The log-normal distribution function

SummaryThere have been, in the past, many fatigue tests carried out on a variety of materials and components. These all indicate a wide scattering in the lives (measured by the number of stress cycles to failure) endured by nominally identical components subjected to nominally identical forces before failure occurs. To interpet this scattering several equations have been suggested as representing the statistical distribution functions that fit the lives obtained for individual types of component. Of these functions the log normal distribution function has been perhaps the most widely used. For the central regions of the probability distribution, i.e. about the mean, the log normal distribution and several others represent experimental results very closely indeed, but engineers and designers of all kinds dare not design on the mean fatigue life. They are concerned with specifications that either exclude the possibility of failure or admit only a very small probability of failure. It is thus with the accuracy with which the “lower tail” of the probability distribution curve fits the experimental results that they are concerned.To assess the fit at this lower end for one type of component, a large number (about 1,000) of aluminium specimens have been tested and the corresponding lives plotted. The results are very interesting. They show clearly that the log normal distribution for this type of component and material is pessimistic for a probability of failure of less than 0·3. This result is felt by the authors to be of very great importance. It has further been shown that the use of the “one-sided censored distribution function”, used previously by one of the authors, gives a curve that will fit the lower results better than the complete log normal distribution would do.It is with the testing procedure adopted, the specimens used, the distribution functions considered and the conclusions obtained therefrom that this paper is concerned.

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Study on the life distribution of microdrills

Proceedings of the Institution of Mechanical Engineers Part B Journal of Engineering Manufacture ◽

10.1243/0954405021519889 ◽

2002 ◽

Vol 216 (3) ◽

pp. 301-305 ◽

Cited By ~ 8

Author(s):

Z Yang ◽

Y Chen ◽

Y Yang

Keyword(s):

Distribution Function ◽

Normal Distribution ◽

Reliability Function ◽

Test Methods ◽

Life Distribution ◽

Machining Processes ◽

Experimental Conditions ◽

Distribution Parameters ◽

Log Normal ◽

Log Normal Distribution

In this paper, the life distribution of microdrills has been studied experimentally. The fit and test methods of the life distribution function of microdrills are described. Under the experimental conditions of the present work, it is concluded that the life of microdrills follows the log-normal distribution and the distribution parameters are different for various workpiece materials. The life distribution function and reliability function of microdrills from the present work provide a base for correct changing of drills in automatic machining processes.

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Peculiarities of data processing for optical measurements of disperse parameters of bimodal media

Computer Optics ◽

10.18287/2412-6179-2019-43-4-692-698 ◽

2019 ◽

Vol 43 (4) ◽

pp. 692-698 ◽

Cited By ~ 1

Author(s):

A.A. Zhirnov ◽

O.B. Kudrjashova

Keyword(s):

Experimental Data ◽

Particle Size ◽

Distribution Function ◽

Particulate Matter ◽

Particle Size Distribution ◽

Normal Distribution ◽

Size Distribution ◽

A Priori ◽

Log Normal ◽

Log Normal Distribution

This study is focused on enhancing the informativity of optical measurement techniques for particulate matter. The problem is that the description of particulate matter with bimodal and multimodal distributions by an a priori defined analytical function of particle size distribution (for example, a log-normal distribution) is not accurate enough. Here, we explore if experimental data can be approximated by a multivariable function of particle size distribution instead of using the a priori defined log-normal distribution. For the comparison of the approximation results, experiments are conducted on standard samples with granulometric compositions OGS-01LM and OGS-08LM separately and jointly in a mix. The experimental data are recorded by a high-selectivity turbidimetric technique in water suspensions of these samples. The purpose of this study is to present the measurement results as a distribution function that enables one to identify more accurately the particle-size distribution profile and the corresponding disperse characteristics of the aerosol in question when measuring parameters of disperse media by optical techniques. The main objective of this work is to develop, implement and verify a search algorithm for the particle-size distribution function by way of a multi-parameter function. We show that the solution to the problem proposed herein is more universal because it allows slow and fast processes in suspensions and aerosols to be examined with a lower error. The algorithm can be applied to the problems which are based on solving first-kind Fredholm equations.

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The aerosol distribution in Europe derived with the Community Multiscale Air Quality (CMAQ) model: comparison to near surface in situ and sunphotometer measurements

Atmospheric Chemistry and Physics Discussions ◽

10.5194/acpd-8-1457-2008 ◽

2008 ◽

Vol 8 (1) ◽

pp. 1457-1503 ◽

Cited By ~ 4

Author(s):

V. Matthias

Keyword(s):

Distribution Function ◽

Air Quality ◽

Normal Distribution ◽

Distribution Functions ◽

Near Surface ◽

Aerosol Mass ◽

Aerosol Distribution ◽

Log Normal ◽

Log Normal Distribution ◽

Distribution In Europe

Abstract. The aerosol distribution in Europe was simulated with the Community Multiscale Air Quality (CMAQ) model system for the years 2000 and 2001. The results were compared with daily averages of PM10 measurements taken in the framework of EMEP and with aerosol optical depth (AOD) values measured within Aeronet. The modelled total aerosol mass is typically about 30–60% lower than the corresponding measurements. However a comparison of the chemical composition of the aerosol revealed a considerably better agreement between the modelled and the measured aerosol components for ammonium, nitrate and sulfate, which are on average only 15–20% underestimated. Sligthly worse agreement was determined for sea salt, that was only avaliable at two sites. The largest discrepancies result from the aerosol mass which was not chemically specified by the measurements. The agreement between measurements and model is better in winter than in summer. The modelled organic aerosol mass is higher in summer than in winter but it is significantly underestimated by the model. This could be the main reason for the discrepancies between measurements and model results. The probability distribution function of the PM10 measurements follows a log-normal distribution at most sites. The model is only able to reproduce this distribution function at non-coastal low altitude stations. The AOD derived from the model results is 20–70% lower than the values observed within Aeronet. This is mainly attributed to the missing aerosol mass in the model. The day-to-day variability of the AOD and the log-normal distribution functions are quite well reproduced by the model. The seasonality on the other hand is underestimated by the model results because better agreement is achieved in winter.

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