On Robe’s restricted problem with a modified Newtonian potential

2020 ◽  
Vol 18 (01) ◽  
pp. 2150005 ◽  
Author(s):  
Elbaz I. Abouelmagd ◽  
Abdullah A. Ansari ◽  
M. H. Shehata
Keyword(s):  
Equilibrium Points ◽  
Equations Of Motion ◽  
Underwater Vehicles ◽  
Newtonian Potential ◽  
Second Primary ◽  
The Moon ◽  
Primary Body ◽  
Out Of Plane ◽  
Robe’S Restricted Problem ◽  
Dust Grain

We analyze the existence of equilibrium points for a particle or dust grain in the framework of unperturbed and perturbed Robe’s motion. This particle is moving in a spherical nebula consisting of a homogeneous incompressible fluid, which is considered as the primary body. The second primary body creates the modified Newtonian potential. The perturbed mean motion and equations of motion are found. The equilibrium points (i.e. collinear, noncollinear and out–of–plane points), along with the required conditions of their existence are also analyzed. We emphasize that this analysis can be used to study the oscillations of the Earth’s core under the attraction of the Moon and it is also applicable to study the motion of underwater vehicles.

scholarly journals Existence and Linear Stability of Equilibrium Points in the Robe’s Restricted Three-Body Problem with Oblateness

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Jagadish Singh ◽  
Abubakar Umar Sandah
Keyword(s):  
Equilibrium Point ◽  
Linear Stability ◽  
Body Problem ◽  
Equilibrium Points ◽  
Oblate Spheroid ◽  
Second Primary ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Out Of Plane ◽  
Three Body

This paper investigates the positions and linear stability of an infinitesimal body around the equilibrium points in the framework of the Robe’s circular restricted three-body problem, with assumptions that the hydrostatic equilibrium figure of the first primary is an oblate spheroid and the second primary is an oblate body as well. It is found that equilibrium point exists near the centre of the first primary. Further, there can be one more equilibrium point on the line joining the centers of both primaries. Points on the circle within the first primary are also equilibrium points under certain conditions and the existence of two out-of-plane points is also observed. The linear stability of this configuration is examined and it is found that points near the center of the first primary are conditionally stable, while the circular and out of plane equilibrium points are unstable.

The motion of a lunar orbiter

1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
Secular Motion ◽  
The Earth ◽  
Close Satellite ◽  
The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

scholarly journals On the Stability of Collinear Points of the RTBP with Triaxial and Oblate Primaries and Relativistic Effects

2021 ◽  
pp. 56-73
Author(s):  
S. E. Abd El-Bar
Keyword(s):  
Characteristic Equation ◽  
Body Problem ◽  
Equilibrium Points ◽  
Equations Of Motion ◽  
Relativistic Effects ◽  
Three Body Problem ◽  
Total Potential ◽  
The Mean ◽  
The Stability ◽  
Three Body

Under the influence of some different perturbations, we study the stability of collinear equilibrium points of the Restricted Three Body Problem. More precisely, the perturbations due to the triaxiality of the bigger primary and the oblateness of the smaller primary, in addition to the relativistic effects, are considered. Moreover, the total potential and the mean motion of the problem are obtained. The equations of motion are derived and linearized around the collinear points. For studying the stability of these points, the characteristic equation and its partial derivatives are derived. Two real and two imaginary roots of the characteristic equation are deduced from the plotted figures throughout the manuscript. In addition, the instability of the collinear points is stressed. Finally, we compute some selected roots corresponding to the eigenvalues which are based on some selected values of the perturbing parameters in the Tables 1, 2.

Energy harvesting from aeroelastic vibrations induced by discrete gust loads

2016 ◽  
Vol 28 (1) ◽  
pp. 47-62 ◽  
Author(s):  
Claudia Bruni ◽  
James Gibert ◽  
Giacomo Frulla ◽  
Enrico Cestino ◽  
Pier Marzocca
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Flow Region ◽  
Rotational Displacement ◽  
Reduced Order ◽  
Piezoelectric Elements ◽  
Out Of Plane ◽  
Gust Loads ◽  
Three Degree Of Freedom ◽  
Euler Bernoulli Beam

This article evaluates the amount of energy that can be extracted from a gust using an aeroelastic energy harvester composed of a flexible wing with attached piezoelectric elements. The harvester operates in a subcritical flow region. It is modeled as a linear Euler–Bernoulli beam sandwiched between two piezoceramics. The extended Hamilton’s principle is used to derive the harvester’s equations of motion and an eigenfunction expansion is used to form a three-degree-of-freedom reduced-order model. The degrees of freedom retained in the model are two flexural degrees for the in-plane and out-of-plane displacements, and a torsional degree for the rotational displacement. Wagner and Küssner functions are used to represent the unsteady aerodynamic and gust loading, respectively. The amount of energy extracted from the system is then compared for two different deterministic gust profiles, 1-COSINE and two sharp-edged gusts forming a square gust, for various magnitudes and durations. The results show that the harvester is able to extract more energy from the square gust profile, although for both profiles the harvester extracts more power after the gust has subsided.

Conditions for the planes of symmetry of an elastically supported rigid body

2009 ◽  
Vol 223 (8) ◽  
pp. 1755-1766 ◽  
Author(s):  
S J Jang ◽  
Y J Choi
Keyword(s):  
Rigid Body ◽  
Eigenvalue Problems ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Mass Matrices ◽  
Out Of Plane ◽  
Modes Of Vibration ◽  
Compliance Matrices ◽  
Special Points ◽  
Elastically Supported

Introducing the planes of symmetry into an oscillating rigid body suspended by springs simplifies the complexity of the equations of motion and decouples the modes of vibration into in-plane and out-of-plane modes. There have been some research results from the investigation into the conditions for planes of symmetry in which prior conditions for the simplification of the equations of motion are required. In this article, the conditions for the planes of symmetry that do not need prior conditions for simplification are presented. The conditions are derived from direct expansions of eigenvalue problems for stiffness and mass matrices that are expressed in terms of in-plane and out-of-plane modes and the orthogonality condition with respect to the mass matrix. Two special points, the planar couple point and the perpendicular translation point are identified, where the expressions for stiffness and compliance matrices can be greatly simplified. The simplified expressions are utilized to obtain the analytical expressions for the axes of vibration of a vibration system with planes of symmetry.

Asymptotic analysis of the forced vibrations of a one-disc rotor on a non-linear flexible base

2010 ◽  
Vol 224 (8) ◽  
pp. 1593-1604 ◽  
Author(s):  
K V Avramov
Keyword(s):  
Asymptotic Analysis ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Forced Vibrations ◽  
Linear Dynamics ◽  
Non Linear ◽  
Out Of Plane ◽  
Symmetric Rotor ◽  
Flexible Base ◽  
Steady Motions

Equations of motion for a four-degree-of-freedom dynamical system describing the vibrations of a one-disc elastic rotor taking into account gyroscopic moments on a non-linear flexural base are derived. A new version of the multiple scales method is developed and applied to analyse the non-linear dynamics of such a system for different resonances. The steady motions of the rotor are analysed. From the asymptotic analysis, it is shown that out-of-plane motions of the disc exist in the symmetric rotor.

ON THE GEOMETRIC EFFECT OF DARK MATTER

2011 ◽  
Vol 20 (supp01) ◽  
pp. 102-109
Author(s):  
ABRAÃO J. S. CAPISTRANO ◽  
PEDRO I. ODON ◽  
LUIS A. CABRAL
Keyword(s):  
Equations Of Motion ◽  
Radial Distance ◽  
Extrinsic Curvature ◽  
Brane World ◽  
Velocity Curve ◽  
Newtonian Potential ◽  
Potential Approximation ◽  
Small Power ◽  
Geometric Effect ◽  
Visible Matter

We study the effect of the extrinsic curvature on the rotation curves within the context of brane-world, in a 5-d bulk with constant curvature. The covariant equations of motion for the brane-world are applied to determine the modified Newtonian potential approximation and the velocity curve. We consider a static disk galaxy which is composed only of ordinary visible matter. Using the Weyl static metric as a model, we find that the velocity curve is given by the square root of a small power of the radial distance to the galactic core.

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