# Dynamic Analysis of the Musical Triangles--Experimental and Numerical Approaches
## 3 Results
### Comparison between Experimental and Numerical Modal Analysis
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Type of**} & \multicolumn{5}{c}{**Average Values of Frequency (Hz)**} \\
**Triangle** & **Excited Side** & \(\mathbf{f_{1}}\) & \(\mathbf{f_{2}}\) & \(\mathbf{f_{3}}\) & \(\mathbf{f_{4}}\) & \(\mathbf{f_{Amax}}\) \\ \hline \multirow{3}{*}{T1} & A & 159.43 & 997.30 & 1531.33 & - & 1657.10 \\  & B & - & 996.97 & - & - & 1657.10 \\  & C & 159.53 & 996.97 & 1531.66 & - & 1657.10 \\  & A & 168.13 & 878.13 & 1053.00 & 1605.80 & 1876.06 \\ \multirow{3}{*}{T2} & B & - & 878.10 & 1052.90 & 1606.00 & 1876.29 \\  & C & 168.20 & 878.17 & 1054.71 & 1597.66 & 1876.57 \\ \multirow{3}{*}{T3} & A & 96.29 & 612.70 & - & 1052.09 & 1052.09 \\  & B & - & 612.83 & - & 1052.24 & 1052.24 \\ \multirow{3}{*}{T3} & C & 96.41 & 612.86 & 899.70 & 1045.47 & 1045.47 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The first natural frequencies obtained experimentally. Since the experimental results do not provide information about the vibration modes of the triangles, the finite element analysis completes these data, especially since the results obtained by the two methods are in good agreement. Figure 6: The spectrograms and the graphs of impact force, for stainless steel triangle T1: (**a**) in case of excitation of side A; (**b**) the plot of excitation force against the time, by hitting side A; (**c**) in case of excitation of side B; (**d**) the plot of excitation force against the time, by hitting side B; (**e**) in case of excitation of side C; (**f**) the plot of excitation force against the time, by hitting side C.  
Figures 10-12 show the main vibration modes of the triangles. Thus, it is observed that in the case of stainless steel triangle T1, the first mode is characterized by vibrations in the plane of the triangle, for 156 Hz (Figure 10a); the next mode is one of torsion, out of the plane of the triangle, for frequency of 162 Hz (Figure 10b). Then, the triangle vibrates along the axis of the bars, obtaining a longitudinal mode at 269 Hz (Figure 10c). For mode 4, torsional vibrations are obtained out of the plane of the triangle, for 805.285 Hz (Figure 10d), then alternating with longitudinal vibrations along the bar at frequency of 978.74 Hz (Figure 10e), and then bending vibration mode around the z-axis is obtained for 1465.83 Hz (Figure 10f). The next modes are mode 7 (1499 Hz), a bending vibration mode (around the Y-axis) (Figure 10g) and mode 8 (1627 Hz)--bending vibration mode around the Z-axis (out of the plane of the triangle) (Figure 10h). Figure 7: The spectrograms and the graphs of impact force, for aluminum triangle T2: (**a**) in case of excitation of side A; (**b**) the plot of excitation force against the time, by hitting side A; (**c**) in case of excitation of side B; (**d**) the plot of excitation force against the time, by hitting side B; (**e**) in case of excitation of side C; (**f**) the plot of excitation force against the time, by hitting side C.  
For the aluminum triangle T2, having dimensions close to the stainless steel triangle (T1), the modal shapes are identical. The values of the natural frequencies differ, being higher than in the case of stainless steel (Figure 11). In the case of the T3 triangle, the same modal shapes occur as for the other two triangles, but at lower frequencies, considering the longer length of the sides (Figure 12).