Melnikov’s method for rigid bodies subject to small perturbation torques

1996 ◽  
Vol 66 (4) ◽  
pp. 215 ◽  
Author(s):  
X. Tong ◽  
B. Tabarrok
Keyword(s):  
Small Perturbation ◽  
Rigid Bodies ◽  
Melnikov's Method ◽  
Melnikov’S Method

Heteroclinic Bifurcations in Rigid Bodies Containing Internally Moving Parts and a Viscous Damper

1999 ◽  
Vol 66 (3) ◽  
pp. 720-728 ◽  
Author(s):  
G. L. Gray ◽  
D. C. Kammer ◽  
I. Dobson ◽  
A. J. Miller
Keyword(s):  
Chaotic Dynamics ◽  
Equations Of Motion ◽  
Major Axis ◽  
Rigid Bodies ◽  
Necessary Condition ◽  
Viscous Damping ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Melnikov Criterion ◽  
Analytical Criterion

Melnikov’s method is used to analytically study chaotic dynamics in an attitude transition maneuver of a torque-free rigid body in going from minor axis to major axis spin under the influence of viscous damping and nonautonomous perturbations. The equations of motion are presented, their phase space is discussed, and then they are transformed into a form suitable for the application of Melnikov’s method. Melnikov’s method yields an analytical criterion for homoclinic chaos in the form of an inequality that gives a necessary condition for chaotic dynamics in terms of the system parameters. The criterion is evaluated for its physical significance and for its application to the design of spacecraft. In addition, the Melnikov criterion is compared with numerical simulations of the system. The dependence of the onset of chaos on quantities such as body shape and magnitude of damping are investigated. In particular, it is found that for certain ranges of viscous damping values, the rate of kinetic energy dissipation goes down when damping is increased. This has a profound effect on the criterion for chaos.

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

scholarly journals Stochastic Stellar Orbits and the Shapes of Elliptical Galaxies

1987 ◽  
Vol 127 ◽  
pp. 477-478
Author(s):  
Ortwin E. Gerhard
Keyword(s):  
Elliptical Galaxies ◽  
The Other ◽  
Melnikov's Method ◽  
Stellar Orbits ◽  
Melnikov’S Method ◽  
Other Hand

Using Melnikov's method to study the appearance of stochastic orbits in perturbed Stäckel potentials, a correlation is found between the observed shapes of elliptical galaxies and the occurence of mainly regular orbits. Some other potential perturbations giving rise to large regions of stochastic orbits, on the other hand, appear to be inconsistent with observations.

A NEW AND SIMPLE CHAOS TOY

2002 ◽  
Vol 12 (08) ◽  
pp. 1843-1857 ◽  
Author(s):  
ERIK M. BOLLT ◽  
AARON KLEBANOFF
Keyword(s):  
Ball Bearing ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Future Application ◽  
Lagrangian Mechanics ◽  
Smale Horseshoe ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Bicycle Wheel

We present two new, and perhaps the simplest yet, mechanical chaos demonstrations. They are both designed based on a recipe of competing nonlinear oscillations. One of these devices is simple enough that using the provided description, it can be built using a bicycle wheel, a piece of wood routed with an elliptical track, and a ball bearing. We provide a thorough Lagrangian mechanics based derivation of equations of motion, and a proof of chaos based on showing the existence of an embedded Smale horseshoe using Melnikov's method. We conclude with discussion of a future application.

Spin Rotor Stabilization of a Dual-Spin Spacecraft with Time Dependent Moments of Inertia

1998 ◽  
Vol 08 (03) ◽  
pp. 609-617 ◽  
Author(s):  
V. Lanchares ◽  
M. Iñarrea ◽  
J. P. Salas
Keyword(s):  
Periodic Function ◽  
Center Of Mass ◽  
The Body ◽  
Body Motion ◽  
Time Dependent ◽  
Moments Of Inertia ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Principal Axes

We consider a dual-spin deformable spacecraft, in the sense that one of the moments of inertia is a periodic function of time such that the center of mass is not altered. In the absence of external torques and spin rotors, by means of the Melnikov's method we prove that the body motion is chaotic. Stabilization is obtained by means of a spinning rotor about one of the principal axes of inertia.

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