Marko Nagode

Reliability approximation using finite Weibull mixture distributions

Abstract The shape of measured or design life distributions of systems can vary considerably, and therefore frequently cannot be approximated by simple distribution functions. The scope of the paper is to prove that the reliability of an arbitrary system can be approximated well by a finite Weibull mixture with positive component weights only, without knowing the structure of the system, on condition that the unknown parameters of the mixture can be estimated. To support the main idea, five examples are presented. In order to estimate the unknown component parameters and the component weights of the Weibull mixture, some of the already existing methods are applied and the EM algorithm for the m-fold Weibull mixture is derived. The fitted distributions obtained by different methods are compared to the empirical ones by calculating the AIC and δC values. It can be concluded that the suggested Weibull mixture with an arbitrary but finite number of components is suitable for lifetime data approximation. For parameter estimation the combination of the alternative and EM algorithm is suggested.

An online algorithm for temperature influenced fatigue life estimation: stress–life approach

Abstract The paper presents an algorithm suitable for stress–time history filtering, rainflow and the Clormann–Seeger cycle counting and damage estimation for moderate but variable temperatures. The algorithm is based on equivalent temperature calculation and recalculation. This provided for the development of an online procedure that is of great significance since time histories entering finite element models can consist of a huge number of data points. The presented algorithm does not take creep and other effects into account, but it enables quick and improved fatigue life estimates.

A general multi-modal probability density function suitable for the rainflow ranges of stationary random processes

An important aspect of fatigue or reliability analysis under random loading is the prediction of load ranges applied to a structural component. Since the usability of currently available probability density functions of load ranges is limited to only some of stationary processes, a new, general multi-modal Weibull distribution with the procedure of unknown parameter estimation is presented in this paper. The proposed distribution function can be used for every stationary process and for any shape of loading spectrum. For unknown parameter estimation only short time history samples are needed. To evaluate the appropriateness of the new distribution function, extensive load measurements on a forklift at various operating conditions have been carried out. The comparison between the measured and hypothetical loading spectra has shown very good agreement.

On a new method for prediction of the scatter of loading spectra

To reliably predict the endurance limit of a randomly loaded structural component, it is necessary to carry out measurements in actual operating conditions. As a result load/stress time histories and corresponding loading/stress spectra are obtained. In the case of short time history samples loading spectra have to be extrapolated. Additionally, in reliability analysis the scatter of loading spectra has to be determined. A new method of the scatter of loading spectra prediction has been developed. Its use is much more general than that of other existing methods. To verify the new method extensive load measurements on a fork-lift at various operating conditions have been carried out. The measured loading spectra has been well within the calculated confidence limits, which proves the appropriateness of the new method.

Parametric modelling and scatter prediction of rainflow matrices

Abstract For service life prediction and stochastic reconstruction of load histories, rainflow matrices have been recently predominately used to describe the scatter of loading. Typically, only limited data are available due to the costs of measurements. As a consequence of this, discrete rainflow matrices have to be modelled and extrapolated. So far non-parametric methods have most frequently been used to transform discrete matrices into smooth functions. In this paper, two appropriate parametric models: a mixture of joint Weibull–normal distributions and a mixture of multi-variate normals, as well as two algorithms for parameter estimation: the EM algorithm and the algorithm developed by Nagode and Fajdiga are thoroughly discussed and compared. Finally, a method to describe the scatter of rainflow matrices is presented.

Generalized renewal process for repairable systems based on finite Weibull mixture

Abstract Repairable systems can be brought to one of possible states following a repair. These states are: ‘as good as new’, ‘as bad as old’ and ‘better than old but worse than new’. The probabilistic models traditionally used to estimate the expected number of failures account for the first two states, but they do not properly apply to the last one, which is more realistic in practice. In this paper, a probabilistic model that is applicable to all of the three after-repair states, called generalized renewal process (GRP), is applied. Simplistically, GRP addresses the repair assumption by introducing the concept of virtual age into the stochastic point processes to enable them to represent the full spectrum of repair assumptions. The shape of measured or design life distributions of systems can vary considerably, and therefore frequently cannot be approximated by simple distribution functions. The scope of the paper is to prove that a finite Weibull mixture, with positive component weights only, can be used as underlying distribution of the time to first failure (TTFF) of the GRP model, on condition that the unknown parameters can be estimated. To support the main idea, three examples are presented. In order to estimate the unknown parameters of the GRP model with m -fold Weibull mixture, the EM algorithm is applied. The GRP model with m mixture components distributions is compared to the standard GRP model based on two-parameter Weibull distribution by calculating the expected number of failures. It can be concluded that the suggested GRP model with Weibull mixture with an arbitrary but finite number of components is suitable for predicting failures based on the past performance of the system.

An improved algorithm for parameter estimation suitable for mixed Weibull distributions

The shape of measured or design load spectra can vary considerably and therefore frequently cannot be approximated by simple distribution functions. The introduction of mixed Weibull distribution has thus been suggested. Compared to the extreme value distribution, the proposed distribution makes the extrapolation of load spectra and its scatter prediction much easier, especially in the case of variable operating conditions. The major drawback of mixed distributions in general is a complicated, and often impossible, calculation of unknown parameters. This article presents an improved algorithm for unknown parameter estimation. The algorithm is distribution-independent and consists of rough parameter estimation and optimization. Each component density is calculated separately. The solving of complicated systems of equations is not necessary. The proposed algorithm has been tested by analyzing load spectra resulting from in-the-field measurements.

An alternative perspective on the mixture estimation problem

The paper presents an alternative perspective on the mixture estimation problem. First, observations are counted into a histogram. Secondly, rough and enhanced parameter estimation followed by the separation of observations is done. Finally, the residue is distributed between the components by the Bayes decision rule. The number of components, the mixture component parameters and the component weights are modelled jointly, no initial parameter estimates are required, the approach is numerically stable, the number of components has no influence upon the convergence and the speed of convergence is very high. The alternative perspective is compared to the EM algorithm and verified through several data sets. The presented algorithm showed significant advantages compared to the competitive methods and has already been successfully applied in reliability and fatigue analyses.

The influence of variable operating conditions upon the general multi-modal Weibull distribution

Abstract For the design spectrum prediction that should be realized within the expected service life, the influence of variable loading conditions is of paramount importance. Further, the results of the measurements must be properly extrapolated and the scatter of loading spectra has to be determined to assure reliable service life prediction. To model load ranges, a general multi-modal Weibull distribution function has recently been proposed. Until now it has been verified only for fixed operating conditions. The scope of this article is to prove that the same distribution model holds for the case of variable operating conditions, too. The influence of variable operating conditions upon the distribution function is demonstrated by a few examples attained by analysing loads acting upon a structure of a fork-lift.

The REBMIX Algorithm and the Univariate Finite Mixture Estimation

The REBMIX algorithm is presented and applied to estimation of finite univariate mixture densities. The algorithm identifies the component parameters, mixing weights, and number of the components successively. Significant improvement is achieved by replacing the rigid restraints with the loose ones, which enables improved modelling of overlapped components. The algorithm is controlled by the extreme relative deviations, total of positive relative deviations, and information criteria. It enables also the modeling of multivariate finite mixtures. However, the article considers univariate normal, lognormal, and Weibull finite mixtures solely. The REBMIX software is available on http://www.fs.uni-lj.si/lavek.