## 4 Analysis of the Crack Effect on the Vibrational Response of the Rotor System
### Nonlinear Vibration and Appearance of Harmonic Components
* Although--due to the presence of one crack--variations in the frequencies and critical speeds of a rotor system exist from a theoretical point of view, these changes are often too small to be considered as reliable indicators for the early detection of defects. This is highlighted in Tables 3 and 4, which give the evolution of the critical speeds and the natural frequencies at rest in the case of a healthy rotor and the three cracked rotor configurations proposed in the present study. It should be noted that the calculations of the natural frequencies at rest are given for an open transverse crack, as illustrated in Figure 1. It has also been shown in previous studies [3, 40, 47] that the detection and identification of small defects in the presence of uncertainties (uncertainties in the vibration measurements or uncertainties in the physical model used) are often not possible if the analysis criteria are based exclusively on these evolutions of natural frequencies or critical speeds. * By comparing the vibration responses of several case studies for rotor systems with small cracks positioned at different locations (i.e., cases 2 and 3 for the present study), it appears that the evolutions of the first-order response (for a chosen vertical or horizontal location on the rotor) over the frequency range are very close. More precisely, the positions of the antiresonances are identical, and the evolution of the vibratory amplitudes between the resonance and antiresonance peaks are similar (see, for example, Figures 3 and 4). It was also verified that this statement is valid for all vertical and horizontal amplitudes over the rotor system. Indeed, the only modification between these various cases comes from the position of the crack, which does not greatly modify the \(1\times\) vibrational response of the rotor system because the latter is mainly governed by the imbalance in the case of a rotor system with a small crack. On the contrary, the evolutions of the higher-order responses and the value of the antiresonance frequencies are dependent on the position of the crack (see Figures 3 and 4 for the \(2\times\), \(3\times\) and \(4\times\) harmonic responses). Consequently, a nondestructive detection technique based on the tracking of antiresonances should be a reliable and effective way for monitoring and identifying the locations of cracks in rotor systems when ensuring their structural health. \begin{table}
\begin{tabular}{c c c} \hline Case & Crack Depth \(\mu\) & Crack Location \(L_{crack}\) \\ \hline
1 & 1 & 0.3625 m \\
2 & 0.1 & 0.2625 m \\
3 & 0.1 & 0.0625 m \\ \hline \end{tabular}
\end{table} 
Table 2: Sets of parameters for the crack element. \begin{table}
\begin{tabular}{c c c c c} \hline  & \(f_{1}\) (Hz) & \(f_{2}\) (Hz) & \(f_{3}\) (Hz) & \(f_{4}\) (Hz) \\ \hline Healthy rotor & 49.0 & 49.2 & 252.3 & 266.1 \\ Case 1 & 48.4 & 49.1 & 249.4 & 264.6 \\ Case 2 & 49.0 & 49.2 & 252.3 & 266.1 \\ Case 3 & 49.0 & 49.2 & 252.2 & 266.1 \\ \hline \end{tabular}
\end{table} 
Table 3: Critical speeds of the healthy or cracked rotor system. \begin{table}
\begin{tabular}{c c c c c} \hline  & \(f_{1}\) (Hz) & \(f_{2}\) (Hz) & \(f_{3}\) (Hz) & \(f_{4}\) (Hz) \\ \hline Healthy rotor & 49.0 & 49.2 & 256.3 & 262.3 \\ Case 1 & 47.7 & 49.0 & 248.0 & 260.8 \\ Case 2 & 48.9 & 49.1 & 256.3 & 262.3 \\ Case 3 & 49.0 & 49.2 & 256.2 & 262.2 \\ \hline \end{tabular}
\end{table} 
Table 4: Natural frequencies of the healthy or cracked rotor system at rest. Figure 3: Vertical amplitudes at 0.1 m from the left side of the rotor system with a breathing crack (case 2—\(\mu=0.1\) and \(L_{crack}=0.2625\) m from the left side) (blue = nonlinear response, black = order 1, red = order 2, magenta = order 3, green = order 4). Figure 2: Vertical amplitudes at 0.1 m from the left side of the rotor system with a breathing crack (case 1—\(\mu=1\) and \(L_{crack}=0.3625\) m from the left side) (blue = nonlinear response, black = order 1, red = order 2, magenta = order 3, green = order 4).