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Nonlinear vibration analysis of compressible constrained layer damping using a frequency- and load-dependent virtual material 基于频率-载荷相关虚拟材料的可压缩约束层阻尼非线性振动分析
IF 4.9 2区 工程技术 Q1 ACOUSTICS Pub Date : 2026-03-31 Epub Date: 2025-12-23 DOI: 10.1016/j.jsv.2025.119626
Yongbu Jin, Dong Wang, Di Yuan, Yihan Du, Qiang Wan
A non-invasive methodology for analyzing the nonlinear dynamic response of compressible constrained layer damping (CCLD) is developed within a finite element framework. The key to the proposed method lies in replacing the joint with a novel frequency- and load-dependent virtual material model and extracting model parameters using geometry-independent mapping equations. The CCLD’s frequency response is strongly influenced by the excitation level and design parameters. These effects are tested and simulated on a structure designated as "Pre-tighten Shear". In the experimental section, the effects of excitation, compression level, and silicone foam thickness on the nonlinear behavior of the CCLD over a wide frequency range are investigated. In the numerical simulation, a proposed finite element–based model is employed to analyze the structure. The validity of the method was verified through experimental comparisons, with the MSE not exceeding 4E-03. In addition, the results show that the use of mapping equations offers higher computational efficiency than the classical approach, achieving faster parameter updates at a rate of >60%. A comparison between the simulation and experimental results indicates that interface sliding reduces the stiffness of the joints, with a maximum change in joint damping of about 65%.
提出了一种在有限元框架下分析可压缩约束层阻尼(CCLD)非线性动力响应的无创方法。该方法的关键在于用一种新的频率和载荷相关的虚拟材料模型代替关节模型,并利用与几何无关的映射方程提取模型参数。CCLD的频率响应受激励电平和设计参数的影响很大。这些影响在一个被称为“预紧剪切”的结构上进行了测试和模拟。在实验部分,研究了激励、压缩水平和硅酮泡沫厚度对CCLD在宽频率范围内非线性行为的影响。在数值模拟中,采用提出的基于有限元的模型对结构进行分析。通过实验对比验证了该方法的有效性,MSE均不超过4E-03。此外,结果表明,使用映射方程比经典方法具有更高的计算效率,参数更新速度更快,达到60%。仿真结果与实验结果对比表明,界面滑动降低了节理刚度,节理阻尼的最大变化约为65%。
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引用次数: 0
Inverse quantification of dynamic strains and damping parameters via piezoresistive inversion in self-sensing materials 自传感材料的压阻反演动态应变和阻尼参数的反量化
IF 4.9 2区 工程技术 Q1 ACOUSTICS Pub Date : 2026-03-31 Epub Date: 2026-01-04 DOI: 10.1016/j.jsv.2026.119636
Julio A. Hernandez , Fabio Semperlotti , Hongfei Zhu , Tyler N. Tallman
Self-sensing via the piezoresistive effect (i.e., having deformation-dependent electrical conductivity) has been widely explored in diverse applications. However, the overwhelming body of literature focuses on static or quasi-static loading, limiting the utility of self-sensing technologies in real-world applications where dynamic loading is prevalent. Moreover, while prior research often reports piezoresistive responses as relative changes in resistance or resistivity, end-users would much rather have direct insight into the underlying mechanical state (i.e., stress or strain) that gives rise to the observed piezoresistive response. This manuscript addresses these gaps by demonstrating 1) self-sensing principles can be used to track transient high-frequency strain wave packets and 2) mechanics such as dynamic strains and material-level properties (i.e., damping coefficients) can be deduced from electrical data. We demonstrate these contributions using carbon nanofiber (CNF)-modified epoxy rods subjected to dynamic end-loading while simultaneously recording electrical resistance data from a distributed surface-mounted electrode network along the rod’s length. A piezoresistivity model was then used to deduce strains from electrical measurements, and Rayleigh damping model parameters were extracted from this data. The piezoresistivity data-derived dynamics showed good agreement with finite element simulations, validating our approach to accurately extract the underlying dynamic mechanical state. These results show that it is indeed possible to quantify the mechanical state of a material from electrical data, thereby opening up exciting new possibilities for self-sensing in highly dynamic applications.
通过压阻效应(即具有与变形相关的导电性)的自传感已经在各种应用中得到了广泛的探索。然而,绝大多数文献都集中在静态或准静态加载上,限制了自传感技术在动态加载普遍存在的实际应用中的应用。此外,虽然先前的研究通常将压阻响应报告为电阻或电阻率的相对变化,但最终用户更愿意直接了解导致观察到的压阻响应的潜在机械状态(即应力或应变)。本文通过展示1)自传感原理可用于跟踪瞬态高频应变波包和2)力学,如动态应变和材料级特性(即阻尼系数)可以从电气数据中推断出来,从而解决了这些空白。我们使用纳米碳纤维(CNF)改性环氧树脂棒进行动态端载,同时记录沿棒长度分布的表面安装电极网络的电阻数据,证明了这些贡献。然后使用压阻模型从电测量中推断应变,并从该数据中提取瑞利阻尼模型参数。压电阻率数据推导的动力学结果与有限元模拟结果吻合良好,验证了我们准确提取底层动态力学状态的方法。这些结果表明,确实有可能从电数据中量化材料的机械状态,从而为高动态应用中的自传感开辟了令人兴奋的新可能性。
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引用次数: 0
On the dynamics of viscoelastic cantilever beams under arbitrary base motion and eccentric damped tip mass via integral transform 基于积分变换的粘弹性悬臂梁在任意基座运动和偏心阻尼质量作用下的动力学研究
IF 4.9 2区 工程技术 Q1 ACOUSTICS Pub Date : 2026-03-31 Epub Date: 2026-01-09 DOI: 10.1016/j.jsv.2026.119640
Gianfranco deM. Stieven , Carlos F.T. Matt , Liviu Nicu , Carolina P. Naveira-Cotta , Renato M. Cotta
A comprehensive hybrid analytical-numerical solution is presented for a viscoelastic cantilever Euler-Bernoulli beam with an eccentric damped tip mass, subjected to external excitation, viscous damping, and an arbitrary base motion that undergoes translation and small rotation. The solution is obtained using the Generalized Integral Transform Technique (GITT), based on the application of an implicit filter and an eigenfunction expansion supported by a biharmonic-type eigenvalue problem, yielding a fast and straightforward implementation. A numerically stabilized eigenproblem formulation is proposed, ensuring robust convergence and accurate eigenfunctions. This hybrid solution, presented in a state-space framework, is validated experimentally against damped and undamped natural frequencies, and verified numerically through time-varying free and forced transverse deflection. A physical analysis is presented through four studies: (i) parametric maps of the first two complex eigenvalues, highlighting the distinct modal roles of viscoelastic damping and viscous damping, tip mass magnitude, and eccentricity; (ii) the combined effect of tip-mass eccentricity and internal damping on free and forced vibration; (iii) the influence of tip-mass damping and viscoelastic damping on free and forced vibration; and (iv) a Frequency Response Function (FRF) evaluation considering viscoelastic damping and viscous damping. The resulting formulation delivers fast-convergent solutions, providing closed-form base actions and frequency-response characterizations. The accompanying time- and frequency-domain results, together with compact eigenvalue maps, supply benchmark-quality references that clarify damping and eccentricity effects and support design, identification, and model assessment in linear vibration.
提出了具有偏心阻尼尖端质量的粘弹性悬臂欧拉-伯努利梁的综合混合解析-数值解,该梁受外部激励、粘性阻尼和任意平移和小旋转基底运动的影响。基于隐式滤波器和双调和型特征值问题支持的特征函数展开式,利用广义积分变换技术(GITT)得到了该问题的解,实现速度快,操作简单。提出了一种数值稳定的特征问题公式,保证了鲁棒收敛性和特征函数的准确性。在状态空间框架中提出的混合解决方案在阻尼和无阻尼固有频率下进行了实验验证,并通过时变自由和强制横向挠度进行了数值验证。通过四项研究提出了物理分析:(i)前两个复特征值的参数映射,突出粘弹性阻尼和粘性阻尼,尖端质量大小和偏心的不同模态作用;(ii)端部质量偏心和内部阻尼对自由和受迫振动的综合影响;(iii)尖端质量阻尼和粘弹性阻尼对自由和强制振动的影响;(iv)考虑粘弹性阻尼和粘性阻尼的频响函数(FRF)评估。由此产生的配方提供了快速收敛的解决方案,提供了封闭形式的基动作和频率响应特性。随附的时域和频域结果,以及紧凑的特征值图,提供基准质量参考,澄清阻尼和偏心效应,并支持线性振动的设计,识别和模型评估。
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引用次数: 0
A Harmonic Balance Method with contact condensation for the frequency-domain computation of self-sustained nonlinear vibration related to railway curve squeal 含接触凝聚的谐波平衡法在铁路曲线噪声自持续非线性振动频域计算中的应用
IF 4.9 2区 工程技术 Q1 ACOUSTICS Pub Date : 2026-03-31 Epub Date: 2025-12-31 DOI: 10.1016/j.jsv.2025.119628
J. Arango Montoya , O. Chiello , J.-J. Sinou , R. Tufano
Railway curve squeal is a highly nonlinear phenomenon involving self-sustained vibration of the wheel/rail system and commonly attributed to friction-related instability. While the occurrence of the noise is often studied through a stability analysis based on the linearization of the contact forces, its nonlinear nature requires methods such as time-integration of the dynamic equations of the system in order to determine the amplitude of the oscillations and hence, the radiated sound levels. This kind of methods are computationally expensive. Furthermore, they are not well adapted for the description of the infinite track behaviour, which represents a major challenge. This paper proposes an approach based on the Harmonic Balance Method (HBM), which aims to overcome these difficulties by assuming multi-harmonic periodic solutions and using a frequency-domain representation of the wheel and rail via their receptances at the contact point. The proposed method, which is directly formulated in the frequency-domain, is applied to a curve squeal model where the wheel is modelled via Finite Elements and the track analytically. Results corresponding to the two main instability mechanisms (falling friction and geometrical instability) are presented. The results are in good agreement with time integration and the computational cost is drastically reduced.
铁路弯道尖叫是一种高度非线性的现象,涉及轮轨系统的自持续振动,通常归因于与摩擦有关的不稳定性。虽然通常通过基于接触力线性化的稳定性分析来研究噪声的发生,但其非线性性质需要诸如系统动力学方程的时间积分等方法来确定振荡的幅度,从而确定辐射声级。这种方法在计算上很昂贵。此外,它们不能很好地适应无限轨迹行为的描述,这是一个主要的挑战。本文提出了一种基于谐波平衡法(HBM)的方法,该方法旨在通过假设多谐波周期解和使用轮轨在接触点通过其接收的频域表示来克服这些困难。该方法直接在频域表示,并应用于通过有限元和轨迹解析建模的车轮曲线尖叫模型。给出了两种主要失稳机制(下降摩擦和几何失稳)的对应结果。计算结果与时间积分结果吻合较好,大大降低了计算成本。
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引用次数: 0
Enhancing acoustic scatterer inversion in closed domains with gradient-constrained deep learning 利用梯度约束深度学习增强封闭域声散射反演
IF 4.9 2区 工程技术 Q1 ACOUSTICS Pub Date : 2026-03-31 Epub Date: 2026-01-06 DOI: 10.1016/j.jsv.2026.119644
Chunlin Jia , Zhanyu Li , Zixuan Yu , Hongkuan Zhang , Gengkai Hu
Accurately reconstructing scatterers within closed regions from sparse acoustic measurements presents a challenging inverse problem. Deep learning techniques are widely regarded as effective tools for solving such complex issues. However, conventional approaches often incur significant computational burdens by relying on massive training datasets to boost prediction accuracy. This paper presents an innovative approach that substantially improves network performance not by data augmentation, but by explicitly incorporating physical knowledge through adjoint-derived gradients. The method involves two synergistic stages: firstly, a physics-informed forward model is constructed by integrating gradient information via the adjoint method, which achieves 87 % higher accuracy in acoustic pressure prediction compared to standard data-driven counterparts on the test set; secondly, utilizing the trained forward network as a surrogate model to generate large-scale synthetic datasets for training a robust inverse estimation network. Results demonstrate superior performance: on independent test data, 99.94 % precision in determining scatterer count and high-precision reconstruction with localization resolution of 1/42 wavelength and radius resolution of 1/401 wavelength. Crucially, the method excels even in challenging acoustic shadow zones, surpassing traditional techniques. As the adjoint method is fundamental to sensitivity analysis across computational physics, this gradient-constrained framework can be readily extended to other inverse problems (including inverse electromagnetic scattering and elastic wave-based nondestructive testing) and gradient-based optimization applications like topology optimization, providing a pathway to enhanced accuracy with reduced data dependency.
从稀疏声测量中精确地重建封闭区域内的散射体是一个具有挑战性的逆问题。深度学习技术被广泛认为是解决此类复杂问题的有效工具。然而,传统的方法往往依赖于大量的训练数据集来提高预测精度,从而带来巨大的计算负担。本文提出了一种创新的方法,该方法不是通过数据增强,而是通过伴随衍生的梯度明确地结合物理知识,从而大大提高了网络性能。该方法包括两个协同阶段:首先,通过伴随方法集成梯度信息构建物理信息正演模型,与标准数据驱动的测试集相比,该模型的声压预测精度提高了87%;其次,利用训练好的正向网络作为代理模型,生成大规模的合成数据集,用于训练稳健的逆估计网络;结果表明:在独立测试数据上,散射体数量的确定精度为99.94%,定位分辨率为1/42波长,半径分辨率为1/401波长,高精度重建。至关重要的是,该方法甚至在具有挑战性的声学阴影区域中表现出色,超越了传统技术。由于伴随方法是计算物理灵敏度分析的基础,这种梯度约束框架可以很容易地扩展到其他逆问题(包括逆电磁散射和基于弹性波的无损检测)和基于梯度的优化应用,如拓扑优化,提供了一种通过减少数据依赖性来提高精度的途径。
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引用次数: 0
Tonal noise produced by a UAV propeller due to unsteady rotational motion 无人机螺旋桨非定常旋转运动产生的噪声
IF 4.9 2区 工程技术 Q1 ACOUSTICS Pub Date : 2026-03-31 Epub Date: 2026-01-09 DOI: 10.1016/j.jsv.2026.119646
Huachen Zhu , Ryan Mckay , Michael Kingan , Xianghao Kong , Jin Xuan Teh , Yusuke Hioka , Gian Schmid
A number of recent studies have shown that unsteady rotational motion of small unmanned aerial vehicle propellers can produce tonal noise. In this paper, time- and frequency-domain methods for calculating this noise are presented and validated against one-another. The unsteady loading on the propeller blades, required for the predictions, is calculated using both blade element momentum theory and computational fluid dynamics simulations. The noise prediction methods are validated against measurements. The propeller unsteady motion during these experiments was measured using a rotary encoder and this measured rotational motion was used as input to the noise prediction methods. The results presented in this paper focus on a case where the electric motor drives the propeller in unsteady rotational motion where the unsteady motion is almost sinusoidal with a frequency equal to 14 times the shaft rotation frequency. Predictions show that this unsteady rotational motion produces high amplitude tones at the frequency of the dominant fluctuation speed and adjacent harmonics of the blade passing frequency — confirming the findings of a previous study. These predictions are shown to be in generally good agreement with measurements. In addition, the polar and azimuthal directivity of this tonal noise is investigated.
近年来的一些研究表明,小型无人机螺旋桨的非定常旋转运动可以产生音调噪声。本文提出了时域和频域计算噪声的方法,并对它们进行了对比验证。利用叶片单元动量理论和计算流体力学模拟计算了预测所需的螺旋桨叶片非定常载荷。通过实测验证了噪声预测方法的有效性。实验过程中螺旋桨的非定常运动采用旋转编码器测量,并将测量到的旋转运动作为噪声预测方法的输入。本文的研究结果集中在电机驱动螺旋桨进行非定常旋转运动的情况下,非定常运动几乎是正弦的,频率等于轴旋转频率的14倍。预测表明,这种非定常旋转运动在主导波动速度的频率和叶片通过频率的相邻谐波上产生高振幅音调,证实了先前研究的发现。这些预测大体上与测量结果一致。此外,还研究了该噪声的极向性和方位指向性。
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引用次数: 0
Corrigendum to “Nonlinear Model Reduction to Random Spectral Submanifolds in Random Vibrations” [J. Sound Vib. 600 (2025), 118923] “随机振动中随机谱子流形的非线性模型化简”[J]。Sound Vib. 600 (2025), 118923]
IF 4.9 2区 工程技术 Q1 ACOUSTICS Pub Date : 2026-03-31 Epub Date: 2025-12-27 DOI: 10.1016/j.jsv.2025.119624
Roshan S. Kaundinya , Alice Marraffa , Zhenwei Xu , Shobhit Jain , George Haller
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引用次数: 0
EMetaNode: Electromechanical friction-induced metamaterial node for broadband vibration attenuation and self-powered sensing EMetaNode:用于宽带振动衰减和自供电传感的机电摩擦感应超材料节点
IF 4.9 2区 工程技术 Q1 ACOUSTICS Pub Date : 2026-03-31 Epub Date: 2025-12-11 DOI: 10.1016/j.jsv.2025.119609
Bao Zhao , Lorenzo Di Manici , Raffaele Ardito , Eleni Chatzi , Andrea Colombi , Songye Zhu
Recent advances in mechanical metamaterials and piezoelectric energy harvesting provide exciting opportunities for directing and converting mechanical energy in electromechanical systems for autonomous sensing and vibration control. However, practical realizations remain rare due to the lack of advanced modeling methods and persistent interdisciplinary barriers. By integrating mechanical metamaterials with power electronics-based interface circuits, this paper achieves a breakthrough by presenting an electromechanical friction-induced metamaterial node, which simultaneously enables self-powered sensing and broadband vibration attenuation. To support this, we introduce a reduced-order modeling framework combined with a numerical harmonic balance method tailored for nonlinear metamaterials. This approach efficiently captures local nonlinearities arising from electromechanical coupling through interface circuits, substantially improving computational efficiency. A key innovation of this work is that it uncovers the role of electromechanical friction, induced by synchronized switching interface circuits, which facilitates energy harvesting and enhanced nonlinear dynamic behavior–manifested through expanded bandgaps and higher-harmonic vibration attenuation. Experimentally, an electromechanical metamaterial node is realized for self-powered sensing of temperature and acceleration data, demonstrating strong potential for structural health monitoring and Internet of Things applications. This study provides a practical pathway toward digitizing structures and systems by uniting smart interface circuitry with mechanical metamaterials to achieve autonomous, energy-aware sensing and control.
机械超材料和压电能量收集的最新进展为在机电系统中引导和转换机械能以实现自主传感和振动控制提供了令人兴奋的机会。然而,由于缺乏先进的建模方法和持续的跨学科障碍,实际实现仍然很少。本文通过将机械超材料与基于电力电子的接口电路相结合,实现了机电摩擦感应超材料节点的突破,同时实现了自供电传感和宽带振动衰减。为了支持这一点,我们引入了一种结合非线性超材料的数值调和平衡方法的降阶建模框架。该方法通过接口电路有效捕获机电耦合引起的局部非线性,大大提高了计算效率。这项工作的一个关键创新是,它揭示了由同步开关接口电路引起的机电摩擦的作用,这有助于能量收集和增强非线性动态行为-通过扩大带隙和高谐波振动衰减表现出来。实验中,实现了一个机电超材料节点,用于温度和加速度数据的自供电传感,显示了结构健康监测和物联网应用的强大潜力。本研究通过将智能接口电路与机械超材料结合起来,实现自主的、能量感知的传感和控制,为数字化结构和系统提供了一条实用的途径。
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引用次数: 0
Dynamic performance investigation of squirrel-cage elastic support structure in high-speed rotor system of aeroengine 航空发动机高速转子系统鼠笼式弹性支撑结构动态性能研究
IF 4.9 2区 工程技术 Q1 ACOUSTICS Pub Date : 2026-03-31 Epub Date: 2026-01-03 DOI: 10.1016/j.jsv.2025.119630
Xu-Yuan Song , Chen-Guang Wang , Chang-Xin Yu , Xu Hao , Qing-Kai Han
As a crucial supporting component, the dynamic behaviour of the squirrel-cage elastic support (SCES) structure is vital for determining the operational stability of high-speed rotor systems in aero engines. However, the SCES structure is often oversimplified as a discrete model with restricted degrees of freedom during the traditional modelling process, making it a significant challenge to capture the dynamic behaviours of the system accurately. To overcome this limitation, this literature proposes a detailed analytical dynamic model of the squirrel-cage elastic support structure by approximating it as a medium-thick cylindrical shell structure with multiple circumferential rectangular cutouts. Firstly, the SCES structure is decomposed into several substructures of open cylindrical panels, and the corresponding kinetic and potential energy expressions are derived via the first-order shear deformation theory. Then, a series of orthogonal displacement functions is constructed as the trial functions of substructures. Meanwhile, the penalty functions are applied to ensure the displacement coordination between the substructures. Subsequently, the dynamic equation of the SCES structure is derived via the Rayleigh-Ritz method. After validating the analytical modelling by finite element simulation and experimental investigation, several complex dynamic performances have been revealed, including the phenomenon of modal density, the evolution of degenerate modes in the SCES structure, and the behaviours under multi-point cyclic excitation. The results indicate that the proposed modelling method provides a theoretical basis for dynamic design optimisation and fault diagnosis of squirrel-cage elastic support structures in aircraft engine rotor systems.
鼠笼弹性支承结构作为航空发动机高速旋翼系统的关键支撑部件,其动力学特性对旋翼系统的运行稳定性至关重要。然而,在传统的建模过程中,SCES结构往往被过度简化为一个具有有限自由度的离散模型,这给准确捕捉系统的动态行为带来了重大挑战。为了克服这一局限性,本文提出了一个详细的鼠笼弹性支撑结构解析动力学模型,将其近似为具有多个周向矩形切口的中厚圆柱壳结构。首先,将SCES结构分解为若干开式圆柱板子结构,并利用一阶剪切变形理论推导出相应的动能和势能表达式;然后,构造一系列正交位移函数作为子结构的试函数。同时,采用罚函数来保证子结构之间的位移协调。随后,采用瑞利-里兹法推导了结构的动力方程。通过有限元仿真和试验研究验证了解析模型的正确性,揭示了结构的复杂动力特性,包括模态密度现象、简并模态演化以及多点循环激励下的性能。结果表明,所提出的建模方法为航空发动机转子系统鼠笼式弹性支撑结构的动态设计优化和故障诊断提供了理论依据。
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引用次数: 0
A symplectic numerical power flow framework based on wave finite-element method for assembled structural systems 基于波动有限元法的组合结构体系的辛数值潮流框架
IF 4.9 2区 工程技术 Q1 ACOUSTICS Pub Date : 2026-03-31 Epub Date: 2026-01-02 DOI: 10.1016/j.jsv.2025.119627
Wenjun Wang , Yu Fan , Jiahui Shi , Qing Wu , Anlue Li , Chuanzhen Wang , Daniele Botto
Identifying the propagation paths of dominant wave modes in complex assembled structure is critical for implementing wave-based vibration and noise control strategies, such as phononic band gaps. This paper presents a symplectic numerical framework to compute the wave-mode power flow in engineering assembled structures based on wave finite element method (WFEM). The power orthogonality among wave modes is explicitly formulated through the symplectic orthogonality (SO) and its adjoint form (SAO), and this formulation is further extended to the Zhong-Williams and λ(φ) symplectic schemes. The generalized symplectic adjoint orthogonality (GSAO) and φSAO are subsequently proposed, providing a physically consistent basis for modal diagonalization and coherent wave propagation within the generalized symplectic eigenspace. These developments enable direct computation of the forced response and power flow entirely within the symplectic space, without reverting to the wave space. Six power-flow formulations are systematically compared and shown to yield consistent results on both beam and cylindrical shell structures. An electric motor housing is used as a case study, in which the proposed approach establishes a wave-mode power flow network. It is noted that the power-flow formulation relies on symplectic orthogonality defined for conservative WFEM systems and therefore cannot be directly applied to non-Hermitian systems.
识别复杂装配结构中主波模式的传播路径对于实现基于波的振动和噪声控制策略(如声子带隙)至关重要。本文提出了一种基于波动有限元法计算工程组合结构波浪型功率流的辛数值框架。通过辛正交性(SO)及其伴随形式(SAO)明确地表述了波模间的幂正交性,并将此公式进一步推广到Zhong-Williams和λ(φ)辛格式。随后提出了广义辛伴正交(GSAO)和φSAO,为模态对角化和波在广义辛特征空间内的相干传播提供了物理上一致的基础。这些发展使得完全在辛空间内直接计算强迫响应和功率流,而不需要返回到波空间。系统地比较了六种功率流公式,并表明在梁和圆柱壳结构上产生一致的结果。以电机外壳为例,该方法建立了一个波浪型潮流网络。值得注意的是,功率流公式依赖于为保守WFEM系统定义的辛正交性,因此不能直接应用于非厄米系统。
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引用次数: 0
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Journal of Sound and Vibration
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