## 1 Introduction  
One of the most challenging problems in the vibration monitoring of rotating machines is to promote effective tools and robust indicators to detect damage at an early stage. Nowadays, rotating machines are designed to operate at increasingly higher efficiencies and under increasingly challenging operating conditions for structural system integrity. As a result, many rotating systems contain rotor shafts that are potentially highly susceptible to transverse fatigue cracks. Certain catastrophic failures of rotating machines are due to the presence of growing cracks in the rotor shaft. This primary accidental risk, which can lead to rotor system downtime and prohibitive economic costs, can be avoided by deploying the early detection of propagating cracks. Many efforts have been made in this field of structural health monitoring to better understand the impact of the presence of a transverse crack on the dynamic behavior of rotating systems, leading to the development of advanced detection of defects to ensure the safety of rotating machines during continuous operation and under potentially fluctuating loading conditions. Generally, three different approaches are attempted to detect the presence of a crack in a rotating structure and possibly identify the position and size of the crack. The first approach is based on monitoring the natural frequencies of the rotor system because the presence of a crack in the rotating shaft reduces the rigidity of the rotor shaft, and therefore, causes a decrease in the natural frequencies of the original uncracked rotor system [1; 2; 3]. This approach has the advantage of being very easy to apply. It is often effective for detecting large faults but has the disadvantage of being incapable of detecting small ones. Consequently, this approach is very limited for the predictive maintenance of rotating systems in a real environment. The second approach focuses on changes in the linear measurements of the frequency response functions (FRF) and certain associated indicators,such as the motion of antiresonances [4; 5; 6; 7] and coupling the vibration measurements of a rotating cracked shaft [8; 9; 10; 11]. It can be noted, however, that some difficulties may arise when seeking a robust identification of damage based on the use of linear analysis: indeed, the crack might open and close alternately during experimental tests [12] due to the real and complex environments of rotating machines during operational conditions. If the presence of an open crack leads only to a loss of physical stiffness, which allows studying linear systems, the development of breathing cracks not only reduces the stiffness of the structure, but tends to make an otherwise linear structure non-linear, because of the evolution of the structure's stiffness characteristics associated with open and closed states. Therefore, considering the fact that the presence of damage can induce more complicated behavior, an alternative approach is to take into account the influence of an active transverse crack in the response of a rotor model by investigating the nonlinear structural vibration responses. As mentioned previously, the concept of an active crack is represented by the opening and closing of the crack (also called the crack breathing phenomenon [13; 14; 15; 16; 17]). Thus, many works based on the tracking of nonlinear dynamic characteristics, such as the presence of \(n\times\) harmonic components or changes in the rotor orbit shape with the occurrence of multi-loops, have been developed and tested by many researchers [18; 19; 20; 21; 22; 23; 24; 25; 26]. It is now generally recognized that the use of nonlinear analysis allows the efficient and robust detection of medium to large defects due to the fact that nonlinear dynamic characteristics are significantly influenced by the presence of a transverse breathing crack in the rotating shaft.