Mathematical model of moving particle by vertical screw in stationary mode

Pylypaka, Serhii; Babka, Vitaliy; Grischenko, I. Yu.; Kresan, Tetiana

doi:10.31548/machenergy2019.04.057

Cited by 2 publications

(3 citation statements)

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“…A spherical line is taken as a guide curve -an analog of Archimedes' spiral on a plane. In [10], the trajectory of the unit radius vector of the helical line, which turns it into a spherical curve, is taken as a guide curve. The approximation result obtained for this case is similar to the previous one.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

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Construction of a mathematical model for approximating the sphere by strips of unfolding surfaces

Nesvidomin,

Ahmed,

Pylypaka

et al. 2023

EEJET

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Approximating non-expandable surfaces by compartments of expandable ones makes it possible to simplify the process of obtaining the required shape without loss of operational properties. There is a known approximation of a sphere using the example of a ball when its surface can consist of polygons. However, this list does not exhaust the possible options for approximating the sphere. Its approximation by truncated cones tangent to parallels or by congruent cylindrical petals tangent to meridians is known. Any line on the surface of a sphere is a line of curvature. This means that the common line of contact of the expanded surface with the sphere will be a line of curvature for the expanded surface as well (rectilinear generatrices of the expanded surface will cross this line at a right angle). When building a sweep of such a surface, the line of contact will be transformed but the rectilinear generatrices will remain perpendicular to it, which simplifies the construction of the sweep. The approximation of the sphere by congruent strips, the number of which can be different, starting from one, is considered. A necessary condition for such an approximation is a common line of contact of adjacent strips. To this end, the line of contact on the sphere or the guide curve must have an appropriate shape. Such a curve is taken as a slope line (a curve whose tangents form a constant angle of inclination to the horizontal plane). The study results are the parametric equations of the strip touching the sphere and its corresponding equations on the sweep. The construction of the strip on the sweep is explained by the invariance of the geodesic curvature of the guide curve when the strip is bent until it aligns with the plane. This explains the difference between the proposed approach and conventional methods of sphere approximation. Approximating the sphere by strips of unfolding surfaces has a practical application in architecture with spherical elements, as well as in religious buildings with domes in the form of a part of the sphere

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Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…Works [7,8] argue about the approximation of the sphere by compartments of surfaces of the same or different shape. In works [6,10], the sphere is approximated by strips of unfolding surfaces, but the lines of contact are other curves, different from the slope line.…”

Section: Construction Of Strips Based On the Built Parametric Equatio...mentioning

confidence: 99%

Construction of a mathematical model for approximating the sphere by strips of unfolding surfaces

Nesvidomin,

Ahmed,

Pylypaka

et al. 2023

EEJET

View full text Add to dashboard Cite

show abstract

“…In work [7], a special grid is created using spherical curves, which is an analog of a rectangular square grid on a plane. Approximation of the sphere by a continuous strip of the unfolding surface is considered in work [8].…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

Constructing geometrical models of spherical analogs of the involute of a circle and cycloid

Nesvidomin,

Pylypaka,

Volina

et al. 2023

EEJET

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The common properties of images on a plane and a sphere are considered in the scientific works by scientists-designers of spherical mechanisms. This is due to the fact that the plane and the sphere share common geometric parameters. They include constancy at all points of the Gaussian curve, which has a zero value for a plane and a positive value for a sphere. Figures belonging to them can slide freely on both surfaces. With unlimited growth of the radius of the sphere, its limited section approaches the plane, and the spherical shape transforms into a plane. Thus, a loxodrome that crosses all meridians at a constant angle is transformed into a logarithmic spiral that intersects at a constant angle the radius vectors that come from the pole. The tooth profile of cylindrical gears is outlined by the involute of a circle. A spherical involute is used for the corresponding bevel gears. Other spherical curves are also known, which are analogs of flat ones. The formation of a cycloid and an involute of a circle are associated with the mutual rolling of a line segment with each of these figures. If the segment is fixed and the circle rolls along it, then the point of the circle describes the cycloid. In the case of a stationary circle along which a segment is rolled, the point of the segment will execute the involute. To move to the spherical analogs of these curves, it is necessary to replace the circle with a cone, and the straight line with a plane. The spherical prototype of the cycloid will be the trajectory of the point of the base of the cone, which rolls along the plane, that is, along the sweep of the cone. The sweep of a cone is a sector, the radius of the limiting circle of which is equal to the generating cone. If this sweep, like a section of a plane, is rolled around a fixed cone, when its top coincides with the center of the sector, then the point of the limiting radius of the sector will execute a spherical involute. This paper analytically implements these two motions and reports the parametric equations of the spherical analogs of the circle involute and the cycloid

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Mathematical model of moving particle by vertical screw in stationary mode

Cited by 2 publications

References 0 publications

Construction of a mathematical model for approximating the sphere by strips of unfolding surfaces

Construction of a mathematical model for approximating the sphere by strips of unfolding surfaces

Constructing geometrical models of spherical analogs of the involute of a circle and cycloid

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