Chaos in a Spacecraft Attitude Maneuver Due to Time-Periodic Perturbations

1996 ◽  
Vol 63 (2) ◽  
pp. 501-508 ◽  
Author(s):  
G. L. Gray ◽  
I. Dobson ◽  
D. C. Kammer
Keyword(s):  
Chaotic Motion ◽  
Chaotic Dynamics ◽  
Equations Of Motion ◽  
Major Axis ◽  
Artificial Satellites ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Attitude Maneuver ◽  
Analytical Criterion ◽  
Time Periodic

We use Melnikov’s method to study the chaotic dynamics of an attitude transition maneuver of a torque-free rigid body in going from minor axis spin to major axis spin under the influence of small damping. The chaotic motion is due to the formation of Smale horseshoes which are caused by the oscillation of small subbodies inside the satellite. The equations of motion are derived and then transformed into a form suitable for the application of Melnikov’s method. An analytical criterion for chaotic motion is derived in terms of the system parameters. This criterion is evaluated for its significance to the design of artificial satellites.

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 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.

Chaos in a model of forced quasi-geostrophic flow over topography: an application of Melnikov's method

1991 ◽  
Vol 226 ◽  
pp. 511-547 ◽  
Author(s):  
J. S. Allen ◽  
R. M. Samelson ◽  
P. A. Newberger
Keyword(s):  
Periodic Orbit ◽  
Saddle Point ◽  
Wind Stress ◽  
Homoclinic Orbits ◽  
Periodic Forcing ◽  
Melnikov's Method ◽  
Geostrophic Flow ◽  
Melnikov’S Method ◽  
Parameter Values ◽  
Time Periodic

We demonstrate the existence of a chaotic invariant set of solutions of an idealized model for wind-forced quasi-geostrophic flow over a continental margin with variable topography. The model (originally formulated to investigate mean flow generation by topographic wave drag) has bottom topography that slopes linearly offshore and varies sinusoidally alongshore. The alongshore topographic scales are taken to be short compared to the cross-shelf scale, allowing Hart's (1979) quasi-two-dimensional approximation, and the governing equations reduce to a non-autonomous system of three coupled nonlinear ordinary differential equations. For weak (constant plus time-periodic) forcing and weak friction, we apply a recent extension (Wiggins & Holmes 1987) of the method of Melnikov (1963) to test for the existence of transverse homoclinic orbits in the model. The inviscid unforced equations have two constants of motion, corresponding to energy E and enstrophy M, and reduce to a one-degree-of-freedom Hamiltonian system which, for a range of values of the constant G = E − M, has a pair of homoclinic orbits to a hyperbolic saddle point. Weak forcing and friction cause slow variations in G, but for a range of parameter values one saddle point is shown to persist as a hyperbolic periodic orbit and Melnikov's method may be applied to study the perturbations of the associated homoclinic orbits. In the absence of time-periodic forcing, the hyperbolic periodic orbit reduces to the unstable fixed point that occurs with steady forcing and friction. The method yields analytical expressions for the parameter values for which sets of chaotic solutions exist for sufficiently weak time-dependent forcing and friction. The predictions of the perturbation analysis are verified numerically with computations of Poincaré sections for solutions in the stable and unstable manifolds of the hyperbolic periodic orbit and with computations of solutions for general initial-value problems. In the presence of constant positive wind stress τ0 (equatorward on eastern ocean boundaries), chaotic solutions exist when the ratio of the oscillatory wind stress τ1 to the bottom friction parameter r is above a critical value that depends on τ0/r and the bottom topographic height. The analysis complements a previous study of this model (Samelson & Allen 1987), in which chaotic solutions were observed numerically for weak near-resonant forcing and weak friction.

Bifurcations and Chaotic Motions of a Class of Mechanical System With Parametric Excitations

2015 ◽  
Vol 10 (5) ◽  
Author(s):  
Liangqiang Zhou ◽  
Fangqi Chen ◽  
Yushu Chen
Keyword(s):  
Mechanical System ◽  
Parametric Excitation ◽  
Chaotic Dynamics ◽  
Stable And Unstable Manifolds ◽  
Chaotic Motions ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Critical Curves ◽  
Unstable Manifolds

Bifurcations and chaotic motions of a class of mechanical system subjected to a superharmonic parametric excitation or a nonlinear periodic parametric excitation are studied, respectively, in this paper. Chaos arising from the transverse intersections of the stable and unstable manifolds of the homoclinic and heteroclincic orbits is analyzed by Melnikov's method. The critical curves separating the chaotic and nonchaotic regions are plotted. Chaotic dynamics are compared for these systems with a periodic parametric excitation or a superharmonic parametric excitation, or a nonlinear periodic parametric excitation. Especially, some new dynamical phenomena are presented for the system with a nonlinear periodic parametric excitation.

Chaos in the Quasiperiodically Forced Helmholtz-Duffing Oscillator With Two-Well Potential

2005 ◽  
Author(s):  
Jing-Jun Lou ◽  
Shi-Jian Zhu ◽  
Qi-Wei He
Keyword(s):  
Hamiltonian System ◽  
Parameter Space ◽  
Homoclinic Orbit ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Duffing Oscillator ◽  
Restoring Force ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
The Asymmetry

The chaotic dynamics of the quasiperiodically excited Helmholtz-Duffing oscillator with two-well potential was investigated. The condition of the existence of homoclinic orbit in the corresponding Hamiltonian system was presented which is asymmetrical resulting from the asymmetry restoring force. It was found that the mechanism for chaos is transverse homoclinic tori and it is illustrated how transverse homoclinic tori give rise to chaos for the Helmholtz-Duffing oscillator with multi-frequency periodic forces. The criterion for the existence of chaos was given utilizing a generalization of the Melnikov’s method. The region in parameter space where chaotic dynamics may occur was given. It was also demonstrated that increasing the number of forcing frequencies increases the area in parameter space where chaotic behavior can occur.

MELNIKOV'S METHOD AND STICK–SLIP CHAOTIC OSCILLATIONS IN VERY WEAKLY FORCED MECHANICAL SYSTEMS

1999 ◽  
Vol 09 (03) ◽  
pp. 505-518 ◽  
Author(s):  
J. AWREJCEWICZ ◽  
M. M. HOLICKE
Keyword(s):  
Numerical Simulation ◽  
Chaotic Dynamics ◽  
Mechanical Systems ◽  
Degree Of Freedom ◽  
Chaotic Oscillations ◽  
Stick Slip ◽  
Melnikov's Method ◽  
Melnikov’S Method

In this paper we predict stick–slip chaotic dynamics in a one-degree-of-freedom very weakly forced (quasiautonomous) oscillator using the Melnikov's technique. Numerical simulation confirms the validity of our approach.

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.

Chaotic Motion of a Functionally Graded Materials Square Thin Plate

2012 ◽  
Vol 531 ◽  
pp. 593-596
Author(s):  
Shuang Bao Li ◽  
Yu Xin Hao
Keyword(s):  
Thin Plate ◽  
Functionally Graded Materials ◽  
Chaotic Motion ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Functionally Graded ◽  
Third Order ◽  
Graded Materials ◽  
Steady State Heat ◽  
Steady State Heat Transfer

Chaotic motion of a simply supported functionally graded materials (FGM) square thin plate under one-to-two internal resonance is studied in this paper. The FGM plate is subjected to the transversal and in-plane excitations. Material properties are assumed to be temperature-dependent and change continuously throughout the thickness of the plate. The temperature variation is assumed to occur in the thickness direction only and satisfy the steady-state heat transfer equation. Based on the Reddy’s third-order plate theory and Hamilton’s principle, the nonlinear governing equations of motion for the FGM plate are derived by using the Galerkin’s method to describe the transverse oscillation in the first two modes Numerical simulations illustrate that there exist chaotic motion for the FGM rectangular plate.

Sign in / Sign up

Export Citation Format

Share Document