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Research Papers

An Optimum Design of Crowned Cylindrical Roller Bearings Using Genetic Algorithms

[+] Author and Article Information
K. Sunil Kumar, P. V. V. N. Prasad

Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India

Rajiv Tiwari1

Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, Indiartiwari@iitg.ernet.in

1

Corresponding author.

J. Mech. Des 131(5), 051011 (Apr 15, 2009) (14 pages) doi:10.1115/1.3116344 History: Received July 18, 2008; Revised December 31, 2008; Published April 15, 2009

The long fatigue life is the one of the most important criterion for the design of rolling bearings, however, due to complex and diverse internal geometries, each type of rolling bearings require a different design formulation. To increase the life of cylindrical roller bearings, the profile (or the crowning) of the roller plays an important role. A flat profile of the rolling element results in the edge stress concentrations at roller ends. A circular crowning of roller eliminates the edge stress concentration at the lower and moderate loads only; however, it develops edge stress concentrations at heavy loads. The logarithmic profile of the roller results in no edge stress concentration at the low, medium, and heavy loads; distribution of contact stresses is also nearly uniform along the length of the roller. A design methodology for the optimum design of cylindrical roller bearings with the logarithmic profile has been outlined. A nonlinear constrained optimization problem has been formulated for the design of cylindrical roller bearings with logarithmic profiles and is optimized by using real-coded genetic algorithms. The change in roller profile has not been accounted for explicitly in the standard definition of the dynamic capacity; hence, for the present case directly the Lundberg–Palmgren life equation has been chosen as an objective function. Design variables include four bearing geometrical parameters and the two logarithmic profile generating parameters are considered. In addition to these, another five design constraint constants are also included, which indirectly affect the fatigue life of cylindrical roller bearings. The five design constraint constants have been given bounds based on the parametric studies through initial optimization runs. The effective length of the roller is taken corresponding to the standard roller diameter, which has standard discrete dimensions. Constraint violation study has been performed to have an assessment of the effectiveness of each of the constraints. A convergence study has been carried out to ensure the global optimum point in the design. A sensitivity analysis of various geometric design parameters has been performed using the Monte Carlo simulation technique, in order to see changes in the fatigue life of the bearing. Illustrations show that the multiplier of the logarithmic profile deviation parameter has more effect on the fatigue life as compared with other geometric parameters.

Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

A typical cylindrical roller bearing cross section

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Figure 2

A logarithmic profiled roller with slicing

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Figure 3

A flow diagram of the optimum design of cylindrical roller bearings

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Figure 4

Variation in L with the population number in the first and final generations for the cylindrical roller bearing NU 202

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Figure 5

Variation in the average fitness with number of generations for the cylindrical roller bearing NU 202

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Figure 6

Variation in the best fitness with number of generations for the cylindrical roller bearing NU 202

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Figure 7

Variation in pitch diameter with the number of generations for the cylindrical roller bearing NU 202

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Figure 8

Variation in the roller diameter with generations for the cylindrical roller bearing NU 202

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Figure 9

Variation in the number of rollers with generations for the cylindrical roller bearing NU 202

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Figure 10

Variation in logarithmic profile parameter (h¯) with generations for the cylindrical roller bearing NU 202

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Figure 11

Variation in the corrective exponent (α) with generations for the cylindrical roller bearing NU 202

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Figure 12

Variation in the logarithmic crown drop along the roller axis for the cylindrical roller bearing NU 202

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Figure 13

Variation in the load distribution along the roller axis for the cylindrical roller bearing NU 202

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Figure 14

Optimum design drawing of a typical NU 202 (all dimensions are in millimeters)

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Figure 15

Constraints violation with population numbers for the last generation of a cylindrical roller bearing with logarithmic profile NU 202 (Constraints 9–16)

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Figure 16

Constraints violation with population numbers for the last generation of a cylindrical roller bearing with logarithmic profile NU 202 (Constraints 17–22)

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Figure 17

A flow diagram of the Monte Carlo simulation for sensitivity analysis

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