Природна криволінійна циліндрична система координат для стержнів із плоскою віссю довільної форми

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Ковальчук, Станіслав Богданович
Горик, Олексій Володимирович
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Одеська державна академія будівництва та архітектури
The article deals with the theoretical model of the curvilinear cylindrical orthogonal coordinate system, in which the construction of a curvilinear bar with a plain axis of an arbitrary form is naturally simulated. Main characteristics and dependences which allow to modify scalar functions as well as their first-order derivatives when passing from the orthogonal spatial system to the natural curvilinear cylindrical coordinate system have been received for the suggested coordinate system. Analytical relations for the angle of planes of tangent lines to coordinate surfaces, which allowed to record the correlation for transformation of vectors and tensors have been received with the help of differential relationships. The methods of natural coordinatization on the given one-parameter family of curves have been developed using the analytical condition of curves family orthogonality. The suggested methods have been tested in obtaining correlations of the famous cylindrical circular coordinate system on the given family of concentric circles. Their possibilities are shown in the construction of a new curvilinear cylindrical coordinate system based on the family of logarithmic curves. The received correlations and the suggested method of the natural coordinate systems construction will be useful in a solution of the mechanical problem of deformation of homogeneous and heterogeneous rods with a curvilinear plain axis of an arbitrary form.
curvilinear rod, plain curvilinear axis, curvilinear cylindrical coordinate system, coordinate surface, family of curves