hw7.sol.Fall.2010x

# hw7.sol.Fall.2010x - ME 452 Machine Design II Solution to...

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1 ME 452: Machine Design II Solution to Homework Set 7 Problem 1 An 02-series ball bearing is to be selected to carry a radial load of 8 kN and a thrust load of 4 kN. The desired life, L D , is to be 5000 hours with the inner ring rotating at 900 rpm. Select the smallest acceptable single-row deep-groove ball-bearing from Table (11-2), page 581, that sustains these loads with 90% reliability? Assume bearing manufacturer number 2 (see the Table of manufacturers on page 608). SOLUTION: The desired life of the bearing in revolutions, see equation (b) page 575, can be written as 6 60 60(5000)(900) 270 10 revs L n = = = × L The desired dimensionless life measure, see Equation (11-4), page 576, and Example 11-3, page 578, can be written as 10 D L x L = From the Table on page 608, the 10 L life (or the rating life) for the bearings from Manufacturer 2 is 6 10 1 10 revs L = × Therefore, the desired dimensionless life measure is 6 6 270 10 270 10 D x × = = The dynamic load rating (or the catalog load rating) for the bearing, see Equation (11-6), page 578, can be written as 1/ 10 1/ 0 0 ( )(ln(1/ )) a D D b D x C F x x R θ = + - where the desired value of the reliability is specified as 0.9 D R = This equation can be re-written into a convenient form: 10 1 a D D C x a F = Where the reliability adjustment factor a 1 is: 1/ 1 0 0 ( )(ln(1/ )) b D a x x R θ = + -

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2 The Weibull parameters for the bearings from Manufacturer 2, see the Table on page 608, are: 0 0.02, 4.459 and 1.483 θ = = = x b where 0 x is the minimum value of the dimensionless life, θ is the characteristic parameter and b is the shape parameter. Therefore: 1/1.483 1 0.02 (4.459 0.02)(ln(1/ 0.9)) 0.9933 a = + - = Notice that the reliability adjustment factor is basically 1.0 for 90% reliability, because the C 10 rating for bearings defined at 90% reliability (10% failures). So for 90% reliability, equation (11-6) page 578 reduces to: 10 a D D C x F = To make an initial selection of the bearing, the catalog load rating can be estimated using the radial load r F only (see Example 11-7, page 589, part b). Therefore, the desired radial load is assumed to be the same as the specified radial load; i.e., 8 kN D r F F = = For a ball bearing, the exponent a , see page 574, is specified as 3 a = Substituting the above values into Equation (11-6), the catalog load rating is 1/ 1/3 10 10 or 8(270) 51.7 kN a a D r D r C x C F x F = = = = Selection of the Single-Row 02-Series Deep-Groove Ball Bearing.
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