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The Tensile strength of an composite materials for aeronautical engineering was tested and its distribution was analyzed based on the two parameter weibull distribution?The maximum normed residual (MNR) test was applied to eliminate abnormal data. A graphical method was used to estimate parameters. Furthermore?the validity of the assumed distribution was examined by the kolmogorov test (KS). The results show that MNR test can be used to exclude abnormal data. It also suggests that the gained two parameter weibull distribution can be used to express the tensile strength and predict its values accurately.

A Bayesian approach is presented for both grouped and ungrouped burn-in test data, to come up with the posterior parameters of the bimodal mixed-Weibull distribution. The concepts of belonging probabilities and fractional ranks are introduced for this approach. A numerical comparison is made by conducting the Kolmogorov-Smirnov goodness-of-fit test on the parameter estimates obtained by Jensen's graphical method, by the Bayesian approach using the conventional separation plotting method and by the Bayesian approach using the fractional rank plotting method . It turns out that the Bayesian approach with the fractional rank plotting method yields the best results.

A variety of shapes of two-Weibull mixtures on Weibull probability paper are explored and classified into six types of Cdf curves (A-F). Types B-D represent Cdf's composed of two well-mixed subpopulations. Type F represents the Cdf composed of two very well-separated subpopulations. Types A and E are in between types B-D and type F. The Kao-Cran graphical parameter estimation method cannot be applied to type A-C, E, and F curves. It is recommended that it not be applied to the type D curve. The theoretical basis is developed for the Jensen-Petersen graphical method for mixtures with well-separated subpopulations, viz, the type F curve. For type A and E curves, the method can be applied. However, for type B-D curves, there is no theoretical basis for the method. >

Many 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 or there may be differences in design and/or material. In these cases, life time data of components would not fall on a straight line on a Weibull probability paper (WPP), that is, the standard 2-parameter Weibull distribution is not an appropriate model. It has been recognized that 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, mixed models have not been widely used. In this paper, a mixed model involving two Weibull distributions is considered. We establish parameter estimation methods for the mixed Weibull model using Nonlinear Least Squares (NLS) theory; and quasi-Newton method is used to solve the optimization problem. A numerical example is given to compare the proposed method with the conventional graphical method.

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2007, Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 2007, Türkiye’nin sahip olduğu yüksek rüzgar enerjisi kaynağının en iyi şekilde değerlendirilmesi için ön çalışmalar ile potansiyeli yüksek bölgeler belirlenmelidir. Bu çalışma kapsamında meteorolojik amaçlarla toplanmış rüzgar ölçümlerinin istatiksel olarak değerlendirilmesi ve potansiyeli yüksek noktaların tespiti için FORTRAN programlama dilinde bir yazılım geliştirilmiştir. Türkiye’nin rüzgar enerjisi potansiyelinin araştırılmasında ülke çapında 224 meteoroloji istasyonundan 2005 yılı boyunca ölçülen saatlik rüzgar hızı verileri kullanılmıştır. 10 m yükselikte ölçülmüş olan veriler rüzgar türbinlerinin 50 m kule yüksekliğine sahip olacağı varsayılarak güç kanunu profili ile bu yüksekliğe uyarlanmıştır. Her istasyonun bulunduğu konum için, Türkiye ortalama sıcaklık normali ve meteroroloji istasyonunun deniz seviyesinden yükseklikleri kullanılarak hava yoğunlukları hesaplanmıştır. Rüzgar verilerinin değerlendirilmesinde istatistiksel yaklaşım olarak Weibull dağılımı kullanılmıştır. Weibull dağılımını belirleyen şekil ve ölçek faktörlerinin hesaplanması için tüm veri kullanılarak maksimum olasılık yöntemi, standart sapma yöntemi, grafik yöntem ve enerji eğilim faktörü yöntemi uygulanmıştır. Farklı yöntemlerin verdiği sonuçlar elde edilen dağılımın gerçek veriyi temsil etme yeteneğine göre karşılaştırılmış ve mevcut veri ile en iyi netice veren maksimum olasılık yöntemi hesaplamalarda kullanılmak üzere seçilmiştir. Rüzgar verisi toplanan her istasyon için yıllık bazda maksimum olasılık yöntemi kullanılarak Weibull şekil ve ölçü faktörlerine bağlı olan standart sapma, ortalama hız, en olası hız, enerji yoğunluğuna en büyük katkıyı sağlayan hız, güç akısı ve enerji akısı hesaplanmıştır., To utilize the great wind energy resources available in Turkey, preliminary research is needed in order to determine the areas with high potential. To perform statistical analysis of the data collected for meteorological purposes and reveal the locations of high potential areas, a software has been developed in FORTRAN programming language. At the research on Turkey’s wind energy potential, hourly wind speed data of the year 2005, which has been collected in 224 meteorological stations throughout the country, has been used. Assuming the wind turbine height as 50 m, the data measured at 10 m height has been oriented to 50 m using the power law profile. Using the Turkey average temperature values and the elevations of the meteorological stations, air densities have been calculated for every station location. For the assessment of wind speed data, Weibull distribution has been chosen as the statistical approach. In order to calculate the shape and the scale factors determining Weibull distribution; maximum likelihood method, standard deviation method, graphical method and the energy pattern factor method have been utilized for all the data available. These methods have been compared by means of their abilities to represent the real data. The maximum likelihood method giving the best results has been chosen to be used in the calculations. For every wind speed measurement station; standard deviation, average speed, most probable speed, the speed with the maximum contribution to the energy density, power density and energy density values, all depending on shape and scale factors, have been calculated on a yearly basis., Yüksek Lisans, M.Sc.

The lifetimes of metal–insulator–metal (MIM) capacitors are predicted from voltage step stress data. Capacitor areas of 400, 5625 and 11,250 μm 2 are investigated. The reliability of capacitors subjected to a single or triple DC pulse prior to a breakdown voltage ramp test is predicted at use conditions (5 V). The method of maximum likelihood is employed in this analysis. With this approach, it is found that the effect of capacitor area on the linear field acceleration parameter, γ , is not statistically significant. Also, it is demonstrated that either a single or a triple DC pulse seriously reduces both the MTTF and γ . Further, the triple DC pulse can change the failure distribution from lognormal to Weibull. The data are also fitted to the reciprocal field model and the predicted failure times under use conditions are too large to be physically reasonable, favoring the linear field model. A graphical method is suggested for discerning between the lognormal and Weibull distributions. This study makes extensive use of statistical methods. One of our goals in this paper is to make the reliability community aware of these powerful techniques so that they may become more widespread.

The paper presents a graphical method for plotting mean cumulative repair function (MCRF) for different capacities or types of distribution transformers. The plots provide information about their failure behavior with time. The intensity function graphs obtained using parameters of Weibull process proportional hazard model confirms the results obtained from nonparametric MCRF graphs. Proportional hazard modeling (PHM) technique is quite helpful to consider the effect of covariates on the failure performance of different transformers with time. The property of proportionality is validated with the help of graphical and analytical methods. The higher values of intensity function for 100 kVA transformers and rural environment transformers show their poor performance compared to less than 100 kVA transformers and urban environment transformers respectively. This information is quite helpful in evaluating maintenance/replacement policies. It is considered that normal life of distribution transformer is about 25 years, for a capacity of 100 kVA or less. The present study indicates that in case of 100 kVA transformers and rural transformers an early replacement is required.

Wind energy is now being used in almost every country of the world as an important and pollution free renewable source of energy. This study deals with the feasibility of utilizing winds in generating electricity in Iraq by using the daily average wind speed data since (1/4/2011 to 1/4/2012) for fifteen stations from different regions have been selected for this purpose and for three height (12,50,100 m)and applied it with wind turbine (3Kw). The wind power and the wind power density for the selected stations have been calculated and the maximum values at Basrah (Albrjsuh), the medium values at Baghdad (Abughraib) and the lowest values in northern region Mosul (Bashiqah). Byusing the graphical method to calculate the Weibull distribution .the highest values foe Weibull parameters at Basrah (Albrjsuh), the medium values at Baghdad (Abughraib) and the lowest values in Mosul (Bashiqah). The regression statistical analysis correlation was found for all stations and the most effective values was in at Basrah (Albrjsuh), the medium values at Baghdad (Abughraib) and the lowest values was Mosul (Bashiqah).

This paper investigates the fatigue-life distributions of steel fiber-reinforced concrete (SFRC) containing different volume fractions of steel fibers for various levels of the applied fatigue stress. An extensive experimental investigation was planned in which 143 flexural fatigue tests and 115 complimentary static flexural tests were conducted on SFRC specimens 100-by-100-by-500 mm in size and containing corrugated steel fibers 30-by-2.0-by-0.6 mm in size in three different volume fractions; that is, 0.5%, 1.0%, and 1.5%. It has been observed that the probabilistic distribution of fatigue life of SFRC, at a particular stress level S, can be approximately modeled by the two-parameter Weibull distribution with statistical correlation coefficient values exceeding 0.90. The shapes of the Weibull distribution for fatigue life of SFRC are different for different levels of the applied fatigue stress. The graphical method, the method of moments, and method of maximum likelihood estimates have been used to obtain the distribution parameters of the Weibull distribution. The two-parameter Weibull distribution has been employed to incorporate failure probability into the S-N relationships of SFRC.

The frequency analysis of flood flows recommended in textbooks yields different results by the graphical method of plotting on probability paper using e.g. the Weibull plotting position formula and the analytical method assuming e.g. log-normal or log Pearson type III distributions. If both methods were correct, the results of the two methods should be identical. The plotting position formulas assume that the plotting position should coincide with the mean (or median) of the probabilities which the observation of a rank m has in a great number of samples, each of the same size N. A study of the problem by using synthetic samples shows that these positions do not coincide with the line of the population. A further study shows, however, that the positions of the probabilities of the expected – mean – values of the largest, next largest etc. observation coincide with the line of the population. Thus, it is not the mean value of p of a rank m, but the value of p of the mean value of log q of the rank m, which agrees with the line of population. Therefore, the author finds it essential that the logarithm of the annual peak flows obtain a plotting probability equal to the probability of the expected value of the logarithm of the observations. A new plotting position formula which corresponds to the probability of the expected values of log q is derived.

Under strong assumptions associated with Weibull distributions for normal and accelerated test conditions, a statistical hypothesis-testing procedure is developed that permits the life of cable insulation material under normal stress conditions to be inferred from accelerated life-test data in which few, if any, breakdown owns occur. The procedure also yields an approximate relation between the number of tests required for testing under normal stress conditions and the number of tests required under accelerated test conditions. This relation, in turn, provides a graphical method for judging the feasibility of proposed accelerated life tests. Subject to the validity of the assumptions these procedures may have application to a broad range of materials, components, and systems.

The time to electric breakdown, and the electric field necessary to result in breakdown of solid insulation, seem to be best represented by a Weibull probability distribution. This tutorial paper reviews the graphical method of estimating the parameters of the Weibull distribution. It also presents the method of Maximum Likelihood as an objective technique, developed by statisticians for improving the estimates of the Weibull parameters, especially for censored experiments. Procedures for calculating confidence intervals are also given. It is shown in an example that the confidence intervals can be very wide for sample sizes commonly used in dielectric life tests.

A graphical method is given for the analysis of repairable systems failing according to a Weilbull process. The integrated hazard rate is plotted in a logarithmic co-ordinate system. If the process is of Weibull type the observed data will be located around a straight line. The plot provides estimates of the parameters. The graphical procedure is completed by confidence bands and variation bands built either around the observed failure numbers at fixed time or vice versa.

It is widely recognized that electron tube failures may be classified into two types: sudden and delayed. A mixture of two Weibull distributions, each representing one type of tube failure, is proposed, and a simple graphical method for estimating the parameters of the mixed Weibull distribution described.

The investigations of candidate distributions that best describe the burst pressure failure probability characteristics of nuclear power steam generator tubes has been continued. To date it has been found that the Weibull distribution provides an acceptable fit for the available data from both the statistical and physical viewpoints. The reasons for the acceptability of the Weibull distribution are stated together with the results of tests for the suitability of fit. In exploring the acceptability of the Weibull distribution for the fitting, a graphical method to be called the ''density-gram'' is employed instead of the usual histogram. With this method a more sensible graphical observation on the empirical density may be made for cases where the available data is very limited. Based on these methods estimates of failure pressure are made for the left-tail probabilities.

The probabilistic fatigue method developed by Nash for least life type distributions is extended to treat intermediate stress levels for which no fatigue data is available. The Weibull distribution is utilised with parameters expressed as functions of the fatigue stress. A new procedure is presented utilising the Gatts equation for smoothing statistical life data for different stress levels. If coupled with statistical data on static properties, use of this method could significantly reduce the number of stress levels needed for statistically significant fatigue testing. The best available statistical fatigue data due to Wagner and Mischke is used. Weibull parameters are determined using several methods, with a combined Maximum Likelihood and graphical method recommended.