## 5 Conclusions  
The present work studied the primary characteristics of the nonlinear responses resulting from the introduction of a transverse breathing crack into a rotor system. Although the \(2\times\) and \(3\times\) harmonic components of the system's response can serve as target observations for the monitoring system, this study also demonstrated that the antiresonance behavior of higher orders can be considered as one common and robust indicator to detect the presence of damage and the location of a small crack. This paper highlighted that using and tracking the nonlinear signature has many advantages for structural health monitoring in rotordynamics and detecting and identifying damage at an early stage during operational conditions. In addition, this work points to several interesting perspectives for future research, as described in the following:  
* It was observed through this numerical study that the location of the crack corresponds to the minimum values of the antiresonance frequencies of the \(n\)th order (with \(n>1\)). However, no formal proof was obtained on the subject in the present paper. To the author's knowledge, there is no theoretical study in the field of rotordynamics on the sensitivity of antiresonance frequencies of the \(n\)th order to structural changes. Additionally, no theoretical study has yet demonstrated that damage location can be effectively found by tracking the antiresonances of the \(n\)th order. It would therefore be interesting to conduct theoretical work in this direction in order to generalize the idea. It can also be pointed out that there is very little research aimed at extending the general higher-order nonlinear analysis of FRF to the detection and damage assessment of general structural systems with breathing cracks [48; 49]. Solid theoretical works on the influences of breathing cracks on the higher-order responses of nonlinear structural systems are of interest due to the strong potential for applications of HOFRF (higher-order vibrational responses) to damage identification and assessment in real structural systems (rotating machines). * It would be interesting to conduct experiments to validate the relevance of the approach proposed to detect crack locations by identifying the positioning of the antiresonance frequencies of higher orders. In a more practical context, where there are only a limited number of sensors and therefore only a limited number of measurement points on the rotor, the question arises as to how to optimally position these sensors to locate the damage. If increasing the distance between the location of the crack and the measurement point leads to an increase in the value of higher orders, then antiresonance frequencies seem to be factors favoring the possibility of robust defect detection in a practical case (by successive adaptation of the placement of the sensors for example). This deserves to be validated by extensive experimental studies. * Many faults that reduce the lifetime of rotating machinery exist, and they significantly affect the dynamics of rotor systems. It would be interesting to better understand the possibility of damage detection for rotor systems with features such as misalignment, bows, and asymmetric shafts. These faults can also generate nonlinear responses, so the resonance-antiresonance pattern of higher orders should therefore be used with caution to avoid irrelevant damage detection on complex industrial rotating machinery. * Even if multi-crack detection in the case of beam-like structures using the antiresonances locus of HOFRFs has been previously investigated by [PERSON] [43], this problem remains completely open and deserves further study for robust and reliable detection of multiple small cracks in rotating systems. This research received no external funding. Not applicable. The author declares no conflict of interest. The following abbreviations are used in this manuscript: