The Weibull distribution is a popular and widely used distribution in reliability and in lifetime data analysis. Since 1958, the Weibull distribution has been modified by many researchers to allow for non-monotonic hazard functions. Many modifications of the Weibull distribution have achieved the above purpose. On the other hand, the number of parameters has increased, the forms of the survival and hazard functions have become more complicated and the estimation problems have risen. This thesis provides an extensive review of some discrete and continuous versions of the modifications of the Weibull distribution, which could serve as an important reference and encourage further modifications of the Weibull distribution. Four different modifications of the Weibull distribution are proposed to address some of the above problems using different techniques. First model, with five parameters, is constructed by considering a two-component serial system with one component following a Weibull distribution and another following a modified Weibull distribution. A new method has been proposed to reduce the number of parameters of the new modified Weibull distribution from five to three parameters to simplify the distribution and address the estimation problems. The reduced version has the same desirable properties of the original distribution in spite of having two less parameters. It can be an alternative distribution for all modifications of the Weibull distribution with bathtub shaped hazard rate functions. To deal with unimodal shaped hazard rates, the third model with four parameters, named as the exponentiated reduced modified Weibull distribution is introduced. This model is flexible, has a nice physical interpretation and has the ability to capture monotonically increasing, unimodal and bathtub shaped hazard rates. It is a generalization of the reduced modified Weibull distribution. The proposed distribution gives the best fit comparing to other modifications of the Weibull distribution including those having similar properties. A three-parameter discrete distribution is introduced based on the reduced distribution. It is one of only three discrete distributions allowing for bathtub shaped hazard rate functions. Four real data sets have applied to this distribution. The new distribution is shown to outperform at least three other models including the ones allowing for bathtub shaped hazard rate functions. The new models show flexibility and can be used to model different kinds of real data sets better than other modified versions of Weibull distribution including those having the same number of parameters. The mathematical properties and statistical inferences of the new models are studied. Based on a simulation study the performances of the MLEs of each model are assessed with respect to sample size n. We find no evidence that the generalized modified Weibull distribution can provide a better fit than the exponentiated Weibull distributionfor data sets exhibiting the modified unimodal hazard function.

An enormous and urgent energy demand is predicted due to the growing global population, increase in power intensive industries, higher living standards, electrification of remote areas, and globalisation (transportation). Moreover, the global consciousness about the harmful effects of traditional methods of power generation on the environment. That, in turn, has created a need to strategically plan and develop renewable and sustainable energy generation systems. This study presents a wind resource assessment of seven locations proximate to the largest industrial hub in the Middle East, Jubail Industrial City, Kingdom of Saudi Arabia, and a Geographic Information System, GIS based model considering a multi-criteria wind farm site suitability approach for the entire Kingdom of Saudi Arabia and elsewhere. The hourly mean wind speed data at 10, 50 and 90 m above the ground level (AGL) over a period of five years was used for a meteorological station at the Industrial Area (Central) of Jubail. At the remaining six sites, the meteorological data were recorded at 10 m AGL only. Five years of wind data were used for five sites and three years of data were available for the remaining one site. At the Industrial Area (East), the mean wind speeds were found to be 3.34, 4.79 and 5.35 m/s at 10, 50 and 90 m AGL, respectively. At 50 and 90 m AGL, the availability of wind speed above 3.5 m/s was more than 75%. The local wind shear exponent, calculated using measured wind speed values at three heights, was found to be 0.217. The mean wind power density values at measurement heights were 50.92, 116.03 and 168.46 W/m2, respectively. After the assessment and comparison of wind characteristics of all seven sites, the highest annual mean wind speed of 4.52 m/s was observed at Industrial Area (East) and the lowest of 2.52 m/s at the Pearl Beach with standard deviations of 2.52 and 1.1 m/s, respectively. In general, at all sites, the highest monthly mean wind speed was observed in February/June and the lowest in September/October. The period of higher wind availability coincides with a high power demand period in the region attributable to the air conditioning load. The wind rose plots show that the prevailing wind direction for all sites was from the north-west. Weibull parameters for all sites were estimated using maximum likelihood, least-squares regression method (LSRM), and WAsP algorithm. In general, at all sites, the Weibull parameter, c, was the highest in the months of February/June and the lowest in the month of October. The most probable and maximum energy carrying wind speed was determined by all three methods. The highest value of most probable wind speed was found to be in the range of 3.2 m/s to 3.6 m/s at Industrial Area (East) and the highest value of maximum energy carrying wind speed was found to be in the range 8.6 m/s to 9.0 m/s at Industrial Area 2 (South) by three estimation methods. The correlation coefficient (R2), root mean square error (RMSE), mean bias error (MBE), and mean bias absolute error (MAE) showed that all three methods represent wind data at all sites accurately. However, the maximum likelihood method is slightly better than LSRM, followed by WAsP algorithm. The wind power output at all seven sites, from five commercially available wind turbines of rated power ranging from 1.8 to 3.3 MW, showed that Industrial Area (East) is most promising for wind farm development. At all sites, based on percentage plant capacity factor, PCF, the 1.8 MW wind turbine was found to be the most efficient. At Industrial Area (East), this wind turbine was found to have a maximum PCF of 41.8%, producing 6,589 MWh/year energy output. The second best wind turbine was 3 MW at all locations except the Al-Bahar Desalination Plant and Pearl Beach. At both of these locations, 3.3 MW was the next best option. The energy output from the 3 MW wind turbine at Industrial Area (East) was found to be 11,136 MWh/year with a PCF of 41.3%. The maximum duration of rated power output from all selected wind turbines was observed to be between 8 to 16.6% at Industrial Area 2 (South). The minimum duration of rated power output, less than 0.3% for all wind turbines, was observed at Pearl Beach. The maximum duration of zero power output of between 35 to 60% was also observed at Pearl Beach. / Thesis (PhD)--University of Pretoria, 2017. / Mechanical and Aeronautical Engineering / PhD / Unrestricted

Kenaf fiber is increasingly getting the attention of the industries due to its excellent mechanical properties, feasibility, growth rate, and ease of cultivation. On the other hand, cellulose nanofibril is one of the important building blocks of all the bast fibers which significantly contributes to their mechanical properties. However, most of the studies in the literature have estimated the value of axial elastic modulus for fiber-bundles which has some unavoidable issues resulting in incorrect modulus. Moreover, the transverse elastic modulus is another important parameter that also needs to be studied. Hence, to compensate for the gap in the literature, a single unit of both kenaf microfiber and cellulose nanofibril have been subjected to nanomechanical characterization to analyze their surface morphology and determine their elastic modulus in the axial and transverse direction. The experiments also pave to a protocol to characterize micro and nanofibrils nanomechanically and determine their elastic moduli.

The intent of this research and these is to describe the development of a series of charts and tables that provide the individual and cumulative probabilities of failure applying to the Weibull statistical distribution. The mathematical relationships are developed and the computer programs are described for deterministic and Monte Carlo models that compute and verify the results. Charts and tables reflecting the probabilities of failure for a selected set of parameters of the Weibull distribution functions are provided.

Prediction intervals are needed in many industrial applications. Frequently in mass production, small subgroups of unknown size with a lifetime behavior differing from the remainder of the population exist. A risk assessment for such a subgroup consists of two steps: i) the estimation of the subgroup size, and ii) the estimation of the lifetime behavior of this subgroup. This thesis covers both steps. An efficient practical method to estimate the size of a subgroup is presented and benchmarked against other methods. A prediction interval procedure which includes prior information in form of a Beta distribution is provided. This scheme is applied to the prediction of binomial and negative binomial counts. The effect of the population size on the prediction of the future number of failures is considered for a Weibull lifetime distribution, whose parameters are estimated from censored field data. Methods to obtain a prediction interval for the future number of failures with unknown sample size are presented. In many applications, failures are reported with a delay. The effects of such a reporting delay on the coverage properties of prediction intervals for the future number of failures are studied. The total failure probability of the two steps can be decomposed as a product probability. One-sided confidence intervals for such a product probability are presented. / Vorhersageintervalle werden in vielen industriellen Anwendungen benötigt. In Massenproduktionen entstehen regelmäßig kleine Untergruppen von unbekannter Größer, welche ein anderes Lebensdauerverhalten als die übrige Population besitzen. Eine Risikoeinschätzung für eine solche Untergruppe besteht aus zwei Schritten: i) der Schätzung der Größe dieser Untergruppe und ii) der Schätzung des Lebensdauerverhaltens dieser Untergruppe. Diese Arbeit behandelt diese beiden Schritte. Eine effiziente Methode zur Schätzung der Größe der Untergruppe wird vorgestellt und mit anderen Methoden verglichen. Vorhersageintervalle unter Vorinformation in Form einer Betaverteilung werden dargelegt. Das Schema wird für die Vorhersage binomialer und negativ binomialer Zufallsvariablen angewandt. Der Effekt der Populationsgröße auf die Vorhersage der Anzahl von zukünftigen Ausfällen wird für eine Weibull Verteilung betrachtet, deren Parameter auf Basis von zensierten Felddaten geschätzt werden. Methoden um Vorhersageintervalle bei unbekannter Populationsgröße zu bestimmen werden dargelegt. In vielen Anwendungen werden Ausfälle mit einem Verzug gemeldet. Die Effekte eines solchen Meldeverzugs auf die Überdeckungseigenschaften von Vorhersageintervallen für die Anzahl an zukünftigen Ausfällen werden untersucht. Die Gesamtausfallwahrscheinlichkeit aus den zwei Schritten kann in eine Produktwahrscheinlichkeit zerlegt werden. Einseitige Konfidenzintervalle für eine solche Produktwahrscheinlichkeit werden dargelegt.

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Reliability experiments provide important information regarding the life of a product, including how various factors may affect product life. Current analyses of reliability data usually assume a completely randomized design. However, reliability experiments frequently contain subsampling which is a restriction on randomization. A typical experiment involves applying treatments to test stands, with several items placed on each test stand. In addition, raw materials used in experiments are often produced in batches. In some cases one batch may not be large enough to provide materials for the entire experiment and more than one batch must be used. These batches lead to a design involving blocks. This dissertation proposes two methods for analyzing reliability experiments with random blocks and subsampling. The first method is a two-stage method which can be implemented in software used by most practitioners, but has some limitations. Therefore, a more rigorous nonlinear mixed model method is proposed. / Ph. D.

Today’s society relies strongly on the mining industry. In this industry, crushing is an essential step of the process of extracting materials. Crushing is a part of the comminution process, where stones are broken down into smaller fragments. One of the key products for this process is the cone crusher. This machine consists of a mantle, which moves eccentrically against two cone surfaces. Along the surface of the mantle parallel to the inner cone, the thickness gradually increases. This thickness is known as the “stroke” or “throw”, which also can be seen as the dimension of the rock’s compression. Hence, the size of the gap decreases, which allows the rocks to break down into smaller pieces as it travels along the mantle through compressive crushing. Each area where the stone gets broken apart is known as a “crushing zone”. The amount of crushing zones increases with the speed of the mantle. The purpose of this thesis project is to evaluate whether weibull analysis is applicable for calibrating breakage models, based on the mechanical properties of the material being crushed. This can be achieved by gathering data through compressive crushing tests. In this project, the tests consist of single-particle breakage tests. The goal is to propose a standardized method, which results in sufficient calibration based on the material and sought after end product.

Electrical and electronic device insulation systems experience multiple stresses while in service. Insulation design engineers usually make attempts to study the behavior of insulation under multiple stresses to keep the size and weight of the electrical insulation at a minimum. In this thesis, magnet wire insulation properties under multiple stresses are studied. Magnet wires are used for insulation in transformers, flyback transformers, solenoids, sensors, motors, adjustable speed drives etc. The magnet wire insulation under study in this work is micrometers thick, yet they are expected to provide high breakdown strength, good windability, and resistance to moisture. Accelerated life tests are performed on two different AWG 43 magnet wires. Combined Weibull Electrical-Thermal life models and Electrical-Thermal-Frequency life models were successfully verified in this work. The statistical information like lifetime characteristics, probability of failures, and lifetime percentiles determined for the two different magnet wire insulation is helpful for a better understanding of their insulation properties.

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