The 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. This study is focused on the effects of unbalance on a parametrically excited rotor operating in the asymptotically stable domain, a research topic which in the scientific literature has not been investigated. Rotor unbalance causes an additional harmonic load acting on flexural deflection, influencing the frequency response together with the parametric excitation, yielding additional combination external resonances. As a first insight into this problem, to study the effects of angular speed independently of variations of the natural frequencies, and to facilitate decoupling of the equations of motion, a simplified model of a distributed-parameter slender rotor is considered, consisting of a homogeneous Euler-Bernoulli beam with circular section, rotating at constant angular speed about its longitudinal axis on isotropic supports. It is affected by unbalance and loaded by an axial end thrust, assumed to have a harmonic time-dependent component. The steady-state response is studied after decoupling the equations of motion, reducing the problem to the analysis of a non-homogeneous single-degree-of-freedom damped Mathieu equation.

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