Construction of geodesic lines as boundary trajectorys of material particles movement on the surface
DOI:
https://doi.org/10.31548/energiya2022.02.094Abstract
Geodetic lines on the surface play the role of straight lines on the plane. From a point on the surface you can draw a bunch of geodetic lines, among which can be straight lines (generating surfaces if the surface is linear) and curves (flat and spatial). An important feature of geodetic lines is that they involve the movement of material particles on surfaces. The greater the speed of movement of a material particle on the surface, the greater its trajectory approaches the geodetic line of the surface.
Finding geodetic lines on the surfaces of tillage bodies and other tools that move the processed material, gives an idea of the possible trajectories of this material. There are practical methods of approximate finding of geodetic lines on the surface in a given direction. To do this, you need to have a model of the surface and a narrow strip of thick paper, which must be pushed in a given direction on the surface so that it does not come off it. The line of contact of the strip to the surface will be a geodetic line.
If there is no model of the surface, but there is its equation, then there are theoretical methods for finding geodetic lines, which are reduced to solving second-order differential equations.
The aim of the research is to find geodetic lines on the surface according to its given parametric equations.
Theoretical methods of finding geodetic lines on a surface given by parametric equations are considered. Differential equations were solved by numerical methods and geodetic lines were constructed on the surface of a hyperbolic paraboloid.
It is established that the middle geodetic line is a rectilinear generating surface, the extreme - flat cross-sections of the surface planes X = 0 and Y = 0, the rest of the geodetic - spatial curves. The reliability of the integration of the differential equation by numerical methods and the error-free visualization of the obtained results are proved.
Key words: geodetic line, trajectory, geodetic curvature, hyperbolic paraboloidReferences
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Gyachev, L. V. (1961). Teoriya lemeshno-otval’noy poverkhnosti [Theory of the share-dump surface]. Zernograd, 1961, 317.
Voitiuk, D. H., Pylypaka, S. F. (2002). Znakhodzhennia traiektorii rukhu materialnoi chastynky po hravitatsiinykh liniichatykh poverkhniakh iz horyzontalnymy tvirnymy [Finding the trajectory of a material particle on gravitational linear surfaces with horizontal generators]. Zbirnyk naukovykh prats Natsionalnoho ahrarnoho universytetu “Mekhanizatsiia silskohospodarskoho vyrobnytstva”, 12, 58-69.
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