Control of chaotic instabilities in a spinning spacecraft with dissipation using Lyapunov''s method

Control of chaotic instability in a simplified model of a spinning spacecraft with dissipation is achieved using an algorithm derived using Lyapunov's second method. The control method is implemented on a realistic spacecraft parameter configuration which has been found to exhibit chaotic instability for a range of forcing amplitudes and frequencies when a sinusoidally varying torque is applied to the spacecraft. Such a torque, may arise in practice from an unbalanced rotor or from vibrations in appendages. Numerical simulations are performed and the results are studied by means of time history, phase space, Poincaré map, Lyapunov characteristic exponents and bifurcation diagrams.     

Nonlinear instabilities in a vibratory gyroscope subjected to angular speed fluctuations

Nonlinear instabilities in a single-axis vibrating MEMS gyroscope that is subjected to periodic fluctuations in input angular rates are investigated. For the purpose of characterizing the bifurcation behavior associated with the steady-state, when the angular rate input is subject to small intensity periodic fluctuations, dynamic behavior of periodically perturbed nonlinear gyroscopic systems is studied in detail. An asymptotic approach based on the method of averaging has been employed for this purpose, and closed-form conditions for the frequency response due to parametric resonances have been obtained. This behavior has been illustrated via amplitude-frequency plots.     

Chaotic Motion in a Spinning Spacecraft with Circumferential Nutational Damper

The dynamics of a simplified model of a spinning spacecraft with a circumferential nutational damper is investigated using numerical simulations for nonlinear phenomena. A realistic spacecraft parameter configuration is investigated and is found to exhibit chaotic motion when a sinusoidally varying torque is applied to the spacecraft for a range of forcing amplitude and frequency. Such a torque, in practice, may arise in the platform of a dual-spin spacecraft under malfunction of the control system or from an unbalanced rotor or from vibrations in appendages. The equations of motion of the model are derived with Lagrange's equations using a generalisation of the kinetic energy equation and a linear stability analysis is given. Numerical simulations for satellite parameters are performed and the system is found to exhibit chaotic motion when a sinusoidally varying torque is applied to the spacecraft for a range of forcing amplitude and frequency. The motion is studied by means of time history, phase space, frequency spectrum, Poincaré map, Lyapunov characteristic exponents and Correlation Dimension. For sufficiently large values of torque amplitude, the behaviour of the system was found to have much in common with a two well potential problem such as a Duffing oscillator. Evidence is also presented, indicating that the onset of chaotic motion was characterised by period doubling as well as intermittency.     

Control of Chaotic Motion in a Spinning Spacecraft with a Circumferential Nutational Damper

Control of chaotic vibrations in a simplified model of a spinning spacecraft with a circumferential nutational damper is achieved using two techniques. The control methods are implemented on a realistic spacecraft parameter configuration which has been found to exhibit chaotic instability when a sinusoidally varying torque is applied to the spacecraft for a range of forcing amplitude and frequency. Such a torque, in practice, may arise in the platform of a dual-spin spacecraft under malfunction of the control system or from an unbalanced rotor or from vibrations in appendages. Chaotic instabilities arising from these torques could introduce uncertainties and irregularities into a spacecraft's attitude and consequently could have disastrous affects on its operation. The two control methods, recursive proportional feedback (RPF) and continuous delayed feedback, are recently developed techniques for control of chaotic motion in dynamical systems. Each technique is outlined and the effectiveness of the two strategies in controlling chaotic motion exhibited by the present system is compared and contrasted. Numerical simulations are performed and the results are studied by means of time history, phase space, Poincaré map, Lyapunov characteristic exponents and bifurcation diagrams.