PL
 |
EN
Preferencje help
Polish version English version Język
Widoczny [Schowaj] Abstrakt
Liczba wyników

Znaleziono wyników: 8

Liczba wyników na stronie
first rewind previous Strona / 1 next fast forward last
Wyniki wyszukiwania
Wyszukiwano:
w słowach kluczowych:  elliptic integrals
help Sortuj według:

help Ogranicz wyniki do:
first rewind previous Strona / 1 next fast forward last
1
Content available 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
|
2023
|
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
Content available Mechanics of infinitesimal gyroscopes on helicoid-catenoid deformation family of minimal surfaces
Kovalchuk Vasyl, Gołubowska Barbara, Mladenov Ivaïlo M.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2021
|
Vol. 69, nr 2
art. no. e136727
EN
In this paper we explore the mechanics of infinitesimal gyroscopes (test bodies with internal degrees of freedom) moving on an arbitrary member of the helicoid-catenoid family of minimal surfaces. As the configurational spaces within this family are far from being trivial manifolds, the problem of finding the geodesic and geodetic motions presents a real challenge. We have succeeded in finding the solutions to those motions in an explicit parametric form. It is shown that in both cases the solutions can be expressed through the elliptic integrals and elliptic functions, but in the geodetic case some appropriately chosen compatibility conditions for glueing together different branches of the solution are needed. Additionally, an action-angle analysis of the corresponding Hamilton-Jacobi equations is performed for external potentials that are well-suited to the geometry of the problem under consideration. As a result, five different sets of conditions between the three action variables and the total energy of the infinitesimal gyroscopes are obtained.
3
Content available Exact solution for large amplitude flexural vibration of nanobeams using nonlocal Euler-Bernoulli theory
Nazemnezhad R., Hosseini-Hashemi S.
Journal of Theoretical and Applied Mechanics
|
2017
|
Vol. 55 nr 2
649--658
EN
In this paper, nonlinear free vibration of nanobeams with various end conditions is studied using the nonlocal elasticity within the frame work of Euler-Bernoulli theory with von K´arm´an nonlinearity. The equation of motion is obtained and the exact solution is established using elliptic integrals. Two comparison studies are carried out to demonstrate accuracy and applicability of the elliptic integrals method for nonlocal nonlinear free vibration analysis of nanobeams. It is observed that the phase plane diagrams of nanobeams in the presence of the small scale effect are symmetric ellipses, and consideration the small scale effect decreases the area of the diagram.
4
Content available remote 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
|
2014
|
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.
5
Content available Boundary-integral model of permanent magnet of a tube segment as shape
Pawluk K., Sulima R.
Archives of Electrical Engineering
|
2011
|
Vol. 60, nr 4
413-432
EN
The magnetic field due to a permanent magnet of a tube-side segment as shape and of radial-oriented magnetization is considered. Such a sheet modelling a single pole of the magnet is used to express the suitable contribution to magnetic quantities. A boundary-integral approach is applied that is based on a virtual scalar quantity attributed to the magnet pole. Such an approach leads to express analytically the scalar magnetic potential and the magnetic flux density by means of the elliptic integrals. Numerical examples of the computed fields are given. The general idea of the presented approach is mainly directed towards designing the magnetic field within the air gap of electric machines with permanent magnets as an excitation source. Other technical structures with permanent magnets may be a subject of this approach as well.
6
Content available remote Zastosowanie odwzorowania Cassiniego-Soldnera do przedstawiania obszaru Polski w wąskich lub szerokich pasach południkowych
Pędzich P.
Roczniki Geomatyki
|
2007
|
T. 5, z. 3
147-158
EN
In the paper, properties of the Cassini-Soldner projection of the area of Poland in narrow and wide zones are presented. Two methods of coordinates calculation in that projection are described: the first one is based on power series and it enables to determine flat rectangular coordinates and projection distortions in narrow . 3.40 projection zones and the second method uses elliptic integrals, which enables to project the area of Poland in one wide zone and even the whole ellipsoid. Moreover, results of calculation of linear, angular and area distortions in the Cassini-Soldner projection are presented. A comparison between Cassini-Soldner and generally used in Poland in geodetic works Gauss- Krüger projection are made. In the area of Poland linear distortions are similar in these two projections; area distortions in Cassini-Soldner projection are much smaller than in Gauss-Krüger projection but there are considerable angular distortions.
7
Content available remote Własności odwzorownania Cassiniego-Soldnera całej elipsoidy
Pędzich P.
Roczniki Geomatyki
|
2007
|
T. 5, z. 2
73-84
EN
Up till present times Cassini-Soldner projection has been used in geodesy and cartography in narrow (2-3°) zones. In this paper, a new approach to construction of Cassiniego-Soldner projection is presented based on elliptic integrals and Jacoby elliptic functions which allows to project the whole ellipsoid. In the paper, development of formulas for coordinates and for distortion are presented. Properties of Cassini-Soldner projection of the whole ellipsoid are also shown. Especially, some peculiarities which occurs in graticule construction are presented, as well as the graticule in Cassini-Soldner projection of the whole ellipsoid. Moreover, maps presenting distortion by ellipses of distortion and isolines are also shown in the paper.
8
Opracowanie odwzorowania Cassiniego-Soldnera całej elipsoidy oraz obszaru Polski w szerokiej strefie odwzorowawczej z zastosowaniem funkcji i całek eliptycznych Jacobiego
Pędzich P.
Prace Naukowe Politechniki Warszawskiej. Geodezja
|
2007
|
z. 42
3-58
PL
Odwzorowanie Cassiniego-Soldnera było dotychczas stosowane w geodezji i kartografii w wąskich (2-3°) strefach odwzorowawczych. Używano uproszczonych formuł odwzorowawczych w postaci rozwinięć w szeregi potęgowe ograniczone do kilku początkowych wyrazów. W niniejszej pracy przedstawiono podstawy teoretyczne tworzenia odwzorowania Cassiniego-Soldnera całej elipsoidy. Opisano nowe podejście do konstrukcji tego odwzorowania, wykorzystujące całki i funkcje eliptyczne Jacobiego. Praca zawiera kompleksowe opracowanie odwzorowania Cassiniego-Soldnera. Przedstawiono w niej rozwiązanie zagadnienia prostego - odwzorowania powierzchni Ziemi w płaszczyznę mapy oraz zadania odwrotnego - znajdowania przeciwobrazu mapy na powierzchni elipsoidy ziemskiej. Zaprezentowano również formuły opisujące zniekształcenia odwzorowawcze kierunków, kątów, długości i pól. Przeprowadzono także badanie własności odwzorowania Cassiniego-Soldnera całej elipsoidy z uwzględnieniem osobliwości występujących na brzegu siatki kartograficznej w odwzorowaniu. Ponadto przebadano własności tego odwzorowania w odniesieniu do obszaru Polski w szerokiej i wąskiej strefie odwzorowawczej. W pracy przedstawiono także pewne koncepcje dotyczące wyznaczania redukcji odwzorowawczych w odwzorowaniu Cassiniego-Soldnera. Zaprezentowano algorytmy i programy komputerowe pozwalające na obliczanie współrzędnych i zniekształceń odwzorowawczych w odwzorowaniu Cassiniego-Soldnera zarówno całego globu, jak również ograniczonego obszaru. Pokazano także, że opracowane algorytmy mogą mieć zastosowanie np. do obliczania długości łuku południka na elipsoidzie lub realizacji zadania przenoszenia współrzędnych na elipsoidzie na duże odległości.
EN
To the present day the Cassini-Soldner projection has been used in geodesy and cartography in narrow (2-3°) zones. Formulas have had the simple form of power series limited to a few starting expressions. In this thesis the theoretical bases for the creation of the Cassini-Soldner projection of a whole ellipsoid are presented. A new approach to the construction of the Cassini-Soldner projection, based on elliptic integrals and Jacoby elliptic functions, are presented. The thesis consists of a complete explanation of the Cassini-Soldner projection. A solution to the specific problem (i.e. the coordinate transformation between ellipsoid and image plane) and the solution of the indirect problem, which allows for the transformation of coordinates from plane to ellipsoid, is presented. The formulas of angular, area and linear distortion are also presented. The properties of the projection of the whole ellipsoid, with its peculiarities occurring on the edge of the graticule, are also shown. Furthermore, the properties of the projection of the area of Poland in wide and narrow zones are considered. In the thesis some ideas concerning projection reductions and corrections are presented. The algorithms and computer programs which enable the calculation of coordinates and distortion in the Cassini-Soldner projection of the whole ellipsoid and limited areas are also presented. Moreover, the possibilities of the application of the elaborated algorithms in the calculation of the length of geodesic lines, the length of meridians and the achievement of a long distance coordinate transformation, along the geodesic line and upon the ellipsoid, are shown.
first rewind previous Strona / 1 next fast forward last
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.