# Study on Crack Identification with Responses Modulated by Nonlinear Energy Sink
## 3 Crack Identification Based on Single-Period Response  
The parameters are set as follows: beam length \(L=1\) m, width \(b=0.02\) m, height \(h=0.01\) m, density \(\rho=2750\) kg/m\({}^{3}\), modulus of elasticity \(E=68.9\) GPa, damping ratio \(\zeta=0.01\), [PERSON]'s ratio \(\
u=0.33\), stiffness coefficient of NES \(k=60000\)N/m, damping coefficient of NES \(\gamma=0.6\)N/(m/s), and mass ratio \(\varepsilon=m_{2}/m_{1}=0.02\). NES is positioned at the beam midpoint and the force is imposed at \(L_{1}=0.3\) m.  
The Runge-Kutta method implemented in MATLAB is used in the numerical simulation. The frequency responses are shown in Figure 3. In this case, \(f=25\) N; the crack depth ratio is set to 0.3; and the position ratio is set to 0.1, 0.3, and 0.5, respectively. As can be seen on the frequency response, for depth ratio \(\xi=0.3\) and with the increase in crack position ratio \(\beta\), the amplitude of the beam both with and without NES gradually increases, while the corresponding frequency of the peak gradually decreases. In comparison with Figure 3a,b, it can be seen that the amplitude of the beam decreases after connecting the NES, and it is seen from Figure 3c that the amplitude of the NES is much greater than that of  
beam and NES, the corresponding frequency at one of their maximum amplitudes is used for excitation. The excitation frequency of the beam without NES attached is 140.5 rad/s and that of the NES is 136 rad/s. Figure 4 shows the phase diagram. It can be seen that the area of the phase diagram of both the beam and NES increases in varying degrees with the increase in the position ratio. The NES phase diagram is better suited for crack identification because of its larger area. In order to identify the crack when the amplitude of the beam is as low as possible, the excitation frequency is chosen as 133 rad/s. The phase diagrams of the beam and NES in the beam-NES system are shown  
Figure 4: Comparison of phase diagrams between beam without NES and NES in the beam–NES system; (**a**) phase diagram of beam without NES at a frequency of 140.5 rad/s, (**b**) phase diagram of NES in the beam–NES system at a frequency of 136 rad/s. Figure 3: Frequency responses of the beam and NES for different crack parameters: (**a**) beam without NES, (**b**) beam with NES, (**c**) NES. in Figure 5. The waveforms of the beam and NES in the beam-NES system with different crack parameters under the excitation of 133 rad/s are given in Figure 6.  
It can be seen in Figure 6 that, for the excitation frequency equal to 133 rad/s, the amplitude of both the beam and NES in the beam-NES system increases with \(\beta\). However, the beam amplitude always remains low, while the NES amplitude increases greatly, resulting in an obvious signal amplification effect. Based on the above analysis, the change in crack parameters inevitably leads to a change in the phase diagram. In addition, the phase diagram includes both the displacement and velocity information, and thus has significant advantages for damage identification with respect to those methods, which only use one of them. In other words, the phase diagram is more sensitive to the emergence and evolution of cracks. The phase diagram of NES is processed when the excitation frequency is 133 rad/s and is segmented with a vertical line \(x=0\) and a horizontal line \(y=0\), as shown in Figure 7.  
The areas of the left and right sides of the phase diagram in Figure 7a and the up- and downsides in Figure 7b are calculated. The ratio of the right part to the left part of the phase diagram area is defined as index \(S_{rl}\) and the area ratio of the downside to the upside is defined as \(S_{du}\). The dependence of these indexes on \(\beta\) and \(\xi\) is shown in Figures 8 and 9. Figures 8 and 9 show that the index \(S_{rl}\) gradually increases, while the index \(S_{du}\) gradually decreases with the increase in either crack depth or location. Therefore, the degree of damage and the location of cracks can be qualitatively estimated using the value of one of the indices.