Variation in Fractal Properties and Non-Gaussian Distributions of Microcontact Between Elastic-Plastic Rough Surfaces With Mean Surface Separation

2005 ◽  
Vol 73 (1) ◽  
pp. 143-152 ◽  
Author(s):  
Jung Ching Chung ◽  
Jen Fin Lin
Keyword(s):  
Fractal Dimension ◽  
Power Spectrum ◽  
Gaussian Distribution ◽  
Experimental Results ◽  
Fractal Parameters ◽  
Spectrum Function ◽  
The Mean ◽  
Non Gaussian ◽  
The Relationship ◽  
Scaling Coefficient

The fractal parameters (fractal dimension and topothesy), describing the contact behavior of rough surface, were considered as constant in the earlier models. However, their results are often significantly different from the experimental results. In the present study, these two roughness parameters have been derived analytically as a function of the mean separation first, then they are found with the aid of the experimental results. By equating the structure functions developed in two different ways, the relationship among the scaling coefficient in the power spectrum function, the fractal dimension, and topothesy of asperity heights can be established. The variation of topothesy can be determined when the fractal dimension and the scaling coefficient have been obtained from the experimental results of the number of contact spots and the power spectrum function at different mean separations. The probability density function of asperity heights, achieved at a different mean separation, was obtained from experimental results as a non-Gaussian distribution; it is expressed as a function of the skewness and the kurtosis. The relationship between skewness and mean separation can be established through the fitting of experimental results by this non-Gaussian distribution. For a sufficiently small mean separation, either the total load or the real contact area predicted by variable fractal parameters, as well as non-Gaussian distribution, is greater than that predicted by constant fractal parameters, as well as Gaussian distribution. The difference between these two models is significantly enhanced as the mean separation becomes small.

A New Method Developed for Fractal Dimension and Topothesy Varying With the Mean Separation of Two Contact Surfaces

2006 ◽  
Vol 128 (3) ◽  
pp. 515-524 ◽  
Author(s):  
Jeng Luen Liou ◽  
Jen Fin Lin
Keyword(s):  
Fractal Dimension ◽  
Power Spectrum ◽  
Contact Spot ◽  
Experimental Results ◽  
Spot Number ◽  
Spectrum Function ◽  
The Mean ◽  
Contact Surfaces ◽  
The Relationship ◽  
Scaling Coefficient

Instead of a general consideration of the fractal dimension (D) and the topothesy (G*) as two invariants in the fractal analysis of surface asperities, these two roughness parameters in the present study are varied by changing the mean separation (d*) of two contact surfaces. The relationship between the fractal dimension and the mean separation is found first. By equating the structure functions developed in two different ways, the relationship among the scaling coefficient in the power spectrum function, the fractal dimension, and topothesy of asperity heights can be established. The variation of topothesy can be determined when the fractal dimension and the scaling coefficient have been obtained from the experimental results of the number of contact spots and the power spectrum function at different mean separations. A numerical scheme is developed in this study to determine the convergent values of fractal dimension and topothesy corresponding to a given mean separation. The theoretical results of the contact spot number predicted by the present model show good agreement with the reported experimental results. Both the fractal dimension and the topothesy are elevated by increasing the mean separation. Significant differences in the contact load or the total contact area are shown between the models of constant D and G* and variable D and G* as the mean separation is reduced to smaller values.

A New Microcontact Model Developed for Variable Fractal Dimension, Topothesy, Density of Asperity, and Probability Density Function of Asperity Heights

2006 ◽  
Vol 74 (4) ◽  
pp. 603-613 ◽  
Author(s):  
Jeng Luen Liou ◽  
Jen Fin Lin
Keyword(s):  
Fractal Dimension ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Gaussian Distribution ◽  
Fractal Theory ◽  
Radius Of Curvature ◽  
The Mean ◽  
Contact Surfaces ◽  
Non Gaussian

In the present study, the fractal theory is applied to modify the conventional model (the Greenwood and Williamson model) established in the statistical form for the microcontacts of two contact surfaces. The mean radius of curvature (R) and the density of asperities (η) are no longer taken as constants, but taken as variables as functions of the related parameters including the fractal dimension (D), the topothesy (G), and the mean separation of two contact surfaces. The fractal dimension and the topothesy varied by differing the mean separation of two contact surfaces are completely obtained from the theoretical model. Then the mean radius of curvature and the density of asperities are also varied by differing the mean separation. A numerical scheme is thus developed to determine the convergent values of the fractal dimension and topothesy corresponding to a given mean separation. The topographies of a surface obtained from the theoretical prediction of different separations show the probability density function of asperity heights to be no longer the Gaussian distribution. Both the fractal dimension and the topothesy are elevated by increasing the mean separation. The density of asperities is reduced by decreasing the mean separation. The contact load and the total contact area results predicted by variable D, G*, and η as well as non-Gaussian distribution are always higher than those forecast with constant D, G*, η, and Gaussian distribution.

A Microcontact Model Developed for Asperity Heights With a Variable Profile Fractal Dimension, a Surface Fractal Dimension, Topothesy, and Non-Gaussian Distribution

2008 ◽  
Author(s):  
Jen Luen Liou ◽  
Jen Fin Lin
Keyword(s):  
Fractal Dimension ◽  
Cross Sections ◽  
Fractal Dimensions ◽  
Surface Fractal Dimension ◽  
Surface Fractal ◽  
Deformation Regime ◽  
Circular Contour ◽  
The Mean ◽  
Non Gaussian ◽  
The Relationship

The cross sections formed by the contact asperities of two rough surfaces at an interference are island-shaped, rather than having the commonly assumed circular contour. These island-shaped contact surface contours show fractal behavior with a profile fractal dimension Ds. The surface fractal dimension for the asperity heights is defined as D and the topothesy is defined as G. In the study of Mandelbrot, the relationship between D and Ds was given as D = Ds+1 if these two fractal dimensions are obtained before contact deformation. In the present study, D, G, and Ds are considered to be varying with the mean separation (or the interference at the rough surface) between two contact surfaces. The D-Ds relationships for the contacts at the elastic, elastoplastic, and fully plastic deformations are derived and the inceptions of the elastoplastic deformation regime and the fully plastic deformation regime are redefined using the equality of two expressions established in two different ways for the number of contact spots (N).

A Microcontact Model Developed for Asperity Heights with a Variable Profile Fractal Dimension, A Surface Fractal Dimension, Topothesy, and Non-Gaussian Distribution

2009 ◽  
Vol 25 (1) ◽  
pp. 103-115
Author(s):  
J. L. Liou ◽  
J. F. Lin
Keyword(s):  
Fractal Dimension ◽  
Contact Area ◽  
Distribution Functions ◽  
Real Contact Area ◽  
Contact Deformation ◽  
Surface Fractal Dimension ◽  
Surface Fractal ◽  
Deformation Regime ◽  
The Mean ◽  
Non Gaussian

AbstractThe cross sections formed by the contact asperities of two rough surfaces at an interference are islandshaped, rather than having the commonly assumed circular contour. These island-shaped contact surface contours show fractal behavior with a profile fractal dimension Ds. The surface fractal dimension for the asperity heights is defined as D and the topothesy is defined as G. In the study of Mandelbrot, the relationship between D and Ds was given as D = Ds + 1 if these two fractal dimensions are obtained before contact deformation. In the present study, D, G, and Ds are considered to be varying with the mean separation (or the interference at the rough surface) between two contact surfaces. The D-Ds relationships for the contacts at the elastic, elastoplastic, and fully plastic deformations are derived and the inceptions of the elastoplastic deformation regime and the fully plastic deformation regime are redefined using the equality of two expressions established in two different ways for the number of contact spots (N). The contact parameters, including the total contact force and the real contact area, were evaluated when the size distribution functions (n) for the three deformation regimes were available. The results indicate that both the D and Ds parameters in these deformation regimes increased with increasing the mean separation (d*). The initial plasticity index before contact deformation (ψ)0 is also a factor of importance to the predictions of the contact load (F*t) and contact area (At*) between the model of variable D and G, non-Gaussian asperity heights and circular contact area and the present model of variable D and G, non-Gaussian asperity heights and fractal contact area.

An iterative method for improved estimation of the mean of peer-group distributions in proficiency testing

2005 ◽  
Vol 43 (4) ◽  
Author(s):  
Kiyoshi Ichihara ◽  
Tadashi Kawai
Keyword(s):  
Computer Simulation ◽  
Iterative Method ◽  
Proficiency Testing ◽  
Conventional Method ◽  
Gaussian Distribution ◽  
Peer Group ◽  
Central Portion ◽  
The Mean ◽  
Non Gaussian ◽  
Main Population

AbstractIn proficiency testing (PT), the peer-group mean is conventionally computed after twice removing values exceeding the mean±3 SD. However, this adjustment fails if there are many outliers. In this study an iterative method was evaluated as a more robust way to estimate the means. The methodology repeatedly removes a proportion of the population (usually those exceeding the mean±1.6 SD), assuming the presence of a Gaussian distribution in the central portion, and reinflates the SD to compensate for the trimming. A computer simulation revealed that the estimated mean of a known Gaussian distribution was less affected by a subpopulation that overlaps the main population than was the conventional method. When the overlapping portions were removed, the iterative method predicted the true mean correctly. The method was applied to external PT results for 44 analytes. Although most peer-group distributions were clearly non-Gaussian, the segment included by the predicted mean±1.6 SD was regarded as Gaussian in 85.9% by the new method and 73.4% by the conventional method. The proposed methodology appears to be an improved way of estimating peer-group means.

scholarly journals REPRESENTATION AND WEAK CONVERGENCE OF STOCHASTIC INTEGRALS WITH FRACTIONAL INTEGRATOR PROCESSES

2009 ◽  
Vol 25 (6) ◽  
pp. 1589-1624 ◽  
Author(s):  
James Davidson ◽  
Nigar Hashimzade
Keyword(s):  
Long Memory ◽  
Moving Average ◽  
Full Range ◽  
Sample Covariance ◽  
Fractional Integrator ◽  
The Mean ◽  
Non Gaussian ◽  
The Time Domain ◽  
The Relationship ◽  
Harmonic Representation

This paper considers the asymptotic distribution of the sample covariance of a nonstationary fractionally integrated process with the stationary increments of another such process—possibly itself. Questions of interest include the relationship between the harmonic representation of these random variables, which we have analyzed in a previous paper (Davidson and Hashimzade, 2008), and the construction derived from moving average representations in the time domain. Depending on the values of the long memory parameters and choice of normalization, the limiting integral is shown to be expressible as the sum of a constant and two Itô-type integrals with respect to distinct Brownian motions. In certain cases the latter terms are of small order relative to the former. The mean is shown to match that of the harmonic representation, where the latter is defined, and satisfies the required integration by parts rule. The advantages of our approach over the harmonic analysis include the facts that our formulas are valid for the full range of the long memory parameters and that they extend to non-Gaussian processes.

Multi-Center Calibration of the Second Reference Material for Thromboplastin, Rabbit, Plain, Coded CRM 149R

1991 ◽  
Vol 65 (03) ◽  
pp. 263-267 ◽  
Author(s):  
A M H P van den Besselaar ◽  
R M Bertina
Keyword(s):  
Reference Material ◽  
Standard Error ◽  
Standard Errors ◽  
Sensitivity Index ◽  
Long Term Stability ◽  
Significant Bias ◽  
The Mean ◽  
The Relationship ◽  
International Sensitivity Index

SummaryIn a collaborative trial of eleven laboratories which was performed mainly within the framework of the European Community Bureau of Reference (BCR), a second reference material for thromboplastin, rabbit, plain, was calibrated against its predecessor RBT/79. This second reference material (coded CRM 149R) has a mean International Sensitivity Index (ISI) of 1.343 with a standard error of the mean of 0.035. The standard error of the ISI was determined by combination of the standard errors of the ISI of RBT/79 and the slope of the calibration line in this trial.The BCR reference material for thromboplastin, human, plain (coded BCT/099) was also included in this trial for assessment of the long-term stability of the relationship with RBT/79. The results indicated that this relationship has not changed over a period of 8 years. The interlaboratory variation of the slope of the relationship between CRM 149R and RBT/79 was significantly lower than the variation of the slope of the relationship between BCT/099 and RBT/79. In addition to the manual technique, a semi-automatic coagulometer according to Schnitger & Gross was used to determine prothrombin times with CRM 149R. The mean ISI of CRM 149R was not affected by replacement of the manual technique by this particular coagulometer.Two lyophilized plasmas were included in this trial. The mean slope of relationship between RBT/79 and CRM 149R based on the two lyophilized plasmas was the same as the corresponding slope based on fresh plasmas. Tlowever, the mean slope of relationship between RBT/79 and BCT/099 based on the two lyophilized plasmas was 4.9% higher than the mean slope based on fresh plasmas. Thus, the use of these lyophilized plasmas induced a small but significant bias in the slope of relationship between these thromboplastins of different species.

Evaluation of the Efficacy of ST2 and NT-proBNP in the Diagnosis and Prediction of Short- Term Prognosis in Heart Failure with Reduced Ejection Fraction

2017 ◽  
Vol 9 (04) ◽  
Author(s):  
Shivananda B Nayak ◽  
Dharindra Sawh ◽  
Brandon Scott ◽  
Vestra Sears ◽  
Kareshma Seebalack ◽  
...  
Keyword(s):  
Heart Failure ◽  
Ejection Fraction ◽  
Cardiac Biomarkers ◽  
Double Blind Study ◽  
Short Term ◽  
Double Blind ◽  
Reduced Ejection Fraction ◽  
The Mean ◽  
Design And Methods ◽  
The Relationship

Purpose: i) To determine the relationship between the cardiac biomarkers ST2 and NT-proBNP with ejection fraction (EF) in heart failure (HF) patients. ii) Assess whether a superiority existed between the aforementioned cardiac markers in diagnosing the HF with reduced EF. iii) Determine the efficacy of both biomarkers in predicting a 30-day cardiovascular event and rehospitalization in patients with HF with reduced EF iv) To assess the influence of age, gender, BMI, anaemia and renal failure on the ST2 and NT-proBNP levels. Design and Methods: A prospective double-blind study was conducted to obtain data from a sample of 64 cardiology patients. A blood sample was collected to test for ST2 and NT-proBNP. An echocardiogram (to obtain EF value), electrocardiogram and questionnaire were also obtained. Results: Of the 64 patients enrolled, 59.4% of the population had an EF less than 40%. At the end of the 30- day period, 7 patients were warded, 37 were not warded, one died and 17 were non respondent. Both biomarkers were efficacious at diagnosing HF with a reduced EF. However, neither of them were efficacious in predicting 30-day rehospitalization. The mean NT-proBNP values being: not rehospitalized (2114.7486) and 30 day rehospitalization (1008.42860) and the mean ST2 values being: not rehospitalized (336.1975), and 30-day rehospitalization. (281.9657). Conclusion: Neither ST2 or NT-proBNP was efficacious in predicting the short- term prognosis in HF with reduced EF. Both however were successful at confirming the diagnosis of HF in HF patients with reduced EF.

Limit tolerances for sums of measurement errors

2018 ◽  
Vol 934 (4) ◽  
pp. 59-62
Author(s):  
V.I. Salnikov
Keyword(s):  
Normal Distribution ◽  
Mean Square Error ◽  
Gaussian Distribution ◽  
Measurement Errors ◽  
Theory And Practice ◽  
Mean Square ◽  
The Mean ◽  
Limiting Values ◽  
Limiting Value

The question of calculating the limiting values of residuals in geodesic constructions is considered in the case when the limiting value for measurement errors is assumed equal to 3m, ie ∆рred = 3m, where m is the mean square error of the measurement. Larger errors are rejected. At present, the limiting value for the residual is calculated by the formula 3m√n, where n is the number of measurements. The article draws attention to two contradictions between theory and practice arising from the use of this formula. First, the formula is derived from the classical law of the normal Gaussian distribution, and it is applied to the truncated law of the normal distribution. And, secondly, as shown in [1], when ∆рred = 2m, the sums of errors naturally take the value equal to ?pred, after which the number of errors in the sum starts anew. This article establishes its validity for ∆рred = 3m. A table of comparative values of the tolerances valid and recommended for more stringent ones is given. The article gives a graph of applied and recommended tolerances for ∆рred = 3m.

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