A Study of the Effect of m and n Coefficients of the Hertzian Contact Theory on Railroad Vehicle Dynamics

2007 ◽  
Author(s):  
J. P. Pascal ◽  
Khaled E. Zaazaa
Keyword(s):  
Contact Area ◽  
Contact Region ◽  
Local Deformation ◽  
Hertzian Contact ◽  
Vehicle Dynamic ◽  
Railroad Vehicle Dynamics ◽  
Original Table ◽  
Data Points ◽  
Railroad Vehicle ◽  
Contact Ellipse

For the wheel/rail contact problem, the Hertz theory for two elastic bodies in contact is commonly used to determine the shape and dimensions of the contact area and the local deformation of the wheel and rail surfaces at the contact region. The shape of the contact area is assumed to be elliptical. The ratio of the contact ellipse semi-axes is equal to the ratio of two non-dimensional contact area coefficients, known as m and n coefficients. Hertz presented a table of these two coefficients, determined as a function of an angular parameter, θ. Most railroad vehicle dynamic codes use this table with online interpolation to determine the contact ellipse semi-axes. Recently, it was found that this original table may be too coarse, and that more data points are needed within the table for solving the wheel/rail contact accurately. This paper discusses the effect of the accuracy of the m and n coefficients in solving for wheel/rail contact, and demonstrates this effect with two numerical examples that show the resulting differences in the dynamic behavior of railroad vehicles dependent on this accuracy. A new table with more data points is presented that is recommended for use in railroad vehicle dynamic codes that employ the Hertzian contact for solving the wheel/rail contact interaction. This modified table was originally derived by Jean-Pierre Pascal as a part of collaborative research between the Federal Railroad Administration (FRA) and the French Ministry of Transportation.

On the Effect of Hydrodynamic Action on the Contact Area of Highly Loaded Line Contacts

1984 ◽  
Vol 106 (1) ◽  
pp. 152-155
Author(s):  
M. O. A. Mokhtar ◽  
A. A. Abdel-Ghany
Keyword(s):  
Contact Area ◽  
Local Deformation ◽  
Hertzian Contact ◽  
Oil Film ◽  
Operating Variables ◽  
Lubricant Oil ◽  
Line Contacts ◽  
Hydrodynamic Solution ◽  
Heavy Loads ◽  
Highly Loaded

The elastically deformed contact area together with the oil film thickness between lubricated plane cylinders in relative motion are experimentally measured and analyzed under various loads and speeds. The measured contact profile has shown to be a function of the operating variables and consequently represents a justifying means in deciding the appropriate solutions to the lubrication problem in concentrated contacts. Under heavy loads and very low speeds, the deformation at the contact zone approaches the flat Hertzian contact, while at increased speeds elastohydrodynamic regimes prevail and relatively thick oil film could form with the presence of a local constriction at the trailing oil exit end. By further speeding up the local deformation diminishes and full (rigid) hydrodynamic solution would govern the lubricant oil behavior with minimum local deformations.

Cytomorphometric characteristics of imprints of the contact area between the gingiva and the dental implant after one-step implantation

2017 ◽  
pp. 77-82
Author(s):  
А.С. Рыбалко ◽  
А.С. Григорьян ◽  
А.А. Орлов
Keyword(s):  
Dental Implants ◽  
Dental Implant ◽  
Contact Area ◽  
Inflammatory Reactions ◽  
Diagnosis And Prognosis ◽  
Dental Implantation ◽  
Data Points ◽  
One Step ◽  
Cytological Method ◽  
Clinical Pictures

Цель исследования состояла в разработке неинвазивного метода диагностики состояния тканей периимплантационной зоны после постановки дентальных имплантатов, а также прогноза течения послеоперационного периода и ранней диагностики послеоперационных осложнений, таких, как мукозит и периимплантит, при одноэтапном методе дентальной имплантации. Методы. В исследовании представлена цитологическая характеристика отпечатков из области контакта имплантата с десной, в которых оценивали состояние эпителиальных клеток при различных проявлениях цитопатологии. Определяли индекс деструкции (ИД) и воспалительно-деструктивный индекс (ВДИ) в клетках воспалительного инфильтрата, которые сопоставляли с клинической картиной течения послеоперационного периода. Сроки забора цитологического материала 1, 5, 15, 25, 30, 60, 120 дней после постановки имплантатов. Количество исследованных цитограмм - 442 от 11 пациентов (по 1-2 имплантата). Общий массив использованных для вычисления интегральных показателей составил »2550. Результаты. Было установлено, что в сроки 1-5 дней после постановки дентальных имплантатов происходит резкое повышение показателей ИД (до 7500 ± 15) и ВДИ (до 80 ± 20), что отражает процесс развития в области имплантатов острых воспалительных реакций. В отдаленные сроки наблюдений отмечалось 2 варианта развития событий. Первый из них характеризовался сохранением во все сроки высоких показателей ИД и ВДИ. Для второго варианта динамики показателей было характерно их падение, что соответствовало развитию процесса остеоинтеграции дентальных имплантатов. The objectives of the investigation were to develop protocols evaluating the validity and effectiveness of the cytological method for diagnosis and prognosis of tissue conditions adjacent to a dental implant. Methods. This study examined the cytological characteristics of imprints of the gingiva-implant contact area, a. the correlation of quantitative ratios of epithelial cells either with or without hystopathological manifestations (an index of destruction, ID); b. ratio of cells of the inflammatory infiltrate (an inflammatory and destructive index, IDI); these data were researched in relation with dynamics of the clinical pictures during a postoperative period.The cytological material was obtained 1, 5, 15, 25, 30, 60, 120 days after dental implantation. Totally 442 cytogramme from 11 patients (1-2 implant per a patient) were assayed. The data array for computation of integrated indicators was composed of ca. 2550 data points. Results. A significant increase of ID (7500 ± 15) and IDI (80 ± 20) indicators, which reflects the development of acute inflammatory reactions in tissues neighboring an implant, was observed within 1-5 days following the insertion of dental implants. Long-term observations revealed two scenarios. The first scenario was characterized by the retention of high levels of ID and IDI over the period of observations. The second scenario featured the reduction of ID and IDI indices, which corresponded to the development of the osseointegration of dental implants.

scholarly journals Improved design criterion for frictionally engaged contacts in overrunning clutches

2021 ◽  
Author(s):  
Nadine Nagler ◽  
Armin Lohrengel
Keyword(s):  
Design Process ◽  
Contact Area ◽  
Axial Load ◽  
State Of The Art ◽  
Tangential Force ◽  
Design Criterion ◽  
Hertzian Contact ◽  
Loss Of Function ◽  
Axial Loads ◽  
Improved Design

AbstractOverrunning clutches, also known as freewheel clutches, are frictionally engaged, directional clutches; they transmit torque depending on the Freewheel clutch rings’ rotation directions. The torque causes a tangential force in the Hertzian contact area. The hitherto “state-of-the-art design criterion” bases on this load situation. In practice, axial loads additionally act on the frictionally engaged Hertzian contact area. This additional axial load can cause the loss of the friction connection and so the freewheel clutch slips. This publication presents an improved design criterion for frictionally engaged contacts in freewheel clutches. It allows to consider tangential as well as axial loads during the design process. Additionally, it offers the possibility to estimate the probability of frictional engagement loss and gross slip based on the freewheel clutch’s application scenario. This publication points out how to use the improved design criterion to design freewheel clutches that are more robust against a loss of function.

A simplified formulation of adhesion problems with elastic plates

2009 ◽  
Vol 465 (2107) ◽  
pp. 2217-2230 ◽  
Author(s):  
Carmel Majidi ◽  
George G. Adams
Keyword(s):  
Potential Energy ◽  
Contact Area ◽  
Contact Region ◽  
Bending Moment ◽  
Total Potential Energy ◽  
Work Of Adhesion ◽  
Contact Radius ◽  
Algebraic Complexity ◽  
Elastic Plates ◽  
Total Potential

The solution of adhesion problems with elastic plates generally involves solving a boundary-value problem with an assumed contact area. The contact region is then found by minimizing the total potential energy with respect to the contact area (i.e. the contact radius for the axisymmetric case). Such a procedure can be extremely long and tedious. Here, we show that the inclusion of adhesion is equivalent to specifying a discontinuous internal bending moment at the contact region boundary. The magnitude of this moment discontinuity is related to the work of adhesion and flexural rigidity of the plate. Such a formulation can greatly reduce the algebraic complexity of solving these problems. It is noted that the related plate contact problems without adhesion can also be solved by minimizing the total potential energy. However, it has long been recognized that it is mathematically more efficient to find the contact area by specifying a continuous internal bending moment at the boundary of the contact region. Thus, our moment discontinuity method can be considered to be a generalization of that procedure which is applicable for problems with adhesion.

Impact on Beams

1936 ◽  
Vol 3 (2) ◽  
pp. A55-A61
Author(s):  
H. L. Mason
Keyword(s):  
Historical Development ◽  
Contact Region ◽  
Local Deformation ◽  
Lateral Vibration ◽  
Transverse Impact ◽  
Concentrated Mass ◽  
Impact Theory ◽  
Inelastic Impact ◽  
Maximum Contact

Abstract This paper deals with transverse impact on beams the mass of which is of importance. Experimental results are presented for comparison with theory. Impacts which appear single to the eye are shown to consist in reality of several blows in quick succession. Section 1 of the paper traces the historical development of this subject by discussing the investigations of Young, Hodgkinson, Cox, Saint Venant, and Timoshenko. Section 2 treats a simplified system in which a concentrated mass strikes a smaller concentrated mass having a “soft” spring restraint. For elastic impact, theory predicts for the struck mass a path composed of sinusoidal elements separated by instantaneous blows. For inelastic impact it predicts a joint harmonic motion. Records of the paths of both masses were obtained experimentally. Section 3 of the paper uses Timoshenko’s method of combining local deformation of the contact region with lateral vibration of the beam. An experimental investigation of maximum contact pressure and of blow duration gives what is believed to be the first confirmation of this theory. Section 4 describes an experimental determination of flexural stresses in elastic and inelastic impact on a 3-in. I-beam by the use of a Westinghouse magnetic strain gage. The indication is that stresses may be higher than those calculated by the usual approximations.

Experimental and numerical study on cavitation under elastohydrodynamic lubrication point contact

2021 ◽  
pp. 135065012110527
Author(s):  
Mingfei Ma ◽  
Wen Wang ◽  
Wenxun Jiang
Keyword(s):  
Contact Area ◽  
Numerical Study ◽  
Ambient Pressure ◽  
Point Contact ◽  
Elastohydrodynamic Lubrication ◽  
Hertzian Contact ◽  
Pressure Area ◽  
Cavitation Pressure ◽  
Experimental Researches ◽  
State 1

As a common phenomenon in elastohydrodynamic lubrication, cavitation has an effect on the completeness of the oil film in the contact area. Many studies have therefore been conducted on cavitation. Experimental researches on cavitation usually rely on optical interference observation, which offers a limited resolution and observation range. In this paper, an infrared thermal camera is used to observe the cavity bubbles on a ball-on-disc setup under sliding/rolling conditions. The results show that the cavity length increases with an increases of the entrainment speed and the viscosity of the lubricants. These observations are explained by a numerical model based on Elrod's algorithm. Effects of entrainment speed and lubricant viscosity on the breakup of cavitation bubbles and the cavitation states are investigated. Both the simulation and experimental results show that a negative pressure area is present behind the Hertzian contact area. The ambient pressure plays a role in maintaining cavitation state 1. The cavitation pressure is close to the vacuum pressure when the entrainment speed is low and to the ambient pressure instead when the entrainment speed is high.

scholarly journals The Elastic Contact of Rough Spheres Investigated Using a Deterministic Multi-Asperity Model

2019 ◽  
Vol 10 (01) ◽  
pp. 1841002 ◽  
Author(s):  
Vladislav A. Yastrebov
Keyword(s):  
Contact Area ◽  
Rough Surfaces ◽  
Contact Problems ◽  
Hertzian Contact ◽  
Contact Radius ◽  
Geometrical Shape ◽  
Elastic Contact ◽  
Deterministic Analysis ◽  
Asperity Model ◽  
Deformation Modes

In this paper, we use a deterministic multi-asperity model to investigate the elastic contact of rough spheres. Synthetic rough surfaces with controllable spectra were used to identify individual asperities, their locations and curvatures. The deterministic analysis enables to capture both particular deformation modes of individual rough surfaces and also statistical deformation regimes, which involve averaging over a big number of roughness realizations. Two regimes of contact area growth were identified: the Hertzian regime at light loads at the scale of a single asperity, and the linear regime at higher loads involving multiple contacting asperities. The transition between the regimes occurs at the load which depends on the second and the fourth spectral moments. It is shown that at light indentation the radius of circumference delimiting the contact area is always considerably larger than Hertzian contact radius. Therefore, it suggests that there is no scale separation in contact problems at light loads. In particular, the geometrical shape cannot be considered separately from the surface roughness at least for approaching greater than one standard roughness deviation.

Hertzian Contact and Adhesion of Elastomers

1969 ◽  
Vol 91 (4) ◽  
pp. 732-737 ◽  
Author(s):  
Richard C. Drutowski
Keyword(s):  
Stress Distribution ◽  
Tensile Stress ◽  
Contact Area ◽  
Hard Sphere ◽  
Continuous Measurement ◽  
Hertzian Contact ◽  
Initial Contact ◽  
Circular Zone ◽  
Outer Annulus ◽  
Contact Size

The contact of a hard sphere with a flat elastomer is examined both analytically and experimentally when adhesive stresses are present. Use of a transparent spherical indenter enables continuous measurement of contact size while the samples are pulled apart. For any combination of load and contact area, the superposition of a Hertz and a Boussinesq stress distribution separates the contact into a circular zone under compression and an outer annulus under tension. During separation, while the contact size decreases and the tensile annulus becomes a larger percentage of the total contact, the average tensile stress remains constant. This average adhesive is a material property which is easily measured and is shown to be invariant with respect to indenter radius and initial contact pressure. An application of this analysis to opaque indenters is described.

Nano-Meter Film Rheology and Asperity Lubrication

2002 ◽  
Vol 124 (3) ◽  
pp. 595-599 ◽  
Author(s):  
Bo Jacobson
Keyword(s):  
Contact Area ◽  
Pressure Fluctuations ◽  
Slip Rate ◽  
Hertzian Contact ◽  
Metal Contact ◽  
Rolling Motion ◽  
Oil Film ◽  
Pure Rolling ◽  
Total Separation ◽  
Molecular Dimensions

It is today possible to manufacture so smooth surfaces that they can elastically conform totally to each other over the whole Hertzian contact area. For pure rolling lubrication such surfaces only need an oil film of molecular dimensions to get total separation. When the rolling motion is combined with sliding, the pressure fluctuations inside the Hertzian contact redistribute the oil and make metal-to-metal contact possible. The redistribution velocity is a function of the slip rate S and the number of asperities N from the inlet to the outlet of the Hertzian contact area. The asperity top oil film thickness decreases with a factor of the order 2NS going from the inlet to the outlet of the Hertzian contact.

On the Contact Search Algorithms for Wheel/Rail Contact Problems

2009 ◽  
Vol 4 (4) ◽  
Author(s):  
Hiroyuki Sugiyama ◽  
Yoshihiro Suda
Keyword(s):  
Vehicle Dynamics ◽  
Line Search ◽  
Point Contact ◽  
Contact Problems ◽  
Search Algorithms ◽  
Railroad Vehicle Dynamics ◽  
Contact Points ◽  
Contact Search ◽  
On Line ◽  
Railroad Vehicle

In this investigation, contact search algorithms for the analysis of wheel/rail contact problems are discussed, and the on-line and off-line hybrid contact search method is developed for multibody railroad vehicle dynamics simulations using the elastic contact formulation. In the hybrid algorithm developed in this investigation, the off-line search that can be effectively used for the tread contact is switched to the on-line search when the contact point is jumped to the flange region. In the two-point contact scenarios encountered in curve negotiations, the on-line search is used for both tread and flange contacts to determine the two-point contact configuration. By so doing, contact points on the flange region given by the off-line tabular search are never used, but rather used as an initial estimate for the online iterative procedure for improving the numerical convergence. Furthermore, the continual on-line detection of the second point of contact is replaced with a simple table look-up. It is demonstrated by several numerical examples that include flange climb and curve negotiation scenarios that the proposed hybrid contact search algorithm can be effectively used for modeling wheel/rail contacts in the analysis of general multibody railroad vehicle dynamics.

Sign in / Sign up

Export Citation Format

Share Document