On the Analysis of Damped Gyroscopic Systems Using Lyapunov Direct Method

In this work, the stability properties of damped gyroscopic systems have been studied using Lyapunov direct method. These systems are generally stable because of the presence of gyroscopic effect. Conditions for determining the stability of the damped gyroscopic systems have been developed. Solution bounds of amplitude and velocity have been obtained for both homogeneous and inhomogeneous cases. An example is given to show how the stability conditions are applied to systems to determine its stability status.

The Influence of Circulatory Forces on the Stability of Undamped Gyroscopic Systems

In this paper, the effects of circulatory forces on undamped gyroscopic systems have been studied. The stability or otherwise of arising MGKN systems have been analysed using eigenvalue method and Lyapunov method. Bottema-Lakhadanov-Karapetyan theorem is also employed to further analyze the systems. Examples are given to illustrate the use of these methods in determining the stability or otherwise of undamped gyroscopic circulatory systems.

Magneto-Aeroelastic Internal Resonances of a Rotating Circular Plate Based on Gyroscopic Systems Decoupling

In this paper, a principal and 3:1 internal resonance of an edge-clamped conductive circular plate rotating in air-magnetic environment is investigated, where the electromagnetic force expressions and a simple empirical aerodynamic model are used in modelling. Based on the transverse displacement assumption with a combination of two degenerate linearized modes, the 2 degree of freedom (2-DOF) magneto-aeroelastic asymmetric nonlinear gyroscopic systems are derived by utilizing the Galerkin approach. The method of multiple scales and the solvable condition for the coefficient matrix of the gyroscopic systems is employed to decouple the 2-DOF nonlinear gyroscopic systems and achieve the nonlinear modulation equations of the steady state responses. The numerical results for the relationship between two eigenfrequencies verify the existence of 3:1 internal resonances of the circular plate rotating in air-magnetic field. In addition, the resonance amplitude varying with detuning parameters and excitation amplitudes are plotted under principal and 3:1 internal resonances, respectively. It is found that the system may lose stability generated by a Hopf bifurcation, which may finally evolve into chaotic state through period-doubling bifurcation of intermittent periodic responses.

A Novel Type of Bi-Gyroscopic System Undergoing Both Rotating and Spinning Motions

In order to explore the influence of combined gyroscopic coupling effect on the gyroscopic system, the dynamics of a beam undergoing both rotating and spinning motions as a bi-gyroscopic system is studied. The natural frequencies, modes, and stability of such a bi-gyroscopic system have been studied by the standard eigenvalue problems. The bifurcation series of frequencies and corresponding modal motions have been presented to show the gyroscopically coupled motions. The complex modes of the proposed bi-gyroscopic systems, such as whirling motions and in-plane reeling motions, have been illustrated.