Mathematical Description for a Particular Case of Ellipse Focus Quasi-Rotation Around an Elliptical Axis

2021 ◽  
Vol 9 (1) ◽  
pp. 39-45
Author(s):  
I. Antonova ◽  
E. Solomonova ◽  
Nina Kadykova
Keyword(s):  
Motion Trajectory ◽  
Geometric Constructions ◽  
Rotation Center ◽  
Geometric Objects ◽  
Zero Radius ◽  
Dupin Cyclide ◽  
Trajectory Equation ◽  
Curve Equation ◽  
Special Case

In this paper is provided mathematical analysis related to a particular case for a point quasi-rotation around a curve of an elliptical axis. The research complements the previous works in this direction. Has been considered a special case, in which the quasi-rotation correspondence is applied to a point located at the elliptical axis’s focus. This case is special, since the quasi-rotation center search is not invariant and does not lead to determination of four quasi-rotation centers, as in the general case. A constructive approach to the rotation center search shows that any point lying on the elliptical axis can be the quasi-rotation center. This feature leads to the fact that instead of four circles, the quasi-rotation of a point lying in the elliptical axis’s focus leads to the formation of an infinite number of circle families, which together form a channel surface. The resulting surface is a Dupin cyclide, whose throat circle has a zero radius and coincides with the original generating point. While analyzing are considered all cases of the rotation center location. Geometric constructions have been performed based on previously described methods of rotation around flat geometric objects’ curvilinear axes. For the study, the mathematical relationship between the coordinates of the initial set point, the axis curve equation and the motion trajectory equation of this point around the axis curve, described in earlier papers on this topic, is used. In the proposed paper has been provided the derivation of the motion trajectory equation for a point around the elliptic axis’s curve.

Mathematical Description for a Particular Case of Ellipse Focus Quasi-Rotation Around an Elliptical Axis

2021 ◽  
pp. 38-44
Author(s):  
I. Antonova ◽  
E. Solomonova ◽  
Nina Kadykova
Keyword(s):  
Motion Trajectory ◽  
Geometric Constructions ◽  
Rotation Center ◽  
Geometric Objects ◽  
Zero Radius ◽  
Dupin Cyclide ◽  
Trajectory Equation ◽  
Curve Equation ◽  
Special Case

In this paper is provided mathematical analysis related to a particular case for a point quasi-rotation around a curve of an elliptical axis. The research complements the previous works in this direction. Has been considered a special case, in which the quasi-rotation correspondence is applied to a point located at the elliptical axis’s focus. This case is special, since the quasi-rotation center search is not invariant and does not lead to determination of four quasi-rotation centers, as in the general case. A constructive approach to the rotation center search shows that any point lying on the elliptical axis can be the quasi-rotation center. This feature leads to the fact that instead of four circles, the quasi-rotation of a point lying in the elliptical axis’s focus leads to the formation of an infinite number of circle families, which together form a channel surface. The resulting surface is a Dupin cyclide, whose throat circle has a zero radius and coincides with the original generating point. While analyzing are considered all cases of the rotation center location. Geometric constructions have been performed based on previously described methods of rotation around flat geometric objects’ curvilinear axes. For the study, the mathematical relationship between the coordinates of the initial set point, the axis curve equation and the motion trajectory equation of this point around the axis curve, described in earlier papers on this topic, is used. In the proposed paper has been provided the derivation of the motion trajectory equation for a point around the elliptic axis’s curve.

Determination of the phase structure of polymerblends by electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 492-493
Author(s):  
Dr. G. Kaemof
Keyword(s):  
Screw Extruder ◽  
Electron Microscopic ◽  
Thin Sections ◽  
Single Screw Extruder ◽  
Transmission Electron ◽  
The Matrix ◽  
Twin Screw ◽  
Special Case ◽  
Microscopic Investigations

A mixture of polycarbonate (PC) and styrene-acrylonitrile-copolymer (SAN) represents a very good example for the efficiency of electron microscopic investigations concerning the determination of optimum production procedures for high grade product properties.The following parameters have been varied:components of charge (PC : SAN 50 : 50, 60 : 40, 70 : 30), kind of compounding machine (single screw extruder, twin screw extruder, discontinuous kneader), mass-temperature (lowest and highest possible temperature).The transmission electron microscopic investigations (TEM) were carried out on ultra thin sections, the PC-phase of which was selectively etched by triethylamine.The phase transition (matrix to disperse phase) does not occur - as might be expected - at a PC to SAN ratio of 50 : 50, but at a ratio of 65 : 35. Our results show that the matrix is preferably formed by the components with the lower melting viscosity (in this special case SAN), even at concentrations of less than 50 %.

The Uniform Flexure of an Incomplete Tore

1949 ◽  
Vol 2 (4) ◽  
pp. 469
Author(s):  
W Freiberger ◽  
RCT Smith
Keyword(s):  
General Solution ◽  
Cross Section ◽  
Harmonic Functions ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Elasticity Problems ◽  
Three Dimensional Elasticity ◽  
Derivatives Of ◽  
Special Case

In this paper we discuss the flexure of an incomplete tore in the plane of its circular centre-line. We reduce the problem to the determination of two harmonic functions, subject to boundary conditions on the surface of the tore which involve the first two derivatives of the functions. We point out the relation of this solution to the general solution of three-dimensional elasticity problems. The special case of a narrow rectangular cross-section is solved exactly in Appendix II.

Analytical Techniques for the Optimisation of Rocket Trajectories

1963 ◽  
Vol 14 (2) ◽  
pp. 105-124 ◽  
Author(s):  
Derek F. Lawden
Keyword(s):  
Gravitational Field ◽  
Artificial Satellite ◽  
Analytical Techniques ◽  
Present Position ◽  
Mayer Problem ◽  
Special Cases ◽  
Minimum Expenditure ◽  
Law Of Attraction ◽  
Special Case

SummaryThe development during the last two decades of analytical techniques for the solution of problems relating to the optimisation of rocket trajectories is outlined and the present position in this field of research is summarised. It is shown that the determination of optimal trajectories in a general gravitational field can be expressed as a Mayer problem from the calculus of variations. The known solution to such a problem is stated and applied, first to the special case of the launching of an artificial satellite into a circular orbit with minimum expenditure of propellant and, secondly, to the general astronautical problem of the economical transfer of a rocket between two terminals in a gravitational field. The special cases when the field is uniform and when it obeys an inverse square law of attraction to a point are then considered, and the paper concludes with some remarks concerning areas in which further investigations are necessary.

scholarly journals The use of parallel computing for the high-resolution determination of earthquake source parameters

2019 ◽  
pp. 68-75
Author(s):  
A. S. Fomochkina ◽  
V. G. Bukchin
Keyword(s):  
Parallel Computing ◽  
Focal Mechanism ◽  
Source Parameters ◽  
Nodal Plane ◽  
Earthquake Parameters ◽  
Earthquake Source Parameters ◽  
Detailed Search ◽  
High Degree ◽  
Special Case

Alongside the determination of the focal mechanism and source depth of an earthquake by direct examination of their probable values on a grid in the parameter space, also the resolution of these determinations can be estimated. However, this approach requires considerable time in the case of a detailed search. A special case of a shallow earthquake whose one nodal plane is subhorizontal is an example of the sources that require the use of a detailed grid. For studying these events based on the records of the long-period surface waves, the grids with high degree of detail in the angles of the focal mechanism are required. We discuss the application of the methods of parallel computing for speeding up the calculations of earthquake parameters and present the results of studying the strongest aftershock of the Tohoku, Japan, earthquake by this approach.

Determination of siblings: A special case report from Halle

2011 ◽  
Vol 3 (1) ◽  
pp. e463-e464
Author(s):  
U.D. Immel ◽  
S. Lutz-Bonengel ◽  
J. Edelmann ◽  
R. Lessig
Keyword(s):  
Case Report ◽  
Special Case

Comparative Evaluation of Analytical Methods for the Determination of Coefficients of Friction and Stresses when Wire Rod Drawing

2018 ◽  
Vol 284 ◽  
pp. 247-252 ◽  
Author(s):  
Sergei I. Platov ◽  
V.A. Nekit ◽  
Nikolay N. Ogarkov
Keyword(s):  
Plastic Deformation ◽  
Comparative Evaluation ◽  
Method Of Characteristics ◽  
Circular Cross Section ◽  
Wire Drawing ◽  
Work Piece ◽  
Relative Height ◽  
Plane Plastic ◽  
Special Case

The article discusses the process of wire drawing of circular cross-section. The study of stresses of wire drawing in conditions of plane plastic flow was held. As theoretical framework the study was adopted the method of characteristics, a special case, having the definitive decision. Stresses during wire drawing are defined by decomposing the decision into two components of plane strain and the superposition of these decisions. The results of theoretical solution of the problem of wire drawing were used to determine the coefficients of friction on the surface of contact of the tool and the work piece during the deformation of steel with a diameter of 5.5 mm. It is recommended to use two-dimensional methods of solution in the analysis of the process of wire drawing in conditions with a high hearth of plastic deformation (with the relative height of the hearth of plastic deformation 2 and bigger). The theoretical dependences between the friction coefficients at the contact surface of the work piece and the tool was obtained. The obtained values of coefficients of friction can be used over solving the task of the wire drawing in conditions with a high hearth of plastic deformation.

Determination of Relative Damping at Various Prestresses

1960 ◽  
Vol 33 (2) ◽  
pp. 282-301
Author(s):  
Th Kempermann ◽  
R. Clamroth
Keyword(s):  
Energy Loss ◽  
Phase Angle ◽  
Loss Factor ◽  
Short Review ◽  
The Other ◽  
Simple Function ◽  
History Of ◽  
Tan Δ ◽  
Special Case

Abstract After a short review of the history of the development of the concept of damping, the definitions—one in words, the other in the form of an equation—given in DIN 53513 are discussed, and their usefulness for measurements at various values of prestress is investigated. It is evident that the mathematical definition given in the Standard applies only in the special case when prestress and alternating stress are equal. For the complete range of prestress, the phase angle δ or some simple function of δ, such as the loss factor, tan δ, should be measured. If it is desired not to relinquish the percentage statement of the energy loss, then a new definition for the relative damping is proposed which is independent of the prestress, just as is tan δ. The dependence upon the prestress of the damping values under discussion is demonstrated in an experimental section, for various types of elastomers.

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