Determination of the Optimal Replacement Age for the Preventive Maintenance of Bearing Assemblies Involving Weibull Failure Probability Distribution Functions

This paper describes the determination of optimal replacement age for bearing systems with failure times described by Weibull distributions. Optimal age replacement policies are determined for bearing assemblies produced two different types of steel. The parameters of the Weibull probability distribution functions for the bearings were determined by non-linear least square analysis from published data on rolling contact fatigue lives. The resulting distribution functions are used as inputs into the standard expression for the maintenance cost of an age replacement policy and manipulated symbolically using the computer program Maple. These yields closed form expressions for the policy costs that invariably exhibit the well-known vase shape characteristic of these types of problems. The resulting expressions can then be easily used to determine the optimal replacement age of the bearing components. The problem of determining the optimal age replacement policy of bearing assembles consisting of independent components arranged in series is also examined. The effect on the optimal age replacement time of using the same type steel to manufacture all the components is compared with that of building the bearing assembly using components made with different steels. As expected, the results clearly show the increased superiority of the higher-quality steel components in the form of much longer optimal replacement ages. However, replacement policy efficiencies depend on replacement time in a complex fashion. Moreover, the results also suggest that building bearing assemblies combining high quality and low quality steel components may be wasteful. Overall, computer experimentation and examination of the behavior of cost functions using symbolic manipulation with Maple can produce useful guidelines for the design of optimal age replacement policies.