Stability analysis of parametrically excited isotropic rotors on anisotropic supports

The stability of slender rotors which are parametrically excited by external loads is studied under the effects of an-isotropic supports. An axisymmetric shaft is considered, described by scaling a spinning Timoshenko beam, carrying a disk and loaded by oscillating axial end thrust and twisting moment. The supports are modelled including ‘principal’ stiffness and damping distributions, able to modify the closely separated modes generated by angular speed (with ‘splitting’ of eigenfrequencies in forward and backward pairs). The proposed model includes all the general features of slender rotors which are relevant for this kind of stability analysis, gyroscopic effects comprised. Stability is studied after discretization of the equations of motion into a set of coupled ordinary differential Mathieu-Hill equations. The influence of angular speed combined with anisotropy in the supports is analyzed with respect to frequency and amplitude of the external loads on stability charts in the form of Ince-Strutt diagrams. The occurrence of different kinds of critical solutions, simple and combination, is investigated, highlighting their dependency on both the degree of anisotropy in the supports and angular speed.