## 4 SmTS Scheme and Its Mechanical Characteristics Analysis
### Mechanical Characteristics Analysis of SmTS
#### 4.4.2 Modal Analysis of Main Beam  
As the key bearing component of the SmTS, the main beam needs to obtain its natural frequency and mode shape through a modal analysis to determine the reliability of its structural design. In addition to the structure of the main beam itself, the natural frequency of the main beam is also related to the stiffness of the wall rock and other SmTS components. Due to the limitation of our computer's computing ability, the scheme of the main beam modal analysis was simplified. The influence of wall rock and other support components on the natural frequency of the main beam were simplified into load and boundary conditions. Its vibration source was the roadheader, and the external excitation frequency was the vibration frequency generated when the roadheader cut coal and rock. The natural frequency obtained from the modal analysis was compared with this vibration frequency to verify whether the main beam would resonate. The modal analysis of the main beam was conducted through the model module of ANSYS Workbench; the first six natural frequencies of the main beam were obtained as shown in Table 6, and the first six vibration modes are shown in Figure 19.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**No.** & **Vibration Frequency (Hz)** \\ \hline
1 & 26.41 \\
2 & 37.16 \\
3 & 40.77 \\
4 & 77.06 \\
5 & 141.92 \\
6 & 152.40 \\ \hline \hline \end{tabular}
\end{table}
Table 6: First six natural frequencies of main beam.  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Working Condition** & **Maximum Deformation (mm)** & **Maximum Stress (MPa)** \\ \hline Working condition 1: uniform loading & 0.082 & 21.26 \\ Working condition 2: loading at both ends & 0.492 & 43.22 \\ Working condition 3: loading on both sides & 0.086 & 22.28 \\ Working condition 4: lateral eccentric load & 5.810 & 210.29 \\ Working condition 5: longitudinal eccentric load & 7.220 & 128.18 \\ Working condition 6: torsional loading & 4.219 & 107.93 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Maximum stress and deformation of main beam.  
According to the first six vibration modes diagram of the main beam, it can be seen that:  
1. The natural frequency of the main beam increased with the increase of the modal order, and the sixth order had the largest natural frequency, which was 152.41 Hz.
2. The deformation of the main beam at the second-, fourth- and sixth-order modes was larger, and the deformation at the sixth order was the largest, with a maximum relative displacement of 2.04.
3. The cross bar and longitudinal bar of the main beam were tubular and were prone to bending and breaking. The maximum deformation at the no. 1, no. 3 and no. 6 natural frequencies was located at the end of the longitudinal beam and transverse beam at one side of the main beam, and the maximum deformation at the no. 2, no. 4 and no. 5 natural frequencies was located at the end of the cross beam at both ends.
4. The working frequency of the roadheader working in parallel with the SmTS was 1.5 Hz. It can be seen from Table 6 that the vibration frequency of each order of the main beam was greatly different from the working frequency of the roadheader, and resonance would not occur.  
Figure 19: First six vibration modes of main beam. (**a**) First vibration mode, (**b**) second vibration mode, (**c**) third vibration mode, (**d**) fourth vibration mode, (**e**) fifth vibration mode, and (**f**) sixth vibration mode.