Research On Weibull Distribution And Its Applications In Mechanical Reliability Engineering

The Weibull distribution is a continuous distribution which can adequately describe observed failure data of many different types of components and phenomena. It has been used for various purposes, such as lifetime analysis, reliability based design, fatigue reliability analysis, maintenance planning, replacement policy evaluation, and so on. The standard Weibull distribution model has two parameters or three parameters. Many different methods can be employed to estimate the parameters of the Weibull distribution. These methods can be classified into two categories, the graphical methods and the statistical methods. The graphical methods include techniques using the empirical cumulative distribution plot, Weibull probability plot, hazard rate plot, and so on. The statistical methods include maximum likelihood estimation, regression method, and so on. In the condition of small sample size, these methods can not obtain proper estimates.For some reasons, failure times of different failure modes or different quality levels are mixed in a single population. This results in a given data set that cannot be modeled adequately by a standard Weibull distribution. Some modified models based on standard Weibull distribution have been developed, which include the mixture models, the competing risk models and the sectional models.In this dissertation, standard Weibull model and mixture model involving two standard Weibull models are considered as main research objects. The aim of this dissertation is to expand the application of standard Weibull model, and to explore valid parameter estimation method on small sample size condition for both the standard model and the mixture model.The contributions of this dissertation are summarized as follows:(1) Parameter estimation method for standard Weibull distribution based on SVRSupport Vector Regression (SVR) is a regression technique based on Statistical Learning Theory. SVR has been proved to have good regression performance under the condition of small sample size. In this dissertation, we use SVR to estimate parameters of two-parameter Weibull distribution when sample size is small. (2) A model for reliability prediction of Fatigue residual lifePrediction of fatigue residual life is an important subject in fatigue reliability research. The normal distribution and lognormal distribution are usually employed to describe fatigue lifetime in existing literatures. However, the Weibull distribution is also an ideal model which can properly describe fatigue life in median life and long life zone. In this dissertation, a fatigue residual life distribution model is established based on a three-parameter Weibull distribution and prior probability theory.(3) Parameter estimation method for P-S-N curves based on Weibull distributionAn S-N curve is a traditional tool for design against fatigue. Because there is often a considerable amount of scatter in fatigue performance of specimens, the S-N curve should be more properly P-S-N curves capturing the probability of failure after a given number of cycles or a certain stress. Most studies focused on S-N curve model with three parameters, and the lognormal distribution and maximum likelihood estimation were employed to estimate the unknown parameters. In this dissertation, a three-parameter Weibull distribution is used to describe the scatter of fatigue life. The relationship among survival probability, stress level and fatigue life which follows Weibull distribution is considered. A new method for estimating parameters of P-S-N curves is proposed. According to this method, three groups of specimens are needed. Each group is submitted to a stress level. The parameters of P-S-N curves can be estimated by solving a set of nonlinear equations.(4) Parameter estimation method for mixed Weibull distributionMany mechanical components exhibit more than one failure mode; and not all components under study have been exposed to similar operating conditions. For example, components may have been used in different operating environments. In these cases, life time data of components would not fall on a straight line on a Weibull probability plot (WPP). That is, the standard 2-parameter Weibull distribution is not an appropriate model. It has been recognized that the mixed Weibull distribution can be used to fit such data properly. However, a mixed Weibull distribution involves more unknown parameters; and due to the difficulty of estimation of these parameters, mixture models have not been widely used. In this dissertation, a mixture model involving two Weibull distributions is considered. We establish parameter estimation methods using Nonlinear Least Squares (NLS) theory; and Levenberg-Marquardt (L-M) method is used to solve the optimization problem.