# Vibroacoustic Optimization Study for the Volute Casing of a Centrifugal Fan
## 5 Vibroacoustic Optimization Strategy  
### Optimization Objective Selection and Volute Panel Thickness Parametrization  
Determining the appropriate optimization objective function is the key problem of optimizing calculations. Section 4.1 of this paper provides two means to control vibroacoustics. Control method (1) can be used to modify the geometrical shape of the volute tongue or wavy edge of the blade to weaken the pressure fluctuations on the volute casing. Control method (2) can be used to modify the model geometry, thickness distribution, and constrained position, and reduce the modal shapes. However, the shapes and natural frequency are determined by the structural thickness distribution, structural stiffness, and the constrained position. The modal shapes can be changed by controlling the thickness distribution, when the geometry, stiffness, and constrained position of the structure are fixed. Concerning the studied centrifugal fan volute with several welding panels, this study developed an optimization method of vibroacoustics by changing different panel thicknesses (FT, BT, and ST) to attenuate the volute vibrational acoustic radiation. Regarding vibroacoustic optimization, we must first determine the optimization target function. From the literature review, it is generally considered that the uniform sound pressure at the received position or the radiated sound power at the structural surface should be set as the optimization function. The selection of the target function depends on the research problems, and the target function is divided into two categories according to different attributes. One is a noise optimization problem (internal noise control problem) applied in a closed domain. In this case, one or a few sound pressure levels of a specific measuring point are taken as the target function [24; 25; 26; 27]. Concerning the noise control problem in the open domain, the external acoustic power on the structural surface was chosen as the optimal target function. This method has been proven by many scholars [24; 25; 37]. For the type of target function, the sound pressure at the arranged receiving position can be clearly determined, but it may result in judgments distortion, such that a low sound pressure value at an arranged position may be obtained, and a high sound pressure value at other points can be presented. Thus, it is very important to choose such target functions. Fortunately, [PERSON] [38] proposed an improved target function (F), which is defined as follows:  
\[F=\widetilde{F}^{\frac{1}{n}}=\left(\frac{1}{\omega_{\text{max}}-\omega_{\text {min}}}{\int_{\omega_{\text{min}}}^{\omega_{\text{max}}}\phi\{p_{l}(\omega) \}d\omega}\right)^{\frac{1}{n}} \tag{23}\]  
In Equation (23), \(\phi\{p_{l}(\omega)\}\) represents the given weighting function, which is replaced by the following equation:  
\[\phi\{p_{l}(\omega)\}=\left\{ \begin{array}{cc}\left(p_{l}-p_{ref}\right)^{n},&p_{l}>p_{ref}\\ 0,&p_{l}<p_{ref}\end{array} \right. \tag{24}\]  
Where \(n=1\), which is the average value of sound pressure in the frequency spectrum; and \(n=2\), which is the root mean square (RMS) value of the sound pressure. For the second type of problem, [PERSON] and [PERSON] [6] proposed an optimization method that takes the external radiated sound power as the target function, and is suitable for the optimal noise control method in this study. They provided the root mean square (RMS) expression of sound pressure in an enclosed space as follows:  
\[W=\int\limits_{S}Re(I_{n})ds=\frac{1}{2}\int\limits_{S}Re(pv_{n}^{*})ds \tag{25}\]  
Where S refers to the structural surface, and the structural surface is discretized based on FEM. At this time, the structural radiated sound power can be regarded as the sum of the individual radiatedsound power on the FEM mesh [39].