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An insight on local defect resonance based on modal decomposition analysis: A two-dimensional case 基于模态分解分析的局部缺陷共振分析:二维案例
IF 4.3 2区 工程技术 Q1 ACOUSTICS Pub Date : 2024-09-04 DOI: 10.1016/j.jsv.2024.118718

The resonance behaviors of defects, specifically local defect resonance (LDR), are useful for defect detection in structures. However, there is a lack of intuitive elucidation regarding the underlying mechanisms governing defect resonance. This study aims to analyze the LDR of a horizontal crack in a plate activated by Lamb waves using the modal decomposition method (MDM). The MDM involves an approximate decomposition of the wavefield into finite constituent Lamb wave modes, including propagating, non-propagating, and inhomogeneous modes. The amplitude of each mode is determined according to the continuous boundary conditions of displacement and stress between the damaged zone and the intact zone. Analytical results reveal specific frequencies at which the damaged zone exhibits significantly stronger vibrations than the intact zone, indicating the emergence of LDR. The corresponding wavefields exhibit various resonance patterns, encompassing both in-plane LDR and out-of-plane LDR types. In contrast to finite element simulations and experimental observations, the MDM unveils the dominant constituents of resonance patterns: out-of-plane LDR results from the combination of reflected and transmitted A0 and A1 modes within the damaged zone, while in-plane LDR is formed by the superposition of reflected and transmitted A0, A1, and S0 modes.

缺陷的共振行为,特别是局部缺陷共振(LDR),对结构中的缺陷检测非常有用。然而,关于缺陷共振的基本机制还缺乏直观的阐释。本研究旨在利用模态分解法(MDM)分析由 Lamb 波激活的板中水平裂纹的 LDR。MDM 包括将波场近似分解为有限组成的 Lamb 波模态,包括传播模态、非传播模态和不均匀模态。每种模式的振幅根据受损区和完好区之间位移和应力的连续边界条件确定。分析结果表明,在特定频率上,受损区的振动明显强于完好区,这表明出现了 LDR。相应的波场表现出各种共振模式,包括平面内 LDR 和平面外 LDR 两种类型。与有限元模拟和实验观察不同的是,MDM 揭示了共振模式的主要成分:面外 LDR 由受损区域内反射和传输的 A0 和 A1 模式组合而成,而面内 LDR 则由反射和传输的 A0、A1 和 S0 模式叠加而成。
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引用次数: 0
Locally resonant metamaterials damped by particles embedded through additive manufacturing 通过增材制造嵌入颗粒阻尼的局部谐振超材料
IF 4.3 2区 工程技术 Q1 ACOUSTICS Pub Date : 2024-09-04 DOI: 10.1016/j.jsv.2024.118715

Damping properties in elastic metamaterials are essential for enhancing vibration suppression within frequency ranges known as bandgaps. However, fabricating metamaterials with high-damping materials through additive manufacturing presents challenges, limiting the design freedom of metamaterials. This study proposes locally resonant metamaterials damped by particles, embedded using powder bed fusion of metals with a laser beam. Particles that remain unfused in the powder bed fusion process are retained as particle dampers, adding damping through friction and collisions between particles. Unlike traditional viscoelastic damping materials, metal particles offer advantages in terms of thermal durability. To investigate the damping effect on wave propagation within the metamaterials, we identified the frequency- and acceleration-dependent damping properties of the particle dampers through vibration testing. These properties were then used to define analytical models for the interaction between the metamaterials and the transition of damping via dispersion analysis. The dispersion analysis indicates that wave attenuation occurs over wider frequency ranges. Additionally, the vibration suppression capability of the proposed metamaterials was experimentally demonstrated through vibration testing of fabricated specimens, resulting in the suppression of vibration responses in the out-of-band gaps frequencies by more than 10 dB. The findings of this study enable the fabrication of elastic metamaterials with damping properties via additive manufacturing, offering a method to suppress vibrations across a wide range of frequencies.

弹性超材料的阻尼特性对于增强带隙频率范围内的振动抑制至关重要。然而,通过增材制造技术制造具有高阻尼材料的超材料是一项挑战,限制了超材料的设计自由度。本研究提出了由微粒阻尼的局部谐振超材料,利用激光束对金属进行粉末床熔融嵌入。在粉末床熔融过程中未熔化的颗粒被保留下来作为颗粒阻尼器,通过颗粒之间的摩擦和碰撞增加阻尼。与传统的粘弹性阻尼材料不同,金属颗粒在热耐久性方面具有优势。为了研究超材料内部波传播的阻尼效应,我们通过振动测试确定了颗粒阻尼器与频率和加速度相关的阻尼特性。然后利用这些特性,通过频散分析确定超材料之间相互作用和阻尼转换的分析模型。频散分析表明,波衰减发生在更宽的频率范围内。此外,通过对制作好的试样进行振动测试,实验证明了所提出的超材料的振动抑制能力,从而将带外间隙频率的振动响应抑制了 10 分贝以上。这项研究的结果使我们能够通过增材制造技术制造出具有阻尼特性的弹性超材料,从而提供了一种在广泛频率范围内抑制振动的方法。
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引用次数: 0
Semi-analytical solutions for steady-state bending-bending coupled forced vibrations of a rotating wind turbine blade by means of Green's functions 利用格林函数对旋转风力涡轮机叶片的稳态弯曲耦合受迫振动进行半解析求解
IF 4.3 2区 工程技术 Q1 ACOUSTICS Pub Date : 2024-09-03 DOI: 10.1016/j.jsv.2024.118714
Wind power has received significant attention in recent years as a renewable energy source. Wind turbine blades, characterized by their long lengths, are prone to damage as a result of aeroelastic instability. This paper aims to derive semi-analytical solutions for steady-state forced vibrations of rotating wind turbine blades, considering the coupling of bending, bending, and unsteady aerodynamic loads. The novelty of this work lies in the use of the Green's function method to solve differential dynamic equations with variable coefficients, which are induced by aerodynamic forces. To model a wind turbine blade, the Euler-Bernoulli beam model is employed, and Greenberg's expressions are taken into consideration. Governing equations for coupled vibrations of the blade are then determined. In order to solve the governing equations, displacement solutions are decomposed into quasi-static displacements and dynamic displacements. The Laplace transformation and Green's function methods are utilized to obtain fundamental solutions of the governing equations. Subsequently, employing the principle of superposition, Fredholm integral equations for steady-state forced vibration of a rotating wind turbine blade are derived. To spatially discretize Fredholm integral equations, compound trapezoid formulae and central finite difference approximations are employed. This discretization process leads to the formation of a system of algebraic equations. Semi-analytical solutions for coupled forced vibrations of the rotating wind turbine blade are obtained by solving the algebraic equations. In the numerical solution part, validation of proposed solutions are verified by comparing them with numerical and finite element solutions in the literature. Influences of some important physical parameters, such as the rotating velocity, the setting angle, the cone angle, the inflow ratio, and damping, on vibration responses of blades are discussed.
近年来,风能作为一种可再生能源备受关注。风力涡轮机叶片的特点是长度较长,容易因气动弹性不稳定而损坏。本文旨在推导旋转风力涡轮机叶片稳态受迫振动的半解析解,其中考虑了弯曲、折弯和非稳态空气动力载荷的耦合。这项工作的新颖之处在于使用格林函数法来求解具有可变系数的微分动力方程,这些系数是由空气动力引起的。为建立风力涡轮机叶片模型,采用了欧拉-伯努利梁模型,并考虑了格林伯格表达式。然后确定叶片耦合振动的支配方程。为了求解支配方程,位移解被分解为准静态位移和动态位移。利用拉普拉斯变换和格林函数方法,得到控制方程的基本解。随后,利用叠加原理,推导出旋转风力涡轮机叶片稳态受迫振动的弗雷德霍姆积分方程。为了对弗雷德霍姆积分方程进行空间离散化,采用了复梯形公式和中心有限差分近似法。这一离散化过程形成了一个代数方程系统。通过求解代数方程,可获得风轮机叶片旋转耦合受迫振动的半解析解。在数值求解部分,通过与文献中的数值和有限元求解进行比较,验证了所提求解的有效性。讨论了一些重要物理参数对叶片振动响应的影响,如旋转速度、设置角、锥角、流入比和阻尼。
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引用次数: 0
Acoustic capsule: A structure that can carry different objects but obtain the same acoustic radiation force 声囊:可承载不同物体但获得相同声辐射力的结构
IF 4.3 2区 工程技术 Q1 ACOUSTICS Pub Date : 2024-09-02 DOI: 10.1016/j.jsv.2024.118711

An acoustic capsule is designed utilizing equivalent anisotropy of density and of sound speed. The designed capsule can carry any object with dimensions smaller than those of its inner cavity. The capsule achieves internal shielding of the sound field, thus ensuring that the capsule and its contents experience the same acoustic radiation force (ARF), regardless of any changes in the contents. An acoustic capsule design method is proposed and its feasibility is verified using a combination of theoretical analysis and finite element simulations. Additionally, optimization strategies to enhance the ARF and enable selection of appropriate material parameters are discussed, and these strategies pave the way toward practical applications of the ARF in life sciences and other fields.

利用密度和声速的等效各向异性设计了一个声囊。所设计的舱体可承载任何尺寸小于其内腔尺寸的物体。舱体实现了声场的内部屏蔽,从而确保舱体及其内装物无论发生何种变化,都能承受相同的声辐射力(ARF)。本文提出了一种声学胶囊设计方法,并通过理论分析和有限元模拟相结合的方法验证了该方法的可行性。此外,还讨论了增强 ARF 和选择适当材料参数的优化策略,这些策略为 ARF 在生命科学和其他领域的实际应用铺平了道路。
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引用次数: 0
Mutual coupling of resonators: The effect of excessive acoustic resistance and its reduction by a local acoustic resistance 谐振器的相互耦合:过大声阻抗的影响以及局部声阻抗对其的削弱
IF 4.3 2区 工程技术 Q1 ACOUSTICS Pub Date : 2024-09-02 DOI: 10.1016/j.jsv.2024.118713

Hybrid resonance structures, characterized by parallel arrangements, play a crucial role in expanding the bandwidth of sound absorption. It involves the assembly of a set of resonators with different resonance frequencies to form broadband sound absorption a peak-dip band structure. While the mechanism behind peak formation is well elucidated, the genesis of dips remains shrouded. In this study, we utilize two coupled resonators to demonstrate that dips arise from anti-resonance, induced by the mutual coupling of resonator impedances with neighboring resonance frequencies. Further investigations uncover that the origin of mutual coupling can be attributed to the velocity dipole response of coupled resonator, resulting in the generation of evanescent waves and the effect of excessive acoustic resistance. Contrary to intuition, we introduce local acoustic resistance by shunt electromechanical diaphragm (SEMD) to mitigate this excessive acoustic resistance of the coupled resonators, exhibiting a promising reduction in overall acoustic resistance at the anti-resonance frequency. Subsequent exploration reveals that this 'acoustic resistance reducing acoustic resistance' approach significantly enhances sound absorption at the dip frequency, particularly under random incidence conditions. Our research encompasses analytical, numerical, and experimental studies, which are in concordance with each other.

以平行排列为特征的混合共振结构在扩大吸声带宽方面发挥着至关重要的作用。它涉及一组具有不同共振频率的谐振器的组装,以形成宽带吸声的峰-洼带状结构。虽然峰值形成背后的机制已被很好地阐明,但凹陷的成因却仍然是个谜。在这项研究中,我们利用两个耦合谐振器证明了骤降源于反共振,而反共振是由相邻谐振频率的谐振器阻抗相互耦合引起的。进一步的研究发现,相互耦合的起源可归因于耦合谐振器的速度偶极子响应,从而产生蒸发波和过量声阻抗的影响。与直觉相反,我们通过分流机电膜片(SEMD)引入局部声阻,以减轻耦合谐振器的过大声阻,在反共振频率处显示出整体声阻的减少。随后的研究发现,这种 "声阻减小声阻 "的方法显著提高了倾角频率的吸声效果,尤其是在随机入射条件下。我们的研究包括分析、数值和实验研究,这些研究相互一致。
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引用次数: 0
Isogeometric modeling and vibro-acoustic analysis of flow-excited irregular cavity-plate-exterior space coupled system 流激不规则空腔-板-外部空间耦合系统的等距建模和振动声学分析
IF 4.3 2区 工程技术 Q1 ACOUSTICS Pub Date : 2024-09-01 DOI: 10.1016/j.jsv.2024.118712

The flow-induced noise has become an important noise source in marine sonar self-noise, which can adversely affect the normal operation of sonar. The marine sonar cabin is simplified as a cavity-plate-exterior space coupled system whose flow-induced vibro-acoustic characteristics are investigated in this paper. An isogeometric vibro-acoustic formulation is proposed in which the cavity with an irregular geometry is precisely described by adjusting the control points and corresponding weights. The flow-induced vibro-acoustic response is obtained by transferring the turbulent pressure data from computational fluid dynamics into the isogeometric vibro-acoustic model. Imposing turbulent pressure into the isogeometric control points is proposed to achieve this objective using a node-based interpolation method. The vibro-acoustic modeling is validated and compared with previous experimental results. These comparisons demonstrate that the developed formulation accurately predicts the vibro-acoustic characteristics of the fluid-excited coupled system. The influences of flow speed, acoustic medium, and cavity shape on flow-excited vibration and sound radiation are discussed. Results show a decrease in radiated acoustic power and radiation efficiency in the exterior space, and a shift in the plate-exterior space coupling modal frequency to lower frequencies when the cavity changes from convex to concave.

流致噪声已成为海洋声纳自噪声中的一个重要噪声源,会对声纳的正常工作产生不利影响。本文将海洋声纳舱简化为一个腔板-外部空间耦合系统,研究其流动诱发的振动声学特性。本文提出了一种等几何振动声学公式,通过调整控制点和相应的权重来精确描述具有不规则几何形状的空腔。通过将计算流体动力学中的湍流压力数据转移到等几何振声模型中,可获得流动诱导的振声响应。为了实现这一目标,建议使用基于节点的插值方法将湍流压力引入等距控制点。振动声学模型与之前的实验结果进行了验证和比较。这些比较结果表明,所开发的公式能准确预测流体激励耦合系统的振动声学特性。讨论了流速、声学介质和空腔形状对流激振动和声辐射的影响。结果表明,当空腔从凸形变为凹形时,外部空间的辐射声功率和辐射效率下降,板-外部空间耦合模态频率向低频移动。
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引用次数: 0
Combining transfer learning and numerical modelling to deal with the lack of training data in data-based SHM 结合迁移学习和数值建模,解决基于数据的 SHM 缺乏训练数据的问题
IF 4.3 2区 工程技术 Q1 ACOUSTICS Pub Date : 2024-09-01 DOI: 10.1016/j.jsv.2024.118710

Structural health monitoring (SHM) involves continuously surveilling the performance of structures to identify progressive damage or deterioration that might evolve over time. Recently, machine learning (ML) algorithms have been successfully employed in various SHM applications, including damage detection. However, supervised ML algorithms often require labelled data for multiple possible damage states of the structure for successful damage identification. Although it may be feasible to gather such data for low-value structures, obtaining damage data for expensive structures such as aircraft could be highly challenging. Herein, this data insufficiency is addressed by combining Finite Element (FE) models with domain adaptation, specifically transfer component analysis (TCA) and joint domain adaptation (JDA). The proposed methodology is showcased in two case studies, a Brake–Reuß beam, where damage scenarios correspond to different torque settings on a lap joint and a wingbox laboratory structure where damage is introduced as saw-cuts. Supervised learning algorithms in the form of Artificial Neural Networks (ANNs) and K-Nearest Neighbours (KNNs) are trained based on FE data after domain adaptation is applied and are then tested with the experimental data. It is shown that even though the performance of classifiers in distinct scenarios of dual, three, four and five-class cases is sensitive to choices in the training stage, the use of TCA or JDA allows for the use of FE data for training and significantly reduces the need for expensive experimental damage data to be used for training. These results can pave the way for a broader use of ML algorithms in SHM of critical and/or expensive structures.

结构健康监测(SHM)是指对结构性能进行持续监测,以识别可能随时间推移而发生的渐进式损坏或劣化。最近,机器学习(ML)算法已成功应用于包括损伤检测在内的各种 SHM 应用中。然而,有监督的 ML 算法通常需要对结构的多种可能损坏状态进行标注数据,才能成功识别损坏。虽然收集低价值结构的此类数据可能是可行的,但获取飞机等昂贵结构的损坏数据却极具挑战性。本文通过将有限元(FE)模型与域自适应相结合,特别是转移分量分析(TCA)和联合域自适应(JDA),解决了数据不足的问题。所提出的方法在两个案例研究中得到了展示,一个是 Brake-Reuß 梁,其损坏情况与搭接接头的不同扭矩设置相对应;另一个是翼盒实验室结构,其损坏情况为锯切。人工神经网络 (ANN) 和 K-Nearest Neighbours (KNN) 形式的监督学习算法在应用域适应后根据 FE 数据进行训练,然后用实验数据进行测试。结果表明,尽管分类器在双级、三级、四级和五级等不同情况下的性能对训练阶段的选择很敏感,但使用 TCA 或 JDA 可以使用 FE 数据进行训练,并大大减少了使用昂贵的实验损坏数据进行训练的需要。这些结果可以为在关键和/或昂贵结构的 SHM 中更广泛地使用 ML 算法铺平道路。
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引用次数: 0
Compliant curved beam support with flexible stiffness modulation for near-zero frequency vibration isolation 具有柔性刚度调节功能的顺应性曲线梁支架,可实现近零频率隔振
IF 4.3 2区 工程技术 Q1 ACOUSTICS Pub Date : 2024-08-30 DOI: 10.1016/j.jsv.2024.118702

Passive isolation of low-frequency vibrations poses a significant challenge due to the requirement of possessing a low stiffness. Herein, a compliant curved beam support (CCBS) for near-zero frequency vibration isolation is proposed and systematically investigated. The CCBS exploits nonlinear negative stiffness provided by an arc-shaped beam support to modulate nonlinear positive stiffness of a spiral-shaped beam support. This nonlinear configuration gives rise to an interesting phenomenon, that is, the stiffness modulation of the CCBS for quasi-zero stiffness (QZS) can be flexibly fulfilled by performing a translation transformation on negative stiffness instead of reshaping it. A thorough static analysis is implemented to reveal this stiffness modulation mechanism. The dynamic governing equation is derived and solved analytically and numerically to calculate the displacement transmissibility, and the effects of excitation amplitude, damping, and applied load are dissected. Finally, various excitation tests are conducted to experimentally evaluate vibration isolation performance. The results demonstrate that the CCBS exhibits a remarkably low resonance frequency of 0.5 Hz and achieves near-zero isolation starting at 0.8 Hz, showcasing excellent performance in isolating low-frequency vibrations. The proposed CCBS provides a novel paradigm for achieving flexible low stiffness modulation in compact QZS isolators, making it highly deserving of promotion.

由于要求具有较低的刚度,因此被动隔离低频振动是一项重大挑战。本文提出了一种用于近零频率振动隔离的顺应性弧形梁支架(CCBS),并对其进行了系统研究。CCBS 利用弧形梁支架提供的非线性负刚度来调节螺旋形梁支架的非线性正刚度。这种非线性配置产生了一个有趣的现象,即通过对负刚度进行平移变换而不是重塑,可以灵活地实现准零刚度(QZS)CCBS 的刚度调制。为了揭示这种刚度调制机制,我们进行了全面的静态分析。推导出动态调节方程,并通过分析和数值求解计算出位移传递率,同时剖析了激励振幅、阻尼和外加载荷的影响。最后,进行了各种激振试验,对隔振性能进行了实验评估。结果表明,CCBS 的共振频率非常低,仅为 0.5 Hz,而且从 0.8 Hz 开始就实现了近乎零的隔振效果,在隔离低频振动方面表现出色。所提出的 CCBS 为在紧凑型 QZS 隔离器中实现灵活的低刚度调制提供了一种新的范例,因此非常值得推广。
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引用次数: 0
Extreme value analysis of multivariate Gaussian processes with wave-passage effects 具有波程效应的多元高斯过程的极值分析
IF 4.3 2区 工程技术 Q1 ACOUSTICS Pub Date : 2024-08-29 DOI: 10.1016/j.jsv.2024.118703

Extreme value analysis is a central aspect of random vibration applications. Most studies focus on a univariate process. System reliability necessitates the extreme value across multiple correlated processes, but analytical methods are scarce and confined to low-dimensional problems. Recently, the authors proposed an analytical method for the extreme analysis of multivariate Gaussian processes. The exact upcrossing rate is derived for the maximum process representing the instantaneous maxima over all processes, and the extreme value distribution is obtained from the Poisson approximation. Nevertheless, for applications involving the wave-passage effect that is commonplace in random vibration, the upcrossings manifest in clumps, rendering the Poisson approximation conservative. The clumping from wave-passage is a complex novel phenomenon, differing from the clumping in narrowband processes. This paper extends the prior work by developing an analytical method for predicting the clump size, thereby providing an accurate prediction of the multivariate extreme value while accounting for the wave-passage effect. The method is powerful as it is fast and amenable to high-dimensional problems. Two examples include the propagation of ocean waves and a multi-span bridge subjected to propagating ground motions. The proposed method is shown to accurately predict the clumping factor and the probability of failure, compared to numerical simulations. In contrast, the Poisson approximation using the exact upcrossing rate noticeably overestimates the failure probability.

极值分析是随机振动应用的一个核心方面。大多数研究侧重于单变量过程。系统可靠性要求对多个相关过程进行极值分析,但分析方法很少,而且仅限于低维问题。最近,作者提出了一种多变量高斯过程极值分析方法。对于代表所有过程瞬时最大值的最大值过程,推导出了精确的上交叉率,并通过泊松近似得到了极值分布。然而,在涉及随机振动中常见的波浪传递效应的应用中,上交表现为团块,使得泊松近似变得保守。波穿越产生的团块是一种复杂的新现象,不同于窄带过程中的团块。本文扩展了之前的工作,开发了一种预测团块大小的分析方法,从而在考虑波穿越效应的同时准确预测多元极值。该方法快速且适用于高维问题,因此功能强大。两个例子包括海浪的传播和多跨桥梁受传播地动的影响。结果表明,与数值模拟相比,所提出的方法能准确预测结块因子和失效概率。相比之下,使用精确上交率的泊松近似法明显高估了失效概率。
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引用次数: 0
Localization of acoustical sources rotating in the cylindrical duct using a sparse nonuniform microphone array 使用稀疏非均匀传声器阵列定位圆柱形管道中旋转的声源
IF 4.3 2区 工程技术 Q1 ACOUSTICS Pub Date : 2024-08-25 DOI: 10.1016/j.jsv.2024.118699

The Doppler effect caused by the rotation of turbine fans brings significant challenges to the identification of fan noise sources inside the nacelle. Current methods of rotating source localization inside the cylindrical duct are executed with tough requirements for the microphone array mounted on the duct wall, which includes a fair enough number of microphones and equal-space distribution of each microphone in the azimuthal direction. A methodology based on nonuniform measurements and duct spin modes superposition (NMDMS) is proposed to identify rotating sources with high spatial resolution and few side lobes even near cut-on frequencies, which requires much fewer microphones and no uniform distribution of each microphone in the circumference. The sparsity in the azimuthal domain of the duct field generated by multiple rotating sources is verified theoretically, after which the azimuthal mode in each duct cross-section is reconstructed by the inhomogeneous measurements through the orthogonal matching pursuit (OMP) method. Followed by the identification of mode amplitude in the rotation frame, the sound pressure distribution as well as the axial acoustic velocity distribution on the source plane is reconstructed through duct modes summation. Numerical simulations and experiments are implemented to validate the method. Localization results indicate that rotating sources identification using the proposed method could not only save more than half the microphones without requirements of equal-space distribution but obtain good accuracy of localization and remarkably fewer side lobes.

涡轮风扇旋转引起的多普勒效应给机舱内风扇噪声源的识别带来了巨大挑战。目前在圆柱形风道内进行旋转声源定位的方法对安装在风道壁上的传声器阵列有严格的要求,其中包括足够数量的传声器和每个传声器在方位角方向上的等空间分布。本文提出了一种基于非均匀测量和管道自旋模态叠加(NMDMS)的方法,用于识别旋转声源,具有较高的空间分辨率和较少的侧叶,甚至在切入频率附近也是如此,这就需要更少的传声器和每个传声器在圆周上的均匀分布。在理论上验证了多个旋转源产生的风道场在方位域的稀疏性,然后通过正交匹配追求(OMP)方法,利用不均匀测量重建每个风道横截面的方位模。在确定旋转框架中的模式振幅后,通过管道模式求和重建声源平面上的声压分布和轴向声速分布。为验证该方法,进行了数值模拟和实验。定位结果表明,使用所提出的方法识别旋转声源,不仅可以节省一半以上的传声器,而且不需要等空间分布,并能获得良好的定位精度和明显较少的侧叶。
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引用次数: 0
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Journal of Sound and Vibration
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