1. Optimal parameters of satellite–stabilizer system in circular and elliptic orbits 2nd International Workshop Spaceflight Dynamics and Control October 9-11, Covilhã, Portugal Sarychev V. A., Seabra A.M. Keldysh Institute of Applied Mathematics, Moscow, Russia Escola Superior de Tecnologia de Viseu, Viseu, Portugal

2. INTRODUTION - Referential frame coordinates Orbital coordinate system Elliptic orbit Satellite: - Centre of mass, mass Stabilizer: -Centre of mass, mass Link: dissipative hinge mechanism, -Oscillations in the orbital plane, eccentricity e angles between, and. Position

3. EQUATIONS OF MOTION Lagrangian formulation of the motion equations will be used derivative with respect to time, t

4. Equations of Motion Elliptic orbit: Using the dimensionless parameters From theory of elliptic motion true anomaly

5. Equilibria in Circular Orbit Equilibria Let us consider small oscillations near the equilibrium position Supposing 4 types of equilibria:

6. Linearized Equations Characteristic Equation Region of Asymptotic Stability Necessary and sufficient conditions of asymptotic stability:, Physical restrictions

7. Optimal Parameters [Borrelli and Leliakov, 1972] For that class of characteristic equations, optimal parameters only can exist if the their roots have one of the 3 configurations in -plane Degree of stability, Maximal degree of stability Minimal duration of the transitional process Characteristic Equation (1) (3) (2) (4) [Sarychev, Sazonov, Mirer, 1976] It is proved that maximum degree of stability of the kind of linear system we have is achieved when the roots of characteristic equation are real and equal.

8. Optimal Parameters Using the first and second equations 1. Upper sign before the root 2. Lower sign before the root First: calculate At last: calculate and

9. Optimal Parameters 1. Upper sign before the root Solutions: 1.1

10. Optimal Parameters 1.2 decreases with

11. Optimal Parameters 2. Lower sign before the root decreases with

12. Optimal Parameters Comments: Investigation of the optimal transitional process of the satellite- stabilizer system will be done, using numerical integration of the exact nonlinear equations. All numerical calculations were made for the configuration of a system with the moment of elastic forces in the hinge. Simulations show [Sarychev, 1970] that optimal transitional process can’t differ very much from analytical results obtained for linear equations. For this system, we suppose that similar results will be obtained.

13. Eccentricity Oscillations in Elliptic Orbit, Study of the forced solution caused by non-uniformity of motion of the centre of mass of the satellite-stabilizer system over the orbit. Search for a forced solution by the small parameter method in the form of series of power of

14. Eccentricity Oscillations in Elliptic Orbit Forced solution of the system Derivatives with respect to variable Equating the coefficients at and a set of four algebraic equations appear Parameters should satisfy the conditions

15. Eccentricity Oscillations in Elliptic Orbit Forced solution of the system Amplitude of eccentricity oscillations of the satellite and of the stabilizer

16. Amplitude of Eccentricity Oscillations in Elliptic Orbit Minimize the function with restrictions

17. Minimal Amplitude of Eccentricity Oscillations in Elliptic Orbit Plane of investigation of : 1)In the interior of the region at fixed Necessary conditions of extreme: 3)Border of the region 2)Border of the region