Instability region classification and control of periodically axially loaded rotor

Parametric resonance can amplify vibrations to levels far beyond what the rotor system would experience under normal operating conditions. This results in excessive wear and tear, reduced operational lifespan, increased maintenance requirements, and even catastrophic failures. Therefore, it is urgently needed to figure out the instability regions of parametrically excited rotors and develop effective methods to control or regulate them. In this paper, our focus is on a damped single-span rotor with periodic axial load, a representative example of a parametrically excited system. The method of multiple scales is applied to solve the governing equations, and then the analytical instability regions' boundaries and forced vibration responses are derived. Based on these analytical results, the instability region classification is carried out. The absence of certain instability regions is rigorously demonstrated. It is found that the primary difference type does not exist, no matter whether the rotor system has isotropic or orthotropic supporting structures. If the supporting structures are isotropic, the secondary difference type also does not exist. In this case as well, the primary and secondary sum types, which are exclusively associated with either forward or backward whirling frequencies, do not manifest. After the instability region classification, the feasibility analysis of introducing piezoelectric shunt damping to stabilize the rotor system is proceeded. In the numerical simulation, it is seen that as the electrical resonant frequency is tuned to a targeted critical speed, the starting points (critical dynamic load coefficients) of instability regions associated with that critical speed are raised with the increase of piezoelectric shunt damping. This means a higher threshold of triggering the uncontrolled parametric resonance.