Gureeva N.A., Klochkov Yu.V., Nikolaev A.P., Klochkov M.Yu. Continuos Parameterization of the Median Surface of an Ellipsoidal Shell and Its Geometric Parameters
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https://doi.org/10.15688/mpcm.jvolsu.2020.1.1
Natalya A. Gureeva
Doctor of Physical and Mathematical Sciences, Associate Professor,
Department of Data Analysis, Decision-Making and Financial Technologies,
Financial University under the Government of the Russian Federation
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Prosp. Leningradsky, 49, 125993 GSP-3, Moscow, Russian Federation
Yuriy V. Klochkov
Doctor of Engineering Sciences, Professor,
Head of the Department of Higher Mathematics,
Volgograd State Agrarian University
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Prosp. Universitetsky, 29, 400002, Volgograd, Russian Federation
Anatoliy P. Nikolaev
Doctor of Engineering Sciences, Professor, Department of Applied Geodesy,
Environmental Management and Water Use,
Volgograd State Agrarian University
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Prosp. Universitetsky, 29, 400002, Volgograd, Russian Federation
Mikhail Yu. Klochkov
Student, Department of Mathematical Modeling and Computer Science,
Lomonosov Moscow State University
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Leninskie Gory, 1, 119991 GSP-1, Moscow, Russian Federation
Abstract. When analyzing the stress-strain state of thin-walled structural elements that have the shape of an ellipsoid, it becomes necessary to calculate the geometric characteristics of the ellipsoidal surface. When using the canonical ellipsoid equation, regions of uncertainty appear in the Cartesian coordinate system at the intersection points of the ellipsoid surface with the horizontal coordinate plane. To exclude these areas of uncertainty, we propose an expression of the radius vector of an ellipsoidal surface whose projections are functions of two parametric representations in mutually perpendicular planes. One of the planes is the vertical plane XOZ, and the other plane is the plane perpendicular to the axis O at the point with the x coordinate. The parameter T of the ellipse obtained from the intersection of the ellipsoid with the XOZ plane was chosen as the argument of the first parametric function. The argument of the second parametric function t is the parameter of an ellipse formed as a result of the intersection of an ellipsoidal surface with a plane perpendicular to the abscissa axis at a distance of x from the origin. The proposed representation of the ellipsoidal surface allowed us to exclude uncertainties at the intersection points of the ellipsoid with the HOWE coordinate plane. By differentiating the proposed radius-vector expression at an arbitrary point on an ellipsoidal surface, we obtain relations for the basis vectors of an arbitrary point and their derivatives represented by components in the same local basis. These relations are necessary for the development of algorithms for numerical analysis of deformation processes of engineering structures that have ellipsoidal surfaces.
Key words: shell, ellipsoid, parameterization, basis vectors, ellipsoidal shell.
Continuos Parameterization of the Median Surface of an Ellipsoidal Shell and Its Geometric Parameters by Gureeva N.A., Klochkov Yu.V., Nikolaev A.P., Klochkov M.Yu. is licensed under a Creative Commons Attribution 4.0 International License.
Citation in English: Mathematical Physics and Computer Simulation. Vol. 23 No. 1 2020, pp. 5-12
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