A Method for Determining the Non-Gaussian Probability Density Functions of Asperity Heights for Two Contact Surface Conditions

2007 ◽  
Author(s):  
Jeng Luen Liou ◽  
Jen Fin Lin
Keyword(s):  
Probability Density ◽  
Contact Surface ◽  
Elastic Recovery ◽  
Normal Load ◽  
Plasticity Index ◽  
Asperity Height ◽  
Engineered Surfaces ◽  
Change In The Mean ◽  
The Mean ◽  
Non Gaussian

Most statistical contact analyses assume that asperity height distributions (g(z*)) follow a Gaussian distribution. However, engineered surfaces are frequently the non-Gaussian with a character dependent upon the material and surface state being evaluated. When two rough surfaces experience contact deformations, the original topography of the surfaces varies with different loads. Two kinds of topographies are considered in the present study. The first kind of topography is obtained during the contact of two surfaces under a normal load. The second kind of topography is obtained from a rough contact surface after the end of the elastic recovery. The g(z*) profile is quite sharp and has a large value at its peak if it is obtained from the surface contacts under a normal load. The g(z*) profile defined for a contact surface after the elastic recovery is quite close to the g(z*) profile before contact deformations occur if the plasticity index is a small value. However, the g(z*) profile for the contact surface after the end of elastic recovery is closer to the g(z*) profile shown in the contacts under a normal load if a large plasticity index is assumed. Skewness (Sk) and kurtosis (Kt), which are the parameters in the probability density function, are affected by the change in the mean separation of two contact surfaces, or the initial skewness (the initial kurtosis is fixed in this study), or the plasticity index of the rough surface are also discussed on the basis of the topography models mentioned above.

A Microcontact Non-Gaussian Surface Roughness Model Accounting for Elastic Recovery

2008 ◽  
Vol 75 (3) ◽  
Author(s):  
Jeng Luen Liou ◽  
Jen Fin Lin
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Contact Surface ◽  
Elastic Recovery ◽  
Normal Load ◽  
Theoretical Method ◽  
Constant Load ◽  
Plasticity Index ◽  
Non Gaussian

Most statistical contact analyses assume that asperity height distributions (g(z*)) follow a Gaussian distribution. However, engineered surfaces are frequently non-Gaussian with the type dependent on the material and surface state being evaluated. When two rough surfaces experience contact deformations, the original topography of the surfaces varies with different loads, and the deformed topography of the surfaces after unloading and elastic recovery is quite different from surface contacts under a constant load. A theoretical method is proposed in the present study to discuss the variations of the topography of the surfaces for two contact conditions. The first kind of topography is obtained during the contact of two surfaces under a normal load. The second kind of topography is obtained from a rough contact surface after elastic recovery. The profile of the probability density function is quite sharp and has a large peak value if it is obtained from the surface contacts under a normal load. The profile of the probability density function defined for the contact surface after elastic recovery is quite close to the profile before experiencing contact deformations if the plasticity index is a small value. However, the probability density function for the contact surface after elastic recovery is closer to that shown in the contacts under a normal load if a large initial plasticity index is assumed. How skewness (Sk) and kurtosis (Kt), which are the parameters in the probability density function, are affected by a change in the dimensionless contact load, the initial skewness (the initial kurtosis is fixed in this study) or the initial plasticity index of the rough surface is also discussed on the basis of the topography models mentioned above. The behavior of the contact parameters exhibited in the model of the invariant probability density function is different from the behavior exhibited in the present model.

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.

Spectral and Probability-Density Nature of Square-Prism Separation-Reattachment Wall Pressures

1978 ◽  
Vol 100 (4) ◽  
pp. 485-492 ◽  
Author(s):  
J. B. Wedding ◽  
J. M. Robertson ◽  
J. A. Peterka ◽  
R. E. Akins
Keyword(s):  
Probability Density ◽  
Wake Flow ◽  
Statistical Nature ◽  
Additional Information ◽  
Density Distributions ◽  
Square Prism ◽  
Pressure Fields ◽  
The Mean ◽  
Non Gaussian ◽  
Probability Density Distributions

The statistical nature of the fluctuating pressures associated with the separation-reattachment flow were studied for a two-dimensional square prism in uniform flow for low (0.33 percent) and high (10.4 percent) turbulence levels. Studies were also made with a splitter plate to inhibit the feedback effect arising from vortex shedding. The nature of the separation reattachment flow was charcterized by use of the measured value of the mean and fluctating pressure fields. Spectral distribution of the unsteady pressures reveals strong energy spikes at the Strenthal frequency which are eliminated by the pressure of the splitler plate. Probability density distributions indicate appreciably non-Gaussian nature only in the wake. Additional information is presented on the variation with angle of the Strouhal frequency for the wake flow.

Hydrodynamic Lubrication of Face Seal in a Turbulent Flow Regime

1996 ◽  
Vol 118 (3) ◽  
pp. 589-600 ◽  
Author(s):  
Jen Fin Lin ◽  
Chih Chung Yao
Keyword(s):  
Porous Material ◽  
Film Thickness ◽  
Hydrodynamic Lubrication ◽  
Normal Load ◽  
Mixed Lubrication ◽  
Turbulent Regime ◽  
Face Seal ◽  
Hydrodynamic Load ◽  
Asperity Height ◽  
The Mean

Models for thermohydrodynamic lubrication in the turbulent regime are developed for a mechanical end face seal with various combinations of asperity height and roughness pattern. A surface wear model, based on deformation, is established for mixed lubrication such that the displacement is at most equal to the mean asperity height. Only normal load is involved in the solution of asperity deformation, and the mean film thickness is determined based on a total volume conservation hypothesis, in conjunction with an elastic-exponential hardening model. The singularity problem, present in the expected form of the Reynolds equation for a seal surface with circumferentially-oriented roughness grain sphere grooves, is avoided by viewing the seal roughness as porous material, thereby introducing roughness permeability. Flow permeability is thus obtained by combining Darcy’s law for porous material with the average flow model developed by Patir and Cheng for mixed lubrication. The hydrodynamic pressure and thereby the hydrodynamic load support are relatively higher from a seal with radially-oriented roughness. Both the mean film thickness and the hydrodynamic load support are substantially elevated by increasing the composite rms roughness, raising the inlet-flow pressure, and decreasing the rotational speed. Good agreement has been obtained from the comparison between the results herein and Lebeck’s experimental results.

Understanding the CoVID-19 pandemic Curve through statistical approach (Preprint)

2020 ◽  
Author(s):  
Ibrar Ul Hassan Akhtar
Keyword(s):  
Probability Density ◽  
Change Point ◽  
Statistical Approach ◽  
Sudden Change ◽  
Global Scale ◽  
Change Point Detection ◽  
Segment Length ◽  
Infected Population ◽  
The Mean ◽  
Point Detection

UNSTRUCTURED Current research is an attempt to understand the CoVID-19 pandemic curve through statistical approach of probability density function with associated skewness and kurtosis measures, change point detection and polynomial fitting to estimate infected population along with 30 days projection. The pandemic curve has been explored for above average affected countries, six regions and global scale during 64 days of 22nd January to 24th March, 2020. The global cases infection as well as recovery rate curves remained in the ranged of 0 ‒ 9.89 and 0 ‒ 8.89%, respectively. The confirmed cases probability density curve is high positive skewed and leptokurtic with mean global infected daily population of 6620. The recovered cases showed bimodal positive skewed curve of leptokurtic type with daily recovery of 1708. The change point detection helped to understand the CoVID-19 curve in term of sudden change in term of mean or mean with variance. This pointed out disease curve is consist of three phases and last segment that varies in term of day lengths. The mean with variance based change detection is better in differentiating phases and associated segment length as compared to mean. Global infected population might rise in the range of 0.750 to 4.680 million by 24th April 2020, depending upon the pandemic curve progress beyond 24th March, 2020. Expected most affected countries will be USA, Italy, China, Spain, Germany, France, Switzerland, Iran and UK with at least infected population of over 0.100 million. Infected population polynomial projection errors remained in the range of -78.8 to 49.0%.

scholarly journals Reconstruction of Diffusion Coefficients and Power Exponents from Single Lagrangian Trajectories

Fluids ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 111
Author(s):  
Leonid M. Ivanov ◽  
Collins A. Collins ◽  
Tetyana Margolina
Keyword(s):  
Black Sea ◽  
Diffusion Coefficients ◽  
Current System ◽  
Mean Square Displacement ◽  
California Current ◽  
The Black Sea ◽  
Mean Square ◽  
California Current System ◽  
The Mean ◽  
Non Gaussian

Using discrete wavelets, a novel technique is developed to estimate turbulent diffusion coefficients and power exponents from single Lagrangian particle trajectories. The technique differs from the classical approach (Davis (1991)’s technique) because averaging over a statistical ensemble of the mean square displacement (<X2>) is replaced by averaging along a single Lagrangian trajectory X(t) = {X(t), Y(t)}. Metzler et al. (2014) have demonstrated that for an ergodic (for example, normal diffusion) flow, the mean square displacement is <X2> = limT→∞τX2(T,s), where τX2 (T, s) = 1/(T − s) ∫0T−s(X(t+Δt) − X(t))2 dt, T and s are observational and lag times but for weak non-ergodic (such as super-diffusion and sub-diffusion) flows <X2> = limT→∞≪τX2(T,s)≫, where ≪…≫ is some additional averaging. Numerical calculations for surface drifters in the Black Sea and isobaric RAFOS floats deployed at mid depths in the California Current system demonstrated that the reconstructed diffusion coefficients were smaller than those calculated by Davis (1991)’s technique. This difference is caused by the choice of the Lagrangian mean. The technique proposed here is applied to the analysis of Lagrangian motions in the Black Sea (horizontal diffusion coefficients varied from 105 to 106 cm2/s) and for the sub-diffusion of two RAFOS floats in the California Current system where power exponents varied from 0.65 to 0.72. RAFOS float motions were found to be strongly non-ergodic and non-Gaussian.

Effects of the Fractional Laplacian Order on the Nonlocal Elastic Rod Response

2017 ◽  
Vol 3 (3) ◽  
Author(s):  
Giuseppina Autuori ◽  
Federico Cluni ◽  
Vittorio Gusella ◽  
Patrizia Pucci
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Coefficient Of Variation ◽  
Fractional Laplacian ◽  
Numerical Procedure ◽  
Elastic Rod ◽  
The Mean ◽  
Distributed Forces ◽  
Nonlinear Fashion

In this paper, we yield with a nonlocal elastic rod problem, widely studied in the last decades. The main purpose of the paper is to investigate the effects of the statistic variability of the fractional operator order s on the displacements u of the rod. The rod is supposed to be subjected to external distributed forces, and the displacement field u is obtained by means of numerical procedure. The attention is particularly focused on the parameter s, which influences the response in a nonlinear fashion. The effects of the uncertainty of s on the response at different locations of the rod are investigated by the Monte Carlo simulations. The results obtained highlight the importance of s in the probabilistic feature of the response. In particular, it is found that for a small coefficient of variation of s, the probability density function of the response has a unique well-identifiable mode. On the other hand, for a high coefficient of variation of s, the probability density function of the response decreases monotonically. Finally, the coefficient of variation and, to a small extent, the mean of the response tend to increase as the coefficient of variation of s increases.

Seven years follow-up of corneal cross-linking (CXL) in pediatric patients: Evaluation of treated and untreated eye

2021 ◽  
pp. 112067212110206
Author(s):  
Iliya Simantov ◽  
Lior Or ◽  
Inbal Gazit ◽  
Biana Dubinsky-Pertzov ◽  
David Zadok ◽  
...  
Keyword(s):  
Visual Acuity ◽  
Pediatric Patients ◽  
Pediatric Population ◽  
Corneal Thickness ◽  
Cross Linking ◽  
Uncorrected Distance Visual Acuity ◽  
Mild Reduction ◽  
Change In The Mean ◽  
The Mean

Background: Retrospective cohort study evaluating long term keratoconus progression amongst cross-linking (CXL) treated pediatric patients in the treated and the fellow untreated eyes. Methods: Data on 60 eyes of 30 patients, 18 years old or younger, who underwent CXL in at least one eye was collected and analyzed. Follow-up measurements taken from the treated and untreated eye up to 7 years after CXL treatment, were compared to baseline measurements. Parameters included uncorrected distance visual acuity (UCDVA), best-corrected spectacle visual acuity (BCSVA), manifest refraction, pachymetry, corneal tomography, and topography. Results: Mean age of patients was 16 ± 2.1 years. For the treated eyes, during follow-up period mean UCDVA had improved (from 0.78 ± 0.22 at baseline to 0.58 ± 0.26 logMAR at 7 years; p = 0.13), as well as mean BCSVA (from 0.23 ± 0.107 at baseline to 0.172 ± 0.05 logMAR at 7 years; p = 0.37). The mean average keratometry showed a significant flattening (from 49.95 ± 4.04 to 47.94 ± 3.3 diopters (D); p < 0.001), However there was no change in the mean maximal keratometry. The mean minimal corneal thickness (MCT) showed a significant mild reduction of 26 µm ( p = 0.006). Although statistically insignificant, the mean manifest cylinder was also reduced to 2D ( p = 0.15). During the follow-up period, eight untreated eyes (26.6%) deteriorated and underwent CXL, while only one treated eye (3.33%) required an additional CXL. Conclusion: CXL is a safe and efficient procedure in halting keratoconus progression in the pediatric population, the fellow eye needs to be carefully monitored but only a 25% of the patients will require CXL in that eye during a period of 7 years.

Governing equations for probability densities of Marcus stochastic differential equations with Lévy noise

2016 ◽  
Vol 17 (05) ◽  
pp. 1750033 ◽  
Author(s):  
Xu Sun ◽  
Xiaofan Li ◽  
Yayun Zheng
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Probability Density ◽  
Lévy Processes ◽  
Planck Equation ◽  
Levy Processes ◽  
Stochastic Dynamical Systems ◽  
Non Gaussian ◽  
Fokker Planck Equations ◽  
Fokker Planck

Marcus stochastic differential equations (SDEs) often are appropriate models for stochastic dynamical systems driven by non-Gaussian Lévy processes and have wide applications in engineering and physical sciences. The probability density of the solution to an SDE offers complete statistical information on the underlying stochastic process. Explicit formula for the Fokker–Planck equation, the governing equation for the probability density, is well-known when the SDE is driven by a Brownian motion. In this paper, we address the open question of finding the Fokker–Planck equations for Marcus SDEs in arbitrary dimensions driven by non-Gaussian Lévy processes. The equations are given in a simple form that facilitates theoretical analysis and numerical computation. Several examples are presented to illustrate how the theoretical results can be applied to obtain Fokker–Planck equations for Marcus SDEs driven by Lévy processes.

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