Analysis of the modal energy distribution of an excited vibrating panel coupled with a heavy fluid cavity by a dual modal formulation

This paper describes the modal interaction between a panel and a heavy fluid cavity when the panel is excited by a broad band force in a given frequency band. The dual modal formulation (DMF) allows describing the fluid–structure coupling using the modes of each uncoupled subsystem. After having studied the convergence of the modal series on a test case, we estimate the modal energies and the total energy of each subsystem. An analysis of modal energy distribution is performed. It allows us to study the validity of SEA assumptions for this case. Added mass and added stiffness effects of the fluid are observed. These effects are related to the non-resonant acoustic modes below and above the frequency band of excitation. Moreover, the role of the spatial coupling of the resonant cavity modes with the non-resonant structure modes is also highlighted. As a result, the energy transmitted between the structure and the heavy fluid cavity generally cannot be deduced from the SEA relation established for a light fluid cavity.     

Response variance prediction in the statistical energy analysis of built-up systems

In the statistical energy analysis (SEA) of high frequency noise and vibration, a complex engineering structure is represented as an assembly of subsystems. The response of the system to external excitation is expressed in terms of the vibrational energy of each subsystem, and these energies are found by employing the principle of power balance. Strictly the computed energy is an average taken over an ensemble of random structures, and for many years there has been interest in extending the SEA prediction to the variance of the energy. A variance prediction method for a general built-up structure is presented here. Closed form expressions for the variance are obtained in terms of the standard SEA parameters and an additional set of parameters alpha(k) that describe the nature of the power input to each subsystem k, and alpha(ks) that describe the nature of the coupling between subsystems k and s. The theory is validated by comparison with Monte Carlo simulations of plate networks and structural-acoustic systems.     

Frequency average loss factors of plates and shells

This paper describes a thorough investigation of the measurement of frequency band average loss factors of structural components for use in the statistical energy analysis (SEA) method of computation of vibration levels. The “traditional” method of measurement is to excite the structure by a random force having a flat spectral density in the frequency band of interest. The force is then cut off and the decay of the modes excited in the band is noted. The average loss factor is deduced from the decay curve. The alternative energy method is the subject of this study. In this test the power input from the band limited random force is measured and the spatial average vibration level of the structure is estimated from several surface accelerometers. It is shown that when the modes in the band have similar loss factors (as is usually the case) the energy method gives a result which is very close to that obtained from the decay method. These in turn are close to the arithmetic average of the loss factors of the individual modes in the band. It is shown that only when the band contains one or two very lightly damped modes amongst several more heavily damped modes is there a difference between the two methods. In this case it is better to use the energy result in the SEA calculations.     

弯管对末端带弹性障板充液管路辐射声能量的影响

基于声固耦合有限元方法建立了末端带弹性障板的充液管路数值模型,重点分析了不同激励下弯管对管口辐射声能量的影响.结果表明:弯管引入的高阶周向模式耦合使结构振动和流体声传播都发生明显改变,以致系统辐射声能量及主要能量贡献源也发生转移,并随激励方式和频率而不同.对本文管路模型,平面波激励下弯管系统在低频的结构辐射声能量明显增...     

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.     

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.     

A quantitative criterion validating coupling power proportionality in statistical energy analysis

The response of two general spring-coupled elements is investigated to develop a unifying approach to the weak coupling criterion in Statistical Energy Analysis (SEA). First, the coupled deterministic equations of motion are expressed in the bases given by the uncoupled elements’ eigenmodes. Then, an iterative solution is expressed as a succession of exchanges between elements, where uncoupled motion provides the start approximation, converging if the ‘coupling eigenvalue’ is less than unity, in which case coupling is said to be weak. This definition is related to whether response is ‘local’ or ‘global’, encompassing a number of previously defined coupling strength definitions, applying for deterministically described structures. A stochastic ensemble is defined by that its members are equal to the investigated structure but the elements have random frequencies. It is required that the coupling eigenvalue be less than unity for all members of the ensemble. This requirement generates the title subject of the article: ‘the modal interaction strength’. It is similar to the previously defined coupling strength criterion characterising the ensemble average energy flow in uni-dimensional waveguides. Finally, SEA models are formulated in terms of the uncoupled elements’ modal data.     

The Fröhlich kinetic equation with additional terms

The Fröhlich kinetic equation (FKE) describes the coherent excitation in a band of polarization waves consisting of normal modes. The FKE involves the one and two quanta terms representing energy supply, energy losses and energy channeling between the modes with the help of the heat bath. The energy channeling between the vibration systems located at different positions and the channeling connected with multiple quanta processes without the help of the heat bath are analyzed. The additional terms of the FKE are derived. The multiple quanta processes may channel the energy from the coherently excited mode (or modes) to higher frequency ones.     

Reduced models for the medium-frequency dynamics of stochastic systems

In this paper, a frequency domain vibration analysis procedure of a randomly parametered structural system is described for the medium-frequency range. In this frequency range, both traditional modal analysis and statistical energy analysis (SEA) procedures well-suited for low- and high-frequency vibration analysis respectively, lead to computational and conceptual difficulties. The uncertainty in the structural system can be attributed to various reasons such as the coupling of the primary structure with a variety of secondary systems for which conventional modeling is not practical. The methodology presented in the paper consists of coupling probabilistic reduction methods with dynamical reduction methods. In particular, the Karhunen-Loeve and Polynomial Chaos decompositions of stochastic processes are coupled with an operator decomposition scheme based on the spectrum of an energy operator adapted to the frequency band of interest.     

Vibrational energy transfer in a protein molecule

Mode coupling in a protein molecule was studied by a molecular dynamics simulation of the intramolecular vibrational energy transfer in myoglobin at near zero temperature. It was found that the vibrational energy is transferred from a given normal mode to a very few number of selective normal modes. These modes are selected by the relation between their frequencies, like Fermi resonance, governed by the third order mode coupling term. It was also confirmed that the coupling coefficients had high correlation with how much the coupled modes geometrically overlapped with each other.     

A hybrid approach for predicting the distribution of vibro-acoustic energy in complex built-up structures

Finding the distribution of vibro-acoustic energy in complex built-up structures in the mid-to-high frequency regime is a difficult task. In particular, structures with large variation of local wavelengths and/or characteristic scales pose a challenge referred to as the mid-frequency problem. Standard numerical methods such as the finite element method (FEM) scale with the local wavelength and quickly become too large even for modern computer architectures. High frequency techniques, such as statistical energy analysis (SEA), often miss important information such as dominant resonance behavior due to stiff or small scale parts of the structure. Hybrid methods circumvent this problem by coupling FEM/BEM and SEA models in a given built-up structure. In the approach adopted here, the whole system is split into a number of subsystems that are treated by either FEM or SEA depending on the local wavelength. Subsystems with relative long wavelengths are modeled using FEM. Making a diffuse field assumption for the wave fields in the short wave length components, the coupling between subsystems can be reduced to a weighted random field correlation function. The approach presented results in an SEA-like set of linear equations that can be solved for the mean energies in the short wavelength subsystems.     

Linear and nonlinear electromagnetic coupling models in vibration-based energy harvesting

This paper investigates the response of an energy harvester that uses electromagnetic induction to convert ambient vibration into electrical energy. A unique aspect of the present study is the comparison of the system's response behavior when either a linear or a physically motivated form of nonlinear coupling is applied. The motivating hypothesis for this work was that nonlinear coupling could be used to improve the performance of an energy harvester by broadening its frequency response. Combined theoretical and numerical studies investigate the harvester's response for both single and multi-frequency base excitation. Our investigations unveil regions in the parameter space where nonlinear coupling is better than linear coupling and regions where the opposite is true. The meaningful conclusion is that nonlinear coupling can sometimes be detrimental, but it can also be beneficial if properly designed into the system.     

Validity diagrams of statistical energy analysis

This paper is concerned with the validity domain of statistical energy analysis (SEA) which is defined in terms of four criteria. The mode count N and the modal overlap M must be high, the normalized attenuation factor and the coupling strength γ must be small. The application of dimensional analysis on the governing equations of plates gives the space of dimensionless parameters in which the validity domain of SEA must be delimited. This domain is discussed on the basis of geometry of the surfaces delimiting it. The diagrams of validity of SEA are introduced and discussed. A numerical simulation on a couple of rectangular plates coupled along one edge illustrates the theoretical approach.     

Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber

This paper deals with the application of the concept of targeted energy transfer to the field of acoustics, providing a new approach to passive sound control in the low frequency domain, where no efficient dissipative mechanism exists. The targeted energy transfer, also called energy pumping, is a phenomenon that we observe by combining a pure nonlinear oscillator with a linear primary system. It corresponds to an almost irreversible transfer of vibration energy from the linear system to the auxiliary nonlinear one, where the energy is finally dissipated. In this study, an experimental set-up has been developed using the air inside a tube as the acoustic linear system, a thin circular visco-elastic membrane as an essentially cubic oscillator and the air inside a box as a weak coupling between those two elements. In this paper, which mainly deals with experimental results, it is shown that several regimes exist under sinusoidal forcing, corresponding to the different nonlinear normal modes of the system. One of these regimes is the quasi-periodic energy pumping regime. The targeted energy transfer phenomenon is also visible on the free oscillations of the system. Indeed, above an initial excitation threshold, the sound extinction in the tube follows a quasi-linear decrease that is much faster than the usual exponential one. During this linear decrease, the energy of the acoustic medium is irreversibly transferred to the membrane and then damped into this element called nonlinear energy sink. We present also the frequency responses of the system which shows a clipping of the original resonance peak of the acoustic medium and we finally demonstrate the ability of the nonlinear absorber to operate in a large frequency band, tuning itself to any linear system.     

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。      

The effects of design modifications on the apparent coupling loss factors in SEA-like analysis

At high frequencies it is often desirable to describe the behaviour of a structure in terms of subsystem energies. The most important method used for high frequency analysis is statistical energy analysis (SEA). Recently, the frequency range in which finite element analysis is applied is being extended to higher frequencies resulting in SEA-like analysis. Methods such as energy distribution modelling can be used to obtain the matrix of energy influence coefficients (EICs); the EIC matrix can be inverted to estimate SEA-like “apparent” coupling loss factors (ACLFs). The ACLFs so estimated depend on details of global modal properties, especially at low and moderate modal overlap. This has implications for design modifications, for example by adding damping treatment to one subsystem, since generally all the EICs change and hence so do all the ACLFs. In principle a full re-analysis is required; this is in contrast to classical SEA. This paper describes these problems and their causes and approximations to the SEA-like parameters of the modified system are proposed. Estimates of the response of the structure after modifications can be found without full re-analysis, leading to a computationally efficient method. The case studies show good agreement between the estimates based on the proposed approaches and the ones based on full re-analysis. The net outcome is that the ACLFs can be estimated after the modification has been made in a manner similar to conventional SEA.     

An investigation into the simultaneous use of a resonator as an energy harvester and a vibration absorber

A mass–spring–damper system is at the core of both a vibration absorber and a harvester of energy from ambient vibrations. If such a device is attached to a structure that has a high impedance, then it will have very little effect on the vibrations of the structure, but it can be used to convert mechanical vibrations into electrical energy (act as an energy harvester). However, if the same device is attached to a structure that has a relatively low impedance, then the device may attenuate the vibrations as it may act as both a vibration absorber and an energy harvester simultaneously. In this paper such a device is discussed. Two situations are considered; the first is when the structure is excited with broadband random excitation and the second is when the structure is excited by a single frequency. The optimum parameters of the device for both energy harvesting and vibration attenuation are discussed for these two cases. For random excitation it is found that if the device is optimized for vibration suppression, then this is also adequate for maximizing the energy absorbed (harvested), and thus a single device can effectively suppress vibration and harvest energy at the same time. For single frequency excitation this is found not to be the case. To maximize the energy harvested, the natural frequency of the system (host structure and absorber) has to coincide with the forcing frequency, but to minimize vibration of the host structure, the natural frequency of the absorber has to coincide with the forcing frequency. In this case, therefore, a single resonator cannot effectively suppress vibration and harvest energy at the same time.     

Analysis of the causes of formation of negative loss factors

I-IatroductionStatisticalEnergyAnalysis(SEA)isakindofeffective,simpleanddirectapproachforan-alyzingvibrationandsound,andithasbeenfoundwidelyapplicationsinanalysisofmechanicalnoiseandvibrationcolltrolsince198osl1-4].However,forgeneralindustrialmachineswhichalwayconsistofcomPlexandheaVystructures,thedeterminationmethodsofSEAparametersintheclassicalSEAtheoryareinapplicable[5]because:(1)SEAparametersofthesekindsofstructuresaredifficulttoobtainfromthetheory(2)theconditionofconservativeandweak…     

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.