Stability and Chaotic Motions of a Symmetric Heavy Gyroscope.

The nonlinear motion of a symmetric heavy gyro mounted on a vibrating base, particularly its long-term dynamic behavior for a wide range of parameters, is investigated. The system exhibits both regular and chaotic motions. Two typical routes to chaos, namely through period-doubling and intermittency, are found in this study. The method of multiple scales is used to analyze the responses and to determine the stability of the trivial fixed point for the system excited by single harmonic force. The stability of the system excited by multiple harmonic forces has been studied by Lyapunov's direct method. As the system is subjected to external disturbance, the Melnikov method is used to show the existence of chaotic motion. The effect of the number of forcing frequencies on the chaotic regions is also established. The numerical results, in which the existence of chaos is demonstrated, are presented in the form of bifurcation diagrams, phase portraits, Poincare maps, and Lyapunov exponents. The effect of the gyroscope's spinning speed is also studied. It is found that increasing the gyroscope's spin speed is the simplest method to overcome the chaotic motion for this system.