On coupled librational dynamics of gravity oriented axi-symmetric satellites
Creators: name: Shrivastava, Shashi Kant
Publisher: name: University of British Columbia
Year: 2011

description: The influence of inertia, eccentricity and atmospheric forces on the attitude dynamics of gravity oriented, non-spinning, axi-symmetric satellites, executing general librational motion is investigated using analytical, numerical and analog techniques. The problem is studied in the increasing order of complexity. For the case of a circular orbit, the autonomous, conservative system represented by constant Hamiltonian yields zero-velocity curves and motion envelopes which identify regions of instability from conditional and guaranteed stable motion. The non-linear, coupled equations of motion are solved using approximate analytical techniques: Butenin’s variation of parameter method and invariant integral approach. A comparison with the numerical response, establishes their suitability in studies involving motion in the small. The invariant integral method maintains reasonable accuracy even for larger, predominantly planar, disturbances. However, for a general motion in the large, the analytical solutions provide only qualitative information and one is forced to resort to numerical, analogic or hybrid procedures. The analysis suggests strong dependence of system response on the in-plane disturbances and satellite inertia. The librational and orbital frequencies are of the same order of magnitude. It also shows that the stable solution, when represented in a three dimensional phase space may lead to 'regular', 'ergodic' or 'island' type regions. The limiting integral manifolds, given here for a few representative values of Hamiltonian, provide all possible combinations of initial conditions, which a satellite can withstand without tumbling. The results, for a range of satellite inertia, are condensed in the form of design plots, indicating allowable disturbances for stable motion. In general, the slender satellites exhibit better stability characteristics. The presence of aerodynamic torque destroys the symmetry properties of the integral manifolds. The stability of the equilibrium configuration, which now deviates from the local vertical, is established through Routh's as well as Liapunov's criteria. As the system is still autonomous and conservative, the Hamiltonian remains constant leading to the bounds of libration. Numerical analysis of the system response indicates increased sensitivity to planar disturbances. The distortion and contraction of the regular, ergodic and island type stability regions show the adverse effects of aerodynamic torque. The design plots suggest that the shorter satellites, normally not preferred from gravity-gradient considerations, could exhibit better stability characteristics in the presence of large aerodynamic torque. An alternate, economical approach to the dynamical analysis of the satellites is attempted using an analog computer. A comparison with the digital data establishes the suitability of the method for design purposes and real time simulation. As the regular surface represents the only usable stability region from design considerations, a detailed study to establish the bound between regular and ergodic type stability was undertaken. The periodic solutions, obtained numerically using variable secant iteration show their spinal character with the body of stability region built around them. Of particular significance is the fundamental periodic solution (two planar oscillations in one out-of- plane cycle) associated with the regular region, suitable for practical operation of a satellite. The remaining periodic solutions represent degeneration of the island-like areas surrounding the mainland. The results lead to a set of fundamental periodic solutions over a wide range of system parameters. Floquet's variational analysis is used to establish the critical disturbance [formula omitted], beyond which no stable motion can be expected. The periodic solutions together with the regular stability region are presented here as functions of Hamiltonian, satellite inertia and aerodynamic torque. The case study of GEOS-A satellite is also included. In elliptic orbit, the Butenin's analysis of coupled forced systems is found to give an approximate solution of good accuracy. However for this non-autonomous situation, where Hamiltonian is no longer a constant of the motion, the concept of integral manifold breaks down. Fortunately, the design plots can still be generated by direct utilization of the response characteristics. In general the stability region diminishes with increasing eccentricity and disappears completely for e &gt; 0.35. The presence of atmosphere adds to the complex behaviour of this non-autonomous system, where even the equilibrium configuration now becomes periodic in character. The stability regions are further reduced with instabilities normally initiating in the planar degree of freedom. Finally, a possibility of using the atmospheric forces in attitude control is explored. The use of a set of horizontal flaps in conjunction with a semi-passive, velocity-sensitive controller appears to be promising. With a suitable choice of system parameters even a large disturbance can be damped in approximately two orbits.