## 5 Acoustic-Structural Coupling Analysis of the Muffler
### Acoustic-Structure Coupling Modal Analysis  
Relevant computations for the sound-structure coupled modes of the split-stream rushing muffler were conducted using the \"Acoustic-Shell Interaction, Frequency Domain\" module within COMSOL. The first 10 coupled modes are presented in Table 5. As shown in Table 5, the frequencies of the second through fifth and the seventh coupled modes are essentially in agreement with the frequencies of the first to fifth structural modes, suggesting that the primary contribution to these coupled modes originates from the structural modes. Conversely, the frequencies of the sixth and eighth coupled modes are essentially in agreement with the frequencies of the second and third acoustic cavity modes, suggesting that this part of coupled modes is primarily governed by the acoustic cavity modes. This indicates that the frequency spectrum of the coupled modes is predominantly governed by the synergistic interaction of structural and acoustic cavity modes, and the coupled system, to some extent, preserves the modal attributes of its constituent subsystems. However, the ninth and tenth coupled modes did not exhibit a frequency correspondence with any individual structural or acoustic cavity mode, demonstrating prominent modal shift characteristics. This phenomenon arises from the enhanced interaction between the structure and the acoustic field within this frequency range, resulting in modal coupling and the emergence of novel modal shapes and frequency distributions that no longer entirely represent the properties of the individual subsystems. This indicates that within a particular frequency range, the coupling effect between the structure and the acoustic field becomes more pronounced, resulting in alterations of varying magnitudes in the inherent modal configurations and frequencies.  
Figures 15 and 16 illustrate comparative diagrams of coupled modes versus structural modes and acoustic cavity modes at proximate frequencies. The observation of both figures reveals that structural modes and coupled modes exhibit identical mode shapes at proximate frequencies. The distinction lies in the fact that the coupled mode shapes display partial torsional deformation in their phase relationship relative to the structural mode shapes. Acoustic cavity modes and coupled modes exhibit a high degree of resemblance in their modal configurations, sound pressure distribution locations, and sound pressure amplitudes at proximate frequencies.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Coupled Modal** & **Coupled Modal** & **Structural** & **Structural Modal** & **Acoustic Cavity** & **Acoustic Cavity Modal** \\
**Order** & **Frequency (Hz)** & **Modal Order** & **Frequency (Hz)** & **Modal Order** & **Frequency (Hz)** \\ \hline
1 & 4.58 \(\times\) 10\({}^{-4}\) & & & & \\
2 & 82.38 & 1 & 82.43 & & \\
3 & 82.39 & 2 & 82.44 & & \\
4 & 139.77 & 3 & 140.12 & & \\
5 & 139.97 & 4 & 140.23 & & \\
6 & 426.62 & & & 2 & 427.04 \\
7 & 554.22 & 5 & 554.24 & & \\
8 & 570.78 & & & 3 & 571.1 \\
9 & 611.47 & & & & \\
10 & 629.54 & & & & \\ \hline \end{tabular}
\end{table}
Table 5: The first 10 acoustic–structural coupled modes of the split-stream rushing muffler.