## 1 Introduction  
Rotors are one of the most important components of rotating machines, which are widely used in many engineering fields. Cracks in rotors are a very dangerous kind of fault that may lead to a sudden and catastrophic failure of a rotating machine. Therefore it is of vital significance to monitor and detect a crack, in order to reduce maintenance cost and avoid failure of a rotating machine. In addition, crack parameter identification can provide a foundation for life prediction of a cracked rotor. A crack in a rotor introduces an additional local flexibility and reduces the overall stiffness. Usually, a fatigue crack in a rotor is a breathing type, which makes the stiffness of such a rotor changing with time and causes the system to become nonlinear. Because of these, a cracked rotor possesses complex dynamic characteristics. Much work on the dynamic behavior of cracked rotors has been performed to identify cracks in rotors, and a number of reviews on this topic [1; 2; 3; 4; 5; 6; 7] have been published. From the literature, vibration-based crack identification methods can be broadly classified as model-basedmethods, signal-based methods, modal parameters, and their derivatives-based methods and pattern recognition-based methods. The approach based on equivalent crack forces is well known as a model-based method for crack identification in rotors, considering the effects of cracks as equivalent forces applied in the intact systems. It has been utilized by [PERSON] [8] to identify the crack locations and depths of a rotor with two cracks. [PERSON] et al. [9] further used the approach for cracks identification of a relatively large experiment rig. There are also some other model-based methods. [PERSON] et al. [10] presented a model-based method based on a proportional-integral observer to identify a crack in an operating rotor. [PERSON] et al. [11] proposed a model-based method to identify a switching crack in a four-degrees-of-freedom Jeffcott rotor utilizing the vibration signal in conjunction with the controller current of the active magnetic bearing. Model-based methods were adopted by [PERSON] et al. [12], and [PERSON] et al. [13] to minimize the difference between real outputs and model outputs to determine the location and depth of a crack in a rotor-bearing system with genetic algorithm (GA). Extended Finite Element methods were developed for cracks and flaws detection and quantification by [PERSON] et al. [14] and [PERSON] et al. [15]. [PERSON] et al. [16] took nonlinear combinational resonances induced by cracks under external diagnostic forces at certain frequencies as an objective function of a differential evolution optimization method to determine the crack location and depth by minimizing the difference between the measured and modelled rotor systems. Short Time Fourier Transform (STFT) [17; 18], Wavelet Transform (WT) [19; 20; 21] and Hilbert-Huang Transform (HHT) [22; 23] as signal-based methods, were used to detect a crack from acceleration response of a rotor by [PERSON] and [PERSON] [24]. Approximated Entropy based on information theory was used by [PERSON] and [PERSON] [25] to detect rotor cracks considering the interference of misalignment. Full-spectrum method was used to detect a breathing crack in a rotor by [PERSON] and [PERSON] [26]. [PERSON] et al. [27] utilized the modulation between crack breathing and vibration response, by using squeezing time-frequency transform (STFT) to extract the periodic oscillation in instantaneous frequency for breathing crack detection. [PERSON] et al. [28] proposed a phase-based spectrum analysis method to extract weak harmonics induced by cracks, to help realize crack detection. For modal parameters and their derivatives based methods, empirical mode decomposition (EMD) and Laplace wavelet finite element combined method was proposed by [PERSON] et al. [29], to obtain the accurate modal parameters to identify a crack in a rotor.