## 4 Analysis of the Crack Effect on the Vibrational Response of the Rotor System
### Nonlinear Vibration and Appearance of Harmonic Components  
Considering the previous equations provided in Section 3, it is obvious that the vibrational response of the rotor system will be composed by only the constant and first harmonic components (i.e., \(\mathbf{\Lambda}_{0}\), \(\mathbf{\Lambda}_{1}\), and \(\mathbf{B}_{1}\)) in the absence of a breathing crack. Indeed, if the rotor is healthy, we have \(\mathbf{\Lambda}_{c}=\mathbf{0}\), and thus the excitation comes from the gravitational and unbalance forces (see Equation (7)). Conversely, due to the presence of a breathing crack (i.e., \(\mathbf{\Lambda}_{c}\
eq\mathbf{0}\)), the vibrational response of the cracked rotor will be composed by not only the constant term and the first harmonic components, but also by the components of higher orders (i.e., \(\mathbf{\Lambda}_{k}\) and \(\mathbf{B}_{k}\) with \(k>1\)). Moreover, by considering the expression of \(\mathbf{\Lambda}_{c}\), it appears that the presence of a breathing crack leads to a direct mutual dependence between the static deflection \(\mathbf{\Lambda}_{0}\) and contributions of the first order (i.e., \(\mathbf{\Lambda}_{1}\)), leading indirectly to a contribution of the static term on the higher orders due to the expression of \(\mathbf{\Lambda}_{c}\) (i.e., see the direct interactions between the coefficients \((\mathbf{\Lambda}_{k-1},\mathbf{B}_{k-1})\), \((\mathbf{\Lambda}_{k},\mathbf{B}_{k})\), and \((\mathbf{\Lambda}_{k+1},\mathbf{B}_{k+1})\) for \(k>1\) in Equation (9)). In practice, this results in the fact that the presence of a breathing crack is characterized by the appearance of super harmonics of the \(j\)th order (with \(j>1\)), leading to amplitude peaks during rotation speeds equal to approximately \(\frac{1}{j-n}\) (with \(n=0,\ldots,j-1\)) of the critical speeds of the dynamic response of the cracked rotor [16; 22]. To briefly illustrate these well-known analyses of fault detection by examining the non-linear vibration signature, three case studies are proposed with the crack parameters specified in Table 2. These studies correspond to a case with a deep crack (i.e., case 1), and the other two cases consider a small crack located at different positions on the rotor shaft (i.e., cases 2 and 3). Figures 2-4 show the non-linear response (i.e., the global response and the harmonic components \(1\times\), \(2\times\), \(3\times\), and \(4\times\)) for the cracked rotor in each case. It is clear that the presence of a breathing crack on the rotor results in the appearance of harmonics of the \(j\)th order with amplitude peaks when passing \(\frac{1}{j}\) critical speeds (see more specifically the marks \((1,4,5)\), \((2,6,7)\), and \((3,8,9)\) for \(4\times\), \(3\times\), and \(2\times\) harmonic components, respectively). To better understand the proposed results and discussion, Table 3 gives the critical speeds of the cracked rotor system in the frequency range \([0;300]\) Hz. Moreover, the marks \((10,11,12)\) also suggest the presence of resonance peaks for each \(j\times\) harmonic component at the main critical speeds. By comparing the results obtained for the three case studies, it can also be concluded that:  
* If the size of the crack is large, the \(j\times\) harmonic responses contribute significantly to the overall dynamic response of the rotor system, resulting in the presence of peaks when passing through \(\frac{1}{j}\) critical speeds with amplitudes greater than the first-order response (see, for example, the marks \((2,3,8)\) in Figure 2). However, in the practical context of a rotor system with a small crack, the presence of the crack is not apparent when inspecting the overall non-linear response alone, and it is necessary to specifically examine the responses of the \(j\times\) harmonic contributions (see Figures 3 and 4). Thus, the evolution of the \(j\times\) harmonic components offers a positive and useful way to diagnose the presence and potential propagation of cracks in a practical way in both preventive and predictive maintenance.