scholarly journals Dynamics of a System of Two Connected Bodies Moving along a Circular Orbit around the Earth

2020 ◽  
Vol 226 ◽  
pp. 02010
Author(s):  
Sergey A. Gutnik ◽  
Vasily A. Sarychev
Keyword(s):  
Circular Orbit ◽  
Orbital Plane ◽  
Radius Vector ◽  
Algebraic Equations ◽  
Real Roots ◽  
Principal Axes ◽  
Special Cases ◽  
Spherical Hinge ◽  
Algebra Method ◽  
Two Bodies

Symbolic–numeric methods are used to investigate the dynamics of a system of two bodies connected by a spherical hinge. The system is assumed to move along a circular orbit under the action of gravitational torque. The equilibrium orientations of the two-body system are determined by the real roots of a system of 12 algebraic equations of the stationary motions. Attention is paid to the study of the conditions of existence of the equilibrium orientations of the system of two bodies refers to special cases when one of the principal axes of inertia of each of the two bodies coincides with either the normal of the orbital plane, the radius vector or the tangent to the orbit. Nine distinct solutions are found within an approach which uses the computer algebra method based on the algorithm for the construction of a Gröbner basis.

scholarly journals Exact statistical solution for the hopping transport of trapped charge via finite Markov jump processes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Andrey A. Pil’nik ◽  
Andrey A. Chernov ◽  
Damir R. Islamov
Keyword(s):  
Charge Transport ◽  
Thin Layers ◽  
Jump Processes ◽  
Dielectric Films ◽  
Algebraic Equations ◽  
Markov Jump ◽  
Statistical Solution ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Thin Dielectric Films

AbstractIn this study, we developed a discrete theory of the charge transport in thin dielectric films by trapped electrons or holes, that is applicable both for the case of countable and a large number of traps. It was shown that Shockley–Read–Hall-like transport equations, which describe the 1D transport through dielectric layers, might incorrectly describe the charge flow through ultra-thin layers with a countable number of traps, taking into account the injection from and extraction to electrodes (contacts). A comparison with other theoretical models shows a good agreement. The developed model can be applied to one-, two- and three-dimensional systems. The model, formulated in a system of linear algebraic equations, can be implemented in the computational code using different optimized libraries. We demonstrated that analytical solutions can be found for stationary cases for any trap distribution and for the dynamics of system evolution for special cases. These solutions can be used to test the code and for studying the charge transport properties of thin dielectric films.

On Graeffe's Method for Complex Roots of Algebraic Equations

1924 ◽  
Vol 22 (2) ◽  
pp. 83-87 ◽  
Author(s):  
S. Brodetsky ◽  
G. Smeal
Keyword(s):  
Nineteenth Century ◽  
Real Axis ◽  
Practical Method ◽  
Algebraic Equations ◽  
Argand Diagram ◽  
Considerable Difficulty ◽  
The Real ◽  
Real Roots ◽  
Bipolar Coordinates ◽  
Complex Roots

The only really useful practical method for solving numerical algebraic equations of higher orders, possessing complex roots, is that devised by C. H. Graeffe early in the nineteenth century. When an equation with real coefficients has only one or two pairs of complex roots, the Graeffe process leads to the evaluation of these roots without great labour. If, however, the equation has a number of pairs of complex roots there is considerable difficulty in completing the solution: the moduli of the roots are found easily, but the evaluation of the arguments often leads to long and wearisome calculations. The best method that has yet been suggested for overcoming this difficulty is that by C. Runge (Praxis der Gleichungen, Sammlung Schubert). It consists in making a change in the origin of the Argand diagram by shifting it to some other point on the real axis of the original Argand plane. The new moduli and the old moduli of the complex roots can then be used as bipolar coordinates for deducing the complex roots completely: this also checks the real roots.

On the Interaction at Anti-Flat Deformation of Stress Concentrators of the Type of Cracks and Stringers with Regard to the Layer Manufactured from Miscellaneous Materials

2019 ◽  
Vol 828 ◽  
pp. 81-88
Author(s):  
Nune Grigoryan ◽  
Mher Mkrtchyan
Keyword(s):  
Integral Transform ◽  
Composite Layer ◽  
Singular Integral Equations ◽  
Original System ◽  
Fourier Integral ◽  
Quadrature Formulas ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Tangential Forces

In this paper, we consider the problem of determining the basic characteristics of the stress state of a composite in the form of a piecewise homogeneous elastic layer reinforced along its extreme edges by stringers of finite lengths and containing a collinear system of an arbitrary number of cracks at the junction line of heterogeneous materials. It is assumed that stringers along their longitudinal edges are loaded with tangential forces, and along their vertical edges - with horizontal concentrated forces. In addition, the cracks are laden with distributed tangential forces of different intensities. The case is also considered when the lower edge of the composite layer is free from the stringer and rigidly clamped. It is believed that under the action of these loads, the composite layer in the direction of one of the coordinate axes is in conditions of anti-flat deformation (longitudinal shift). Using the Fourier integral transform, the solution of the problem is reduced to solving a system of singular integral equations (SIE) of three equations. The solution of this system is obtained by a well-known numerical-analytical method for solving the SIE using Gauss quadrature formulas by the use of the Chebyshev nodes. As a result, the solution of the original system of SIE is reduced to the solution of the system of systems of linear algebraic equations (SLAE). Various special cases are considered, when the defining SIE and the SLAE of the task are greatly simplified, which will make it possible to carry out a detailed numerical analysis and identify patterns of change in the characteristics of the tasks.

scholarly journals Equivalence among orbital equations of polynomial maps

2018 ◽  
Vol 29 (09) ◽  
pp. 1850082 ◽  
Author(s):  
Jason A. C. Gallas
Keyword(s):  
Equations Of Motion ◽  
Algebraic Equations ◽  
Nonlinear Transformations ◽  
Unique Representation ◽  
Real Roots ◽  
Polynomial Maps ◽  
Infinite Sequences

This paper shows that orbital equations generated by iteration of polynomial maps do not necessarily have a unique representation. Remarkably, they may be represented in an infinity of ways, all interconnected by certain nonlinear transformations. Five direct and five inverse transformations are established explicitly between a pair of orbits defined by cyclic quintic polynomials with real roots and minimum discriminant. In addition, infinite sequences of transformations generated recursively are introduced and shown to produce unlimited supplies of equivalent orbital equations. Such transformations are generic and valid for arbitrary dynamics governed by algebraic equations of motion.

An Approximate Method for Determining Areal Sweep Efficiency and Flow Capacity in Formations with Anisotropic Permeability

10.2118/16-pa ◽  
1961 ◽  
Vol 1 (04) ◽  
pp. 277-286 ◽  
Author(s):  
M. Mortada ◽  
G.W. Nabor
Keyword(s):  
Principal Axis ◽  
Bedding Plane ◽  
Sweep Efficiency ◽  
Simple Method ◽  
Anisotropic Permeability ◽  
Preferred Direction ◽  
Flow Capacity ◽  
Principal Axes ◽  
Special Cases ◽  
Directional Permeability

Abstract The effects of anisotropic or directional permeability on the areal sweep efficiency and the flow capacity are examined. The paper points out the importance of taking directional permeability into consideration in planning a flood. It analyzes the two-dimensional flow pattern associated with the skewed line drive for a unit mobility ratio. The direct and staggered line drives are treated as special cases of the skewed line drive. Analytical expressions are developed for the areal sweep efficiency at breakthrough and the flow capacity. They are related to the spacing between like wells, the distance between a row of injectors and the nearest row of producers, and the degree of skewness of the line drive. The latter quantity is defined such that it is equal to zero for the direct line drive and equals one-half for the staggered line drive. The a real sweep efficiency and the flow capacity depend also on the orientation of the flood pattern with respect to the principal axis of anisotropy. The paper provides a simple method for determining the a real sweep efficiency and the flow capacity for a formation in which the permeability in the bedding plane is anisotropic. Introduction Directional or anisotropic permeability is manifested by the ability of the formation to conduct fluids more readily along certain preferred directions. This situation occurs in many producing formations and is usually attributed to depositional features in which the sand grains are oriented in a preferred direction. In some cases it results from the formation of a major and a minor fracture system. Directional permeability should be taken into account in many phases of the production and exploration activities. Recognizing its existence in the formation of interest and planning accordingly can lead to increased recovery and substantial savings. For instance, the areal sweep efficiency in a water flood depends to a great extent on the orientation of the flood pattern with respect to the principal axis of permeability. Anisotropic permeability is specified by the directions of its three principal axes and the permeability along each axis. The principal axes of permeability are mutually perpendicular. This paper deals with the areal sweep efficiency at breakthrough and the flow capacity for formations with anisotropic permeability. The flood pattern considered consists of alternate rows of injecting and producing wells. The rows of wells are parallel and form a developed, skewed line drive which is illustrated in Fig. 1. The staggered and direct line drives are treated as special cases of the skewed line drive.

Singularities at the Tip of a Crack Terminating Normally at an Interface Between Two Orthotropic Media

1996 ◽  
Vol 63 (2) ◽  
pp. 264-270 ◽  
Author(s):  
J. C. Sung ◽  
J. Y. Liou
Keyword(s):  
Characteristic Equation ◽  
Complex Form ◽  
Stress Singularities ◽  
Real Form ◽  
Antisymmetric Mode ◽  
Real Roots ◽  
Complex Roots ◽  
Principal Axes ◽  
Orthotropic Media ◽  
Set Up

The order of stress singularities at the tip of a crack terminating normally at an interface between two orthotropic media is analyzed. Characteristic equation in complex form for the power of singularity s, where 0 < Re{s} < 1, is first set up for two general anisotropic materials. Attention is then focused on the problem that is composed by two orthotropic media where one of them (say, material #2 ) the material principal axes are aligned while the other one (say, material #1) the principal axes can have an angle γ relative to the interface. For such a problem, a real form of the characteristic equation is obtained. The roots are functions of γ in general. Two real roots exist for most values of γ; however, there are possible ranges of γ that the complex roots will occur. The roots s are found to be independent of γ when material #1 has the property that δ(1) = 1. When γ = 0, two roots are always real. Furthermore, each of these two roots is associated with symmetric or antisymmetric mode and they become equal when Δ = 1. Many other features of the effects of the material parameters on the behaviors of the roots s are further investigated in the present work, where the six generalized Dundurs’ constants, expressed in terms of Krenk’s parameters, play an important role in the analysis.

Orbital and Attitude Stability of Space Shuttles in Parking Orbits

1975 ◽  
Vol 26 (1) ◽  
pp. 20-24
Author(s):  
R Arho
Keyword(s):  
Gravitational Field ◽  
Stationary State ◽  
Circular Orbit ◽  
Principal Axis ◽  
Orbital Plane ◽  
Moments Of Inertia ◽  
Attitude Stability ◽  
Principal Moments Of Inertia ◽  
Bounded Perturbation ◽  
The Stability

SummaryA unified treatment is given of the orbital and attitude stability of space shuttles in parking orbits (in vacuo) in the earth’s gravitational field. A shuttle in a circular orbit with a principal axis aligned with the horizontal in the orbital plane is found to be in stationary geostatic equilibrium. The demand for stability leads to a condition which must be satisfied by the principal moments of inertia. The stability which is achieved is not asymptotic without control. The stationary state is a stable centre about which a bounded perturbation oscillation without damping may exist.

scholarly journals An Analysis of Flight Dynamics of a Space Debris Collector Transferring from its Orbital Plane to the Orbital Plane of a Debris Fragment

2020 ◽  
pp. 63-71
Author(s):  
S.V. Arinchev
Keyword(s):  
Space Debris ◽  
Specific Impulse ◽  
Orbital Plane ◽  
Base Station ◽  
Radius Vector ◽  
Orbit Plane ◽  
Zonal Harmonic ◽  
Orbit Inclination ◽  
Analysis Of Dynamics ◽  
Propulsion Unit

A space debris collector and a debris fragment move along random noncoplanar orbits ranging from 400 to 2000 km in height. The space collector leaves the base station, transfers into the orbital plane of a debris fragment, aligns itself with and approaches the fragment, grabs it and returns to the base station. The execution time of the flight mission is 24 hours. This paper examines only the stage when the debris collector transfers from its orbital plane to the fragment’s orbital plane. Dampening is provided by repeated activation of a cruise propulsion unit with the thrust of no less than 20000 N and the fuel specific impulse of no less than 15000 m/s. An analysis of dynamics of the orbital flight is performed by numerically integrating the equations of orbital movement of the debris collector and the debris fragment using the fourth order Runge-Kutta methods. The change of the scalar product sign of the vector of the orbit area integral of the debris fragment and the radius-vector of the debris collector is the criterion for intersecting the final orbit plane. Fuel depletion and the nonsphericity of the Earth’s gravitational field in the second zonal harmonic are taken into account, and an example of the calculations is given. Convergence estimates for the integration procedure with regard to the final orbit inclination relative to the orbit’s eccentricity are provided.

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