## 4 Dynamic Response Characteristics of Gear Transmission with Gear Breakage
### Summary of Dynamic Characteristics of Gear Pair with Broken Tooth  
Faulty gears with broken teeth at the vibration exciter have significant vibration characteristics. Some key parameter values in the time domain and frequency domain are, respectively, listed in Table 3 and Figure 14. \(F_{n}\_{n}\), \(F_{n}\_{c}\), \(F_{n}\_{3}/4\), \(F_{n}\_{1}/2\), and \(F_{n}\_{1}/4\) are defined as the normal meshing force of normal gear, faulty gears with a full broken tooth, 3/4 broken tooth, 1/2 broken tooth, and 1/4 broken tooth, respectively. This definition rule is also applicable to other parameters. Some conclusions are summarized as follows: \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Indicators**} & \multicolumn{4}{c}{**Full Broken Tooth**} & \multicolumn{4}{c}{**1/2 Broken Tooth**} & \multicolumn{4}{c}{**1/4 Broken Tooth**} \\ \cline{2-10}  & **AVG** & **MAX** & **RMS** & **AVG** & **MAX** & **RMS** & **AVG** & **MAX** & **RMS** \\ \hline \(F_{n}\) (N) & 2851.5 & 15,815.4 & 3718.7 & 2846.3 & 13,822.5 & 3757.0 & 2880.6 & 15,413.3 & 3773.4 \\ AX(mm/s\({}^{2}\)) & & 43,630.2 & 10,630.5 & & 46,774.4 & 10,999.2 & & 46,058.5 & 10,574.2 \\ AX(mm/s\({}^{2}\)) & & 52,317.3 & 11,565.0 & & 49,027.1 & 11,439.4 & & 52,572.6 & 11,575.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters in time domain in different fault states. Figure 13: (a) Schematic diagram of 1/2 broken tooth and normal meshing force with its zoom magnified image of position I; (b) frequency spectrum. 1. As shown in Table 3, compared with the fault-free state, the AVG and RMS of the normal meshing force with a broken tooth are smaller, and the internal impact of the gear pair is generally weakened. 2. As shown in Table 3, compared with the fault-free state, the MAX of the vibration acceleration in the x and y directions with a broken tooth is greater; however, the RMS  
Figure 14: (**a**) Eight eigenvalues and (**b**) amplitudes of main frequency components in different fault states: F\({}_{n}\), AX, AY, and AZ. of the acceleration in the x, y, and z directions is smaller, indicating that the broken tooth leads to the weakening of the overall vibration intensity of the gear transmission. It can be seen that it is difficult to diagnose the broken tooth directly through the vibration amplitude. 3. Eight eigenvalues are calculated from meshing force and vibration acceleration in the x, y, and z directions under different states, respectively: mean value (AVG), root mean square (RMS), peak value (Xp), peak index (C), waveform index (S), pulse index (I), kurtosis index (K), and margin index (L). Statistical results show that the eight eigenvalues of meshing force and vibration acceleration in the x and y directions have no obvious regularity relative to the damage degree. However, some regularity can be discovered from vibration acceleration in the z direction, as shown in Figure 14a: the S and I of AZ increase with the increase of damage degree. After the introduction of 3/4 broken teeth, the following laws still hold: S_c > S_3/4 > S_1/2 > S_1/4 > S_n, I_c > I_3/4 > I_1/2 > I_1/4 > I_n, which demonstrates that the law is universal. As a result, the conclusion can be drawn that S and I of vibration acceleration in the z direction are sensitive to damage degree, which can be used as identification indicators. 4. Figure 14b reveals that the amplitudes of Fr, GMF, 2 GMF, and 3 GMF of the meshing force of the faulty gear are less than those components of normal gears, while the amplitude of 4 GMF is greater than that of normal gear. Among them, 3 GMF is difficult to find because it is covered by side frequency. In addition, this paper also analyzed the fault state with 3/4 broken tooth and proves that these conclusions are still valid, which are not listed in detail. These fault features can be used for fault diagnosis. 5.