# Vibroacoustic Optimization Study for the Volute Casing of a Centrifugal Fan
## 6 Results and Discussion  
### Results of Sensitivity and Parametric Analysis  
In order to investigate the influence of each design variable on the optimization objectives, it is necessary to perform the sensitivity analysis for the design variables. Sensitivity analysis is performed by studying the correlations between the design variables and the objectives. Figure 24 shows the  
Figure 23: The flow chart of multi-objective optimization.  
correlation distribution of design variables and optimization objectives (Kirchhoff SPW, volute mass on the volute structure surface); the correlation theory and definition referred to the reference [52]. The positive value indicates that the optimization objective is proportional to the design variable, and otherwise is an inverse relationship. The closer the absolute value of the coefficient is to one, the higher the degree of correlation. It can be seen from Figure 24 that ST had the largest influence on all the objectives, followed by FT, then BT. Concerning radiated sound power, ST and FT present an inverse proportionality to the radiated sound power, which means that the larger the volute panel thickness (ST and FT), the smaller the radiated sound power.  
However, the relationship diagram between each volute panel (ST, BT, FT) and objective functions (the Kirchhoff SPW or the total mass of the volute) in Figure 25 shows that BT and FT present nonlinearity characteristics for the acoustic power of the volute surface. With an increase of BT value, the vibrational radiated sound power of the volute surface increases first, and then decreases, and FT value presents the opposite change. However, the volute radiated sound power decreases with ST value if the ST does not exceed nine mm. This means that each panel thickness (BT, ST, FT) has an optimal value that satisfies the radiated sound power minimum. However, Figure 25 only shows the influence of each single thickness variable on the radiated sound power (the other two variables remain unchanged; the relationship between the other two variables and the noise radiation is determined). To determine the relationship between each variable combination (BT-ST; FT-ST; FT-BT) and the acoustic radiation power, Figure 26 gives the influence of each two-panel thickness variation on the radiated sound power. It can be observed that the FT-ST and BT-ST combinations present similar distribution characteristics. Therefore, the Kirchhoff SPW basically remained a constant smaller value when the ST was greater than 7.0 mm, and the BT was less than 8.5 mm or the FT was less than 7.0 mm. In addition, as the FT is larger and the BT is smaller or the FT and BT are both smaller (less than 5.5), the Kirchhoff SPW is observed to have a smaller value as shown in Figure 26c. Besides, linear relationships could be seen in Figure 25b, which means that the thicker the thickness of each volute panel, the greater the total mass of the casing.  
Figure 24: The correlation distribution of design variables and responses (Kirchhoff SPW and volute mass on the volute structure surface).  
Figure 26: The relationship between each panel thickness (BT-ST; FT-ST; FT-BT) and radiated sound power of solute surface. (**a**) BT-ST; (**b**) FT-ST; (**c**) FT-BT.  
Figure 25: The relationship profile between each panel thickness and objective function (the Kirchhoff SPW or the total mass of the solute). (**a**) The relationship of solute panel thickness and Kirchhoff SPW. (**b**) The relationship of solute panel thickness and total mass.  
In conclusion, the variation in ST is the most sensitive to the sound radiation power of the volute structure surface, while FT is the second-most sensitive, and BT is the least. Therefore, it is possible to obtain the smallest value of radiated sound power of the volute surface by optimizing the panel thickness combination (increasing the ST value and decreasing the FT and BT values).