# Prediction of Vibration-Mode-Induced Noise of Structure-Acoustic Coupled Systems
## 4 Results and Discussion
### Acoustic Mode Prediction  
Mode decomposition was applied to the time histories of nine acceleration and five sound pressure levels obtained through numerical analysis to extract the vibration and acoustic modes. Figure 5 shows the decomposed time history of the vibration and acoustic modes as a power spectrum of the 10-70 Hz band through the Fourier transform [41]. Figure 5a shows the spectrum of the vibration mode (acceleration), which represents a stable mode decomposition. The shape of the spectrum does not change significantly because the vibration mode is not considerably influenced by the sound pressure. This phenomenon can be explained considering Equation (9): When the coupling effect is small (\(g_{s}=0\)), \(F_{a}(s)\) is eliminated from the numerator and denominator, and only the single degree transfer function remains. \begin{table}
\begin{tabular}{c c c} \hline
**Shell Only** & **Acoustic Only** & **Coupled** \\ \hline
16.714 + 0.48985i & 25.227 + 0.10865i & **16.686 + 0.48736i** \\
26.053 + 0.67818i & 32.718 + 0.11787i & 25.207 + 0.12469i \\
39.543 + 1.0955i & 36.003 + 0. Figure 5b shows the spectrum of the acoustic mode. The solid and dashed lines represent the acoustic- and vibration-mode-induced pressure, respectively. Since the damping ratio of the acoustic mode was smaller than that of the vibration mode, the spectrum appeared sharp. The sound pressure spectrum was not smooth at frequency bands exceeding 40 Hz because the vibration and acoustic modes were close to each other and cannot be completely decomposed using the mode decomposition method adopted in this study. Moreover, in the sound pressure spectrum shown in Figure 5b, both the vibration and acoustic modes appeared because, as indicated in Equation (9), the acoustic mode was expressed as a two degrees-of-freedom system, which was then decomposed into two single degree-of-freedom modes (vibration and acoustic modes), contributing to the sound pressure. Figure 3: The aspect of sound pressure propagation: Time series of the propagating sound pressure level after the impact point (impact hammer) was shocked. The vibration was converted into sound pressure and the sound were scattered gradually as propagation whole room. The sound pressure level also lessens with the lapse of time. As mentioned previously, the vibration mode affected the nearby acoustic mode. In practice, the vibration mode affected all acoustic modes. However, if the natural frequencies of the two modes were highly deviated, the influence diminished drastically, as shown in Figure 1. Therefore, only the acoustic mode near the vibration mode was considered in this analysis. An analysis was performed to verify if the acoustic mode could be predicted from the decomposed vibration mode. The 6 th acoustic mode (47.871 + 1.4390i) from the \"Acoustic only\" column in Table 3, which was near the 5 th vibration mode (47.871 + 1.4390i) from the \"Shell only\" column, was selected as the analysis object. The two modes generated the coupled eigenvalues (47.822 + 1.4333i, 48.829 + 0.14539i) owing to the coupling effect. Since the time history of the vibration mode was known, it was introduced in the differential equation (Equation (8)) to determine the time history of the acoustic mode. This value was compared with the decomposed acoustic mode to evaluate the prediction performance. The findings indicated that the proposed predictive formula can reliably predict the heavy-weight sound, which could not be accurately predicted (large deviations between 0.46 and 0.9) in the existing studies. A previous study reported a high performance in predicting the impact sound with the FDTD approach [22; 23]. The structures considered in this study were different from those in the previous study. The model performance can be enhanced if the damping coefficient condition is known. Moreover, the vibration data are less influenced by the boundary conditions.