Mixed Weibull Distribution Research Articles

Research on the P‐S‐N Curve Fitting Method of Notched Specimens Considering Small Sample Properties

ABSTRACTAiming at the issue of fatigue test data for large‐scale mechanical components of building steel are very limited, a method for fitting P‐S‐N curves under small sample data of notched specimens is proposed to predict fatigue life. First, a fatigue life subsample augmented and its reliability assessment method are established, based on Bayesian hierarchical modeling and modified Monte Carlo method. Second, a clustering combination weighting method is proposed, to define weights of hidden variables of the binomial mixture Weibull distribution, and the expectation–maximization algorithm is used to determine probability density function of the distribution. Finally, the P‐S‐N curves under various failure probabilities are fitted with Weibull distributed life models, and the convergence and prediction accuracy of the different models are compared. The results show that the fatigue data of small samples can be predicted better by using mixed Weibull distribution, and the fitting P‐S‐N curve is more reliable and accurate.

Constant false alarm rate detection method in mixed Weibull distribution sea clutter

In maritime surveillance, sea clutter represents a major obstacle to target detection accuracy due to missed or false alarms. Several methods have been proposed to find the most efficient. However, most of these methods have only been tested with simulated data, and their applicability and success in a real environment cannot be guaranteed. This work proposes constant false alarm rate (CFAR) detection techniques tailored to mixed Weibull-distributed sea clutter. The mixed Weibull distribution was used to model a mixture of Gaussian and non-Gaussian data. For CFAR detection in mixed Weibull distribution and Weibull clutter, the log t CFAR (logt-CFAR), Weber Haykin CFAR (WHCFAR), Geometric Mean Order Statistic CFAR (GMOS-CFAR), Trimmed Mean Order Statistic CFAR (TMOS-CFAR), Inclusion/Exclusion CFAR (IE-CFAR), and Weber Haykin Order Statistic CFAR (WHOS-CFAR) algorithms were performed. Both synthetic and actual ice multi-parameter imaging X-band (IPIX) radar data were used for experimental verification of this distribution. Simulation studies demonstrated that the proposed detectors maintain the CFAR property in Weibull-distributed clutter, making them effective in controlled scenarios, and confirming how better is the detection performance compared to other detection methods. This research bridges a critical gap in radar signal processing by significantly improving the detection of targets in challenging clutter environments.

A P–S–N curve fitting method based on mixed Weibull distribution and expectation-maximization algorithm

In practical engineering, there are many factors affecting the fatigue life of actual components, and a single Weibull distribution is insufficient to accurately describe the impact of various factors on fatigue life. In order to solve the above problems, 316L stainless steel is taken as the experimental object to study the effects of phase difference and stress levels on fatigue life based on the mixed Weibull distribution model. Firstly, the mixed Weibull distribution model is converted into the mixed extreme value distribution model, and used Euler's constant to calculate relevant parameters of extreme value distribution and Expectation-Maximization (EM) algorithm to determine the probability density function of the mixed Weibull distribution. Secondly, the probability density function is integrated to obtain the distribution function and the fatigue life under different failure probabilities is calculated. Finally, the influence of non-proportional additional reinforcement and stress amplitude on fatigue life is studied by taking them as hidden variables of mixed Weibull distribution, which is used to obtain P-S-N curves for various failure probabilities. Through the proposed method, the influence of different factors on fatigue life can be considered from the perspective of statistics, and the accuracy of fatigue life prediction can be promoted.

Time to capsize for damaged passenger ships in adverse weather conditions. A Multi-modal analysis

After an accident in open seas, the final fate for a damaged ship could be the loss of stability/floatability and consequently capsize/sinkage. The latter may occur even in calm water, but is more critical and probable in adverse weather conditions, i.e., in irregular waves. Identifying a possible capsize event and determining the time that it takes for the ship to capsize is extremely important for safety and risk assessment, meaning whether it would be possible or not to evacuate the ship in a specific scenario and the possibility of loss of life. In this respect, and notwithstanding the impact of many other factors, model tests or time-domain simulations could be used to provide answers to this question. However, even in this case, dealing with irregular waves, entails that both approaches are affected by the random nature of phase spectral components, which leads to a different time to capsize determination at each run/simulation or to the identification of cases where the vessel is not capsizing in the given time window. Here, a dedicated study is provided to describe the time to capsize in irregular waves for critical damages. Simulations performed on a passenger ship at different wave heights and different metacentric heights highlights the appearance of more than one capsize mode for the same damage case. A model based on Mixed-Weibull distributions has been developed to describe the multi-modal behaviour of the time to capsize distributions for the analysed damage cases.

Three-Parameter Estimation Method of Multiple Hybrid Weibull Distribution Based on the EM Optimization Algorithm

The hybrid Weibull distribution model can describe the failure rules of electromechanical products more accurately than the single Weibull distribution model, and it can improve the accuracy of reliability analysis. However, the hybrid Weibull distribution model is also more complex, and the multi-parameter estimation is more difficult. In this paper, a reliability mathematical model based on the two-fold three-parameter hybrid Weibull distribution model was established, an EM optimization algorithm was derived for its solution, and a practical initial parameter selection scheme was designed. The validity of the model and the algorithm were verified, and goodness-of-fit tests were conducted through an arithmetic example. The results showed that the initial value selection scheme proposed in this paper and the corresponding solution algorithm could solve all the parameters and weight coefficients to be estimated for each sub distribution, and the obtained failure probability fitting curve had a high fit with the actual sample data, which effectively solved the multi-parameter estimation problem of the multiple mixed Weibull distribution model.

A probability distribution model based on the maximum entropy principle for wind power fluctuation estimation

Volatility is an inherent characteristic of wind power. The normal, Weibull, and other distributions are commonly used to describe the characteristics of wind power. However, many of these single distribution models are not suitable to characterize wind power, particularly those converging from multiple wind farm groups. In this study, the maximum entropy distribution model is utilized as probability density function to fit wind power fluctuations. To evaluate the effectiveness and performance of the model, we compared the probability density function curve and cumulative distribution function curve fitting with the mixed Gaussian, the mixed logistic, and the mixed Weibull distribution models using simulation experiments. We developed four scenarios with different numbers of wind farm groups and calculated the fitting error, determination coefficient, and K–S test value for the four models. The simulation results show that the maximum entropy distribution model has the best fitting effect in all four scenarios. Compared with the other three distribution models, the maximum entropy distribution model has clear advantages and is more suitable to describe the fluctuation characteristics of wind power. The parameter estimation of the maximum entropy distribution model can be transformed into a linear optimization problem to significantly reduce calculation complexity.

Weibull Analysis Applications in Unexpected Electrical Breakdown of Ion Thruster

The unexpected electrical breakdown (UEBD) of ion thruster is a critical problem. It directly affects the reliability and performance of the ion thruster. The basic UEBD characteristics and Weibull theory are introduced. The self-consistency and applicability of Weibull analysis for UEBD study are discussed. The UEBD period is a proper characteristic variable for Weibull analysis. The approaches to study the factors affecting the UEBD period are presented: (i) distinguishing the effect of coupled factors with the shape parameter of a mixed Weibull distribution, and (ii) studying the effect of a single factor with its respective parameters of the Weibull distribution. The factors discussed include electric field strength, grid material, and ion density. Further aspects to investigate involving the coupling of these factors and mechanisms are prospected.

Mixed Weibull distributions for the Bayesian analysis of reliability when failures are progressively censored

The progressive type-II censoring has a wide range of applications in reliability studies and survival analysis. The analysis of the mixture models under progressively type-II censored samples has recently been introduced. The reliability estimation from the mixture model under the optimal progressive censoring schemes is still lacking in the current literature. We have discussed the estimation of parameters and reliability characteristics from the Weibull mixture model when failures are progressively censored. The Weibull mixture is proposed for this case as it is S-shaped, which can appropriately handle the mixture data. We have addressed the problem of determination of optimal censoring schemes and estimation of expected test times using the said mixed model. The optimal censoring schemes were found to be less sensitive hence robust. As the corresponding posterior distributions do not provide explicit solutions, we have considered Lindley’s approximation for the approximate numerical solutions.

Statistical properties of group height and group length in combined sea states

The statistical properties of ocean wave groups are extremely important for the analysis of resonant effects on ships and marine structures. The patterns in combined sea states have attracted increasing attention recently. This study focuses on the probability distributions of the group height and group length as defined by an envelope approach in sea states with bimodal spectra. Wave data from laboratory experiments and numerical simulations are used. The envelope spectra in combined sea states might exhibit a double-peaked structure. Although the bimodal envelope spectrum has slight effect on the group height distribution, the group length might have a multimodal and broader distribution. As a result, the joint distribution might be characterised by a multimodal shape. Further, a mixed Weibull distribution of the marginal distributions of the group parameters and a mixed copula model of their bivariate distributions are constructed. For comparison, a conventional Weibull distribution and a single copula model are also applied to fit the distributions. The results suggest that the single models can be used only to fit the group height distribution. By contrast, the mixed models can provide satisfactory descriptions of both the marginal distributions and joint distributions.

Mixed Weibull distribution model of DC protection system based on entropy weight method

By analyzing the blocking event of DC converter station of the State Grid Corporation in recently ten years, it can be concluded that the sampling system failure, secondary circuit fault, board fault, host fault, etc. of the protection system caused more than 10% of the DC system blockings, which have seriously threatened the normal operation of DC system. Towards this end, this paper makes a statistical analysis of the operation defects of high-voltage direct current (HVDC) transmission protection in Hubei Company in the past five years. A mixed Weibull distribution model method based on entropy weight method is proposed. By using this method, the reliability curve and average defect interval time (MTBF) of the DC protection system are obtained. Combined with the actual situation of the converter station, the DC protection system of the converter station is evaluated, which provides an important guiding basis for the subsequent maintenance, system transformation, and replacement of the converter station.

Reliability evaluation of the servo turret with accurate failure data and interval censored data based on EM algorithm

The servo turret is a complex electromechanical hydraulic component that is the most likely to fail in a numerical control lathe. Reliability evaluation is used to make statistical inferences about the reliability characteristics of products according to all the information related to product reliability. Failure data is the basis of reliability evaluation; however, it is very difficult to collect many accurate failure data for reliability evaluation. In this paper, the reliability of servo turret is evaluated based on failure data that contains accurate failure data and interval censored data. First, a mixture Weibull distribution is chosen for fitting the reliability model. Then, expectation-maximization algorithm is used for estimating the parameters of the distribution which contains hidden variable, and the confidence interval of parameters is constructed using the delta method. In the simulation, different percentages of accurate data and interval data are used and compared with data containing only accurate data. The accuracy of this method is evaluated by mean square error. Finally, the method is applied to the failure data of servo turret and the parameters of mixture Weibull distribution are determined. For possibly simplifying the mixed Weibull distribution, the hypothesis of shape or scale parameters being equal is tested. The hazard property and mean time between failure are then estimated and associated 95 % confidence intervals are obtained.

Oceanic Surface Environmental Severity

Abstract More than 30 years of temperature and humidity data from bodies of water across the globe were compared to a standard benchmark environment of 85 °C and 85% Relative Humidity (RH). The Peck Temperature Humidity Model provided this linkage. Results were organized by global region and severity to create a severity distribution of the natural oceanic temperature and humidity environments applicable in product testing. The severity distribution was achieved through a four-subpopulation mixed-Weibull distribution fit to the data. The result is the Oceanic Severity Model. The Oceanic Severity Model was compared to prior research that used land-based environmental data to create a similar percent-severity approach. The ocean-based model was less severe at the extremes, but overall similar to the results found in the land-based assessment. Several military-standard-defined environments were compared to the Oceanic Severity Model. One of the military environments was found to be extremely severe when considered as representative of a 10-year natural environment, such that it is highly improbable that the environment would occur in nature.

Nonlinear Least Squares Estimation for Parameters of Mixed Weibull Distributions by Using Particle Swarm Optimization

Mixed Weibull distributions are widely used in lifetime modeling of products with multiple failure modes. It is difficult to estimate parameters of the mixed Weibull distribution since it contains multiple parameters. A parameter estimation model for the mixed Weibull distributions is proposed based on nonlinear least squares estimation (LSE). An approach of determining parameters' approximate values and rough bounds is presented for selecting good starting points used in the particle swarm optimization (PSO) procedure. The PSO solution of the nonlinear LSE method is proposed by a step-by-step procedure. A case study is given to illustrate the accuracy and efficiency of our proposed method. Compared with the genetic algorithm (GA)-based nonlinear LSE method, our method shows advantages in both accuracy and efficiency. And compared with the maximum likelihood estimation (MLE) method, our method shows significant advantages in efficiency.

Distribution Characterisation of Coil Spring Strain Histories Using Mixed Weibull Analysis

This paper investigates the scatter of strain histories obtained from coil spring of a vehicle suspension system. Statistical characterisation is essential in fatigue analysis due to the random nature of fatigue process. The element of uncertainty can be dealt with appropriately by applying probabilistic approach. In this study, four different road profiles were used to obtain strain signal data. Initial description of the strain histories was achieved by computing the global statistics values. The strain range data was calculated from the counted fatigue cycle and distribution fitting was performed using the Anderson Darling test. Four different types of distribution; the exponent, Gamma, 3P-Weibull, and 2P-Weibull; were compared to find the appropriate fit to model the strain range data. The results showed that strain range data are highly skewed with thick-tailed indicating a non-Gaussian distribution. All four tested distribution turned out to be insignificance, with the closest p-value of 0.01 produced by the 2P-Weibull distribution at significance level of 0.05. Due to the non-straight probability plot of data, the mixed Weibull distribution was chosen to model the data. The campus area and highway profiles follow this distribution with 2 sub-populations while rural and housing area resulted in 3 sub-populations. Finally, this distribution is found suitable for analysing the strain range data of vehicle coil spring and hence can be used in time-domain fatigue life evaluation.

Reliability Estimation of Aero-engine Based on Mixed Weibull Distribution Model

Aero-engine is a complex mechanical electronic system, based on analysis of reliability of mechanical electronic system, Weibull distribution model has an irreplaceable role. Till now, only two-parameter Weibull distribution model and three-parameter Weibull distribution are widely used. Due to diversity of engine failure modes, there is a big error with single Weibull distribution model. By contrast, a variety of engine failure modes can be taken into account with mixed Weibull distribution model, so it is a good statistical analysis model. Except the concept of dynamic weight coefficient, in order to make reliability estimation result more accurately, three-parameter correlation coefficient optimization method is applied to enhance Weibull distribution model, thus precision of mixed distribution reliability model is improved greatly. All of these are advantageous to popularize Weibull distribution model in engineering applications.

Understanding the distribution characteristics of bus speed based on geocoded data

Data-driven traffic management and control has attracted much attention recently. This paper conducts a series of coherent analyses based on geocoded data to understand the distribution characteristics of bus operational speed and to explore the potential applications of speed distributions. First, an original bipartite model is adopted for capturing instantaneous speed where the suspended and moving states are considered separately and a two-component mixed Weibull distribution is used to model the speed distribution in moving states. The mixed Gaussian distribution with variable components is found to be capable of expressing the speed distribution patterns of different road sections. Second, elaborate analyses on the basis of speed distribution modelling are conducted: (i) regression analyses are conducted to explore the correlations between parameters of instantaneous speed distributions and traffic related factors; (ii) a powerful clustering method using Kullback-Leibler divergence as the distance measure is proposed to grade the road sections of a bus route. These results can be utilized in fields such as bus operations management, bus priority signal control and infrastructure transformation aiming to improve the efficiency of bus operations systems.

Application of the mixed Weibull distribution in machine reliability analysis for a cell formation problem

PurposeThe purpose of this paper is to understand the consequence of the use of mixed Weibull distribution in the cell formation problem. In reliability theory, a mixed distribution is used for more than one hazard cause, and the Weibull distribution can be used for ascendant, monotonous and descendant failure rate. Here, the authors mixed these two theme and use it in a common problem in group technology.Design/methodology/approachIn this paper, the authors made a non-polynomial-hard mathematical model based on past research and solved it with an exact algorithm. The algorithm is coded and solved in GAMS to illustrate the model, and the authors use simulation. A common numerical example is solved with the model, and the results are compared.FindingsReliability analysis model based on the mixed Weibull distribution approach will give options to a user to select the suitable failure rate and modes for a specific situation. If the user uses the exponential or Weibull distribution instead of the mixed Weibull distribution, the calculated cost and reliability are wrong; therefore, it leads to user making wrong decisions.Originality/valueThe model the authors use is the one used in past research, but in the past, researchers did not use the mixed distribution for explaining failure time. Therefore, the model can be considered as a new and more complete model.

Statistical evaluation of very high cycle fatigue property of high carbon chromium bearing steel under axial loading

Axial loading test was performed to investigate the S–N characteristics and the failure mechanisms of a high carbon chromium bearing steel under very high cycle fatigue (VHCF). Compared with the experimental results under rotating bending, the bearing steel under axial loading also exhibited two failure modes including surface failure and interior inclusion-induced failure. In the life regime beyond about 106 cycles, the fine granular area can occur around the inclusion. But under the axial loading with the uniform stress distribution on the cross-section of specimen, the duplex S–N characteristics of bearing steel was not so distinct. In view of the limitation of common statistical method, a statistical method based on mixed Weibull distribution and parameter optimisation was proposed to evaluate the VHCF property of bearing steel corresponding to two failure modes. Based on the proposed statistical method, the P–S–N curves of bearing steel under axial loading can be better established.This paper is part of a Themed Issue on Recent developments in bearing steels.

Determining the Behaviour of Fatigue Strain Histories of Vehicle Coil Springs by Using Statistical Inferences

The evaluation of fatigue behavior of real components under service loading is important in reliability analysis. The present paper investigates the characteristics of two strain signals spectrum by using statistical inferences. The data used in this study are obtained from strain gauges installed on coil spring component of car suspension system driven over two different road surfaces. The coil springs are made of SAE 5160 carbon steel materials. The strain signals are explored to produce the summary statistics (i.e. root-mean-square, kurtosis, skewness etc.) and the rainflow cycle counting is performed to obtain total number of cycles and damage per cycle. Further, distribution fitting is applied to the cycle-counted strain ranges data. The results show that both signals fit well to a mixed Weibull distribution with three subpopulations.

Fluctuations in a quasi-stationary shallow cumulus cloud ensemble

Abstract. We propose an approach to stochastic parameterisation of shallow cumulus clouds to represent the convective variability and its dependence on the model resolution. To collect information about the individual cloud lifecycles and the cloud ensemble as a whole, we employ a large eddy simulation (LES) model and a cloud tracking algorithm, followed by conditional sampling of clouds at the cloud-base level. In the case of a shallow cumulus ensemble, the cloud-base mass flux distribution is bimodal, due to the different shallow cloud subtypes, active and passive clouds. Each distribution mode can be approximated using a Weibull distribution, which is a generalisation of exponential distribution by accounting for the change in distribution shape due to the diversity of cloud lifecycles. The exponential distribution of cloud mass flux previously suggested for deep convection parameterisation is a special case of the Weibull distribution, which opens a way towards unification of the statistical convective ensemble formalism of shallow and deep cumulus clouds. Based on the empirical and theoretical findings, a stochastic model has been developed to simulate a shallow convective cloud ensemble. It is formulated as a compound random process, with the number of convective elements drawn from a Poisson distribution, and the cloud mass flux sampled from a mixed Weibull distribution. Convective memory is accounted for through the explicit cloud lifecycles, making the model formulation consistent with the choice of the Weibull cloud mass flux distribution function. The memory of individual shallow clouds is required to capture the correct convective variability. The resulting distribution of the subgrid convective states in the considered shallow cumulus case is scale-adaptive – the smaller the grid size, the broader the distribution.