Method for calculating the perturbed trajectoryof a two-impulse flight between the halo orbit in the vicinity of the L2 point of the Sun — Earth systemand the near-lunar orbit

2020 ◽  
Author(s):  
Zhou Rui
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Continuation Method ◽  
Initial Approximation ◽  
Halo Orbit ◽  
Lunar Orbit ◽  
The Sun ◽  
Zonal Harmonic ◽  
The Earth ◽  
Parameter Continuation

The paper introduces a new method for solving the problem of calculating the perturbed trajectory of a two-impulse flight between a near-lunar orbit and a halo orbit in the vicinity of the L2 point of the Sun — Earth system. Unlike traditional numerical methods, this method has better convergence. Accelerations from the gravitational forces of the Earth, the Moon and the Sun as point masses and acceleration from the second zonal harmonic of the geopotential are taken into account at all sections of the trajectory. The calculation of the flight path is reduced to solving a two-point boundary value problem for a system of ordinary differential equations. The developed method is based on the parameter continuation method and does not require the choice of an initial approximation for solving the boundary value problem. The last section of the paper provides examples and results of the analysis based on this method.

scholarly journals The Lunar orbit paradox

2013 ◽  
Vol 40 (1) ◽  
pp. 135-146
Author(s):  
Aleksandar Tomic
Keyword(s):  
Inertial Mass ◽  
Lunar Orbit ◽  
The Sun ◽  
The Moon ◽  
The Earth ◽  
Three Body ◽  
Origin Of The Moon ◽  
Force Intensity ◽  
Critical Consideration

Newton's formula for gravity force gives greather force intensity for atraction of the Moon by the Sun than atraction by the Earth. However, central body in lunar (primary) orbit is the Earth. So appeared paradox which were ignored from competent specialist, because the most important problem, determination of lunar orbit, was inmediately solved sufficiently by mathematical ingeniosity - introducing the Sun as dominant body in the three body system by Delaunay, 1860. On this way the lunar orbit paradox were not canceled. Vujicic made a owerview of principles of mechanics in year 1998, in critical consideration. As an example for application of corrected procedure he was obtained gravity law in some different form, which gave possibility to cancel paradox of lunar orbit. The formula of Vujicic, with our small adaptation, content two type of acceleration - related to inertial mass and related to gravity mass. So appears carried information on the origin of the Moon, and paradox cancels.

Laplacian structure mirroring surface topography in determining the gravity potential by successive approximations

2021 ◽  
Author(s):  
Petr Holota ◽  
Otakar Nesvadba
Keyword(s):  
Boundary Value Problem ◽  
Laplace Operator ◽  
Boundary Value ◽  
Successive Approximations ◽  
Gravity Potential ◽  
Geodetic Boundary Value Problem ◽  
Physical Surface ◽  
The Earth ◽  
Geodetic Boundary ◽  
The Laplace Operator

<p>Similarly as in other branches of engineering and mathematical physics, a transformation of coordinates is applied in treating the geodetic boundary value problem. It offers a possibility to use an alternative between the boundary complexity and the complexity of the coefficients of the partial differential equation governing the solution. In our case the Laplace operator has a relatively simple structure in terms of spherical or ellipsoidal coordinates which are frequently used in geodesy. However, the physical surface of the Earth and thus also the solution domain substantially differ from a sphere or an oblate ellipsoid of revolution, even if optimally fitted. The situation becomes more convenient in a system of general curvilinear coordinates such that the physical surface of the Earth is imbedded in the family of coordinate surfaces. Applying tensor calculus the Laplace operator is expressed in the new coordinates. However, its structure is more complicated in this case and in a sense it represents the topography of the physical surface of the Earth. The Green’s function method together with the method of successive approximations is used for the solution of the geodetic boundary value problem expressed in terms of the new coordinates. The structure of iteration steps is analyzed and if useful and possible, it is modified by means of integration by parts. Subsequently, the iteration steps and their convergence are discussed and interpreted, numerically as well as in terms of functional analysis.</p>

Fluid Dynamics of the Elliptic Porous Slider

1978 ◽  
Vol 45 (2) ◽  
pp. 435-436 ◽  
Author(s):  
L. T. Watson ◽  
T. Y. Li ◽  
C. Y. Wang
Keyword(s):  
Boundary Value Problem ◽  
Moving Objects ◽  
Stokes Equations ◽  
Boundary Value ◽  
Initial Approximation ◽  
Frictional Resistance ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Transformation Of Variables ◽  
Lift And Drag

Fluid cushioned porous sliders are useful in reducing the frictional resistance of moving objects. This paper studies the elliptic slider. After a transformation of variables, the Navier-Stokes equations reduce to a nonlinear two-point boundary-value problem. This boundary-value problem was solved by a homotopy-type method, which did not require a good initial approximation to the solution. The problem was solved for several Reynolds numbers and ellipse eccentricities. Lift and drag calculations show that an elliptic porous slider should be operated along the minor axis.

Index to Volume 277

1975 ◽  
Vol 277 (1274) ◽  
pp. 649-649
Keyword(s):  
Orbital Motion ◽  
Lunar Orbit ◽  
Artificial Satellites ◽  
Time Interval ◽  
The Sun ◽  
The Moon ◽  
Slight Effect ◽  
The Earth ◽  
Westerly Direction ◽  
Complete Revolution

Dr R. R. Newton has notified the following correction to his contribution. The paragraph at the bottom of page 16 and the top of page 17 should read: The node of the lunar orbit rotates in a westerly direction around the plane of the ecliptic, making a complete revolution in about 18.61 years. This motion, and this time interval, are important in eclipse theory, as we shall discuss in the next section. This motion results almost entirely from the perturbation of the Sun’s gravitation on the Moon’s orbital motion. The Earth’s equatorial bulge, which is almost entirely responsible for the motion of the nodes of artificial satellites near the Earth, has only a slight effect on a satellite as distant as the Moon.

Remark on Hilbert's boundary value problem for Beltrami systems

1984 ◽  
Vol 98 (3-4) ◽  
pp. 305-310 ◽  
Author(s):  
Heinrich Begehr
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
A Priori ◽  
Continuation Method ◽  
Elliptic Systems ◽  
Constructive Proof ◽  
Nonlinear Elliptic Systems ◽  
Nonlinear Elliptic ◽  
Hilbert Boundary Value Problem ◽  
Positive Index

SynopsisThe Schauder continuation method for nonlinear problems is based on appropriate a priori estimates for related linear equations. Recently, in a paper by the present author and G. C. Hsiao, the Hilbert boundary value problem with positive index for nonlinear elliptic systems in the plane was solved by this method but the constructive derivation of the a priori estimate necessarily required a restriction on the ellipticity condition. This is because the norm of the generalized Hilbert transform in the case of positive index is too big. Here, as in a forthcoming paper by G.C. Wen, an indirect and therefore non-constructive proof of the a priori estimate is given which does not require any further restrictions and allows the Hilbert boundary value problem to be solved for nonlinear elliptic systems in general.

scholarly journals The continuation method for calculation of continuous beams of arbitrary bending rigidity

2021 ◽  
Vol 350 ◽  
pp. 00017
Author(s):  
Andrei Verameichyk ◽  
Mikhael Mazyrka ◽  
Vitaliy Khvisevich
Keyword(s):  
Optimal Design ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Continuation Method ◽  
Numerical Implementation ◽  
Bending Rigidity ◽  
Multipoint Boundary ◽  
Continuous Beams ◽  
Multipoint Boundary Value Problem

This article presents an effective mathematical continuation method for the numerical implementation of the multipoint boundary value problem, to which the calculation of a beam of arbitrary rigidity at any of its supports is reduced. The problem can be treated as a direct one in the matter of constructing an optimal design based on beam systems. A test example of the calculation is given.

Determination of the initial conditions by solving boundary value problem method for period-one walking of a passive biped walking robots

Robotica ◽  
2015 ◽  
Vol 35 (1) ◽  
pp. 166-188 ◽  
Author(s):  
Masoumeh Safartoobi ◽  
Morteza Dardel ◽  
Mohammad Hassan Ghasemi ◽  
Hamidreza Mohammadi Daniali
Keyword(s):  
Boundary Value Problem ◽  
Limit Cycles ◽  
Initial Conditions ◽  
Boundary Value ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Basin Of Attraction ◽  
Initial Guess ◽  
Walking Robots ◽  
Nonlinear Equations Of Motion

SUMMARYWith regard to the small basin of attraction of the passive limit cycles, it is important to start from a proper initial condition for stable walking. The present study investigates the passive dynamic behaviors of two-dimensional bipedal walkers of a compass gait model with different foot shapes. In order to find proper initial conditions for stable and unstable period-one gait limit cycles, a method based on solving the nonlinear equations of motion is presented as a boundary value problem (BVP). An initial guess is required to solve the related BVP that is obtained by solving an initial value problem (IVP). For parametric analysis purposes, a continuation method is applied. Simulation results reveal two, period-one gait cycles and the effects of parameters variation for all models.

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