## 3 Investigation of the Effects of Crack Parameters on Super-Harmonic Characteristics  
After validating the model of the cracked rotor, to select the most sensitive features to establish the Kriging surrogate model for crack identification, the effects of crack parameters on the super-harmonic components at 1/3 and 1/2 critical speed of the rotor were investigated.  
Responses at node 12 were obtained by the established finite element model. The crack location was made to vary from element 10 to 52, which covers the part of the shaft between the two bearings, and crack depth ranges from 0 to \(0.45D\) with step size of \(0.05D\). The numerical simulation results are illustrated in Figure 5 below.  
Figure 4: Whirl orbits of the cracked rotor in the neighborhood of 1/3 (top row) of and 1/2 (bottom row) of the critical rotating speed.  
From Figure 5a,c above, one can see that the amplitudes of 2\(\times\) component at 1/3 and 1/2 critical speeds increased monotonously with the increase of crack depths, and they were also distinct between different crack locations. In contrast, the amplitudes of 3\(\times\) component at 1/3 and 1/2 critical speeds were not monotonous with the increase of crack depths, which can be seen from Figure 5b,d. If the features are monotonous with a crack parameter, then it will be easier to distinct a similar feature value, which will be helpful to reduce the identification errors. Therefore, in view of crack quantification, 2\(\times\) components at 1/3 and 1/2 critical speed are better features for crack parameters identification than the 3\(\times\) ones.  
As the purpose of this paper is to propose a crack identification method for rotors and investigate the feasibility of the method based on a Kriging surrogate model, to simplify the problem, and reduce the demand of input features, only 2\(\times\) component at 1/3 critical speed was utilized for crack identification. However, it is difficult to identify a crack from the rotor response at just one measure point, because such a solution may not be unique. To tackle the problem, more measurement points are needed. Considering the difficulties of setting up sensors for rotating rotors, two measurement points were used to determine the crack parameters. Two measurement points should be enough, when the model that maps crack parameters to amplitudes of 2\(\times\) components from two measurement points is accurate enough. In view of the accessibility of sensors in real applications, the two measurement points A and B were chosen to be node 12 and node 50 respectively in the numerical experiment, which were near the two bearings. Here, it can be said that it is possible to identify the crack in rotors by using the 2\(\times\) components from two measurement points. However, the cost of experiment or computation is another important issue which will affect the crack identification efficiency. As the nonlinear breathing crack modelled in this work was response-dependent, computation cost of the simplified beam model was still heavy. Therefore, it is not efficient to utilize the beam model directly combined with an  
Figure 5: Normalized amplitudes of super-harmonic components with different crack locations and depths (**a**) 2\(\times\) at 1/3 critical rotating speed; (**b**) 3\(\times\) at 1/3 critical rotating speed; (**c**) 2\(\times\) at 1/2 critical rotating speed; (**d**) 3\(\times\) at 1/2 critical rotating speed.  
optimization method, as the computationally expensive FE models will be called repeatedly during the optimization process. To reduce the cost of experiment or computation in real applications and to avoid the repeated analysis of computationally expensive FE models during the optimization process, a Kriging surrogate model was adopted to establish the accurate relationship with fewer samples, which will be elaborated in the following sections.