Three-parameter Weibull Model Research Articles

Time-dependent mechanical behavior analysis using Weibull models of cellulose hollow fiber membranes produced by green solvent

The mechanical behavior of sustainable (“green”), bio-based cellulose hollow fiber membranes (hereafter, hollow fiber) produced using green solvent, which is potential for desalination and solvent-resistant nanofiltration, can be evaluated either in dry or water-conditioned (wet) states. However, the effect of conditioning time on the mechanical behavior of hollow fibers has not been conclusive. In this work, an extensive evaluation involving an experimental campaign and rigorous application of statistical models for cellulose hollow fibers produced using green solvent (1-ethyl-3-methylimidazolium diethyl phosphate or [EMIM][DEP]) via a dry-jet spinning technique is reported. First, we evaluated the microstructures and separation performance (water permeation, solute rejection) of hollow fibers made of different cellulose concentrations (10–15 wt%) in both pristine and conditioned states. Then, the strength, stiffness, and ductility of hollow fiber samples of different gauge lengths (25 and 50 mm) were measured by a standard tensile test device. Here, the effect of conditioning time on mechanical properties was evaluated after the hollow fiber samples were water-immersed at different times (from 30 s to 2 months). The strength distribution of conditioned hollow fibers was modeled and analyzed using two-parameter and three-parameter Weibull models. Our results showed that the cellulose content of 15 wt% in the hollow fibers produced a denser microstructure, providing the lowest molecular weight cut-off (MWCO). The cellulose content and gauge length slightly modified the mechanical properties of hollow fibers when they were evaluated in pristine, dry conditions. The mechanical properties of hollow fibers were drastically reduced after 10 min of water immersion, beyond which their properties were stabilized. Weibull modeling of mechanical reliability showed that two- or three-parameter Weibull could estimate the failure probability of hollow fibers quite well. Nonetheless, the three-parameter Weibull is better than the two-parameter one in terms of accuracy in determining the threshold strength located at the “lower-end tail.” Our analysis provides a rigorous demonstration of Weibull modeling in evaluating the time-dependent mechanical behavior of ductile, cellulose-based hollow fiber membranes, which is essential to support the design and development of future hollow fiber membranes in desalination industries.

Three-Parameter Weibull for Offshore Wind Speed Distribution in Malaysia

It is usual practice to employ the probability density function in order to ascertain the wind energy potential. In this paper, the three-parameter Weibull model was suggested to distribute the low wind speed data at a site off the coast of Kijal, Terengganu. The maximum likelihood approach was used to determine the value of the distribution parameters. The Matrix Laboratory (MATLAB), better known for its user-friendly programming language, is used to carry out all of the analysis. The three-parameter Weibull was compared with the two-parameter Weibull and Rayleigh using the R-squared and the root mean square error (RMSE). These three distributions statistically model hourly ERA-5 reanalysis wind speeds. The three-parameter Weibull has a good performance, since it appears between the best R2 closest to 1 and RMSE the closest to 0. The R2 for the three-parameter Weibull is 0.9898, the two-parameter Weibull is 0.9874, and Rayleigh is 0.5153. While the RMSE Three-parameter Weibull is 0.0070, the two-parameter Weibull is 0.0082, and Rayleigh is 0.0677.

Enhancing Wind Energy Potential Assessment with Three-Parameter Weibull Distribution: A Comparative Analysis using MATLAB

To determine the wind energy potential, the probability density function is typically used. For data distribution with modest wind speeds, this paper developed a three-parameter Weibull model. The distribution factors were determined using the maximal likelihood technique. The world renowned, user-friendly programming language Matrix Laboratory (MATLAB) processes all data that needs analysis. A comparison was made between the 3-factor Weibull, the 2-factor Weibull, and Rayleigh through R2 and root mean square error (RMSE). The ECMWF Reanalysis v5 (ERA 5) reanalysis's hourly wind speeds are statistically represented by these three distributions. Due to its placement between the optimal R2 and RMSE, the three-parameter Weibull model achieves good results. Weibull with three parameters has a R2 of 0.9898, Weibull with two parameters has a R2 of 0.9886, and Rayleigh has a R2 of 0.5162. The root-mean-squared errors (RMSEs) for Rayleigh, 2-factor and 3-factor Weibull, are 0.0082 and 0.0070, respectively.

Reliability analysis of wind turbine subassemblies based on the 3-P Weibull model via an ergodic artificial bee colony algorithm

The Weibull distribution is the most widely used model for the reliability evaluation of wind turbine subassemblies. Considering the important role of the location parameter in the three-parameter (3-P) Weibull model and its rare application in wind turbines, this study conducted a reliability analysis of wind turbine subassemblies based on field data that obeyed the 3-P Weibull distribution model via maximum likelihood estimation (MLE). An improved ergodic artificial bee colony algorithm (ErgoABC) was proposed by introducing the chaos search theory, global best solution, and Lévy flights strategy into the classical artificial bee colony (ABC) algorithm to determine the maximum likelihood estimates of the Weibull distribution parameters. This was validated against simulation calculations and proved to be efficient for high-dimensional function optimization and parameter estimation of the 3-P Weibull distribution. Finally, reliability analyses of the wind turbine subassemblies based on different types of field failure data were conducted using ErgoABC. The results show that the 3-P Weibull model can reasonably evaluate the lifetime distribution of critical wind turbine subassemblies, such as generator slip rings and main shafts, on which the location parameter has a significant effect.

Group Acceptance Sampling Plan Based on New Compounded Three-Parameter Weibull Model

In this study, we introduce a new compounded model called the complementary Bell–Weibull model and use it to address the problem of a group acceptance sampling plan predicted on a truncated life test. The median lifespan is used as a quality index to obtain the design constraints, namely sample size and approval number, under a predefined consumerś risk and test termination period. Additionally, two real data applications are presented, and unknown parameters are estimated using the maximum likelihood approach.

Generalized hierarchical Bayesian inference for fatigue life prediction based on multi-parameter Weibull models

This article proposes the use of generalized hierarchical Bayesian inference for fatigue life prediction based on general multi-parameter Weibull models. First, a five-parameter Weibull model for corrosion-stress-life (C-S-N) is introduced; neglecting the influence of corrosion, the model degenerates into a three-parameter Weibull model for stress-life (S-N). To predict the fatigue life based on the observation data, a three-layer hierarchical Bayesian structure for these Weibull models is established, and the posterior joint PDF of the parameters and hyperparameters in the generalized hierarchical Bayesian model (GHBM) is derived. Gibbs sampling is employed to obtain posterior samples for parameters and hyperparameters using their full conditional distribution, which is simplified using a Markov blanket based on the probabilistic dependences of the parameters and hyperparameters. Subsequently, three groups of fatigue data for steel wires with different corrosion degrees are used to validate the GHBM, considering both noninformative and informative priors. When considering conventional fatigue data (i.e., without corrosion), the fatigue life prediction obtained from the GHBM and the maximum likelihood estimation (MLE) are compared, along with the results obtained from traditional lognormal models. The results indicate that the scatter in fatigue life prediction for the corroded specimens using the GHBM becomes smaller when the informative priors for the parameters are considered in the Weibull model for C-S-N. For conventional fatigue data, the fatigue life predictions using the Weibull model for S-N are similar for both the GHBM with the noninformative priors and the MLE, which are more conservative when compared with the results obtained from the traditional lognormal models. The proposed method in this paper can be used as a general approach in fatigue life prediction.

Grey-based approach for estimating Weibull model and its application

Weibull distribution is widely used in engineering because of its flexibility to take on many different shapes. For different application fields, people put forward different estimation methods, including grey estimation method. The purpose of this paper is to improve the existing grey estimation method of Weibull distribution as its some serious shortcomings. Firstly, a grey three-parameter Weibull model is proposed in a discrete form. Then, by means of the grey Weibull model, a two-stage hybrid method for estimating the three-parameter Weibull distribution is proposed, which is composed of the linear and nonlinear least square principles. To demonstrate the feasibility of the proposed grey Weibull model, a simulation study was conducted and compared the results with the modified MLE method, and a comparison study on the frequency analysis of four breakdown data sets of epoxy resins is also carried out. Moreover, as a practical application, Weibull model based on the grey estimation approach is applied to the frequency analysis of monthly wind speed of Hohhot in 2018, and compared with a series of existing estimation methods. Results show that the proposed grey Weibull estimation improves the existing grey estimation and can be well applied to the frequency analysis of Weibull data.

Parameter estimation of three-parameter Weibull probability model based on outlier detection.

The Weibull probability model used in statistical analysis has become more popular in the inconsistency evaluation of used Li-ion batteries due to its flexibility in fitting asymmetrically distributed data. However, despite its better fitting of data with a non-zero minimum, the three-parameter Weibull model is less used because of its complicated calculation. Additionally, the Weibull family is likely to overfit and shows inference from outliers. Although conventional estimation methods for Weibull parameters based on dispersion and symmetry of the overall distribution lead to derivation from the actual data features, there is little research into methods to solve the contradiction between estimation accuracy and proper outlier detection. In this study, a Weibull parameter estimation method was proposed that features simplified computation and eliminates the interference from outliers. The outliers were identified based on the obtained Weibull parameters and excluded from the sample data. The method was implemented for fitting the capacity distribution of Li-ion batteries, which was verified by a chi-square test at a confidence of 95% and the Anderson-Darling test. It showed a higher goodness-of-fit and less error than the results of the maximum likelihood estimated Weibull model as well as the normal distribution. The optimal presetting of column number and peak reference point selection were determined by parameter discussion.

Experimental investigation on the corrosion and corrosion fatigue behavior of butt weld with G20Mn5QT cast steel and Q355D steel under dry–wet cycle

Corrosion and corrosion fatigue experiments were conducted on a butt weld constructed of G20Mn5QT cast steel and Q355D steel in a dry–wet cycle with 3.5 wt% NaCl solution. The microstructural characterization was analyzed using SEM. The butt weld suffered from pitting corrosion in a dry–wet cycle. Chloride ion erosion and defects (welding defects and casting defects) have important influences on pitting corrosion. Internal defects and corrosion pits are the main cause of corrosion fatigue failure of the butt weld specimens. Drying time accelerates pitting corrosion and corrosion fatigue progression without changing the corrosion and corrosion fatigue mechanism in 3.5 wt% NaCl solution. The three-parameter Weibull model has good applicability for evaluating the corrosion fatigue behavior of G20Mn5QT cast steel and Q355D steel butt welds in simulated seawater.

Performance analysis of optical wireless communications with aperture averaging over exponentiated Weibull turbulence with pointing errors

Aperture averaging (AA) is a potential technique to reduce the effect of atmospheric turbulence in optical wireless communications (OWC). Although there is significant research on point-like detectors, there is a gap in the study on the analytical performance evaluation of OWC systems with AA over atmospheric turbulence and pointing errors. In this paper, we analyze the performance of a single-link OWC system by considering the three-parameter exponentiated Weibull (EW) model for the atmospheric turbulence and the zero boresight fading model for pointing errors. We derive analytical expressions for moments of signal-to-noise ratio (SNR), ergodic capacity, outage probability, average bit-error-rate (BER), and average energy consumption under the combined effect of atmospheric turbulence and pointing errors in terms of system parameters. To provide insights on the system behavior, we develop asymptotic bounds at high SNR on the performance of the considered system. We also derive analytical expressions for the performance of OWC system under the atmospheric turbulence with negligible pointing errors. We demonstrate the performance of OWC system over EW turbulence with a comparison to the performance obtained using the Gamma–Gamma (GG) model and demonstrate the tightness of the derived performance bounds through numerical and simulation analysis.

Assessment and modelling the antibacterial efficacy of vapours of cassia and clove essential oils against pathogens causing foodborne illness

The present study deals with assessment of antibacterial activity of essential oil vapours (EOV) of cassia (Cinnamomum cassia) and clove (Syzygium aromaticum) using dose-response equation, and modelling the inactivation pattern of Escherichia coli, Salmonella enterica serovar Typhi and Yersinia enterocolitica under the influence of EOVs using three-parameter Weibull model. The fitted inhibition (%) data to dose-response equation estimated 0.98 and 3.63 μL/cm3 as concentration of cassia and clove EOV that could inhibit the growth of E. coli up to 90%, whereas estimated IC90 for cassia and clove EOV against S. enterica ser. Typhi was 0.52 and 1.33 μL/cm3, respectively. However, Y. enterocolitica exhibited more sensitivity under the influence of cassia and clove EOVs with IC90 of 0.2 and 0.79 μL/cm3, respectively. The inactivation model estimated 3.1 and 4.4 h required by cassia and clove EOV for 1-log10 reduction of E. coli, respectively. Similarly, 1-log CFU reduction of Y. enterocolitica under the influence of cassia and clove EOV were estimated to be 3.6 and 5.1 h, respectively, and S. enterica ser. Typhi showed highest δ-value for both the selected EOVs. Susceptibility of the bacteria strains to the EOVs can be attributed to the antibacterial efficacy of the chemical components present.

Impression deformation of carbon-microsphere and polytetrafluoroethylene composite

Carbon-based materials have been extensively used in the devices and systems for energy storage, catalysis, etc. The mechanical behavior of the carbon-based materials plays an important role in the performance and reliability of the devices and systems. In this work, we use impression test to investigate the deformation behavior of carbon-microsphere and polytetrafluoroethylene composite, in which the carbon microspheres are fabricated from high-fructose corn syrup via hydrothermal process. The three-parameter Weibull model is used to analyze the contact modulus, maximum impression depth and energy dissipation per impression cycle. The impression modulus, maximum impression depth and dissipated energy increase with the increase of the impression load and holding time, which are due to impression-induced consolidation of the carbon-sphere-based composite. A simple relation between the Young’s modulus of the carbon-microsphere-based composite and the ratio of the maximum impression depth to the sample thickness is developed and supported by the experimental results.

Probabilistic S-N curves for CFRP retrofitted steel details

Experimental fatigue data of non-cracked open-hole steel specimens retrofitted with bonded carbon fiber reinforced polymer (CFRP) are collected from the literature and assessed in this paper. The specimens are representative of steel plates in tensile riveted or bolted connections. The CFRP substantially extends the fatigue life of specimens at stress range Δσ<220MPa where the fatigue data points converge towards a fatigue limit. These are promising results since most of the ancient metallic riveted assemblies are subjected to stress levels below 150 MPa. The fatigue life extension ratio decreases when the stress level increases due to the fatigue degradation of the bonding joint and/or the CFRP. The fatigue data of specimens retrofitted with CFRP follow a nonlinear trend and code-based linear S-N curves with a slope m = 3 do not capture this behavior. The commonly used methods that assume the fatigue life extension ratio as constant are not adequate to predict the full fatigue life of CFRP retrofitted specimens. To assess the fatigue behavior of the CFRP retrofitted specimens, probabilistic methods are adopted in this study using: the Normal distribution; the two-parameter Weibull; the Gumbel; and the three-parameter Weibull model developed by Castillo & Fernández-Canteli (CFC). Regardless of the assumed distribution function, linear probabilistic S-N curves (P-S-N) are not adequate to model the complex behavior of retrofitted specimens with CFRP. The nonlinear CFC model fitted fairly well the different trends of the fatigue data including the fatigue life at high cycle regime and captured the transition between medium–high cycle range. P-S-N curves derived using the CFC model are proposed for CFRP retrofitted specimens.

Statistical analysis of bamboo scrimber tensile mechanical property based on different models and experiments

PurposeBamboo scrimber is a new natural fiber-reinforced composite material in the modern industry. In this paper, the strength characteristics of this material were chosen as the object of study.Design/methodology/approachFirst, axial tensile fracture experiments were conducted on different test specimens to determine the corresponding strength data. Then three commonly used distribution models were applied to analyze the relationship between the strength and survival rate.FindingsThe results show that the tensile data demonstrate obvious dispersion. In addition, the three-parameter Weibull model fits the data best and may be used as the fundamental model for future analysis.Originality/valueSpecial statistical analysis research of the bamboo material experimental results has rarely been discovered. This paper provides a solid foundation for better understanding the mechanism of the bamboo scrimber strength phenomenon.

Prediction of Hydraulic Automatic Transmission Reliability Using Failure Data Based on Exponential Decay Oscillation Distribution Model

Aiming at the problem that the traditional reliability models of mechanical products are used to predict the reliability of hydraulic automatic transmission and the expected result is relatively large, firstly, the empirical distribution model line is used to statistically analyze the failure distribution law of the hydraulic automatic transmission; then, the Fourier transform is used to perform frequency domain analysis on experience distribution; on this basis, comprehensively consider the characteristics of experience distribution and frequency domain characteristics of experience distribution, constructs the reliability model of exponential decay oscillation distribution and the corresponding reliability, failure efficiency and average life calculation model; meanwhile, studies the influence of attenuation coefficient, oscillation amplitude, oscillation angle frequency, and other parameters on the probability distribution characteristics. On this basis, the established probability distribution models are adopted to fit the failure time data of hydraulic automatic gearbox carried by a forklift, and the fitting results are compared with exponential distribution models, three‐parameter Weibull models, and “bathtub curve” models. The comparing results show that the established exponential decayed oscillation distribution model can better describe the probability distribution characteristics of the fault‐free working time of automatic transmission, and the use of this model can obtain a smaller root mean square error. Simultaneously, the research conclusions of this paper can provide meaningful guidance and reference for the analysis of the life distribution model of mechanical products with exponentially attenuated oscillation probability density change law.

Analysis of Occupants’ Air-Conditioning Opening Behaviour Based on the Survival Model

It is essential to predict building energy consumption through more accurate simulation of building energy consumption, and then put forward suggestions for building energy conservation. Therefore, it is a very important issue to study the variable, random and complex air conditioning usage mode. In the previous studies on air conditioning, it could be found that whether the air conditioning is on or off, it is only a mathematical function about environmental parameters. However, when we arrive at the office and feel uncomfortable, we don't open the window immediately. Instead, we put up with it for a while. In view of the above shortcomings, this study proposed a survival model based on Weibull function to predict the air-conditioning on behavior. Through the verification of the model, we found that the accuracy of the air-conditioning regulation model based on the survival model is more than 74%. We compared and verified the common three-parameter Weibull model with the survival model-based Weibull model, and found that the accuracy of the common three-parameter Weibull model was slightly higher than that of the survival model. At the same time, we analyzed the death event (tolerance temperature) of the survival model, and further improving the tolerance temperature is of great help to the accuracy of the model.

Estimation of the wind energy potential for coastal locations in India using the Weibull model

Abstract Wind energy has exhibited the fastest growth of all renewable energy sources. Available wind energy potential for use by wind turbines has been found to be highly overestimated by existing methodologies when the wind energy potential is assessed from the total wind speed data because the wind turbine operates between cut-in and cut-out wind speeds. While applying existing methodologies, wind power density is overestimated on average by nearly 25% compared to the actual wind power available to a wind turbine. Hence, for estimating wind energy potential, availability factors and wind energy between cut-in and rated wind speeds should be properly estimated using Weibull models. The appropriateness of different methods of estimating Weibull parameters are site specific. In this article, a novel method has been developed for estimating the actual wind power available to the wind turbine. The parent two-parameter Weibull model can be used to determine the availability factor, whereas when determining the available wind energy between the cut-in and rated wind speeds, wind speed data should be refitted in the range defined by the cut-in and rated wind speeds using a three-parameter Weibull model, where the location parameter can be equated to the cut-in wind speed.

Inventory optimisation in a three-parameter Weibull model under a prepayment system

Negotiated conditions for payment of goods and services may involve one or a combination of three things: buyers pay for inventory when it is delivered, before it is delivered or after it is delivered. The payment options have significant impact on inventory-system total cost, and should be accounted for when analysing an inventory system. The inventory literature is mostly devoted to analysis of problems with payment at the time of delivery and credit payment. Following the latest development of a class of economic order-quantity models with prepayment included, we propose a generalised prepayment inventory model that subsumes a wide range of inventory problems. Optimisation results are presented for the cycle time and the unit selling price of an item. The results indicate that optimal strategies can exist for such problems. Numerical analyses of businesses with prepayment inventory systems are presented. A sensitivity analysis was conducted and the results show that buyers can strategically adjust those input parameters to which the optimal solution is insensitive in order to increase profit. Furthermore, the analysis indicates that buyers should focus on reducing the proportion of the purchase cost to the total prepayments.

Estimation of the reliability parameter for three-parameter Weibull models

• Proposal of correct likelihood for three-parameter Weibull models and inferences about their reliability parameter. • New insight for the relationship between the smallest observation and the threshold parameter. • Profile likelihood inferences for reliability parameter when Weibull models have common shape and threshold parameters. • Novel Bootstrap inferences for reliability parameter of threeparameter Weibull distributions with all parameters unknown. • Proposed likelihood can also be used with other densities having singularities and threshold parameter. For X and Y independent three-parameter Weibull random variables , the estimation of the reliability parameter δ = P ( Y < X ) is important in applications of failure times and stress-strength situations in industry, medicine, extreme value theory, hydrology, and environmetrics. Estimation problems arise when the likelihood function is not defined properly, since the three-parameter Weibull distribution is non regular and its density has singularities. Here, a correct likelihood for the three-parameter Weibull case is proposed for the first time, in the spirit of the original definition of likelihood. It takes into account the occurrence of the smallest observation with respect to the threshold parameter, as well as the fact that all measuring instruments necessarily have a finite precision. Possible repeated observations are immediately explained. When the Weibull distributions of X and Y have common threshold and shape parameters, the profile likelihood can be used for inferences about δ . For the case of all three Weibull parameters unknown and arbitrary, inferences about δ are obtained via a novel Bootstrap approach . An example previously analyzed under alternative inferential approaches is presented to illustrate the convenience of the proposal.

A Note on the Existence of the Location Parameter Estimate of the Three-Parameter Weibull Model Using the Weibull Plot

The Weibull model is one of the widely used distributions in reliability engineering. For the parameter estimation of the Weibull model, there are several existing methods. The method of the maximum likelihood estimation among others is preferred because of its attractive statistical properties. However, for the case of the three-parameter Weibull model, the method of the maximum likelihood estimation has several drawbacks. To avoid the drawbacks, the method using the sample correlation from the Weibull plot is recently suggested. In this paper, we provide the justification for using this new method by showing that the location estimate of the three-parameter Weibull model exists in a bounded interval.