# Study on Crack Identification with Responses Modulated by Nonlinear Energy Sink
## 1 Introduction  
Nonlinear energy sink (NES) is a type of device that can realize one-way irreversible energy transfer and has various designs [1, 2]. It has received much attention owing to its broadband vibration absorption characteristics and lightweight property [3, 4]. The conventional NES transfers energy through a purely nonlinear elastic component and dissipates it via damping. It has been both theoretically and experimentally verified that the conventional NES can efficiently suppress vibrations of many kinds of primary structures [5, 6, 7, 8]. Since its discovery, the most important role of the NES has been vibration absorption. Consequently, the main focus of research is its performance and robustness enhancement for vibration control. Therefore, various improved versions such as vibrop-impact NES [9, 10], multi-stable NES [11, 12], lever-type NES [13], inertial NES [14], and piecewise stiffness/damping NES [15, 16, 17] have been proposed.  
Owing to its broadband energy transfer properties, the NES is suitable for use in energy harvesting and has also exhibited various other useful properties. Different kinds of piezoelectric and magnetoelectric energy harvesters combined with NES have been proposed [18, 19, 20, 21]. The NES was also used in a viscoelastic vibration isolation system, achieving favorable results [22]. Generally speaking, the NES produces large amplitudes during its operation, which is usually considered to be a drawback because it increases the dimensions of the device. However, in this paper, this property is used to assist the crack identification of structures.  
Crack is a common structural damage that results in the deterioration of mechanical properties. The evolution of a crack can lead to severe hidden vulnerabilities of engineeringstructures. Indices based on the linear vibration parameters such as natural frequency and mode were used to detect cracks at an early age [23; 24; 25]. The harmonic response is currently the most widely used nonlinear index [26; 27], and other characteristics such as bifurcation and nonlinear strength [28; 29] were also developed. The orbits in phase diagrams are sensitive to the change in mechanical properties of structures and are usually used to qualitatively analyze cracks [30; 31; 32], as they contain displacement and velocity information and show strongly nonlinear dynamic characteristics. Recently, phase diagrams have also been employed to quantitatively determine crack parameters. Identification methods based on orbital eccentricity, offset deviation [33; 34], and area [35] of the phase diagrams were developed.  
In this work, the dynamic equation of a beam with a breathing crack and attached to an NES is derived. The influences of the NES on the dynamics of the crack beam are analyzed. The effect of indirect signal amplification and the induction of the strongly modulated response are used for crack identification. The identification indices based on the area of the phase diagram are defined and their variations with crack parameters are obtained. The position and depth of the crack are estimated from the intersection point of contour lines.