# Nonlinear Vibration Analysis of a Beam with a Breathing Crack
## 1 Introduction  
Beams are one of the basic structural members and are widely used in various fields, such as aerospace, machinery, automobile and structure industries [1; 2; 3; 4]. Most beam-like structures are subjected to dynamic loading. Due to various causes such as corrosion and temperature fluctuation, cracks may result in failure [5]. If these cracks are not detected in time, a sudden structural failure can lead to catastrophic damages [6; 7]. Due to the various types of crack in different structures, currently, industries have a number of different damage-detection methods, such as ultrasonic methods [8; 9; 10], X-ray methods and vibration-based damage identification methods. Owing to the advancement of reliable fault detection and diagnosis techniques, vibration-based damage identification methods have been developed for various engineering applications, such as the rotating blades of engines [11], submarine pipes [12], rotating rotors [13] and atomic force microscope (AFM) probes [14]. In the early stage, most researchers assumed that cracks keep opening during vibration, and thus a linear system was adopted. In this case, the dynamic characteristics of the linear system, such as natural frequencies and mode shapes, are used to determine the crack severity. However, these linear response features are not valid when detecting fatigue cracks due to the crack breathing effect in reality [15]. For example, [PERSON] [16] experimentally investigated the crack closure effect on the vibrations of a cracked beam. The results showed that the natural frequencies measured from the experiment are different from those obtained from analysis that does not consider the effect of crack closing. Therefore, in the case of the closing crack (also known as breathing crack), the dynamicsystem is nonlinear because of the time-varying stiffness caused by opening and closing effects of the crack [17]. Over the last decades, many scholars have exploited the different nonlinear characteristics of beams with breathing cracks by using analytical methods [18; 19; 20], numerical methods [21; 22] and experimental approaches [23; 24]. Their studies showed that nonlinear vibration characteristics have a higher sensitivity to breathing cracks, especially the presence of sub- and super-harmonic resonances [25]. For example, [PERSON] et al. [18] presented an approximate analytical method to calculate the relative amplitude of the dominant harmonics of a rod-like structure with a closing crack at sub- and super-harmonic resonances, and they showed that the acceleration amplitude of the second harmonic is more sensitive than that of the fundamental harmonic. [PERSON] et al. [19] developed the dynamic model of a beam with a closing crack which considered the damping change due to crack propagation. The relationship of the amplitude of dominant harmonics with the force application point and sensor location at sub-resonance and super-resonance was investigated. As the crack depth increases, the super-harmonic resonance in the frequency response is distinct, meaning that the second order becomes dominant. [PERSON] et al. [21; 22] considered the closing crack interfaces as a contact problem and studied the forced harmonic response of the cracked beam by using the finite analysis software ADINA. The simulation results demonstrated that the phenomena of sub- and super-harmonic resonances exist in the cracked beam. Later, the same authors [24] also conducted an experimental investigation into the forced harmonic response of a cracked beam with a breathing crack. They observed that several sub or super- harmonic components are present in the spectra. However, the phenomena causing these several harmonics to emerge in the experiment have not been explained fully. The main reason for this is that the nonlinear vibration of a beam with a breathing crack is not fully understood, and that the inherent relation between the sub- and super-harmonic resonance responses and the parameters of the cracked beam require further investigation.