# Vibroacoustic Optimization Study for the Volute Casing of a Centrifugal Fan
## 1 Introduction
[PERSON] [11] calculated the vibrational sound radiation of a volute casing of the same centrifugal fan using a fluid-structure-acoustic coupling method. Indeed, this BEM method discretizes the Lighthill equations by applying a free-field Green function integral. At present, the Green function integral method can only solve problems with simple geometric boundaries, and those complex boundaries must be simplified in the free field. Without doubt, this simplification does not consider reflection and scattering effects in the noise propagation. Based on the above advantages, the finite element method (FEM) for solving noise radiation has been recognized by scholars. [PERSON] [12] predicted the structural acoustics of automotive vehicles through using the FEM model. However, there is a computational disadvantage when the finite element method solves the structural acoustic problem of a closed domain. To overcome this disadvantage, automatically matched boundary layers (AMLs) were introduced to simulate the unbound boundary of the exterior computational domain. The outermost layer exposed to the AML surface that satisfied the Sommerfeld radiation condition was defined as a non-reflecting boundary. Based on the FEM method, [PERSON] [13] performed the aerodynamic noise of the centrifugal fan using the FEM method,and achieved higher prediction accuracy while using less computing resources. To reveal and reduce the noise primarily generated by the freezer fan unit, [PERSON] [14] investigated the vibration and acoustic interactions between the structure and the cavity inside the freezer cabinet, and the FEM method was also used. [PERSON] [15] performed a vibroacoustic analysis of a centrifugal compressor with connecting piping systems, in which the sound was induced by the unsteady flow in the centrifugal compressor and pipes, and the same FEM method was used. These studies have been of benefit in promoting the development of the research of vibrational sound radiation on structural casing, allowing for a deeper understanding of vibrational noise during the fan operation, and have provided a useful reference for the noise reduction of such machinery. The purpose of vibrational noise research is to explore the generation mechanism of vibrational noise, and then propose targeted methods of vibration and noise reduction. Concerning vibration and noise control, there are certain means: controlling the vibrational source, such as vibration absorption and vibration isolation [16]; dynamic vibration absorption; damping vibration control [17]; and structural vibration control [18; 19; 20; 21; 22; 23; 24; 25; 26; 27]. At present, a structural vibration control method that meets specific requirements by modifying the dynamic characteristics of the controlled object without adding any subsystem is a research hotspot. Moreover, current structural vibration control is focused on structural optimization. However, a centrifugal fan casing belongs to a thin-casing structure, and the vibrational sound power of the thin casing is a quadratic function of the structural vibration velocity [18; 19]. As optimization reduces the structural vibration speed of a fan casing, it can be concluded that the sound power radiation must be reduced within a specific range. The optimal design of a thin-casing structure usually uses the panel thickness as the design variable and the square sum of the vibration velocities of the nodes on the wall as the optimization target function [20; 21]. Adopting the mentioned method, [PERSON] et al. [22] and [PERSON] et al. [23] implemented the optimization study on structure vibration control and noise reduction for the T9-19 No.4A centrifugal fan. By reducing fan casing vibration, an effect of casing noise reduction is achieved using the aforementioned optimization method, which sets the vibration (node vibrational speed) as the target function. However, this method does not consider the propagation of sound waves and sound boundary influences on the calculation results; thus, deviation is nearly inevitable.