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1

Recurrent nets for tribology solutions

Wierzcholski K.

Journal of KONES

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2013

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Vol. 20, No. 2

401--408

EN

The contemporary problems of numerical calculations ocurring in powertrain tribology and transport problems demand the more and more exactness for obtained results.Moreover in performed calculations very important is the convergence, stability and reliability of the gained numerical values. The main scientific topic of the presented paper concerns the method of the determination of the optimum net for numerical calculations of partial difference and recurrence equations. The abovementioned optimum difference and recurrence method is referring to the stability of obtained particular and general numerical solutions and assures the convergence process of obtained calculation values. The Unit Net Region (UNR) was assumed at first for Laplace Operator. The optimum of the nod geometry localization was examined at first for UNR. The optimization index is defined and derived for UNR to determine the most useful net among the various geometries of the nods localization during the difference methods performances of partial recurrence numerical calculations. In the next considerations had been proved the corollary, where taking into account the optimum UNR, we can create optimum nets for other numerous partial difference and recurrence equations in discrete spaces. For example the numerous calculation results of presented optimum net for recurrent calculations are applied for numerical solutions of a Reynolds partial recurrence equation with variable coefficients in curvilinear orthogonal coordinates for curvilinear boundary conditions, and for other numerical problems occurring in applied and hydrodynamics.

2

3D hydrodynamic pressure in gap height direction for cylindrical bearing viscoelastic lubrication

Wierzcholski K.

Journal of KONES

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2013

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Vol. 20, No. 1

367--374

3

Slide bearing wear prognosis formulated as the solutions of recurrence equations

Wierzcholski K.

Tribologia

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2012

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nr 5

199--206

EN

In this paper was elaborated the simulation method of determination of anticipated wear of HDD micro-bearings by virtue of Bharat Bhushan and Chizhik experiments. Measurements of micro-scale and macro-scale wear depth show that values on both scales of all the silicon samples are about the same among different silicon materials and higher than of SiC. In both materials the wear depends of normal load, number of time cycles and dynamic conditions. On the basis of experiments it is visible that the tendencies of wear depth increases versus time periods or time cycles are different for various materials. On the ground of mentioned measured values and using the recurrence dependencies, was elaborated the algorithm of wear values in HDD microbearing in arbitrary long time period after exploitation beginning. Coefficients of recurrence equation are obtained in experimental way. These coefficients contain wear corrections resulting from the vibrations, unsteady load and surface roughness asperities measured from the nominal mean level. The obtained solutions of wear recurrence equation are assigned to the regions of coefficient-values for which the wear process converges to stability limit wear value or tend to larger and larger wear values during the exploitation process. This fact enables to see a connection between roughness, material properties in one side and the wear values or wear increases in succeeding time units of exploitation process in other side. Moreover on the basis of mentioned analysis we can anticipate and optimize the least wear of slide HDD micro-bearing.

PL

Zużycie mikrołożysk w napędach HDD w kolejnych okresach eksploatacyjnych, takich jak kolejne miesiące, lata jest bardzo istotnym problemem procesu eksploatacji komputerów. Jak wynika z badań doświadczalnych, wartość zużycia zależy od wartości obciążenia, rodzaju materiału trących się powierzchni oraz ich chropowatości, środków smarujących, a także czasu trwania eksploatacji. W zależności od wymienionych parametrów rozwój zużycia odbywa się mniej lub bardziej intensywnie. Niniejsza praca za pomocą zależności rekurencyjnych przedstawia algorytm symulujący zużycie mikrołożysk HDD w dowolnym czasie trwania eksploatacji, mając do dyspozycji pomierzone wartości zużycia po dwóch kolejnych latach lub miesiącach pracy łożyska oraz dysponując wiedzą o zmianach i wzroście wartości zużycia zaczerpniętą z dokonanych pomiarów doświadczalnych. Współczynniki rozpatrywanych zależności i równań rekurencyjnych są ściśle uzależnione od wymienionych parametrów eksploatacyjnych.

4

Solutions of recurrence and summation equations and their applications in slide bearing wear calculations

Wierzcholski K.

Journal of KONES

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2012

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Vol. 19, No. 2

543-550

EN

The investigations under the paper include a derivation of a non-homogenous recurrence equation of the second and higher orders with variable coefficients, whose particular solutions, with the boundary conditions set and obtained from experimental measurements, will be the sequences of the wear value of the bearing in the successive years of operation. Owing to these investigations, it will be possible to predict for example the wear values of slide bearings. Moreover, this paper presents the some particular applications of recurrence equations with regard to the calculation prognosis of micro-bearing parameters such as friction forces, friction coefficients and wear. Recurrence equations are presented in a form of difference equations where the unknown functions occur as the main terms of the sequence of inquired values. In this paper, the properties of the particular values of recurrence equations are defined. Possibility of modulation and control of mentioned problem belong to the artificial intelligence of HDD microbearing. Presented problem describes not continuous relations hence determines the mathematical and numerical solutions in discrete spaces. Properly in the case of continuous functions, the mentioned recurrence equations have the same meaning as differential equations. Recurrent equations for discrete function correspond to differential equations for the continuous function. In final conclusions, the application of presented theory in this paper contains the numerical solutions referring the wear values of HDD bearing system in the indicated period of operating time.

5

Operating parameters of slide micro – bearings with consideration of oil temperature changes and micro - grooves on sleeve surface

Miszczak A.

Journal of KONES

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2012

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Vol. 19, No. 2

329-336

EN

In this paper author presents results of numerical calculations of hydrodynamic pressure distribution in bearing gap, load-carrying capacity, friction force and friction coefficient of slide micro-bearing considering the influence of lubricating oil temperature changes and also taking into account the influence of micro-grooves, which occur on sleeve internal surface. The micro-grooves on that surface are in longitudinal direction. The equation, which describes a bearing gap with micro-grooves on sleeve surface, was adopted from prof. K Wierzcholski's investigations. In very thin gap height of cylindrical micro-bearings, large gradients of temperature can be observed. This causes significant changes of oil dynamic viscosity in the gap height direction. According to this, oil flow velocity, friction forces, and a hydrodynamic pressure during the micro-bearing operation are changing. Up to now the influence of temperature on oil viscosity changes and due to this, on hydrodynamic pressure and on load carrying capacity in cylindrical micro-bearing gap in numerical way were not considered yet. The numerical calculations were performed with the use of Mathcad 14. The finite differences method and own computational procedures were implemented. The calculations were begun by solving the Reynolds' equation, assuming, that the dynamic viscosity is constant. After calculating the hydrodynamic pressure distribution, the temperature distribution in lubricating oil was determined. The obtained function of temperature was used to describe the viscosity changes with temperature. Next step involved determining the hydrodynamic pressure distribution taking into account the viscosity dependence on temperature, and then new distribution of temperature and again new values of viscosity were calculated. Calculations were repeated until assumed convergence and accuracy were reached. The friction force depends on pressure gradient and rotational motion of bearing journal. Part of friction force, which resulting from the pressure gradient, is determined for the area, where the oil film occurs, i.e. from omega p to omega k. Part of friction force, which is related to journal motion, is determined for full wrap angle, i.e. from 0 to 2 pi. The results were presented in the form of graphs, for eccentricity ratio gamma from 0.1 to 0.9, for dimensionless length of the bearing L1=1/4. In numerical calculations were used the theoretical considerations and solutions presented in papers of K. Wierzcholski and A. Miszczak.

6

Summation equation tools for slide microbearing systems wear prognosis

Wierzcholski K.

Journal of KONES

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2011

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Vol. 18, No. 2

463-470

EN

Efficient functioning of slide micro-bearings systems especially HDD micro-bearing require to recognition and modulation of the proper values of the friction forces and wear during the exploitation time. Possibility of modulation and control of mentioned problem belong to the artificial intelligence of HDD micro-bearing. This paper presents the some applications of summation equations regard to the calculation prognosis of micro-bearing parameters such as friction forces, ,friction coefficients and wear .Summation equations are presented a new form of difference and recurrence equations where the unknown function occurs as the argument of the reciprocal unified operator of summation (UOS) Presented problem describes not continuous relations hence determines the mathematical and numerical solutions in discrete spaces. Properly in the case of continuous functions, the mentioned summation equations have the same meaning as integral equations. This paper will present the transformation method of summation equations to recurrent equations. Recurrent equations for discrete function correspond to differential equations for the continuous function. The Application of presented theory in this paper contains the numerical solutions referring the wear values of HDD bearing system in the indicated period of operating time.