80	3.55	8.06	3.65
	17	40	12	0.6	19.5	34	9.56	4.50	9.95	4.75
	20	47	14	1.0	25	41	12.7	6.20	13.3	6.55
	25	52	15	1.0	30	47	14.0	6.95	14.8	7.65
	30	62	16	1.0	35	55	19.5	10.0	20.3	11.0
	35	72	17	1.0	41	65	25.5	13.7	27.0	15.0
	40	80	18	1.0	46	72	30.7	16.6	31.9	18.6
	45	85	19	1.0	52	77	33.2	18.6	35.8	21.2
	50	90	20	1.0	56	82	35.1	19.6	37.7	22.8
	55	100	21	1.5	63	90	43.6	25.0	46.2	28.5
	60	110	22	1.5	70	99	47.5	28.0	55.9	35.5
	65	120	23	1.5	74	109	55.9	34.0	63.7	41.5
	70	125	24	1.5	79	114	61.8	37.5	68.9	45.5
	75	130	25	1.5	86	119	66.3	40.5	71.5	49.0
	80	140	26	2.0	93	127	70.2	45.0	80.6	55.0
	85	150	28	2.0	99	136	83.2	53.0	90.4	63.0
	90	160	30	2.0	104	146	95.6	62.0	106	73.5
	95	170	32	2.0	110	156	108	69.5	121	85.0

Accepted Answer

[latex]R=\sqrt{0.96}=0.980[/latex]

[latex]T=12\left(240 \cos 20^{\circ} ight)=2706 lbf \cdot in[/latex]

[latex]F=\frac{2706}{6 \cos 25^{\circ}}=498 lbf[/latex]

In xy-plane:

[latex]\Sigma M_{O}^{z}=-16(82.1)-30(210)+42 R_{C}^{y}=0[/latex]

[latex]\begin{aligned}&R_{C}^{y}=181 lbf \&R_{O}^{y}=82.1+210-181=111.1 lbf\end{aligned}[/latex]

In xz-plane:

[latex]\begin{aligned}&\Sigma M_{O}^{y}=16(226)-30(451)-42 R_{C}^{z}=0 \&R_{C}^{z}=-236 lbf \&R_{O}^{z}=226-451+236=11 lbf \&R_{O}=\left(111.1^{2}+11^{2} ight)^{1 / 2}=112 lbf \&R_{C}=\left(181^{2}+236^{2} ight)^{1 / 2}=297 lbf \&x_{D}=\frac{50000(300)(60)}{10^{6}}=900\end{aligned}[/latex]

[latex]\begin{aligned}\left(C_{10} ight)_{o} &=1.2(112)\left\{\frac{900}{0.02+4.439[\ln (1 / 0.980)]^{1 / 1.483}} ight\}^{1 / 3} \&=1860 lbf =8.28 kN\end{aligned}[/latex]

[latex]\begin{aligned}\left(C_{10} ight)_{C} &=1.2(297)\left\{\frac{900}{0.02+4.439[\ln (1 / 0.980)]^{1 / 1.483}} ight\}^{1 / 3} \&=4932 lbf =21.9 kN\end{aligned}[/latex]

Bearing at O: Choose a deep-groove 02-17 mm.
Bearing at C: Choose a deep-groove 02-35 mm.

Question 11.34: A gear-reduction unit uses the countershaft depicted in the ...

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