NUMERICAL STUDIES ON FAN CASING VIBRATION AND NOISE RADIATION 风扇外壳振动和噪声辐射数值研究
Jian-Cheng CAI*, Da-Tong QI, Fu-An LU 蔡剑诚*、齐大同、吕福安Institute of Fluid Machinery, School of Energy & Power Engineering, Xi'an JiaoTong University 西安交通大学能源与动力工程学院流体机械研究所Xi'an , Shaanxi, P.R.China, 710049 中国陕西西安, 710049Tel: +86-13201613744, E-Mail: 763421532@qq.com Tel: +86-13201613744, E-Mail：763421532@qq.com
Abstract 摘要
This paper presents a method for predicting casing vibration and noise radiation of a centrifugal fan due to interna fluctuating pressure fields at the blade passing frequency (BPF). Using computational fluid dynamics (CFD) technique based on the finite volume method (FVM), a three-dimensional unsteady simulation of internal flow was carried out by solving the Reynolds averaged Navier-Stokes equations. The oscillating pressure distribution at the volute casing was taken as the excitation of the finite element analysis (FEA) model of the casing structure. Time-domain nodal forces on the casing were obtained by integrating the pressure and viscous stress over the element surfaces. Fast Fourier transform (FFT) was applied to these time series of nodal forces in order to extract the BPF components. Harmonic response analysis was carried out to the casing structure. Sound radiation was finally calculated by the indirect boundary element method (IBEM) with the vibration results as velocity boundary condition. This study shows that it is feasible to use fluid-structure weakly coupled simulations for the prediction of flow-induced casing vibration and noise radiation of centrifugal turbomachinary. 本文介绍了一种预测离心风机在叶片通过频率（BPF）下由于内部波动压力场引起的机壳振动和噪声辐射的方法。利用基于有限体积法（FVM）的计算流体动力学（CFD）技术，通过求解雷诺平均纳维-斯托克斯方程，对内部流动进行了三维非稳态模拟。涡壳处的振荡压力分布被作为壳体结构有限元分析（FEA）模型的激励。通过对元素表面的压力和粘性应力进行积分，获得了套管上的时域节点力。对这些节点力的时间序列进行快速傅立叶变换 (FFT)，以提取 BPF 分量。对套管结构进行了谐响应分析。最后采用间接边界元法（IBEM）计算了声辐射，并将振动结果作为速度边界条件。这项研究表明，使用流固弱耦合模拟来预测离心透平机械的流动诱导套管振动和噪声辐射是可行的。
Centrifugal fans, blowers and compressors are widely employed for industrial and civilian use. Their noise has been studied extensively, and can be classified into aerodynamic noise and structural noise. Most studies and literatures about fan noise focus on aeroacoustics. This is due to the fact that aerodynamic noise of centrifugal fans is usually louder than the structural version in normal operation. A thorough literature survey on experimental studies of aerodynamic noise reduction of centrifugal fans is found in Neise [1]. Experimental and analytical studies were conducted to understand the acoustic characteristics of turbomachinery. The results reveal that BPF noise in relation to the impeller rotation plays an important role in sound production [2-4]. Liu et al. [5] numerically studied the aerodynamic noise of a centrifugal fan using the generalized Lighthill's theory. They concluded that to subsonic centrifugal fans, aerodynamic dipole sources contribute mainly to the aeroacoustic noise, especially those near the volute tongue. Similar approach can also be found in Ref. [6, 10]. Unsteady flow fields were simulated first, and aeroacousitc sources (mainly dipole sources to subsonic centrifugal fans) were extracted. Sound radiation was calculated using these sources. All these authors calculated the sound fields outside the machines due to the inner aeroacoustic sources without regarding the presence of the casing, i.e. aeroacoustic sources emit sound to the free field directly. 离心风机、鼓风机和压缩机广泛应用于工业和民用领域。人们对它们的噪声进行了广泛的研究，可将其分为空气动力噪声和结构噪声。大多数关于风机噪声的研究和文献都侧重于空气声学。这是因为离心风机在正常运行时的气动噪声通常比结构噪声大。Neise [1] 对离心风机空气动力降噪实验研究进行了全面的文献调查。为了解透平机械的声学特性，进行了实验和分析研究。研究结果表明，与叶轮旋转有关的 BPF 噪声在产声中起着重要作用 [2-4]。Liu 等人[5]利用广义 Lighthill 理论对离心风机的空气动力噪声进行了数值研究。他们得出结论，对于亚音速离心风机，气动偶极子源是气声噪声的主要来源，尤其是涡舌附近的气动偶极子源。类似的方法也见于参考文献[6, 10]。[6, 10].首先模拟非稳态流场，然后提取气声源（主要是亚音速离心风机的偶极子源）。利用这些声源计算声辐射。所有这些作者都计算了由内部气声源引起的机器外部声场，而没有考虑机壳的存在，即气声源直接向自由场发射声音。
Problems will arise when concerning the presence of the volute casing. It will affect the sound field outside by two ways: one is to impede the transmission of aeroacoustic noise inside the casing to the environment by scattering and reflecting the generated sound; the other is that its own vibration will lead to sound radiation. The present work tries to answer the following questions. What's the magnitude of contribution of fan casing vibroacoustic radiation to the overall sound power radiated by the fan? What does the pattern of casing vibration look like? (This can serve as a reference for further treatment of the fan casing with a damping material if necessary.) This work was part of a series of studies aiming for designing low noise centrifugal fans. The first step was to study various noise sources. This can afford information on what is the major cause of noise and what action should be taken to reduce it. It can also supply useful information about noise in the design of new fans. 涡壳的存在会带来一些问题。它将通过两种方式影响外部声场：一种是通过散射和反射产生的声音，阻碍机壳内的气声噪声向环境传播；另一种是其自身的振动将导致声音辐射。本研究试图回答以下问题。风扇机壳振动声辐射对风扇整体辐射声功率的贡献率有多大？机壳振动的模式是怎样的？(如有必要，这可作为进一步使用阻尼材料处理风扇外壳的参考）。这项工作是旨在设计低噪音离心风机的一系列研究的一部分。第一步是研究各种噪声源。这可以提供有关噪音的主要原因以及应采取何种措施来降低噪音的信息。它还可以为新风机的设计提供有关噪声的有用信息。
An industrial centrifugal fan (the same fan as used in Ref. [5]) was picked as the research target. The sketch of the casing is shown in Fig.1.The fan works at a rotating speed of 2900 with 12 forward curved blades. The outlet diameter of the impeller is . The blade-passing frequency is . Fan casing vibration can be caused by the internal unsteady flow fields and vibration transmitted by other components such as the electric motor. Here we consider only the former excitation based on the following reason. From experimental results, a typical spectrum of normal vibration at one specific location on the volute casing of the running fan is plotted in Fig.2. One can observe that the BPF component is pronounced. This component of vibration is mainly ascribed to the internal unsteady flow fields, because the excitation from the electric motor affects mainly to the rotational frequency component. According to the "China Standards GB/T2888-91: Methods of noise measurement for fans, blowers, compressors and roots blowers", the measured spectrum of sound pressure (1 meter away from the fan outlet) is shown in Fig.3. The sound pressure of BPF component is predominant over other frequencies. 研究对象是一台工业离心风机（与参考文献 [5] 中使用的风机相同）。风扇的外壳简图如图 1 所示。风扇的转速为 2900 ，有 12 个前弯叶片。叶轮的出口直径为 。叶片通过频率为 。风扇外壳的振动可由内部不稳定流场和其他部件（如电机）传递的振动引起。基于以下原因，我们只考虑前者。根据实验结果，图 2 绘制了运行中风机涡壳上某一特定位置的典型法向振动频谱。可以看出，BPF 分量非常明显。该振动分量主要归因于内部不稳定流场，因为来自电机的激励主要影响旋转频率分量。根据 "中国标准 GB/T2888-91：风机、鼓风机、压缩机和罗茨鼓风机噪声测量方法》，测量到的声压频谱（距离风机出口 1 米处）如图 3 所示。从图中可以看出，BPF 部分的声压高于其他频率。
Therefore attention was paid to the BPF component of casing vibration and sound radiation. This means that we can solve frequency-domain wave equation rather than solving time-domain version as was done in Ref. [5, 6, 10]. A great benefit of doing this is that one can obtain the overall sound field and sound power conveniently. This paper studied casing tonal vibroacoustic noise. Study of aeroacoustic tonal noise was being under way, and the results would come out soon. 因此，我们关注了套管振动和声辐射的 BPF 部分。这意味着我们可以求解频域波方程，而不是像参考文献[5、6、10]那样求解时域波方程。[5, 6, 10].这样做的一大好处是可以方便地获得整体声场和声功率。本文研究的是机壳音调振动声学噪声。目前正在对航空声学音调噪声进行研究，很快就会有结果。
Fully transient internal flow fields in the centrifugal fan was simulated by commercial CFD code-ANSYS CFX. Time series of the pressure fluctuations at the volute wall were obtained. Time-domain nodal forces on the casing were obtained by integrating the pressure and viscous stress over the element surfaces through the interface of ANSYS CFX to ANSYS. Simulation of the volute casing dynamic response at BPF was carried out using FEA technique in ANSYS. The results were imported to the BEA acoustic radiation model in LMS SYSNOISE as velocity boundary condition, where noise emission due to casing vibration was finally solved. 商用 CFD 代码--ANSYS CFX 模拟了离心风机中的全瞬态内部流场。获得了涡壁压力波动的时间序列。通过 ANSYS CFX 与 ANSYS 的接口对元件表面上的压力和粘性应力进行积分，获得了机壳上的时域节点力。在 ANSYS 中使用有限元分析技术对 BPF 时的涡壳动态响应进行了模拟。仿真结果作为速度边界条件被导入到 LMS SYSNOISE 中的 BEA 声辐射模型中，最终解决了由套管振动引起的噪声排放问题。
Fig. 1 Schematic of the volute casing 图 1 涡壳示意图
SIMULATION OF INTERNAL FLOW FIELDS 内部流场模拟
A three dimensional numerical simulation of the complete unsteady flow on the whole impeller-volute configuration was carried out using the computational fluid dynamics (CFD) code ANSYS CFX. Because of the low Mach number (Ma<0.3) flow of this centrifugal fan, the fluid was assumed to be incompressible. Thus, the continuity equation (1) and the momentum equation (2) could be solved independent of the 使用计算流体动力学（CFD）代码 ANSYS CFX 对整个叶轮-叶片配置上的完整非稳定流进行了三维数值模拟。由于离心风机的低马赫数（Ma<0.3）流动，流体被假定为不可压缩。因此，连续性方程 (1) 和动量方程 (2) 的求解与流体的流速无关。
energy equation. Due to the low pressure rise, we further regarded the flow as isothermal (air at ). Therefore only the continuity equation and the momentum equation were solved in the simulation. 能量方程。由于压力上升较低，我们进一步将气流视为等温气流（空气温度为 ）。因此，模拟中只求解了连续性方程和动量方程。
(a)
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Fig. 2 Typical fan casing acceleration spectrum: (a) Location of measuring point; (b) Spectrum 图 2 典型的风机外壳加速度频谱：（a）测量点位置；（b）频谱
Fig. 3 Spectrum of sound pressure 图 3 声压频谱
where and are fluid velocity, density, body force, pressure and dynamic viscosity respectively, and denotes time. The rate-of-deformation tensor of fluid is defined as 其中， 和 分别为流体速度、密度、体力、压力和动态粘度， 表示时间。流体 的变形率张量定义为
Since the flow in the centrifugal fan is typically threedimensional turbulent, it is appropriate to solve the Reynolds averaged Navier-Stokes equations (RANS) economically without loss of resolving the main characteristics of flow fields such as mean pressure fluctuations. CFD technology has been widely employed in simulating the fully 3-D unsteady flow in centrifugal turbomachinery[5-9], and good agreement was found between the numerical and the experimental results[8,9]. We were trying to simulate the flow fields using almost the same techniques, such as boundary conditions, described in Ref.[9] with the anticipation to obtain good results. 由于离心风机中的流动是典型的三维湍流，因此宜采用经济的雷诺平均纳维-斯托克斯方程（RANS）求解，同时又不失对平均压力波动等流场主要特征的解析。CFD 技术已被广泛应用于模拟离心透平机械中的全三维非稳态流动[5-9]，并发现数值结果与实验结果之间具有良好的一致性[8,9]。我们尝试使用与文献[9]中描述的几乎相同的技术（如边界条件）来模拟流场，期望获得良好的结果。
For the three dimensional calculations, structured hexahedral cells were used to define the inlet zone, the impeller and the volute (with a total of cells). Some mesh crosssections are shown in Fig.4. Concerning the numerical simulation parameters, velocity inlet according to a specific flow rate and pressure outlet boundary conditions were applied at the inlet and outlet. The turbulence characteristics of the flow were modeled by the standard equations. The scalable wall-function was used to describe the near wall velocity in stead of standard wall-function. One of the major drawbacks of the standard wall-function approach is that the predictions depend on the location of the point nearest to the wall and are sensitive to the near-wall meshing; refining the mesh does not necessarily give a unique solution of increasing accuracy. Scalable wall functions overcome this drawback in that they can be applied on arbitrarily fine meshes. A second order high resolution discretization of advection terms was applied, and the transient terms were approximated by second order backward Euler scheme. The sliding mesh technique was applied to the interfaces in order to allow the unsteady interactions between the impeller and the volute casing. The RMS residue is lower than . 在三维计算中，使用了结构化六面体单元来定义入口区、叶轮和涡槽（共 个单元）。部分网格横截面如图 4 所示。关于数值模拟参数，在入口和出口处应用了根据特定流速的入口速度 和出口压力边界条件。流动的湍流特性由标准的 方程建模。用可扩展壁面函数代替标准壁面函数来描述近壁速度。标准壁面函数方法的主要缺点之一是预测结果取决于最靠近壁面的点的位置，并且对近壁网格很敏感；细化网格并不一定能得到精度越来越高的唯一解。可扩展壁面函数克服了这一缺点，因为它们可以应用于任意精细的网格。对平流项采用二阶高分辨率离散化，对瞬态项采用二阶后向欧拉方案近似。为了考虑叶轮和涡壳之间的非稳态相互作用，对界面采用了滑动网格技术。残差有效值低于 。
The CFD simulation process began with a steady flow calculation using the frozen-rotor approach. In this case, the relative position of the impeller and the casing does not change during the calculation. For unsteady calculation, the grids change their relative positions during the calculation according to the angular velocity of the impeller. A complete impeller revolution was divided into 512 time steps, i.e., each time step spans 60/2900/512=4.041e-5s. The chosen time step is related to the rotational speed of the impeller and it is small enough to CFD 模拟过程首先使用冻结转子方法进行稳定流计算。在这种情况下，叶轮和机壳的相对位置在计算过程中不会发生变化。对于非稳态计算，网格在计算过程中会根据叶轮的角速度改变其相对位置。一个完整的叶轮旋转被分为 512 个时间步，即每个时间步的跨度为 60/2900/512=4.041e-5s。所选的时间步长与叶轮的旋转速度有关，它要足够小，以满足以下要求
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Fig. 4 Sketch of CFD mesh: (a) Mesh of circumference at midspan of volute; (b) Mesh at meridional plane 图 4 CFD 网格草图：（a）涡流中跨圆周网格；（b）子午线网格
get the necessary time resolution and to capture the phenomena of the pressure fluctuations. A total of 2048 time steps were calculated. 以获得必要的时间分辨率并捕捉压力波动现象。总共计算了 2048 个时间步长。
The results of simulation were satisfying. The phenomenon of leakage flow from the volute into the impeller at the shroud side was captured (see Fig.5), so was the secondary flow. In the calculation, some pressure monitoring points were set near the shroud. The locations and evolution of pressure fluctuations with time are shown in Fig.6. For convenience of interpretation, only three pressure fluctuations were plotted. From Fig. 6 (b), one can see the quasi-periodicity of pressure fluctuations. The shapes are sinusoidal in nature, with the BPF as the primary frequency. A typical pressure distribution on the volute wall at a certain time step is shown in Fig.7. From Fig.7, it can be seen that at a specific time step, pressure at the volute wall increases along the radial direction outward, and the pressure distribution around the volute tongue is rather complicated. Fig. 8 plots the pressure fluctuations of five 模拟结果令人满意。在护罩一侧捕捉到了从涡槽进入叶轮的泄漏流现象（见图 5），二次流也是如此。计算中，在护罩附近设置了一些压力监测点。压力波动的位置和随时间的变化如图 6 所示。为便于理解，只绘制了三个压力波动点。从图 6 (b) 可以看出，压力波动具有准周期性。其形状呈正弦曲线，以 BPF 为主频。图 7 显示了某一时间步长下涡流壁上的典型压力分布。从图 7 可以看出，在特定的时间步长内，涡壁的压力沿径向向外增加，涡舌周围的压力分布比较复杂。图 8 显示了五个时间步长的压力波动情况。
Fig. 5 Leakage flow between volute and impeller 图 5 涡轮和叶轮之间的泄漏流量
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Fig. 6 (a) Pressure monitoring points' locations; (b) Evolution of pressure fluctuations with time steps 图 6 (a) 压力监测点的位置；(b) 压力波动随时间步长的变化情况
monitoring points around the volute tongue. The locations are shown in Fig. 6 (a). Periodicity is still obvious, but the fluctuations of some points have notable second and third harmonics of the BPF component, rather than sinusoidal. Besides, the amplitudes of fluctuations are much larger than those of points on the front volute wall. (The strong pressure fluctuations at the tongue contribute a major part of discrete frequency aeroacoustics.) 涡舌周围的监测点。监测点位置如图 6 (a)所示。周期性仍然很明显，但某些点的波动明显带有 BPF 分量的二次和三次谐波，而不是正弦波。此外，波动的振幅比前涡壁上的点要大得多。(舌部的强烈压力波动是离散频率气动声学的主要部分）。
The pressure fluctuations and viscous stress were converted into time-varying forces which could be applied to each node of the FEA model by integrating the pressure and viscous stress over the element surfaces through the interface to ANSYS multiphysics module provided by ANSYS CFX. We were not trying to solve the structural dynamic response in time domain by exerting these fluctuating forces. Due to the small damping of the casing structure, long duration numerical integration would be required to achieve the steady state of structural vibration. This would introduce large numerical errors if special techniques were not used. We, on the contrary, solved the volute structural vibration in frequency domain. This was also based on the fact that only the BPF component of vibration was interested in. FFT technique was applied to each node's time-domain forces of 2048 time steps, in order to obtain the spectra of the forces. Fig. 9 shows one node's normal direction time-varying force (one revolution) and its spectrum. (Its location is shown in Fig.2(a).) From the spectrum, it can be seen the BPF harmonic component has the largest amplitude (counting out the direct current component, i.e. ). The tangential force is about 2 or 3 orders of magnitude smaller than the normal force, and is rather irregular. 通过 ANSYS CFX 提供的 ANSYS 多物理场模块接口，将压力波动和粘性应力转换为时变力，并将其施加到有限元分析模型的每个节点上。我们并没有试图通过施加这些波动力来解决时域结构动态响应问题。由于机壳结构的阻尼很小，因此需要进行长时间的数值积分才能实现结构振动的稳定状态。如果不使用特殊技术，这将带来很大的数值误差。相反，我们在频域中解决了涡管结构振动问题。这也是基于我们只对振动的 BPF 分量感兴趣这一事实。为了获得力的频谱，我们对每个节点 2048 个时间步的时域力采用了 FFT 技术。图 9 显示了一个节点的法线方向时变力（一圈）及其频谱。(其位置如图 2（a）所示）。从频谱可以看出，BPF 谐波分量的振幅最大（不包括直流分量，即 ）。切向力大约比法向力小 2 或 3 个数量级，而且相当不规则。
Fig. 7 Pressure distribution on the volute wall 图 7 涡流壁上的压力分布
All the time-domain node forces were converted into frequency-domain counterparts. Real and imaginary parts of node forces at BPF were obtained, and were used as external excitation to the volute casing when executing harmonic response analysis. Thus a uni-directional fluid-structure interaction (one way coupling) was realized. Jiang et al. [11] used one-way coupling technique to study the flow-induced casing vibration of a pump, without considering the influence of the structural vibration on the internal flow. They justified this scheme basing on the following two arguments. The vibration of structure can act on the flow through two possible ways. One is the movement of the fluid boundary, and the other is the stimulated elastic wave, i.e. a hydroacoustic field. However, the displacement of wall surface created by elastic vibration is very small and the flow is incompressible. Furthermore the characteristic Mach number is much smaller than one. Although this case involves an air fan, rather than a water pump, these two conditions are met correspondingly. 所有时域节点力都被转换为对应的频域力。获得了 BPF 节点力的实部和虚部，并在进行谐波响应分析时将其用作涡壳的外部激励。这样就实现了单向流固耦合。Jiang 等人[11]使用单向耦合技术研究了流量引起的泵壳振动，而没有考虑结构振动对内部流动的影响。他们基于以下两点论证了这一方案。结构振动可以通过两种可能的方式对流动产生影响。一种是流体边界运动，另一种是受激弹性波，即水声场。然而，弹性振动产生的壁面位移非常小，而且流动是不可压缩的。此外，特征马赫数远小于 1。虽然本例涉及的是风扇而不是水泵，但这两个条件也相应地得到了满足。
Fig. 8 Pressure fluctuations at the volute tongue 图 8 涡舌处的压力波动
SIMULATION OF CASING VIBRATION 套管振动模拟
The volute structure system is considered as a linear timeinvariant (LTI) system. This is because its dynamic response is small and the structure (mass, rigidity etc.) doesn't change at operation. The assumption of a LTI system guarantees that modal analysis and harmonic response analysis can be applied to the volute casing. The dynamic response of linear elastic structure is subject to 涡流结构系统被视为线性时不变（LTI）系统。这是因为其动态响应较小，且结构（质量、刚度等）在运行时不会发生变化。LTI 系统的假设保证了可以对涡壳进行模态分析和谐波响应分析。线性弹性结构的动态响应受制于
where and are material's density, Young's modulus and Poisson's ratio, respectively. and are the structure's displacement and body force respectively. Besides the usual displacement and force boundary conditions of the structure, extra interface conditions should be fulfilled at the interface of fluid and structure, 其中 和 分别为材料的密度、杨氏模量和泊松比。 和 分别为结构的位移和体力。除了结构的常规位移和力边界条件外，流体和结构的界面还需要满足额外的界面条件、
Displacement compatibility: 排量兼容性：
Traction equilibrium: 牵引平衡
where and are unit normal vector, fluid stress tensor and structure stress tensor at the fluid-structure interface correspondingly. The displacement compatibility condition might not be satisfied strictly because the fully coupled fluidstructure interaction problem was reduced to a weakly coupled version, as done in Ref.[11]. However the traction equilibrium from fluid to structure is still satisfied. 其中 和 分别为流固界面上的单位法向量、流体应力张量和结构应力张量。位移相容条件可能无法严格满足，因为完全耦合的流固耦合问题被简化为弱耦合版本，如参考文献[11]所做的那样。但是，流体与结构之间的牵引平衡仍然得到了满足。
After using FEA technique, equation (4) becomes discrete in matrix form: 使用有限元分析技术后，方程 (4) 变为离散矩阵形式：
where and are structural mass, damping and stiffness matrix (damping is taken into account here). The nodal acceleration, velocity and displacement vectors are and , respectively, and the applied load vector is . 其中 和 为结构质量、阻尼和刚度矩阵（此处已考虑阻尼）。节点加速度矢量、速度矢量和位移矢量分别为 和 ，外加载荷矢量为 。
To modal analysis, ignoring the damping, equation (7) is reduced to 在模态分析中，忽略阻尼，方程 (7) 简化为
For a linear system, free vibration will be harmonic of the form 对于线性系统，自由振动的谐波形式为
is the eigenvector representing the mode shape of the th natural frequency, and is the th circular frequency (radians per unit time). Thus, equation (8) becomes: 是代表 th 固有频率模态形状的特征向量， 是 th 圆周频率（单位时间内的弧度）。因此，方程 (8) 变为
This equation has non-trivial solution when the determinant of is zero, 当 的行列式为零时，该方程有非微分解、
This is an eigenvalue problem which may be solved for up to values of and eigenvectors. Rather than natural frequencies in radians per second, we prefer natural frequencies , cycles per unit time, i.e. . 这是一个特征值问题，最多可求解 和 的特征向量值。与以每秒弧度为单位的自然频率相比，我们更倾向于以每单位时间的周期为单位的自然频率 ，即 。
In harmonic response analysis, all the applied loads vary sinusoidally at the same frequency, although not necessarily in phase. Equation (7) becomes, 在谐波响应分析中，所有施加的负载都以相同的频率正弦变化，但不一定相位相同。公式 (7) 变为
where, is the imposed circular frequency, is the square root of -1 . and are real and imaginary displacement vectors, and are real and imaginary force vector. 其中， 是施加的圆周频率， 是 -1 的平方根。 和 是实位移矢量和虚位移矢量， 是实力矢量和虚力矢量。
In the FEA model, a total of 34,896 elements of SHELL63 type (ANSYS) were used to construct the volute structure. SHELL63 has both bending and membrane capabilities. Both in-plane and normal loads are permitted. The element has six degrees of freedom at each node: translations in the nodal , and directions and rotations about the nodal , and -axes. The material is steel, so the density , Young's modulus , and Poisson's ratio . The volute casing was fixed to the supporting stand by four fastening bolts at the casing backside. Three translational degrees of freedom of nodes at bolts were restricted to zero. 在有限元分析模型中，共使用了 34 896 个 SHELL63 类型的元素（ANSYS）来构建涡流结构。SHELL63 具有弯曲和膜能力。允许承受平面内载荷和法向载荷。该元素在每个节点上有六个自由度：在节点 和 方向上的平移以及绕节点 和 轴的旋转。材料为钢，因此密度 、杨氏模量 和泊松比 。涡壳通过壳体背面的四个紧固螺栓固定在支架上。螺栓处节点的三个平移自由度被限制为零。
In order to validate the casing FEA model, modal analysis was performed firstly and the results were compared to those of experimental modal analysis (EMA). From numerical modal analysis, the first 7 modes and 8 modes whose frequencies are nearby BPF were extracted. Modal impact experiment was conducted on the casing. A total of 201 points on the volute casing surface were impacted by moving the hammer. Three accelerometers were placed at three fixed locations to measure the responses. Therefore, a total of frequency response functions were obtained. Modal parameters (natural frequencies and damping) were estimated by the least squares complex exponential method [12]. The computed natural frequencies are compared with the measured ones in Table 1 and Table 2. 为了验证套管有限元分析模型，首先进行了模态分析，并将分析结果与实验模态分析（EMA）结果进行了比较。从数值模态分析中提取了前 7 个模态和频率在 BPF 附近的 8 个模态。对套管进行了模态冲击实验。通过移动锤子对涡壳表面的 201 个点进行了冲击。在三个固定位置放置了三个加速度计来测量响应。因此，总共获得了 频率响应函数。模态参数（固有频率和阻尼）是通过最小二乘复指数法估算得出的[12]。表 1 和表 2 将计算出的自然频率与测量值进行了比较。
From Table 1, good agreement is found between FEA and EMA results. The maximal relative error among these natural frequency pairs of FEA and EMA results is . From Table 2, natural frequencies of FEA and EMA around BPF agree well. Thus the FEA model of the casing is accurate to certain degree and can be used for other further dynamic response analysis. Here, it should be noted that, from modal impact results another two low natural frequencies were identified, and . Judging from the EMA results of modal shapes, we regarded these two modal shapes mainly rigid modals, i.e. the volute casing as a whole can sway. These two modals had not been obtained in FEA, because we assumed that the supporting stand fully rigid. 从表 1 中可以看出，有限元分析和 EMA 结果之间的一致性很好。有限元分析和 EMA 结果的这些固有频率对之间的最大相对误差为 。从表 2 中可以看出，FEA 和 EMA 在 BPF 周围的固有频率吻合得很好。因此，套管的有限元分析模型在一定程度上是准确的，可用于其他进一步的动态响应分析。这里需要指出的是，从模态冲击结果中还确定了另外两个较低的固有频率，即 和 。从模态振型的 EMA 结果来看，我们认为这两个模态振型主要是刚性模态振型，即涡壳作为一个整体可以摇摆。这两个模态没有在有限元分析中得到，因为我们假定支撑架是完全刚性的。
The damping is a rather subtle parameter. It usually includes hysteretic or material damping, damping in structural joints, acoustic radiation damping, air pumping and aerodynamic damping [13]. Estimated damping ratios were about of the poles around BPF in the stabilization plots of modal analysis, therefore we took the damping ratio to 阻尼是一个相当微妙的参数。它通常包括滞后阻尼或材料阻尼、结构连接处的阻尼、声辐射阻尼、气泵和空气动力阻尼[13]。在模态分析的稳定图中，BPF 周围极点的估计阻尼比约为 ，因此我们将阻尼比 取为
account for all the damping effects on the harmonic response at BPF. Nodal solutions of displacement are plotted in Fig.10. We can see the vibration of petal-like local deformation at BPF The petals behave as sound radiation sources. The maximum amplitudes of each petal are on the order of several micrometers. 图 10 中绘制的是位移的节点解。图 10 中绘制了位移的节点解。 我们可以看到在 BPF 处的花瓣状局部变形振动。每个花瓣的最大振幅约为几微米。
Table 1: Comparison between computed natural frequencies and measured ones of the first 7 modes 表 1：前 7 个模态的计算固有频率与测量固有频率的比较
FEA(Hz)
51.1
62.0
90.3
118
142
187
199
EMA(Hz)
51.7
61.9
83.0
115
131
178
194
Rela. Er. Rela.Er.
1.2
0.16
8.8
2.6
8.4
5.1
2.6
Table 2: Comparison between computed natural frequencies and measured ones around 表 2：计算的自然频率与周围测量频率的比较
FEA(Hz)
530.35
547.32
549.98
563.51
EMA(Hz)
530.55
546.90
552.74
574.08
FEA (Hz)
588.22
594.68
631.13
649.51
EMA(Hz)
593.98
615.37
645.69
666.49
SIMULATION OF SOUND RADIATION 声辐射模拟
The governing equation of sound fields generated by vibrating structures is the wave equation, 振动结构产生声场的控制方程是波方程、
where and are the speed of sound and sound pressure respectively. At the vibrating structure boundary, the relationship between the normal acoustic pressure gradient of the fluid and the normal acceleration of the structure is: 其中 和 分别为声速和声压。在振动结构边界，流体的法向声压梯度与结构的法向加速度之间的关系为
(a) where is the mean fluid density. The time-domain wave equation can be transformed into Helmholtz equation in frequency-domain after Fourier transform, (a) 其中 为平均流体密度。经过傅里叶变换后，时域波方程可转化为频域亥姆霍兹方程、
where is wave number, defined as . And equation (14) becomes: 其中 是波数，定义为 。方程（14）变为
This is equivalent to give the second (Neumann) boundary condition to the Helmholtz equation. Equation (16) can also be written as: 这相当于为 Helmholtz 方程给出了第二个（Neumann）边界条件。方程 (16) 也可以写成
where is the normal fluid velocity at the interface, which is equal to the normal structural velocity due to the velocity compatibility condition. The Sommerfeld radiation condition must be satisfied at the boundary surface located at infinity, in order to ensure that all acoustic waves propagate freely towards infinity and that no reflections occur at this boundary, 其中 是界面处的流体法向速度，由于速度相容条件，它等于结构法向速度。位于无穷远处的边界面必须满足索默费尔德辐射条件，以确保所有声波都能自由地向无穷远处传播，并且在该边界上不会发生反射、
The boundary element method (BEM) has been used extensively in numerical acoustics for computing noise radiated from vibrating objects. The basic concept of BEM is to describe the physical phenomena in the interior of a volume by suitable quantities on the surface of the volume (i.e. on the boundary). Montgomery et al. [14] used BEM to predict the sound radiation from a vibrating compressor housing based on experimental velocity response data. Koopmann et al. [15] also employed BEM together with experimental vibration data of the casing to compute the casing noise radiation of a centrifugal fan. 在数值声学中，边界元法（BEM）被广泛用于计算振动物体辐射的噪声。BEM 的基本概念是用体积表面（即边界）上的合适量来描述体积内部的物理现象。Montgomery 等人[14] 根据速度响应实验数据，使用 BEM 预测了振动压缩机外壳的声辐射。Koopmann 等人[15]也利用 BEM 和机壳的实验振动数据来计算离心风机的机壳噪声辐射。
We employed indirect boundary element method (IBEM) to 我们采用间接边界元素法（IBEM）
(b)
Fig. 10 Amplitudes of nodal total displacement at 580Hz: (a) Real part; (b) Imaginary part 图 10 580Hz 时节点总位移的振幅：（a）实部；（b）虚部
calculate the acoustic fields outside the centrifugal fan (due to the fact that the volute casing didn't constitute a closed boundary surface), and the velocity response of structure was obtained by FEM rather than by measured results. 在计算离心风机外部声场时（由于涡壳不构成封闭的边界面），结构的速度响应是通过有限元而不是测量结果获得的。
To acoustic problems, the acoustic pressure and the normal velocity constitute the primary variables for the direct boundary element method, while for indirect method they are double layer and single layer potentials related to jump of pressure and normal velocity respectively. The formulation relating the pressure at the point of an acoustic field to the distributions of a single layer potential and a double layer potential on the boundary surface is 对于声学问题，声压和法向速度构成直接边界元法的主要变量，而对于间接法，它们分别是与压力和法向速度的跃迁有关的双层和单层势。声场点的压力与边界面上的单层势 和双层势 分布有关的公式 是
where and denote the locations of acoustic field and acoustic boundary respectively, and is the Green's kernel function, 其中 和 分别表示声场和声边界的位置， 是格林核函数、
The single layer and double layer potentials are the difference in normal pressure gradient and the pressure difference between both sides of the boundary surface , respectively, 单层势和双层势分别是法向压力梯度差和边界表面两侧的压力差 、
The positions and indicate the boundary surface positions at the positive and negative side of the normal direction respectively. 和 分别表示法线方向 正负两侧的边界曲面位置。
The boundary condition involved in sound radiation of the fan volute casing was the Neumann boundary condition, and by assuming the volute shell a thin boundary surface, it gives 风扇涡壳的声辐射所涉及的边界条件是诺伊曼边界条件，假设涡壳为薄边界面，则可得出
The results of harmonic response analysis in ANSYS were imported into SYSNOISE. SYSNOISE is a program for modeling acoustic wave behavior in fluids, fluid-structure interaction, using implementations of the finite element and boundary element methods. Through equations (16) and (17), the structural displacement in ANSYS was converted into velocity boundary conditions in SYSNOISE. Because the structural FEA mesh and the acoustic IBEM mesh are not compatible, a geometrical interpolation algorithm was used to project the structural results onto the acoustic mesh. And the normal velocities of each acoustic element were obtained. The interpolated result of normal velocity at BPF is shown in Fig.11. The maximum frequency, for which the discretization of the acoustic mesh is valid, is based on the 'minimum six elements per wavelength' criterion according to the sound velocity of air [16]. This choice is explained by considering one wavelength (see Fig.12): to identify the locations of peaks and troughs with reasonable accuracy, seven points can be spaced-out from start to finish of the wave. Zero pressure jump of pressure conditions were applied to the free edges. A spherical acoustic field mesh, whose radius is to the center of the volute, was constructed. Firstly, the double layer potential on the boundary surface was calculated. Then, the sound pressure distributions on the field mesh were obtained using equation (19), and the results are shown in Fig. 13 . ANSYS 中的谐波响应分析结果被导入 SYSNOISE。SYSNOISE 是使用有限元和边界元方法对流体中的声波行为和流固耦合进行建模的程序。通过公式 (16) 和 (17)，ANSYS 中的结构位移被转换为 SYSNOISE 中的速度边界条件。由于结构有限元网格和声学 IBEM 网格不兼容，因此采用几何插值算法将结构结果投影到声学网格上。然后得到每个声学元素的法向速度。BPF 处的法向速度插值结果如图 11 所示。根据空气声速[16]的 "每波长最少六个元素 "标准，声学网格离散有效的最大频率为 。考虑到一个波长（见图 12），就可以解释这种选择：为了以合理的精度确定波峰和波谷的位置，从波的起点到终点可以间隔七个点。自由边缘采用零压力跃变条件。构建了一个球形声场网格，其半径为 ，以涡流中心为圆心。首先，计算边界面上的双层势。然后，利用公式 (19) 求出声场网格上的声压分布，结果如图 13 所示。
In Fig.10, there are several petal-like local vibrations on both the front and back sides of the volute casing. This leads to complicated directivity and several local maximums and minimums of sound pressure on the acoustic field spherical 在图 10 中，涡壳的前后两侧都有几个花瓣状的局部振动。这导致声场球面上出现复杂的指向性和多个局部声压最大值和最小值。
Fig. 12 Spacing of nodes at maximum frequency 图 12 最高频率下的节点间距
Fig. 13 Sound pressure distributions at away from the center of the volute (BPF) 图 13 远离涡流中心的声压分布（BPF）
mesh. The maximum sound pressure amplitude at BPF on the field sphere mesh is , i.e. . The computed radiated sound power is , i.e. . 网格。场球网格上 BPF 处的最大声压幅值为 ，即 。计算得出的辐射声功率为