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1

Influence of pressure changes on the viscosity of lubricating oil in CFD simulations of conical bearing hydrodynamic lubrication

Czaban Adam, Miszczak Andrzej

Journal of KONBiN

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2022

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Vol. 52, iss. 3

259--277

EN

The aim of this study is to consider the effect of pressure on the viscosity of lubricating oil in the adopted model of hydrodynamic lubrication and on the calculated flow parameters and operating parameters of a conical slide bearing. The numerical analysis consisted in solving the Reynolds type equation for the process of stationary hydrodynamic lubrication.

PL

Celem niniejszej pracy jest zbadanie, w jakim stopniu uwzględnienie wpływu ciśnienia na lepkość oleju smarnego, w przyjętym modelu hydrodynamicznego smarowania, oddziałuje na obliczane parametry przepływowe i eksploatacyjne analizowanego stożkowego łożyska ślizgowego. W badaniach wykorzystano znane z literatury równanie typu Reynoldsa, określające proces stacjonarnego hydrodynamicznego smarowania.

2

Estimation of random bio-hydrodynamic lubrication parameters for joints with phospholipid bilayers

Wierzcholski Krzysztof, Miszczak Andrzej

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2021

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Vol. 69, nr 1

art. no. e135834

EN

This paper presents a new form of a mathematical estimation of stochastic bio-hydrodynamic lubrication parameters for real human joint surfaces with phospholipid bilayers. In this work, the authors present the analytical and stochastic considerations, which are based on the measurements of human joint surfaces. The gap is restricted between two cooperating biological surfaces. After numerous experimental measurements, it directly follows that the random symmetrical as well as unsymmetrical increments and decrements of the gap height in human joints influence the hydrodynamic pressure, load-carrying capacity, friction forces, and wear of the cooperating cartilage surfaces in human joints. The main focus of the paper was to demonstrate the influence of variations in the expected values and standard deviation of human joint gap height on the hydrodynamic lubrication parameters occurring in the human joint. It is very important to notice that the new form of apparent dynamic viscosity of synovial fluid formulated by the authors depends on ultra-thin gap height variations. Moreover, evident connection was observed between the apparent dynamic viscosity and the properties of cartilage surface coated by phospholipid cells. The above observations indicate an indirect impact of stochastic changes in the height of the gap and the indirect impact of random changes in the properties of the joint surface coated with the phospholipid layers, on the value of hydrodynamic pressure, load carrying capacity and friction forces. In this paper the authors present a synthetic, comprehensive estimation of stochastic bio-hydrodynamic lubrication parameters for the cooperating, rotational cartilage bio-surfaces with phospholipid bilayers occurring in human joints. The new results presented in this paper were obtained taking into account 3D variations in the dynamic viscosity of synovial fluid, particularly random variations crosswise the film thickness for non-Newtonian synovial fluid properties. According to the authors’ knowledge, the obtained results are widely applicable in spatiotemporal models in biology and health science.

3

An Analysis of Hydrodynamic Pressure Distribution and Load Carrying Capacity of a Conical Slide Bearing Lubricated with Non-Newtonian Second Grade Flui

Miszczak Andrzej, Czaban Adam

Tribologia

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2019

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nr 2

83--95

EN

In this paper, the authors present the equations of the hydrodynamic lubrication theory for conical slide bearings lubricated with the oil with properties described by the Rivlin-Ericksen model. It is assumed, that the considered lubricating oil shows non-Newtonian properties, i.e. it is an oil for which, apart from the classic dependence of oil viscosity on pressure, temperature and operating time, there is also a change in dynamic viscosity values caused by the changes of shear rate. The method of a small parameter was used to solve the conservation of momentum, stream continuity, and energy conservation equations. The small parameter method consists in presenting the sought functions (pressure, temperature, components of the velocity vector) in the form of a uniformly convergent series expansion in powers of a constant small parameter. These functions are substituted into the system of basic equations, and then the series are multiplied by the Cauchy method. By a comparison of the coefficients with the same powers of a small parameter, we obtain systems of partial differential equations, from which the subsequent approximations of unknowns of the sought functions are determined. The small parameter method separates the non-linear system of partial differential equations and creates several linear systems of equations. The aim of this work is to derive the equations describing and allowing the determination of the temperature distribution, hydrodynamic pressure distribution, velocity vector components, load carrying capacity, friction force and friction coefficient in the gap of conical slide bearing, lubricated with the oil of the properties described by the Rivlin-Ericksen model, taking into account its viscosity changes due to time of operation.

PL

W artykule autorzy przedstawiają równania hydrodynamicznej teorii smarowania olejem o modelu Rivlina-Ericksena stożkowego łożyska ślizgowego. Olej ten charakteryzuje się nienewtonowskimi właściwościami, czyli jest to olej, dla którego, oprócz klasycznych zależności lepkości oleju od ciśnienia, temperatury i czasu eksploatacji, występuje dodatkowo zmiana lepkości dynamicznej od szybkości ścinania. Do rozwiązania równań zachowania pędu, ciągłości strugi i zachowania energii wykorzystano metodę małego parametru. Metoda ta polega na przedstawieniu poszukiwanych funkcji (ciśnienia, temperatury, składowych wektora prędkości) w formie jednostajnie zbieżnego szeregu potęgowego rozwiniętego względem stałego małego parametru. Funkcje te podstawia się do układu równań podstawowych, a następnie wymnaża te szeregi metodą Cauchy’ego. Porównując współczynniki przy jednakowych potęgach małego parametru, otrzymuje się układy równań różniczkowych cząstkowych, z których wyznacza się kolejne przybliżenia niewiadomych, poszukiwanych funkcji. Metoda małego parametru rozprzęga nieliniowy układ równań różniczkowych cząstkowych, tworząc kilka liniowych układów równań. Celem niniejszej pracy jest wyprowadzenie równań umożliwiających wyznaczenie rozkładu temperatury, rozkładu ciśnienia hydrodynamicznego, składowych wektora prędkości, siły nośnej, siły tarcia i współczynnika tarcia w szczelinie poprzecznego łożyska ślizgowego smarowanego olejem o modelu Rivlina-Ericksena z uwzględnieniem zmian lepkości od czasu eksploatacji oleju.

4

Simulations of the influence of the heat flux at the shaft surface of the conical slide bearing on its hydrodynamic lubrication and operating parameters

Czaban Adam, Miszczak Andrzej

Journal of KONES

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2019

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Vol. 26, No. 4

29--37

EN

The aim of this work is to investigate, how in the adopted model of hydrodynamic lubrication of a conical slide bearing, the change of the heat flux value at the bearing shaft, affects bearing operating parameters. In this research, the authors use, the known from the literature, Reynolds type equation, describing the stationary hydrodynamic lubrication process of a conical slide bearing. The analytical, solutions, that determine the components of the lubricating oil velocity vector and the equation (analytical solution of energy equation) determining the threedimensional temperature distribution in the lubrication gap, was also adopted from previous works. In order to obtain numerical solutions, the Newton’s method was used, and the derivatives in the Reynolds type equation were approximated by the finite differences. An application of the method of subsequent approximations allowed considering the influence of temperature, pressure and shearing rate on the viscosity of lubricating oil. The considerations were performed by adopting the Reynolds condition of the hydrodynamic oil film. It was tested, how the assumed value of the heat flux on the bearing shaft surface affects the values of the obtained operating parameters, i.e. the transverse and longitudinal component of the load carrying capacity, friction force and coefficient of friction.

5

Operating parameters of a slide bearing with parabolic-shaped slide surfaces with consideration of the stochastic changes in the lubrication gap height

Miszczak Andrzej, Wierzcholski Krzysztof

Journal of KONES

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2019

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Vol. 26, No. 4

171--178

EN

In this article, the authors present the equations of the hydrodynamic theory for a slide bearing with parabolicshaped slide surfaces. The lubricating oil is characterized by non-Newtonian properties, i.e. an oil for which, apart from the classic oil viscosity dependence on pressure and temperature, also an effect of the shear rate is taken into account. The first order constitutive equation was adopted for considerations, where the apparent viscosity was described by the Cross equation. The analytical solution uses stochastic equations of the momentum conservation law, the stream continuity and the energy conservation law. The solution takes into account the expected values of the hydrodynamic pressure EX[p(ϕ,ζ)], of the temperature EX[T(ϕ,y,ζ)], of the velocity value of lubricating oil EX[vi(ϕ,y,ζ)], of the viscosity of lubricating oil EX[ηT(ϕ,y,ζ)] and of the lubrication gap height EX[εT(ϕ,ζ)]. It was assumed, that the oil is incompressible and the changes in its density and thermal conductivity were omitted. A flow of lubricating oil was laminar and non-isothermal. The research concerned the parabolic slide bearing of finite length, with a smooth sleeve surface, with a full wrap angle. The aim of this work is to derive the stochastic equations, that allow to determine the temperature distribution, hydrodynamic pressure distribution, velocity vector components, load carrying capacity, friction force and friction coefficient, in the parabolic sliding bearing, lubricated with nonNewton (Cross) oil, including the stochastic changes in the lubrication gap height. The paper presents the results of analytical and numerical calculation of flow and operating parameters in parabolic sliding bearings, taking into account the stochastic height of the lubrication gap. Numerical calculations were performed using the method of successive approximations and finite differences, with own calculation procedures and the Mathcad 15 software.

6

Estimation of random friction forces on the microbearing cooperating surfaces

Wierzcholski Krzysztof, Miszczak Andrzej

Journal of KONES

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2019

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Vol. 26, No. 2

167--174

EN

Presented paper concerns a new mathematical form of stochastic theory of hydrodynamic friction forces occurring on the real cooperating surfaces in computer micro bearing fan and computer microbearing in hard disc driver HDD. This paper presents particularly a new-review of stochastic analytical considerations realized by the authors for friction forces estimation during hydrodynamic lubrication performed on the ground of the measurements of the gap height between two roughness surfaces. After numerous experimental measurements directly follows that the random unsymmetrical increments and decrements of the gap height of computer microbearings have important influence on the load carrying capacities and finally on the friction forces and wear of cooperating surfaces. The main topic demonstrates the influence of the variations of expectancy values and standard deviation of the computer microbearing gap height on the friction forces occurring in the HDD and microbearing fan. Moreover, it is observed the evident connection between the apparent dynamic viscosity and the features of the microbearing superficial surface. Hence after abovementioned remarks follows the corollary that the influence of the microbearing gap height stochastic variations connected with the surface roughness tend moreover indirect from the apparent viscosity into the friction forces variations. The synthetic, complex elaboration of obtained results indicates finally the influence of the random roughness in micro and nano level of microbearing surfaces on the friction forces distribution. A new results contained in this paper are obtained taking into account 3D variations of dynamic viscosity of the non-Newtonian lubricant, particularly variations crosswise the film thickness and influences of surface superficial layer features on the lubricant apparent viscosity.

7

The effect of a surface roughness on load carrying capacity in slide journal bearing lubriceted with oil of non-newtonian properties

Miszczak Andrzej

Journal of KONBiN

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2019

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Vol. 49, iss. 3

497--526

EN

The Author presents the method that taking account an influence of surface roughness on both flow and operating parameters of a slide journal bearing in the paper. For this purpose, the expected value for the gap height function should be determined. The expected value is then the most probable value of the height of the lubrication gap. There are presented the results of analytical and numerical calculations of flow and operating parameters in slide journal bearings. Numerical calculations were performed using the finite difference method as well as own calculation procedures and the Mathcad 15 software.

PL

W prezentowanej pracy Autor przedstawia metodę uwzględniania wpływu chropowatości powierzchni na parametry przepływowe i eksploatacyjne łożysk ślizgowych. W tym celu wyznacza się wartość oczekiwaną dla funkcji wysokości szczeliny. Wartość oczekiwana jest wtedy najbardziej prawdopodobną wartością wysokości szczeliny smarnej. W pracy przedstawione są wyniki obliczeń analitycznych i numerycznych parametrów przepływowych i eksploatacyjnych w poprzecznych łożyskach ślizgowych. Obliczenia numeryczne wykonano wykorzystując metodę różnic skończonych, własne procedury obliczeniowe i oprogramowanie Mathcad 15.