## 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
This has the advantage of providing a clear visualization of the evolution of the low-level amplitudes as a function of the  
Consequently, the high amplitude peaks that correspond to the fundamental critical speeds and \(\frac{1}{j-n}\) (with \(n=0,\ldots,j-1,j>1\)) of the critical speeds are indicated by the red contour lines, and the antiresonance frequencies are indicated by the blue contour lines. In addition, the two dotted red horizontal lines indicate the rotor element on which the crack is located. The red circle corresponds to the identification carried out for the crack location based on the minimum of the antiresonance frequencies (note that this criterion for detecting the damage location has already been mentioned for the locations of small cracks on beam systems [42; 43] and will be explained in the following for application to rotor systems). In addition, the red cross corresponds to the identification of the crack location based on the minimum of the second antiresonance frequencies (if this identification is carried out successfully). In the following discussion, the notion of resonance frequency will generally be preferred to the exact notion of critical speed (i.e., undamped natural frequency of the rotor system due to imbalance), in order not to make the discussion too complex and to facilitate the understanding of the analysis of the results. Based on the numerical results and additional studies which have been conducted by the author (but for which the results are not shown for the sake of conciseness), several conclusions can be proposed regarding the behavior of the antiresonances and their use for detecting and identifying a breathing crack in rotating machines:  
* The antiresonance frequencies of higher orders can be defined as local characteristics of the system that depend on both the driving point and the measurement point. This observation is of primary importance because the driving point is only non-null at the crack position for higher orders, since the excitation undergone by the rotor for the \(n\times\) harmonic components (with \(n>1\)) is due to the breathing behavior of the crack (for memory, see the expressions of Equations (10) and (11)). * On the contrary, not surprisingly, the resonance frequencies and super-harmonic resonances (i.e., the fundamental critical speeds and \(\frac{1}{j-n}\) of the critical speeds with \(n=0,\ldots,j-1\), and \(j>1\)) are global quantities whose values do not depend on the measurement point. * The minimum value of the antiresonance frequencies of orders 2 and 3 is located at the crack location on the rotor system. It should be noted that this result is also valid for the \(n\)th orders with \(n>3\), which are not shown in this study for the sake of conciseness. * Focusing on the vicinity of the antiresonance frequencies of orders 2 and 3, and increasing the distance between the crack location (i.e., excitation points for the 2 nd and  
Figure 7: Case 3—Identification of the crack location based on the antiresonances of orders 2 and 3 (results are drawn for the vertical amplitudes in log-scale). 3 rd orders, and the shaft element with reduced stiffness) and the measurement point, will lead to higher antiresonance frequency values. * The resonance-antiresonance pattern is quite similar in both the vertical and horizontal directions. This can be explained by the fact that the only difference between the vertical and horizontal responses corresponds to the small dissymmetry in stiffness due to the breathing crack mechanism and the structural properties of the rotor supports of the rotor system under study. It should be noted that this commentary is based on the complete results of the different cases treated, which have been analyzed by the author but are not illustrated by figures in the study for the sake of brevity. * The resonance-antiresonance pattern is quite similar for cases where the crack size is different but the crack position is the same.