Optimal parameters of satellite

1
Optimal parameters of satellitestabilizer 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

• 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
- Referential frame
- Referential frame
Position
coordinates
angles between , and
.
3
EQUATIONS OF MOTION
derivative with respect to time, t
Lagrangian formulation of the motion equations
will be used
4
Equations of Motion
Elliptic orbit
From theory of elliptic motion


true anomaly
Using the dimensionless parameters
5
Equilibria in Circular Orbit
Equilibria
Supposing
4 types of equilibria
Let us consider small oscillations near the
equilibrium position
6
Region of Asymptotic Stability
Linearized Equations
Characteristic Equation
Necessary and sufficient conditions of asymptotic
stability
Physical restrictions
,
7
Optimal Parameters
Degree of stability
,
Maximal degree of stability Minimal
duration of the transitional process
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
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.
Characteristic Equation
8
Optimal Parameters
Using the first and second equations
1. Upper sign before the root
2. Lower sign before the root
First calculate
and
At last calculate
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 cant 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
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
2)Border of the region
3)Border of the region