# Vibroacoustic Optimization Study for the Volute Casing of a Centrifugal Fan
## 6 Results and Discussion
### Results of Optimization. After optimization, the vibrational radiated sound power of the volute surface is greatly reduced. Table 4 shows the massless constrained optimization results (the \"optimized\" represents the calculation results of the approximate model, and the \"numerical validated\" represents the vibroacoustic analysis with the optimum values). It can be seen that the radiated sound power on the volute structure surface decreases by 9.4 dB when ST increases to a maximum, while BT and ST locate at minimum values; unfortunately, at the same time, the volute total mass increases by 18.13%. Table 5 shows the optimization results with the volute total mass invariable. Even though the volute total mass remains the same, the radiated sound power on the surface of the volute will also be weakened by an average of 6.3 dB. In addition, the radiated sound power spectrum of the volute structural surface shown in Figure 27 also shows that the single-objective optimization significantly improves the radiated sound power on the volute surface at the fundamental frequency (BPF). In order to better understand the improvement effect of optimum panel thickness combinations on vibroacoustic performance before and after optimization, the comparison of distribution of vibrational sound radiation before and after the total mass invariable optimization at the fundamental frequency are given in Figures 28 and 29. From the comparison, it can be concluded that the sound pressure of the vibrational radiation close to the tongue on the side volute is greatly reduced by optimization, and \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Table Title** & **FT (mm)** & **ST (mm)** & **BT (mm)** & **W\({}_{\text{s}}\)/(dB)** & **M\({}_{\text{T}}\) (kg)** \\ \hline Original & 6.000 & 5.000 & 6.000 & 63.78 & 86.592 \\ Optimized & 4.000 & 10.00 & 4.000 & 54.38 & 102.29 \\ Numerically validated & 4.000 & 10.00 & 4.000 & 56.25 & 102.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Massless constrained optimization results. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Table Title** & **FT (mm)** & **ST (mm)** & **BT (mm)** & **W\({}_{\text{s}}\)/(dB)** & **M\({}_{\text{T}}\) (kg)** \\ \hline Original & 6.000 & 5.000 & 6.000 & 63.78 & 86.592 \\ Optimized & 4.335 & 7.470 & 4.000 & 57.52 & 86.592 \\ Numerically validated & 4.335 & 7.470 & 4.000 & 58.68 & 86.592 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The optimization results with the volute total mass invariable. Figure 27: The radiated sound power spectrum of the volute structural surface. the other sound pressure of the vibrational radiation on the other area is also weakened to varying degrees. In addition to these results, the optimization also changed the directivity distribution of acoustical radiation, and very strong directivity was produced on the back panel side of the volute. Figure 28: The distribution of vibrational sound radiation of original volute at the fundamental frequency. (**a**) The whole volute body; (**b**) XY section; (**c**) ZY section. Figure 29: Distribution of vibrational sound radiation after the total mass invariable optimization at the fundamental frequency. (**a**) The whole volute body; (**b**) XY section; (**c**) ZY section. The multi-objective optimization results, which took the volute surface vibrational radiated sound power and the total mass of the volute as the objective function of the optimization model, conformed to the actual needs. Multi-objective optimization solutions are not unique, and are usually presented in the form of solutions (Pareto front solutions). According to the distribution of Pareto front solutions in Figure 30, it can be seen that the radiated sound power on the volute surface and the total mass of the volute show an approximately linear inverse proportional relationship, and a smaller vibrational sound power can be obtained, when the volute mass (mass increase control: \(\pm\) three kg) changes within a small range.