A steel ball bearing consisting of an inner race, an outer race, and 12 balls is shown in the following Figure. (Material properties of balls and race E=210 GPa, v=0.29)
A rated load of P_{0}=3600 N is given in the manufacturer's data sheet for this ball bearing when operated at 2500 rpm. An empirical relation
[Ref-1] is used to determine the load P on the topmost ball that bears the largest portion of the load; P = 5P_{0}/n =
1500 N in which n is the number of balls.

At the region of contact between the inner race and topmost ball, determine the maximum shear stress, maximum Hertzian contact pressure and dimensions of the area of contact.

INPUT PROPERTIES SUMMARY | |||

Parameter | Symbol | Value | Unit |

Poisson's ratio of ball |
v_{1} |
0.29 | --- |

Poisson's ratio of race |
v_{2} |
0.29 | --- |

Elastic modulus of ball |
E_{1} |
210 | GPa |

Elastic modulus of race |
E_{2} |
210 | GPa |

Radius of ball |
R_{1} |
4.90 | mm |

Inner race radius |
R_{2} |
5 | mm |

Inner race radius |
R_{3} |
25 | mm |

Maximum load on a bearing ball | P | 1500 | N |

RESULTS | ||||

Parameter | Symbol | Obj-1 | Obj-2 | Unit |

Maximum Hertzian contact pressure |
p_{max} |
2193.6 | MPa | |

Maximum shear stress |
τ_{max} |
684.9 | ||

Rigid distance of approach of contacting bodies | d | 0.012 | mm | |

Semimajor axis of contact ellipse | a | 2.101 | ||

Semiminor axis of contact ellipse | b | 0.156 |

Contact stresses of steel ball bearing have been calculated. The maximum Hertzian contact pressure and shear stress are calculated as 2193.6 MPa and 684.9 MPa respectively.

- Boresi A.P. , Schmidt R.J.. (2003) . Advanced Mechanics of Materials . 6th edition. John Wiley & Sons, Inc.

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