The regular and chaotic motions of a symmetric heavy gyroscope with harmonic excitation

Abstract

The non-linear motion of a symmetric gyro mounted on a vibrating base is investigated, with particular emphasis on its long-term dynamic behaviour for a wide range of parameters. A single modal equation is used to analyze the qualitative behavior of the system. External disturbance appears as vertical harmonic motion of the support point and linear damping is assumed. The complete equation of motion is a non-linear non-autonomous one. The stability of the system has been studied by damped Mathieu equation theory and the Liapunov direct method. As the system is subjected to external disturbance, the Melnikov method is used to show the existence of chaotic motion. Finally, the bifurcation of the parameter dependent system is studied numerically. The time evolutions of the non-linear dynamical system responses are described in phase portraits via the Poincare map technique. The occurrence and the nature of chaotic attractors are verified by evaluating Liapunov exponents and average power spectra. The effect of the gyroscope's spinning speed is also studied, and it is shown that the gyroscope's spin velocity oz has a significant effect on the dynamic behavior of the motion. (C) 1996 Academic Press Limited