## 4 Results and Discussions
### Effects of Surface Crack Propagation on Meshing Stiffness and Vibration Characteristics
#### 4.2.3 Effects of the Surface Crack Propagation Progress on Acceleration Response  
The effects of surface crack propagation on the time domain of acceleration response are displayed in Figure 14. We can see that the spalling fault has an obvious impact compared with the crack fault, which results from the larger changes in meshing stiffness when the spalling gear just enters and exits the spalling area. It can also be seen that the influence of the tooth surface crack fault on the time domain of acceleration responses is also not obvious before the crack evolves into a spalling fault. Figure 15 shows the influence of surface crack propagation on acceleration spectra. It can be found that the side frequency components also appear near the mesh frequency and its harmonic frequencies in the acceleration spectra of four fault cases. The side frequency bands change significantly when the crack developed into a spalling fault. Meanwhile, compared with the DTE spectra, the increased amplitudes of the side frequencies are not obvious before the surface crack propagates into the spalling fault. Figure 14: Effects of surface crack propagation on time domain of acceleration response. Figure 13: DTE spectra of different fault cases: (**a**) case #1, (**b**) case #2, (**c**) case #3, (**d**) case #4. From the above results, we can find that sidebands of the spectra resulting from the crack propagation are more sensitive than the time domain responses. Thus, the sideband amplitude ratio (BAR) is introduced to further quantitatively analyze the effects of the sideband and sideband amplitudes. Figure 15: Acceleration spectra of different fault cases: (**a**) case #1, (**b**) case #2, (**c**) case #3, (**d**) case #4.  
tooth surface crack propagation on the spectra of DTE and acceleration. The calculation formula of the BAR is as follows [35],  
\[BAR=\sum_{i=1}^{k}\frac{A_{f_{m}-if_{n}}+A_{f_{m}+if_{n}}}{2A_{f_{m}}} \tag{21}\]  
where \(A\) represents the amplitude of the \(i\)th side frequency on either side of the harmonic frequency \(f_{m}\). \(k\) denotes the number of side frequency. From Equation (21) we can learn that the higher the BAR is, the richer the sideband frequency components are. The BAR values of DTE and acceleration under different fault cases are computed and displayed in Table 3. It can be observed that the average BARs of DTE and acceleration tend to increase with the surface crack propagation, and reaches maximum after the spalling failure occurs. From Table 3, it can be calculated that the growth rates of the average DTE BARs under case #2, case #3, and case #4 are 1.00, 2.98, and 500.19, respectively, when compared to case #1, while the average BARs of acceleration are 0.26, 1.00, and 30.62, respectively. It can be concluded that the BAR of DTE increases faster and changes more obviously than that of the acceleration during the surface crack propagation, which is more conducive to the surface crack fault diagnosis. Figure 16 shows the BAR changes of DTE and acceleration under different harmonic orders and fault cases. It can be found that BARs of DTE and acceleration tend to increase with the surface crack propagation at each harmonic frequency. The BAR value of DTE increases obviously at each harmonic frequency, while the BAR value of acceleration increases significantly only at the harmonic frequency of order 5-10. Therefore, compared with the acceleration response, the side band variations of DTE during the surface crack propagation are more propitious to the fault detection of a tooth surface crack. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\ \begin{tabular}{c} **Response** \\ **Type** \\ \end{tabular}  } & \multirow{2}{*}{**Fault Case**} & \multicolumn{10}{c}{**BAR (dB)**} \\ \cline{3-14}  & & & \multicolumn{10}{c}{**Harmonic Order**} & & & & & & & & & **Value** \\ \cline{3-14}  & & \(f_{m}\) & \(2f_{m}\) & \(3f_{m}\) & \(4f_{m}\) & \(5f_{m}\) & \(6f_{m}\) & \(7f_{m}\) & \(8f_{m}\) & \(9f_{m}\) & \(10f_{m}\) & \(11f_{m}\) & Value \\ \hline \multirow{4}{*}{