# Nonlinear Vibration Analysis of a Beam with a Breathing Crack
## 7 Results and Comparisons
### Experimental Results  
The dimensions in mm of the cracked cantilever beam are shown in Figure 8. The distance from the crack location to the fixed end is 176 mm. The crack depth and width are 3.8 mm and 12 mm, respectively. From the free vibrations of the cracked beam excited by a hammer impact, the first natural frequency \(f_{1}\) is found to be 57.4 Hz.  
The forced vibration consists of three groups. The external harmonic force of each test is 20 N, and the excited frequency ratios \(\lambda=\)\(f_{\omega}\)/\(f_{1}\) are considered to be \(\lambda=\) 1/3, 1/2, 2. The sampling frequency is set as 10,000 Hz. The spectrum domain of the acceleration signals for the three cases \(\lambda=\) 1/3, 1/2, and 2 are depicted in Figures 9-11, respectively. The spectrum analysis is conducted by fast Fourier transform (FFT). To evaluate the nonlinear characteristics, a relative harmonic amplitude is introduced [22]:  
\[h_{r}=\frac{H_{s}}{H_{f}} \tag{27}\]  
where \(H_{s}\) is the magnitude of the sub-harmonic or super-harmonic component, and \(H_{f}\) is the FFT magnitude of the harmonic component corresponding to the exciting frequency.  
Figure 8: The dimensions of the experimental cantilever beam.  
Figure 9: Fast Fourier transform (FFT) spectrum of the transverse acceleration at the tip of the cracked beam with \(\lambda=\) 1/3.  
Figure 9 shows that when \(f_{\omega}=1/3f_{1}=19.1\) Hz, the spectrum exhibits rich harmonic contents. The harmonic at \(f_{1}=3f_{\omega}=\) has the largest amplitude, and the corresponding value of \(h_{r}\) is 132%. The magnitude of the harmonic at the exciting frequency \(f_{\omega}\) (1/3\(f_{1}\)) is slightly smaller than that of the third harmonic component. Obviously, the several components at the frequency \(2f_{\omega}\) (\(f_{1}-f_{\omega}\)), 3\(f_{\omega}\) (\(f_{1}\)), 4\(f_{\omega}\) (\(f_{1}+f_{\omega}\)), and 5\(f_{\omega}\) (\(f_{1}+2f_{\omega}\)) coincide with the frequent factors \(n\omega\pm\omega_{1}\) (\(n=1,2\)), which validate the correctness of the analytical results from the multiple-scale method.  
Figure 10 shows that when \(f_{\omega}=1/2f_{1}=28.7\) Hz, the response also contain several harmonics. The amplitude reaches its maximum at the second harmonic \(2f_{\omega}\) (\(f_{1}\)), which is higher than the harmonic  
Figure 11: FFT spectrum of the transverse acceleration at the tip of the cracked beam with \(\lambda=2\).  
Figure 10: FFT spectrum of the transverse acceleration at the tip of the cracked beam with \(\lambda=1/2\).  
component at the exciting frequency \(f_{\omega}\) (1/2\(f_{1}\)). In this case, the corresponding \(h_{r}\) is 179%. Besides this, the harmonics \(3f_{\omega}\) (\(f_{1}+f_{\omega}\)) and \(4f_{\omega}\) (\(f_{1}+2f_{\omega}\)) are also observed. It should be noticed that these harmonic components coincide with the frequent factors \(n\omega\pm\omega_{1}\) (\(n=1,2\)).  
Figure 11 shows that when \(f_{\omega}=2f_{1}=114.8\) Hz, the response has only two dominant harmonics in the frequency range of consideration. The peak at the sub-harmonic with \(f_{1}\) (1/2\(f_{\omega}\)) is smaller than the excitation harmonic component at the exciting frequency \(f_{\omega}\). The corresponding \(h_{r}\) is 23%. The experimental results show that the sub-harmonic resonance occurs at \(f_{\omega}=2f_{1}\).