Outer cylindrical spaces of a hollow cylinder of finite length

DOI: 10.47026/1810-1909-2023-4-15-23

УДК [517.956.225:517.982.43]:621.313-465

ББК В161.68:К500.131

Aleksandr A. AFANASYEV

Key words

hollow and solid cylinders, Laplace equation, boundary conditions, electromechanical devices with radial and axial gaps

Abstract

Introduction. The solution of the Laplace equation by the method of separation of Fourier variables for cylindrical domains makes it possible to study electromechanical devices in 3D format.

The purpose of the study is to obtain analytical expressions for magnetic potentials and inductions of external three-dimensional spaces of electromechanical devices in a cylindrical coordinate system.

Materials and methods. The unknown constants in the expressions for the regions under consideration are found from the boundary conditions of the magnetic field at the common boundaries of the electromechanical device with its outer space: the scalar magnetic potentials and magnetic inductions are the same, the current sheets experience a jump. When obtaining analytical expressions, the methods of mathematical physics were used.

Research results. To find the Fourier constants, the outer three-dimensional spaces of classical electromechanical devices (generators, motors) are considered, which in a cylindrical coordinate system are represented by both hollow and solid cylinders joining the active regions of the devices with their cylindrical or end surfaces. General expressions for magnetic potentials and inductions of external spaces are obtained based on the solution of the Laplace equation as the Sturm–Liouville problem by the Fourier method of separation of variables. The magnetic fields of the outer spaces of the original hollow cylinder are analyzed: an outer cylinder with an infinitely large radius; internal solid cylinder; end cylinders of finite length. These data make it possible to increase the required number of equations to find the required number of Fourier constants in the analytical calculation of electromechanical devices by the method of separation of variables.

Findings. Analytical expressions for the boundary values of magnetic potentials and inductions of the external space adjacent to the active regions of electromechanical devices with air gaps of both radial and axial form are obtained.

References

1. Afanasyev A.A., Ivanova N.N. Reshenie uravneniya Laplasa metodom razdeleniya peremennykh dlya pologo tsilindra konechnoi dliny [Solution of the Laplace equation by the method of separation of variables for a length hollow cylinder]. Vestnik Chuvashskogo universiteta, 2023, no. 2, pp. 32–40. DOI: 10.47026/1810-1909-2023-2-32-40.
2. Bogolyubov A.N., Kravtsov V.V. Zadachi po matematicheskoi fizike [Problems in mathematical physics]. Moscow, Moskow University Publ., 1998, 350 p.
3. Bronshtein I.N., Semendyaev K.A. Spravochnik po matematike [Handbook of Mathematics]. Moscow, Nauka Publ., 1980, 975 p.
4. Goloskokov D.P. Uravneniya matematicheskoi fiziki. Reshenie zadach v sisteme Maple [Equations of mathematical physics. Solving problems in the Maple system]. St. Petersburg, Piter Publ., 2004, 539 p.
5. Dyakin V.V., Kudryashova O.V., Raevskii V.Y. To the calculation of the field of a finite magnetic cylinder. Russian Journal of Nondestructive Testing, 2019, vol. 55, no. 10, pp. 734–745.
6. Kuznetsov D.S. Spetsial’nye funktsii [Special features]. Moscow, Vysshaya shkola Publ., 1965, 423 p.
7. Mizokhata S. Teoriya uravnenii s chastnymi proizvodnymi [Theory of partial differential equa-tions]. Moscow, Mir Publ., 1977, 504 p.
8. Polyanin A.D. Spravochnik po lineinym uravneniyam matematicheskoi fiziki [Handbook of Linear Equations of Mathematical Physics]. Moscow, Fizmatlit Publ., 2001, 576 p.
9. Abromowitz M., Sregun I., eds. Handbook of mathematical functions with formulas, graphs and math-ematical tables Issued. National Bureau of Standarts, 1964, 470 p. (Russ. ed.: Spravochnik po spetsial’nym funktsiyam. Moscow, Nauka Publ., 1979, 832 p.).
10. Farlow S.J. Partial differential egnations for scientists and engineers. Wileu, 1982 (Russ. ed.: Uravneniya s chastnymi proizvodnymi dlya nauchnykh rabotnikov i inzhenerov. Moscow, Mir Publ., 1985, 384 p.).
11. Zubov V.I., comp. Funktsii Besselya [Bessel functions]. Moscow, MFTI Publ., 2007, 51 p.
12. Shaitor N. Elektromekhanicheskie struktury slozhnykh konfiguratsii [Electromechanical structures of complex configurations]. Vologda, Infra-Inzheneriya Publ., 2023, 168 p.

Information about the author

Aleksandr A. Afanasyev – Doctor of Technical Sciences, Professor, Department of Automation and Control in Technical Systems, Chuvash State University, Russia, Cheboksary (afan39@mail.ru).

For citations

Afanasyev A.A. OUTER CYLINDRICAL SPACES OF A HOLLOW CYLINDER OF FINITE LENGTH. Vestnik Chuvashskogo universiteta, 2023, no. 4, pp. 15–23. DOI: 10.47026/1810-1909-2023-4-15-23 (in Russian).

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