We investigate the relative equilibria of a gyrostat with a spring-mass-dashpot damper to gain new insights into the dynamics of spin-stabilized satellites. The equations of motion are developed using a Newton-Euler approach, resulting in equations in terms of system momenta and damper variables. Linear and nonlinear stability methods produce stability conditions for simple spins about the nominal principal axes. We use analytical and numerical methods to explore system equilibria, including the bifurcations that occur for varying system parameters for varying rotor momentum and damper parameters. The equations and bifurcations for zero rotor absolute angular momentum are identical to those for a rigid body with an identical damper. For the more general case of non-zero rotor momentum, the bifurcations are complex structures that are perturbations of the zero rotor momentum case. We examine the effects of spring stiffness, damper position, and inertia properties on the global equilibria. Stable equilibria exist for many different spin axes, including some that do not lie in the nominally principal planes. Some bifurcations identify regions where a jump phenomenon is possible. We use Liapunov-Schmidt reduction to determine an analytic relationship between parameters to determine if the jump phenomenon occurs. Bifurcations of the nominal gyrostat spin are characterized in parameter space using two-parameter continuation and the Liapunov-Schmidt reduction technique. We quantify the effects of rotor or damper alignment errors by adding small displacements to the alignment vectors, resulting in perturbations of the bifurcations for the standard model. We apply the global bifurcation results to several practical applications. We relate the general set of all possible equilibria to specific equilibria for dual-spin satellites with typical parameters. For systems with tuned dampers, where the natural frequency of the spring-mass-damper matches the gyrostat precession frequency, we show numerically and analytically that the existence of certain equilibria are related to the damper tuning condition. Finally, the global equilibria and bifurcations for varying rotor momentum provide a unique perspective on the dynamics of simple rotor spin-up maneuvers. / Ph. D.

Active magnetic bearings have several potential applications in spacecraft design. Based on the gyrostat model, we develop equations that describe the internal torques and forces that occur between the body and one of the attached wheels. We evaluate the transverse torques for the torque--free gyrostat and a gyrostat undergoing attitude maneuvers using momentum wheels and external torques. We then apply these internal forces to a model of an active magnetic bearing system and discuss their effects on the force limit, the actuator slew rate and the equivalent stiffness and damping parameters. As a basis for this study we use the Distributed Spacecraft Attitude Control System Simulator (DSACSS) with a Revolve MBRotor active magnetic bearing system. The results of several numerical simulations show that the magnitude and frequency of the internal torques remain small over the estimated range of motion of the DSACSS--MBRotor gyrostat. As such, the transverse torques caused by the rotational motion remain less than the discussed performance limits. We show that the magnitude of the internal torques can also be minimized by reducing the axial moment of inertia of the wheel. Furthermore, we discuss the equivalent Jeffcott model. By applying a standard Proportional--Integral--Derivative controller to the active magnetic bearing both the equivalent stiffness and damping parameters remain constant. / Master of Science

This dissertation investigates attitude transition maneuvers of a gyrosat-satellite between relative equilibria. The primary challenge in transitioning between relative equilibria is the proper adjustment of the system angular momentum so that upon completing the transition maneuver the gyrostat-satellite will satisfy all the requirements for a relative equilibrium. The system angular momentum is a function of the attitude trajectory taken during the transition maneuver. A new concept, the constant orbital momentum equilibrium trajectory or COMET, is introduced as a means to a straight-forward solution to a subset of the possible transitions between relative equilbria. COMETs are a class of paths in SO(3) that a gyrostat-satellite may travel along that maintain a constant system angular momentum. The primary contributions of this dissertation are the introduction and analysis of COMETs and their application to the problem of transitioning a gyrostat-satellite between two relative equilibria. The current work introduces, defines, and analyzes COMETs in detail. The requirements for a path in SO(3) to be a COMET are defined. It is shown via example that COMETs are closed-curves in SO(3). Visualizations of families of COMETs are presented and discussed in detail. A subset of COMETs are shown to contain critical points that represent isolated relative equilibrium attitudes or furcations of the COMET. The problem of transitioning between two relative equilibria is split into the sub-problems of transitioning between relative equilibria on the same COMET and transitioning between relative equilibria on different COMETs. For transitions between relative equilibria on the same COMET, an open-loop control law is developed that drives a gyrostat-satellite along the COMET until the target relative equilibrium is reached. For transitions between relative equilibria on different COMETs, an open-loop control law is developed that transfers a gyrostat-satellite from the initial relative equilibrium to a relative equilibrium that resides on the same COMET as the target relative equilbrium. Acquisition of the target relative equilibrium is then accomplished via the application of the open-loop control law for transitions between relative equilibria on the same COMET. The results of numeric simulations of gyrostat-satellites executing these transitions are presented. / Ph. D.