Hybrid power flow analysis using coupling loss factor of SEA for low-damping system—Part I: Formulation of 1-D and 2-D cases

This paper proposes a hybrid method for the prediction of vibrational and acoustic responses of low-damping system in the medium-to-high frequency ranges by using the power flow analysis (PFA) algorithm and statistical energy analysis (SEA) coupling concepts. The main part of this method is the application of the coupling loss factor (CLF) of SEA to the boundary condition of PFA in reverberant system. First, for hybrid PFA, the hybrid boundary conditions on 1-D and 2-D cases were derived in the general form. To verify the derived boundary conditions, numerical analyses for each case were performed. The hybrid PFA solutions using derived boundary conditions were compared with the classical PFA solutions with various reverberance factors including the effects of the characteristic length, excitation frequency and group velocity besides damping loss factor of the subsystem. Additionally, the hybrid PFA on 3-D case and the hybrid power flow finite element method (PFFEM) for hybrid PFA of built-up structures are described in the other companion paper.     

中频声振耦合的波函数统计能量法

为了提高中频声振耦合的计算效率,提出了波函数-统计能量法的结构-声学耦合方法,该方法从波动理论的角度出发,将波函数法(WBM)和统计能量法(SEA)结合,通过在耦合面分别施加声压激励和速度边界条件,推导了耦合面参数理论计算公式。将该方法用到长方体声腔和钢板耦合的模型中,并对100~1000 Hz的计算结果进行了实验验证。WBM-SEA模型与参考FEM-SEA模型以及实验模型的频响曲线对比结果表明,WBM-SEA与FEM-SEA以及实验结果吻合很好,验证了混合WBM-SEA的有效性。通过收敛性分析发现混合WBM-SEA方法计算时间比混合FEM-SEA方法更少。从而可以得出结论:混合波函数-统计能量法方法对中频声振耦合预测是有效的,且比FEM-SEA更加高效。     

非保守耦合系统的统计能量分析原理

传统的统计能量分析(SEA)理论不能解决非保守耦合系统的能量分析问题。本文在非保守耦合振子的能量分布与功率流特征的研究基础上,推导了互不相关随机激励条件下非保守耦合系统的功率平衡方程式及各有关功率项的计算式,建立了非保守耦合系统的统计能量分析理论。研究结果表明,保守耦合仅是非保守耦合的一个特例,耦合阻尼对非保守耦合系统的能量分布和功率流的特征有着显著的影响,只有在耦合阻尼远小于系统内阻尼时这种影响才可近似忽略。作为理论的一个应用实例,本文对非保守耦合板的能量问题进行了理论和实验研究。     

Power flow and energy distribution in conservatively or non-conservatively coupled oscillators under correlative excitations

As a foundation of SEA for consevatively or non-conservativelycoupled systems under correlative excitations,power flow and energy distribu-tion in two coupled osillators under proportionally correlative excitations arestudied mainly in this paper.Theoretical relationship among power flow andtime averaged vibational energy of the osillators,as well as power balance equa-tions and energy ratio or them,are derived.Further along,power flow,energydistribution and power balance equations of the systems in conservative ornon-conservative coupling and under proportional correlative excitations arediscussed in detail,with comparing with characteristics of those innon-correlative excitations.Numerical resu1ts supporting the basic theory aregiven,too.Results show that the power flow and energy distribution of theosillators under correlative excitations,either conservative or non-conservativecoupling,are much different to those in the condition of non-correlativeexcitations.     

Prediction of averaged energy for moderately damped systems with strong coupling

In this paper, the prediction of averaged energy of damped structural-acoustic systems is investigated. A simplified energy finite element method, referred to as EFEM⁰, is developed based on the energy flow analysis (EFA) equations and implemented using the finite volume method. The resulting formulations can be incorporated into statistical energy analysis (SEA) software and extends SEA application to moderately damped systems with strong coupling. The formulations are verified against analytical solutions for a single beam and coupled beams with both strong and weak coupling. A hybrid technique consisting of the EFEM⁰ approximation superimposed on a direct field is used to model moderately damped plates. For lightly damped systems, both methods produce acceptable results. For moderately damped two-dimensional systems, the EFEM⁰ method augmented with the direct field component produces significantly improved results.     

SEA and contribution analysis for interior noise of a high speed train

This paper presents a detailed Statistical Energy Analysis (SEA) and contribution analysis of the interior noise of a high-speed train through extensive simulations and measurements. The SEA model was developed based on the actual geometrical parameters of a benchmark high-speed coach. Sound transmission loss levels of the structural components of the car body, which are required in the SEA model, were tested in a dedicated acoustic laboratory following international standard ISO 140-3:1995. Modal densities of these structural components were derived from measured frequency response functions using the modal counting method. Damping loss factors were obtained using the half-power bandwidth method and the vibration attenuation method. By considering the relationship between sound radiation and power transmission, coupling loss factors between structures and cavities were estimated. Source inputs to the SEA model were derived from field experiment data. Interior noise due to those sources was predicted using the SEA model and the prediction was generally in good agreement with measurement. Contribution analysis was then performed using this validated model through parametric study, and this analysis was further examined experimentally. In conclusion, for the coach that was investigated in this paper, the key factors for interior noise are sidewall vibration, bogie area noise, and floor sound transmission loss. Based on this and other engineering considerations, an interior noise control strategy can be defined.     

三维随机粗糙面上导体目标散射的解析-数值混合算法

提出三维导体目标与导体粗糙面复合散射的解析-数值混合迭代算法,推导出三维目标与粗糙面的耦合积分方程,以及粗糙面散射的Kirchhoff近似(KA)计算式.粗糙面的KA解析计算大大降低了粗糙面求解的复杂度,与目标矩量法的混合迭代保证了计算结果的精度,使得三维体-面目标复合散射计算变得可行.由于体-面两者的高阶耦合作用明显减小,保证了该混合迭代算法的收敛性.与镜像Green函数方法的比较表明该混合算法的有效性,并讨论了粗糙面长度选择对计算结果的影响.结合Monte-Carlo方法,数值分析了理想导体Gauss 关键词： 复合散射 Kirchhoff近似 共轭梯度法 互耦迭代     

二维单步交替方向隐式时域有限差分法吸收边界性能分析

给出了两种适用于二维单步交替方向隐式时域有限差分(2-D Leapfrog ADI-FDTD)方法的吸收边界：Mur边界和卷积完全匹配层(CPML)边界。单步交替方向隐式时域有限差分(Leapfrog ADI-FDTD)方法是一种无条件稳定的全隐式差分算法,由于二维空间Leapfrog ADI-FDTD的迭代同时存在显式和隐式方程,故而不同电磁分量的边界条件也存在差异。从原理出发,推导了适用于2-D Leapfrog ADI-FDTD方法的CPML边界条件,并与一阶Mur边界进行比较,利用自由空间的反射误差来表征两种边界的吸收性能,简要总结了两种吸收边界的优缺点。     

Vibration of L-shaped plates under a deterministic force or moment excitation: a case of statistical energy analysis application

Analytical and closed form solutions are presented in this paper for the vibration response of an L-shaped plate under a point force or a moment excitation. Inter-relationships between wave components of the source and the receiving plates are clearly defined. Explicit expressions are given for the quadratic quantities such as input power, energy flow and kinetic energy distributions of the L-shaped plate. Applications of statistical energy analysis (SEA) formulation in the prediction of the vibration response of finite coupled plate structures under a single deterministic forcing are examined and quantified. It is found that the SEA method can be employed to predict the frequency averaged vibration response and energy flow of coupled plate structures under a deterministic force or moment excitation when the structural system satisfies the following conditions: (1) the coupling loss factors of the coupled subsystems are known; (2) the source location is more than a quarter of the plate bending wavelength away from the source plate edges in the point force excitation case, or is more than a quarter wavelength away from the pair of source plate edges perpendicular to the moment axis in the moment excitation case due to the directional characteristic of moment excitations. SEA overestimates the response of the L-shaped plate when the source location is less than a quarter bending wavelength away from the respective plate edges owing to wave coherence effect at the plate boundary.     

Extension of SEA model to subsystems with non-uniform modal energy distribution

In order to widen the application of statistical energy analysis (SEA), a reformulation is proposed. Contrary to classical SEA, the model described here, statistical modal energy distribution analysis (SmEdA), does not assume equipartition of modal energies.Theoretical derivations are based on dual modal formulation described in Maxit and Guyader (Journal of Sound and Vibration 239 (2001) 907) and Maxit (Ph.D. Thesis, Institut National des Sciences Appliquées de Lyon, France 2000) for the general case of coupled continuous elastic systems. Basic SEA relations describing the power flow exchanged between two oscillators are used to obtain modal energy equations. They permit modal energies of coupled subsystems to be determined from the knowledge of modes of uncoupled subsystems. The link between SEA and SmEdA is established and make it possible to mix the two approaches: SmEdA for subsystems where equipartition is not verified and SEA for other subsystems.Three typical configurations of structural couplings are described for which SmEdA improves energy prediction compared to SEA: (a) coupling of subsystems with low modal overlap, (b) coupling of heterogeneous subsystems, and (c) case of localized excitations.The application of the proposed method is not limited to theoretical structures, but could easily be applied to complex structures by using a finite element method (FEM). In this case, FEM are used to calculate the modes of each uncoupled subsystems; these data are then used in a second step to determine the modal coupling factors necessary for SmEdA to model the coupling.     

三维粗糙面电磁双站散射的直接型区域分解计算

提出三维粗糙面双站电磁散射的直接型有限元-区域分解方法.首先建立含有迭代Robin边界条件(IRBC)的区域分解法耦合模型,再用内视法导出高度稀疏分块的分区耦合矩阵,之后给出缩减耦合矩阵带宽的子区域排序方法和IRBC的FFT加速算法.用有限元-完全匹配层和未分区的有限元-IRBC方法验证数值结果.     

Corrected Statistical Energy Analysis Model in a Non-Reverberant Acoustic Space

Statistical Energy Analysis (SEA) is a well-known method to analyze the flow of acoustic and vibration energy in a complex structure. This study investigates the application of the corrected SEA model in a non-reverberant acoustic space where the direct field component from the sound source dominates the total sound field rather than a diffuse field in a reverberant space which the classical SEA model assumption is based on. A corrected SEA model is proposed where the direct field component in the energy is removed and the power injected in the subsystem considers only the remaining power after the loss at first reflection. Measurement was conducted in a box divided into two rooms separated by a partition with an opening where the condition of reverberant and non-reverberant can conveniently be controlled. In the case of a non-reverberant space where acoustic material was installed inside the wall of the experimental box, the signals are corrected by eliminating the direct field component in the measured impulse response. Using the corrected SEA model, comparison of the coupling loss factor (CLF) and damping loss factor (DLF) with the theory shows good agreement.     

涡轮三维叶栅的气热耦合数值模拟

本文借助数值模拟技术分析了三维跨音速涡轮导向叶栅不同壁面边界条件对流场流动状况的影响,壁面边界分别采用绝热、等温、耦合换热三种条件。通过对计算结果的分析表明,不同叶片表面边界条件对流场温度场有很大影响,对流场内的压力场及其它参数也有明显影响,因此在气冷涡轮设计中使用多场耦合技术进行数值模拟是非常必要的。     

Vibro-acoustic analysis of complex systems

A general method is presented for predicting the ensemble average steady-state response of complex vibro-acoustic systems that contain subsystems with uncertain, or random, properties. The method combines deterministic and statistical techniques to produce a non-iterative hybrid method that incorporates equations of dynamic equilibrium and power balance. The method is derived explicitly without reference to statistical energy analysis (SEA); however, it is seen that the wave approach to SEA can be viewed as a special case of the proposed method. The proposed method provides a flexible way to account for necessary deterministic details in a vibro-acoustic analysis without requiring that an entire system be modeled deterministically. The method therefore provides a potential solution to the mid-frequency problem (in which a system is neither entirely deterministic nor entirely statistical). The application of the method is illustrated with a numerical validation example.     

相关激励形式下保守和非保守耦合振子的能量分布与功率流

作为相关输入形式下保守或非保守耦合系统统计能量分析的基础,本文主要研究耦合振子在比例相关输入条件下的能量分布与功率流.在推导了相关输入条件下耦合振子间的功率流与振子能量的基本关系式和功率平衡方程式及振子能量比的表达式之后,本文分别详细讨论了保守耦合振子和非保守耦合振子系统的能量分布和功率流问题,并与互不相关输入时耦合振子的能量分布与功率流的特点进行了比较,对于各种耦合阻尼情况下的振子能量分布与功率流也进行了数值计算,研究结果表明,相关输入下耦合振子的能量分布与功率流的特征与互不相关输入时的特征有着显著的差别。     

相似耦合系统的统计能量分析

根据相关或互不相关激励条件下非保守耦合系统的统计能量分析原理,建立了两个相似的SEA系统之间的能量关系,提出了由已知系统预估未知相似系统的SEA方法。通过对保守和非保守耦合情况下板-梁和板-边框两个相似耦合系统的研究,分别从理论和实验上论证了这种方法的合理性。作为方法的应用,本文研究了两个相似切削系统的切削噪声问题,研究结果表明,理论分析和实验结果有很好的一致性,该方法可以有效地用于复杂耦合系统动态问题的研究。     

Triple-decker: Interfacing atomistic–mesoscopic–continuum flow regimes

Multiscale flow phenomena in microfluidic and biomedical applications require the use of heterogeneous modeling approaches. In this paper we present a hybrid method based on coupling the Molecular Dynamics (MD) method, the Dissipative Particle Dynamics (DPD) method, and the incompressible Navier–Stokes (NS) equations. MD, DPD, and NS are formulated in separate subdomains and are coupled via an overlapping region by communicating state information at the subdomain boundaries. Imposition of boundary conditions in the MD and DPD systems involves particle insertion and deletion, specular wall reflection and body force terms. The latter includes a boundary pressure force in order to minimize near-boundary density fluctuations, and an adaptive shear force which enforces the tangential velocity component of boundary conditions. The triple-decker algorithm is verified for prototype flows, including simple and multi-layer fluids (Couette, Poiseuille, and lid-driven cavity), using highly accurate reference solutions. A zero-thickness interface is also possible if it is aligned with the flow streamlines.     

Efficient parametric uncertainty analysis within the hybrid Finite Element/Statistical Energy Analysis method

This paper is concerned with the development of efficient algorithms for propagating parametric uncertainty within the context of the hybrid Finite Element/Statistical Energy Analysis (FE/SEA) approach to the analysis of complex vibro-acoustic systems. This approach models the system as a combination of SEA subsystems and FE components; it is assumed that the FE components have fully deterministic properties, while the SEA subsystems have a high degree of randomness. The method has been recently generalised by allowing the FE components to possess parametric uncertainty, leading to two ensembles of uncertainty: a non-parametric one (SEA subsystems) and a parametric one (FE components). The SEA subsystems ensemble is dealt with analytically, while the effect of the additional FE components ensemble can be dealt with by Monte Carlo Simulations. However, this approach can be computationally intensive when applied to complex engineering systems having many uncertain parameters. Two different strategies are proposed: (i) the combination of the hybrid FE/SEA method with the First Order Reliability Method which allows the probability of the non-parametric ensemble average of a response variable exceeding a barrier to be calculated and (ii) the combination of the hybrid FE/SEA method with Laplace's method which allows the evaluation of the probability of a response variable exceeding a limit value. The proposed approaches are illustrated using two built-up plate systems with uncertain properties and the results are validated against direct integration, Monte Carlo simulations of the FE and of the hybrid FE/SEA models.     

A novel hybrid ES-FE-SEA for mid-frequency prediction of Transmission losses in complex acoustic systems

The widely-used numerical modeling approaches such as the finite element method (FEM) and statistical energy analysis (SEA) often have limited applicability to the transmission loss prediction in mid-frequency range. In this paper, a novel hybrid edge-based smoothed FEM coupled with statistical energy analysis (ES-FE-SEA) method is proposed to further improve the accuracy of “mid-frequency” transmission loss predictions. The application of ES-FEM will “soften” the well-known ‘‘overly-stiff’’ behavior in the standard FEM solution and reduce the inherent numerical dispersion error. While the SEA approach deals with the physical uncertainty in the relatively higher frequency range. The plate of interest is appropriately described by an ES-FEM model, due to its relative robustness to perturbations. Its adjacent reverberation cavities are modeled by employing the SEA approach, because of their high model density. The coupling and interaction between SEA subsystems and the FE subsystem is governed by the “reciprocity relationship” theorem. A standard numerical example for benchmarking is examined and excellent agreement was achieved between the prediction and reference results. The proposed ES-FE-SEA is also verified by various numerical examples. The method is finally applied to the modeling a complicated engineering problem–acoustic fields on both sides of the front windshield in a passenger car.     

棒板耦合结构的振动能量传输研究

本文以棒、板耦合结构为例,从理论和实验上证明和验证了统计能量方法对强耦合结构仍然适用;进而又对该结构的耦合损耗因子进行了较为详细的计算和讨论,并与弱耦合条件下的统计能量方法结果进行了比较,得出了该方法适用的条件以及在低模态密度情况下耦合损耗因子的表示式;最后通过大量实验对所得结论进行了验证,结果是令人满意的。