Dynamics of multibody chains in circular orbit: non-integrability of equations of motion

2016 ◽  
Vol 126 (4) ◽  
pp. 297-311 ◽  
Author(s):  
Andrzej J. Maciejewski ◽  
Maria Przybylska
Keyword(s):  
Circular Orbit ◽  
Equations Of Motion

scholarly journals THE STABILITY OF QUASI-CIRCULAR ORBITS NEAR THE CENTRAL BODY

2020 ◽  
Vol 70 (2) ◽  
pp. 155-159
Author(s):  
М.Е. Abishev ◽  
◽  
С. Тоktarbay ◽  
А.Z. Таlkhat ◽  
A.Zh. Abylayeva ◽  
...  
Keyword(s):  
Circular Orbit ◽  
Stability Problem ◽  
Central Body ◽  
Equations Of Motion ◽  
Test Body ◽  
Circular Orbits ◽  
The Stability ◽  
The Hill

In the problem of relational bounded three bodies, the stability of quasicircular orbits close to the central body was investigated. In the case when it is not relational, the orbits of the test body can be described through the Hill surfaces. The location of the central body corresponds to the origin, the second body moves in a circular orbit that is around the central (first) body. The equations of motion of the problem of bounded, relational three bodies were investigated for circular orbits. Using these equations of relational motion, the stability problem of the relational quasicircular orbits of the test body in regions close to the central body was investigated.

Coupled Librational Motion of an Axi-Symmetric Satellite in a Circular Orbit

1969 ◽  
Vol 73 (704) ◽  
pp. 674-680 ◽  
Author(s):  
V. J. Modi ◽  
S. K. Shrivastava
Keyword(s):  
Circular Orbit ◽  
Equations Of Motion ◽  
Orbital Plane ◽  
Gradient Field ◽  
Gravity Gradient ◽  
Transverse Motion ◽  
Complex Nature ◽  
Coupled Equations ◽  
Librational Motion ◽  
Zero Velocity Curves

The review of the literature suggests that the planar motion of a rigid satellite in a gravity gradient field has been the subject of considerable investigation during the past ten years. In contrast the dynamic study of a satellite executing librational motion out of the orbital plane has received comparatively little attention. Such an investigation is important because as pointed out by Kane, for large amplitudes the transverse motion is strongly coupled with that in the plane. The lack of information may be due to the complex nature of the problem. The non-linear, non-autonomous, coupled equations of motion involving a large number of parameters are not amenable to simple, concise analysis. Some simplification may be achieved by restricting the satellite motion to a circular orbit. For this case, as indicated by Auelmann, closed zero-velocity curves exist under certain conditions which limit the amplitude of motion.

scholarly journals The uniform circular motion with invariable normal spin of a rigidly and uniformly electrified sphere, IV

1937 ◽  
Vol 159 (899) ◽  
pp. 570-591 ◽  
Keyword(s):  
Electromagnetic Field ◽  
Angular Velocity ◽  
Circular Orbit ◽  
Normal Force ◽  
Equations Of Motion ◽  
Circular Motion ◽  
Classical Electrodynamics ◽  
Quadratic Polynomials ◽  
Total Reaction

In order to exhibit clearly and fully the possibilities inherent in classical electrodynamics when it is developed rigorously without approximations or unnecessary restrictions I have in this paper worked out completely the case in which the centre of the sphere describes a circle with any uniform speed less than that of light whilst it is spinning about a diameter normal to the plane of the circle with an invariable angular velocity unrestricted in magnitude or sense. It is to be noted that, although the speed of the centre is assumed for the sake of simplicity to be less than that of light, that of points on the surface of the sphere (other than the ends of the axis of spin) can be as large as we please. In §§ 2–5 general expressions for the tangential and normal force constituents and the couple constituent of the total reaction on the sphere of its own electromagnetic field are obtained from the general expressions given in paper III (§ 8), and the resulting equations of motion are written down (cf. (2·1)–(2·3) and (5·7)). There are two points to be noticed: (1) the tangential and normal force constituents are quadratic polynomials in the spin p with coefficients depending on the speed cβ of the centre and the radius R of its orbit, whilst the couple constituent is linear in p ; these results are true for any orbit with invariable spin. (2) The couple is found in § 5 to vanish identically, i. e. for all values of p, β and R , in the case of a circular orbit, owing to its symmetry with respect to a diameter; for this reason the result is probably peculiar to this class of orbit.

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.

On the motion of comet P/d’Arrest

1974 ◽  
Vol 22 ◽  
pp. 145-148
Author(s):  
W. J. Klepczynski
Keyword(s):  
Equations Of Motion ◽  
Variational Equations ◽  
Partial Derivatives

AbstractThe differences between numerically approximated partial derivatives and partial derivatives obtained by integrating the variational equations are computed for Comet P/d’Arrest. The effect of errors in the IAU adopted system of masses, normally used in the integration of the equations of motion of comets of this type, is investigated. It is concluded that the resulting effects are negligible when compared with the observed discrepancies in the motion of this comet.

Validation of a Belt Model for Prediction of Hub Forces from a Rolling Tire4

2009 ◽  
Vol 37 (2) ◽  
pp. 62-102 ◽  
Author(s):  
C. Lecomte ◽  
W. R. Graham ◽  
D. J. O’Boy
Keyword(s):  
Internal Pressure ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Prediction Method ◽  
Analytical Models ◽  
Beam Model ◽  
Impedance Boundary ◽  
Bending Waves ◽  
Tire Rotation ◽  
Three Dimensional Models

Abstract An integrated model is under development which will be able to predict the interior noise due to the vibrations of a rolling tire structurally transmitted to the hub of a vehicle. Here, the tire belt model used as part of this prediction method is first briefly presented and discussed, and it is then compared to other models available in the literature. This component will be linked to the tread blocks through normal and tangential forces and to the sidewalls through impedance boundary conditions. The tire belt is modeled as an orthotropic cylindrical ring of negligible thickness with rotational effects, internal pressure, and prestresses included. The associated equations of motion are derived by a variational approach and are investigated for both unforced and forced motions. The model supports extensional and bending waves, which are believed to be the important features to correctly predict the hub forces in the midfrequency (50–500 Hz) range of interest. The predicted waves and forced responses of a benchmark structure are compared to the predictions of several alternative analytical models: two three dimensional models that can support multiple isotropic layers, one of these models include curvature and the other one is flat; a one-dimensional beam model which does not consider axial variations; and several shell models. Finally, the effects of internal pressure, prestress, curvature, and tire rotation on free waves are discussed.

Calculation of Dynamic Tire Forces from Aircraft Instrumentation Data

2010 ◽  
Vol 38 (3) ◽  
pp. 182-193 ◽  
Author(s):  
Gary E. McKay
Keyword(s):  
System Performance ◽  
Equations Of Motion ◽  
Test Laboratory ◽  
Excellent Correlation ◽  
Friction Behavior ◽  
Aircraft Landing ◽  
Data Scatter ◽  
Tire Forces ◽  
Six Degree Of Freedom ◽  
Tire Friction

Abstract When evaluating aircraft brake control system performance, it is difficult to overstate the importance of understanding dynamic tire forces—especially those related to tire friction behavior. As important as they are, however, these dynamic tire forces cannot be easily or reliably measured. To fill this need, an analytical approach has been developed to determine instantaneous tire forces during aircraft landing, braking and taxi operations. The approach involves using aircraft instrumentation data to determine forces (other than tire forces), moments, and accelerations acting on the aircraft. Inserting these values into the aircraft’s six degree-of-freedom equations-of-motion allows solution for the tire forces. While there are significant challenges associated with this approach, results to date have exceeded expectations in terms of fidelity, consistency, and data scatter. The results show excellent correlation to tests conducted in a tire test laboratory. And, while the results generally follow accepted tire friction theories, there are noteworthy differences.

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