Stability analysis of parametrically excited gyroscopic systems

This study is aimed at analyzing and clarifying gyroscopic effects on the stability of parametrically excited rotor systems, a topic which in the literature is not sufficiently investigated. As case–study giving rise to a set of coupled differential Mathieu–Hill equations with both gyroscopic and damping terms, a balanced shaft is considered, modelled as a spinning Timoshenko beam loaded by oscillating axial end thrust and twisting moment, with possibility of carrying additional inertial elements. After discretization of the equations of motion into a set of coupled ordinary differential Mathieu-Hill equations, stability of Floquet solutions is studied via eigenproblem formulation, obtained by applying the harmonic balance method. A numerical algorithm is then developed for computing global stability thresholds in presence of both gyroscopic and damping terms, aimed at reducing the computational load. Finally, the influence on stability of the main characteristic parameters of the shaft is analyzed with respect to frequency and amplitude of the external loads on stability charts in the form of Ince-Strutt diagrams. As a result, it has been demonstrated that gyroscopic terms produce substantial differences in both critical solutions and stability thresholds: the former are generally non-periodic limited-amplitude functions, and modifications induced on stability thresholds consist of shifts and merging of unstable regions, depending on the separation of natural frequencies into pairs of forward and backward values.