# Nonlinear Vibration Analysis of a Beam with a Breathing Crack
## 7 Results and Comparisons
### Numerical Analysis  
To validate the proposed model, the forced response obtained from the proposed model are compared to the experiment results. The structure of the experimental beam sketched in Figure 8 are also discretized in seven beam elements with three DOFs. The lengths of the beam elements are, \(l_{1}=40\) mm, \(l_{2}=40\) mm, \(l_{3}=60\) mm, \(l_{4}=70\) mm, \(l_{5}=70\) mm, \(l_{6}=70\) mm, and \(l_{7}=56\) mm, respectively. The breathing crack is set on element 4. The damping is assumed to be a proportional model with a damping ratio of \(\xi=0.05\), which is estimated by applying the half-power bandwidth method to the forced response. The boundary strain is \(\varepsilon_{k}=0.004\). The forced response is obtained by solving Equation (26) through the MATLAB program by adopting the fourth-order Runge-Kutta method. The forced responses which are obtained from the proposed model for different excited frequencies are plotted with a dashed line in Figures 9-11. It can be seen that the numerical results agree well with the experimental results. The numerical results have the same harmonics as those in the experimental results. The harmonic component frequencies obtained from the numerical results also coincide with the analyzed frequent factors, which validates the correctness of the analytical results.  
To determine the differences between the force responses from the proposed model and those from the experiment, Table 1 compares the relative amplitude of the dominant harmonic \(h_{r}\) obtained from proposed model and those from the experimental results. For the three forcing frequencies \(f_{w}=1\)/3  
* The study shows that the nonlinear vibration of a cracked beam is influenced not only by the parameters of its structure, but also by the crack parameters and excited force parameters. Under some circumstances, this system presents third-order super-harmonic resonance, second-order super-harmonic resonance, 1/2 th-order sub-harmonic resonance, combination resonance, and multiple resonance.
* The frequent factors show that several sub- and super-harmonic components exist in the dynamic response, whose frequencies related to the natural frequencies are \(n\omega\)\(\pm\)\(\omega_{r}\) (\(r=1\), \(2\),..., \(q\); \(n=1\), \(2\),..., \(N\)). The analytical results agree well with the experimental data and numerical results. Therefore, the frequent factor can be used as a qualitative damage indicator for crack detection.
* When the beam is excited at one of those frequency factors, the spectrum exhibits obvious nonlinear characteristics. In particular, when the cracked beam is excited by the forcing frequency \(f_{\omega}=1/2f_{1}\), the nonlinear behavior is most visible, indicating that this frequency is the most efficient one to reveal damage.
* The relative harmonic amplitudes calculated by the proposed method agree well with the experimental values for the three cases under consideration. This shows that the proposed stiffness model is reliable.  
Conceptualization, Y.L.; funding acquisition, Y.L.; investigation, H.L.; methodology, H.L. and K.L.; validation, H.L.; writing--original draft preparation, H.L.; writing--review and editing, H.L. and K.L.  
This research work was funded by the National Natural Science Foundation of China (51375500), and the Science and Technology Project of Shaoguan (2018 sn052).  
The authors declare no conflict of interest.