# Vibroacoustic Optimization Study for the Volute Casing of a Centrifugal Fan
## 4 **Volute Vibroacoustic Model, FEM Validation, and Simulation**
### Volute Vibracoustics Estimation Method  
The vibroacoustic simulation was performed using the LMS Virtual Acoustics commercial code, and the volute acoustical FEM model is shown in Figure 13. It was similar to the acoustical finite element mesh that was used for aerodynamic noise calculations [13]. Taking into account the characteristic of radiated vibrational noise, the volute's inlet and outlet were completely enclosed. More importantly, according to the requirements on element size driven by maximum frequency, the computational acoustic mesh had to satisfy each wavelength corresponding to six elements. An acoustical mesh with a maximum element size of 15 mm was applied in the sound computation, and guaranteed a spatial resolution at the maximum frequency of 3236 Hz of six points per wavelength. Atmospheric boundary layers (AMLs) were introduced to simulate the unbound boundary of the exterior fluid domain. The outermost layer exposed to the AML surface that satisfied the Sommerfeld radiation condition was defined as a non-reflecting boundary. Then, a field point mesh based on standard ISO3744 [35] that enclosed the entire calculation domain was established using an approximate free-field engineering method. Figure 11: The compassion of numerical and experimental amplitude of normal acceleration. Figure 12: The spectrum of acceleration on the three different measuring points. Figure 14 shows the numerical evaluation method of the solute vibroacoustic coupling. It can be seen that the one-way fluid-structure-acoustic coupling method is divided into three main steps. The first involves the acquisition of the vibrational source of the solute based on the unsteady flow calculation on the centrifugal fan, and then transformation of the extracted time-domain fluctuation data into frequency-domain data through FFT, providing basic data for the next vibration response and vibroacoustic calculation. The second steps involves the interpolation of the frequency-domain node pressure of the fluid into the corresponding structural FEM nodes according to Equation (5) (where \(P_{i}\) (\(i\) = 1, 2, 3, 4) is the source node pressure load, \(P_{\rm A}\) is the target node pressure load, and \(d_{i}\) (\(i\) = 1, 2, 3, 4) is the distance from the source node to the target node; Figure 15 is a sketch of the geometric interpolation algorithm), assignment of the interpolated node pressure of the structure to the boundary loads of vibroacoustics, and then application of the structural FEM to obtain the modal participation factor of the solute. The third step involves loading the modal participation factor and vibroacoustic boundary loads that were obtained during the second step in order to calculate the volute vibrational sound radiation using the modal superposition vibroacoustic method. \[P_{\rm A}=\frac{P_{1}\frac{1}{d_{1}}+P_{2}\frac{1}{d_{2}}+P_{3}\frac{1}{d_{3} }+P_{4}\frac{1}{d_{4}}}{\frac{1}{d_{1}}+\frac{1}{d_{2}}+\frac{1}{d_{3}}+\frac {1}{d_{4}}} \tag{22}\]  
Figure 14: The flow chart of the numerical evaluation method of volute vibroacoustic coupling. Figure 13: The volute acoustical FEM model. Figure 16 presents the spectrum chart of the vibrational sound radiation of volute casing, and the vibrational noise at the fundamental frequency (BPF) is obvious. Besides, the distribution of the vibrational sound radiation and the normalized velocity of the volute casing surface at the fundamental frequency is presented in Figures 17 and 18. It can be observed that the distribution shape of the surface sound pressure and surface normal velocity on the volute have identical characteristics, and the outlet of the volute side panel near the volute tongue region and the volute back panel at 180\({}^{\circ}\) from the tongue presented very strong vibrational acoustic radiation values. In addition, the previous study [36] showed that the normal vibration velocity of the volute was the decisive factor that determined the volute surface acoustic radiation.