## 1 Introduction  
Fault detection is an important branch in rotor dynamics and has attracted extensive attention from many scholars. A lot of research on dynamic characteristics analysis and uncertainty quantification of faulty rotor systems has been conducted by many domestic and international scholars. The main purpose is to deepen understanding of faulty rotor systems and provide a theoretical basis and technical support for rotor fault detection in different environments. Cracks are common faults in rotor systems. If a crack is not found in time, it will expand deeper, resulting in excessive rotor vibration, affecting the safe operation of the unit, and it may cause casualties in serious cases. Therefore, it is necessary to study crack rotor fault detection method. The research on crack fault detection methods is mainly divided into numerical and experimental studies. It is an economical and feasible idea to establish a numerical model to study crack fault detection methods based on numerical signals. In 2002, [PERSON] et al. [1] proposed a multi-rotor fault detection method based on a numerical model. The fault parameters were detected by the least square method for multi-order harmonic components, and a residual map was introduced to strengthen the visualization process. In 2004, [PERSON] et al. [2] proposed a numerical model-based fault detection approach for cracked rotors. The modal expansion method was used to map the response at the finite measuring point to the whole model. Then, the equivalent fault force and torque were calculated. The fault parameters were detected based on the least square method. In 2004, [PERSON] et al. [3] used a method based on a numerical model to diagnose the location and depth of rotor cracks and used the least square method to achieve the best matching of fault parameters in the time domain. The diagnostic effect of this method was verified by numerical simulation signals. The influence of the number of measuring points, bearing calibration deviation and model error on the diagnostic accuracy was also studied. Subsequently, [PERSON] et al. [4] detected the location and depth of double crack faults on the rotor by numerical signals. In the same year, [PERSON] et al. [5] adopted the wavelet finite element method to establish three curves with the first three vibration frequencies of the crack shaft as input, and the crack location and depth were diagnosed according to the intersection of the curves. In 2008, [PERSON] et al. [6] proposed a crack diagnosis technology based on an amplitude deviation curve by extracting the axial amplitude and slope of the cracked rotor and completed the crack location identification in a double-cracked rotor. In 2013, [PERSON] et al. [7] used numerical methods to analyze the response characteristics of cracked rotors and pointed out that the fault parameters of rotor cracks could be estimated by using the second harmonic components of the system. Finally, the estimation values of crack location and depth were given. In 2014, [PERSON] et al. [8] proposed a rotor crack fault detection method based on orthogonal natural frequency. In this method, the extracted features have sharp notch peaks at the crack location rather than circular peaks at the crack location. According to different peak shapes, crack fault detection can be realized. In 2016, [PERSON] et al. [9] proposed a crack fault detection method based on numerical signals by combining support vector machine and wavelet analysis. The research results show that this method has stronger robustness and can effectively deal with interference and noise. In 2020, [PERSON] et al. [10] generated the vibration signals of the rotor system under unbalanced force and crack fault by a numerical method, built an artificial neural network, and realized the classification of unbalanced faults and crack faults. In 2020, [PERSON] et al. [11] used the finite element method (FEM) to simulate the transient response of a slant cracked rotor during shutdown.