## 4 Analysis of the Crack Effect on the Vibrational Response of the Rotor System
### Crack Location Based on the Antiresonances of Higher Order Vibrational Responses  
Based on the analysis carried out previously, the following section dealt with a robust approach to meet the challenges of predictive maintenance and thus promotes reliable monitoring of the condition of rotating systems by identifying the presence of a small crack. The main objective is to be able to identify one crack in the early state and consequently to prevent equipment failures that lead to costly downtime and repairs. The methodology proposed is based on using the antiresonances of higher-order responses for small crack detection and identifying the positions of cracks in rotor systems. Firstly, a concise discussion on the linear equations of higher orders in the frequency domain is proposed. Through Equations (4), (6), and (9), it appears that the parametric terms due to the breathing behavior of the crack induce an additional internal force only at the crack location for all the harmonic components of the rotor response. As previously stated, the gravitational force is defined by Fourier series with only constant components, and the imbalance is exactly characterized by Fourier series with periodic components of the first order in the frequency domain. Therefore, it follows that the higher-order Fourier coefficients \(\mathbf{A}_{k}\) and \(\mathbf{B}_{k}\) (with \(k>1\)) are directly excited only via the parametric terms contained in the matrix \(\mathbf{\Lambda}_{c}\) due to the presence of the breathing crack. More precisely, it follows that the linear equations of the unknown coefficients \(\mathbf{A}_{k}\) and \(\mathbf{B}_{k}\) (with \(k>1\)) can be rewritten in the following form:  
\[\left[ \begin{array}{cc}\mathbf{K}-k^{2}\omega^{2}\mathbf{M}-\frac{1}{2} \mathbf{K}_{\mathrm{c}}&k\omega\mathbf{D}\\ -k\omega\mathbf{D}&\mathbf{K}-k^{2}\omega^{2}\mathbf{M}-\frac{1}{2}\mathbf{K }_{\mathrm{c}}\end{array} \right]\left[ \begin{array}{c}\mathbf{A}_{k}\\ \mathbf{B}_{k}\end{array} \right]=\frac{\mathbf{K}_{\mathrm{c}}}{4}\left[ \begin{array}{c}\mathbf{A}_{k-1}+\mathbf{A}_{k+1}\\ \mathbf{B}_{k-1}+\mathbf{B}_{k+1}\end{array} \right] \tag{10}\]  
with  
\[\mathrm{diag}(\mathbf{K}_{\mathrm{c}})=\delta_{i}\mathbf{K}_{crack} \tag{11}\]  
where \(\delta_{i}=1\) for the dof of the \(i\)th element (i.e., at the location of the crack in the shaft rotor) and otherwise null. Thus, the right side of Equation (10) clearly indicates that the excitation contribution for the \(k\)th harmonic components is due only to the presence of the breathing crack, and so the excitation forces for the higher orders are located only at the crack location. It is also necessary to note that these excitations depend indirectly on the contributions of the gravitational and unbalance forces through the calculations of lower orders and the interconnections between orders via matrix \(\mathbf{\Lambda}_{c}\) (see Equations (6) and (9)). Figure 4: Vertical amplitudes at 0.1 m from the left side of the rotor system with a breathing crack (case 3—\(\mu=0.1\) and \(L_{crack}=0.0625\) m from the left side) (blue = nonlinear response, black = order 1, red = order 2, magenta = order 3, green = order 4). These observations lead us to assume that the antiresonance behavior of higher-order responses could be very effective for detecting and identifying cracks in rotor systems. In the following, the numerical results associated with the three cases previously discussed in Section 4.1 are analyzed to consider the possibility of using the placement of antiresonance frequencies of higher orders to identify the crack location. Figures 5-7 illustrate the vertical displacements for all the rotor shaft positions for the 2 nd and 3 rd orders by considering the three cases of cracked rotor discussed previously. The contour lines indicate the isolines of the vertical amplitudes for the 2 nd and 3 rd orders. To facilitate the reader's comprehension, twenty contour lines are uniformly distributed between the minimum and maximum values of the amplitudes set in logarithmic scale.