【作者】 凌丹；

【导师】 黄洪钟；

【作者基本信息】 电子科技大学 ， 机械电子工程， 2011， 博士

【摘要】 威布尔分布是一种连续分布,它能够描述各种类型机械零部件失效数据的分布规律,在寿命数据分析、可靠性设计、疲劳可靠性分析、维修决策、保修策略制定等方面得到了一定程度的应用。标准的威布尔分布有二参数和三参数两种形式。常用来估计威布尔分布参数的方法可以分为两大类,图解法和解析法。图解法包括经验分布图法、威布尔概率图法和风险率统计图法等;解析法包括极大似然估计法和回归估计法等。这些传统的参数估计方法在样本数据较少时,难以获得较好的结果。在一些样本数据较为复杂的条件下,比如不同失效模式的数据、不同质量的产品失效数据混合在一起时,传统的威布尔分布不能得到令人满意的拟合结果,传统的参数估计方法也不适用。在可靠性和统计学文献中,提出了在威布尔分布基本模型基础之上的改进模型,如混合模型、分段模型和竞争风险模型等,对于这三种模型的应用和参数估计方法目前也有了一些研究。本文以威布尔分布模型和威布尔混合分布模型为研究对象,研究在小样本条件下,威布尔分布标准模型的参数估计方法,拓展威布尔模型的应用领域,探讨威布尔混合模型的参数估计方法。本文的主要研究成果如下:(1)基于支持向量回归机(Support Vector Regression,SVR)的威布尔分布参数估计。支持向量回归机是在统计学习理论基础上发展的用于预测输入输出量之间函数关系的一种方法,可以实现线性回归和非线性回归,特别适合于小样本情况。本文建立威布尔分布的线性回归估计模型,用支持向量回归机求解回归模型中的未知参数,从而获得威布尔分布的形状参数和位置参数的估计值,并讨论支持向量回归机在小样本和大样本条件下的适用性及在小样本条件下的优势。(2)疲劳剩余寿命可靠性建模。疲劳可靠性中的一个重要问题就是疲劳剩余寿命预测,现有的研究大多是假设疲劳寿命服从正态分布或对数正态分布。威布尔分布在中、长寿命区都适用,也是描述疲劳寿命分布的理想模型。本文假定零件的疲劳寿命服从三参数威布尔分布,建立了已知工作时间时的疲劳剩余寿命可靠性模型。(3)疲劳寿命服从威布尔分布的P-S-N曲线参数估计。在抗疲劳设计中常用到S-N曲线,考虑疲劳寿命可靠度的一组S-N曲线称为P-S-N曲线,它是进行疲劳可靠性设计的基础工具。获取P-S-N曲线的传统方法是成组试验法,将各种应力水平下疲劳寿命分布曲线上可靠度相等的点用光滑曲线连接而成。本文假定在各应力水平下,疲劳寿命为独立同分布随机变量,分别服从一个三参数威布尔分布,S-N曲线采用三参数方程来描述,通过寻找疲劳寿命可靠度函数与S-N曲线方程之间的关系,建立非线性方程组,其中的未知数即为S-N曲线方程的三个参数。通过求解非线性方程组,可以获得各种可靠度时的S-N曲线方程系数,从而可以求解P-S-N曲线方程的表达式。(4)威布尔混合分布模型的参数估计方法及其应用。威布尔混合分布用来考虑机械零件多种失效模式、不同质量产品失效观测数据混合在一起时的情况。威布尔混合分布模型的应用尚不广泛,主要原因是其分布参数较多,参数估计过程繁琐。本文利用非线性最小二乘法在处理非线性回归问题方面的能力,建立两重威布尔混合分布参数估计的非线性回归模型,并采用Levenberg-Marquardt(L-M)算法求得其参数估计值。更多还原

【Abstract】 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.更多还原