## 1 Introduction
Since it is not possible to give an exhaustive list of all the work of interest carried out on structural health monitoring and damage detection, the reader can refer to the various state-of-the-art papers that have been proposed in the past for crack detection based on the measurement of linear and nonlinear vibrations [27; 28; 29; 30; 31; 32; 33; 34]. More recently, some studies have shown that the presence of uncertainties might cause classical linear and nonlinear approaches to be less robust for detecting small cracks [35; 36; 37; 38; 39; 40; 41]. In order to overcome this difficulty, recent works have suggested looking at the zeros of the higher-order frequency response function (HOFRF), i.e., the antiresonance frequencies of HOFRF, for the detection and location of small cracks in beam systems [42; 43]. It can be noted that the use of antiresonances from linear measurements has rarely been considered in structural health monitoring [4; 5; 6; 7], whereas the antiresonance frequencies of measured FRFs provide useful information on the dynamic properties of a mechanical structure [44] and can be used for robust model updating [45; 46]. For linear systems, it is well known that the magnitude of the FRF is characterized by the resonance-antiresonance pattern in the frequency domain and the antiresonance behavior due to the relationship between the driving and measurement points of the mechanical system. Indeed, the resonance frequencies can be defined as global quantities whose values do not depend on the given driving point and chosen measurement point. On the contrary, the antiresonance frequencies are typical of each FRF and depend on the location of both the driving force and the measurement point. One of the most well-known results regarding resonance-antiresonance behavior is that the resonances and antiresonances alternate continuously only for FRFs where the excitation points and the measurement points coincide. In this context, this study proposes a preliminary analysis of the supplementary information that can be obtained from the antiresonances of the higher-order responses for structures with local nonlinearities. More specifically, it considers the potential of using the locations of these antiresonances for the detection and identification of damage in rotating machines. Indeed, when considering the nonlinear response of a rotor system with a breathing crack, antiresonances of higher orders could become an attractive alternative in structural damage assessments. Due to the fact that antiresonances are very sensitive to small structural changes, the use of antiresonances of \(n\)th orders (with \(n>1\)) seems to potentially meet several requirements for early damage detection in rotating machines. In this paper, the aim is to achieve this objective by proposing a complete numerical study to examine the nonlinear responses and the tracking of the antiresonance frequencies of \(n\)th orders for different configurations of cracked rotors. This work continues from two previous works [42; 43] that addressed the problem of crack detection in a pipeline beam by using the non-linear vibrations and the antiresonances of HOFRF. Here, we propose an extension of this approach to the problem of structural health monitoring for rotating machines. This paper is organized as follows: A brief reminder of the general description and characteristics of the cracked rotor system under study is first presented. The second section presents the general formulation of predicting the dynamic responses of the cracked rotor system. Then, the most relevant results on the crack's effect on the vibrational response of the rotor system are discussed. More specifically, the impact of the presence of a breathing crack on the vibrational response of the cracked rotor is analyzed, focusing more particularly on the appearance of harmonic components.