Frequency analysis of dynamic systems loaded by both parametric and external excitations, with application to rotor dynamics

Abstract The analysis of critical conditions of dynamic systems under combined parametric and external excitations is a research field of theoretical interest with important applications in several branches of physics and engineering. Parametric excitation occurs in the presence of time-varying coefficients in the dynamic equilibrium equations, and an extensive body of literature exists in the case of time-periodic coefficients. Dynamic systems under the effects of both parametric and external excitations are worthy of great attention as well, since they represent a rather common occurrence. This study is focused on the development of a theoretical method of analysis of general validity, however applied to rotor dynamics. Parametrically excited rotors constitute a research field in which theoretical developments are still needed to fully understand problems of practical relevance such as instability and resonant behaviour, which can cause issues ranging from anomalous noise and wear to catastrophic failures in several classes of machines. The additional effect of unbalance (always present in real rotors) causes external harmonic loads acting on flexural deflection, which can affect the dynamic responses of the machines in their stable operational fields. The aim of this study, therefore, is the frequency analysis of dynamic systems under the effects of both parametric and external excitations in stable working conditions. Basic models of distributed-parameter slender rotors are considered to facilitate decoupling of the equations of motion, and the analysis of steady-state responses, including both cases in which natural frequencies are independent of, and dependent on, angular speed, the latter case due to gyroscopic effect. Steady-state responses are studied by reducing the problem to the analysis of non-homogeneous single-degree-of-freedom damped Mathieu-Hill equations. It is shown that, even in the linear problem, a theoretically infinite sequence of resonances is related to each modal coordinate of the decoupled system, due to the combination of parametric and external excitation frequencies, and that, in the presence of parametric excitation, flexural critical speeds can occur at lower values than in the same, non-parametrically excited rotors.