“…The natural characteristics in this paper specifically refer to the critical speeds and mode shapes of rotor systems, which are the fundamental points to be taken care of in designing and dynamic analysis. Further, the most concerned is the vibration responses of the systems under imbalance excitations and they are analyzed from different perspectives in the time domain and frequency domain as well as their sensitivities [208]. In practice, the vibrations of a rotor system should not exceed the allowed maximum amplitude to (b) (a) Fig.…”