Some recent advances in the dynamic analysis of parametrically excited continuous rotor systems

Dynamic analysis of parametrically excited rotors is a research field of great interest and practical importance, since instability and resonant behavior can cause issues ranging from anomalous noise and wear to catastrophic failures. An overview is presented of some advances recently proposed by the authors on the dynamic analysis of parametrically excited continuous rotor systems, including results regarding stability analysis (parametric resonances) and preliminary insights into resonant behavior in the asymptotically stable domain, due to unbalance (external resonances). An axisymmetric shaft described by a spinning Timoshenko beam is studied, loaded by oscillating axial end thrust and twisting moment, carrying additional inertial elements like discs. Both isotropic and anisotropic supports are considered, as well as gyroscopic effects and different kinds of damping distributions (both external and internal), which represents a model including all the general features of slender rotors which are relevant for their dynamic analysis. Stability is studied after discretization of the equations of motion into a set of coupled ordinary differential Mathieu-Hill equations. Stability maps in the form of Ince-Strutt diagrams are discussed to highlight the occurrence of simple and combination critical solutions, as well as the influence of angular speed, damping, and anisotropy in the supports. Steady-state response is studied in the asymptotically stable domain under the effect of unbalance, yielding an additional external harmonic load, acting on flexural deflection. As a first insight into this problem, to study the effects of angular speed independently to variations of natural frequencies and to facilitate decoupling of the equations of motion, the Timoshenko model is simplified into the Euler-Bernoulli model, neglecting the gyroscopic effects, additional discs, anisotropy in the supports and twisting moment at the ends of the shaft.