1 / 28

On Orbital Stability of Periodic Motions in Problems of Satellite Dynamics

On Orbital Stability of Periodic Motions in Problems of Satellite Dynamics. Universidad Publica de Navarra , Navarra, 2010. Bardin B.S. Moscow Aviation Institute Dept. of Theoretical Mechanics. Orbital Stability. P ( t 1 ). P ( t 0 ). M ( t 1 ). M ( t 0 ). P ( t 0 ). P ( t 1 ).

tayten
Download Presentation

On Orbital Stability of Periodic Motions in Problems of Satellite Dynamics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. On Orbital Stability of Periodic Motions in Problems of SatelliteDynamics Universidad Publica deNavarra, Navarra, 2010 Bardin B.S. Moscow Aviation Institute Dept. of Theoretical Mechanics

  2. Orbital Stability P(t1) P(t0) M(t1) M(t0) P(t0) P(t1) M(t1) M(t0)

  3. Orbital Stability P(t0) P(t0+T) M(t0)=M(t0+T)

  4. Definition of orbital stability • Periodic orbit • Neighborhood of periodic orbit • Def. A periodic orbit G is orbitally stable iff for all e>0 there exist d>0 such that if y(t0) belong to N(G,d) then y(t) belong to N(G,e) for all t>t0 .

  5. Orbital Stability N(G,e) G

  6. Orbital Stability of Hamiltonian Systems • Autonomous Hamiltonian system • Periodic solution • Transformation: • Periodic solution in the variables xi hi

  7. Isoenergetic reduction • The Hamiltonian in the variables xi hi is periodic in x1 • Isoenergetic reduction on the level H=0: • Whittaker Equations • Stability of the solution xj=hj=0, (j=2,…,n)

  8. Stability of the reduced system • Hamiltonian of the reduced system • Linear system • Normalization of K • Stability analyses

  9. Orbital stability of planar periodic motions of a satellite • We consider motions of a satellite about its center of mass. • Satellite is a rigid body which moments of inertia satisfy С=А+В (plate) • The mass center moves in a circular orbit • Unperturbed motion: axes of the minimal moment of inertia is perpendicular to the orbital plane. • Two types of planar periodic motions: oscillations and rotations

  10. Reference frames • OXYZ–orbital frame: axes OX, OY, OZ are directed along the radius vector of the mass center, the normal and transversal of the orbit • Oxyz – principal axes frame, fixed in satellite. Axes directed along the ellipsoid inertia axes. • Euler angels y,q,j define orientation of the satellite

  11. Hamiltonian of the problem • Equation of motion: Hamiltonian: • QA = A/B,QC= C/B;From С=A+B => QA = QC-1

  12. Unperturbed motion. Action-angle variables • Planar motions (unperturbed motion): q=p/2, j=0, pq=pj=0 • Equations for the unperturbed motion: • General solution (oscillation): • Action-angle variables: • Unperturbed motion in the action-angle variables

  13. Perturbed motion, Isoenergetic reduction • Let us introduce perturbations : • Hamiltonian of the perturbed motion: • Isoenergetic reduction (level H=0): New independent variable: w=w1(I0)n

  14. Hamiltonian of the reduced system

  15. Linear system • Linear system • Matrix of fundamental solutions (k=1,2; j=1,2,3,4). Initial conditions: X(0)=E

  16. Study of stability of the linear system • Characteristic equation • Conditions of stability in linear approximation • Characteristic exponents • Linear normalization

  17. Nonlinear study of stabilityNonresonant case • Nonlinear normalization: • Hamiltonian normal form where • Criteria of stability for most of initial conditions • Criteria of formal stability: Formm(r1,r2) is definite for all

  18. Nonlinear study of stabilityResonant case • Resonant cases: • Hamiltonian normal form • Criteria of stability in third approximation

More Related