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1

Comparison of main geometric characteristics of deformed sphere and standard spheroid

Kovalchuk Vasyl, Mladenov Ivaïlo M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2023

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Vol. 71, nr 5

art. no. e147058

EN

In the paper we compare the geometric descriptions of the deformed sphere (i.e., the so-called λ-sphere) and the standard spheroid (namely, World Geodetic System 1984’s reference ellipsoid of revolution). Among the main geometric characteristics of those two surfaces of revolution embedded into the three-dimensional Euclidean space we consider the semi-major (equatorial) and semi-minor (polar) axes, quartermeridian length, surface area, volume, sphericity index, and tipping (bifurcation) point for geodesics. Next, the RMS (Root Mean Square) error is defined as the square-rooted arithmetic mean of the squared relative errors for the individual pairs of the discussed six main geometric characteristics. As a result of the process of minimization of the RMS error, we have obtained the proposition of the optimized value of the deformation parameter of the λ-sphere, for which we have calculated the absolute and relative errors for the individual pairs of the discussed main geometric characteristics of λ-sphere and standard spheroid (the relative errors are of the order of 10−6 – 10−9). Among others, it turns out that the value of the,sup> flattening factor of the spheroid is quite a good approximation for the corresponding value of the deformation parameter of the λ-sphere (the relative error is of the order of 10−4).

2

On efficient evaluation of integrals entering boundary equations of 3D potential and elasticity theory

Rybarska-Rusinek L., Jaworski D., Linkov A.

Journal of Mathematics and Applications

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2014

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Vol. 37

85--96

EN

The paper presents recurrent formulae for efficient evaluation of all the integrals needed for colving static 3D potential and elasticity problems by the boundary elements method. The power-type asymptoties for the density at edges of the boundary are accounted for explicitly.

3

Integrals of Lipschitz-Hankel type in analysis of magnetostatic fields

Pawluk K.

Journal of Technical Physics

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2003

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Vol. 44, no 2

133-144

EN

A boundary-integral model of the static magnetic field due to cylindrical permanent magnets that is put in free space is considered. Magnetic scalar potential quantities created by a virtual quantity "surface magnetic charge density" is expressed by means of Lipschitz-Hankel integrals that for the considered case are reducible (by the way of hypergeometric series) to some algebraic expressions, in which elliptic integrals of various kinds occur. This approach seems to be more effective than that can be reached by the use of a typical professional software for the field problems in which numerical integration, being not quite conform to the considered case, is common. The magnet subjected to the analysis has to be virtually subdivided in some number of elementary pieces, inside of which the uniform distribution of the inherent magnetization is supposed.

4

Magneto-static field distribution due to inherent magnetisation of permanent magnets

Pawluk K.

Journal of Technical Physics

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2002

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Vol. 43, no 4

443-452

EN

A boundary-integral model of the static magnetic field due to cylindrical permanent magnets that is put in free space is considered. Magnetic scalar potential quantities created by a virtual quantity "surface magnetic charge density" is expressed by means of Lipschitz-Hankel integrals that for the considered case are reducible (by the way of hypergeometric series) to some algebraic expressions, in which elliptic integrals of various kinds occur. This approach seems to be more effective than that can be reached by the use of a typical professional software for the field problems in which numerical integration, being not quite conform to the considered case, is common. The magnet subjected to the analysis has to be virtually subdivided in some number of elementary pieces, inside of which the uniform distribution of the inherent magnetization is supposed.

5

Analysis of the state of plastic deformation in the upper layer after surface burnishing.

Jezierski Józef, Mazur Tomasz, Tubielewicz Krzysztof

Advances in Manufacturing Science and Technology

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2001

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Vol. 25, nr 1

59-74

EN

The paper presents the state of stresses at the contract of two bodies and general formulas for depth of plastic deformations under the burnishing roller along the axis of the effective pressure force F, onto the roller. The origin of the development od plastic deformation at Belaev point is also presented. These deformations propagate and form plastic deformation zone in the surface layer of parts being burnished. The results of numerical calculations of the plastic deformation depth have been obtained for four values of the yield point and for four values of roller radius.

PL

W pracy przedstawiono stan naprężeń przy styku dwóch ciał i podano ogólne wzory na obliczanie głębokości zalegania odkształceń plastycznych pod rolką nagniatającą wzdłuż osi działania zredukowanej siły nacisku F na rolkę. Omówiono również początek rozwoju (zalążek) odkształceń plastycznych w punkcie Bielajewa. Odkształcenia te rozprzestrzeniają się tworząc w warstwie wierzchniej nagniatanej części strefę odkształceń plastycznych. Zamieszczono także przykład obliczeń numerycznych głębokości zalegania odkształceń plastycznych dla czterech wartości granicy plastyczności i czterech wartości promienia rolki nagniatającej.