Low Modal Overlap Research Articles

Transmission loss of plates with multiple embedded acoustic black holes using statistical modal energy distribution analysis

Acoustic black holes (ABHs) in beams and plates have been extensively studied as a passive method for vibration attenuation and noise reduction. However, most research to date has focused on analyzing the behavior of a single ABH structural element, using numerical or semi-analytical deterministic approaches. If ABHs are to be exploited for practical industrial applications, there is a need to characterize their performance in complex built-up structures and to describe them with statistical methods in the mid-high frequency range. This paper presents a first step towards this goal by employing statistical modal energy distribution analysis (SmEdA) to evaluate the transmission loss of ABH panels separating two air cavities. SmEdA splits vibroacoustic systems into subsystems and establishes power balance equations between the modes belonging to different subsystems. This avoids the energy equipartition assumption of traditional statistical energy analysis (SEA) and extends it to low modal overlap systems. In this work, the benefits of embedding ABHs on plates for noise reduction between cavities are predicted with SmEdA. The role played by the size, shape and number of ABH indentations on the plate are inspected, as well as the influence of other parameters like the truncation thickness, ABH order and damping. The effects of added mass and stiffness of the damping layer are investigated and it is observed that, with proper design, ABH plates can exhibit substantial transmission loss improvement in the vicinity of the critical frequency of uniform plates. Both resonant and non-resonant transmission are considered in the analysis.

Statistical Energy Analysis with fuzzy parameters to handle populations of structures

Statistical modal Energy distribution Analysis (SmEdA) approach was developed to enlarge the application field of Statistical Energy Analysis (SEA) when equipartition of modal energies is not achieved. SmEdA gives more precise results than standard SEA when compared to exact energy response of a deterministic system in the case of low modal overlap, heterogeneous systems or point excitation. The present paper was initiated by this question: when considering a population of similar structures, each of them being described by SmEdA, do the ensemble averaged energies of subsystem and injected power tend to satisfy SEA equations? In other terms, despite the non-equipartition of energy observed on each element of the population of structures does the ensemble averaging leads to SEA equation where equipartition of energy is assumed? The response to that question that rises from this paper is yes, if the terms of the SEA equation are fuzzy numbers. It results that the energy response given by the model can be interpreted using fuzzy numbers theory.

A graph theory approach to identify resonant and non-resonant transmission paths in statistical modal energy distribution analysis

Statistical modal energy distribution analysis (SmEdA) extends classical statistical energy analysis (SEA) to the mid frequency range by establishing power balance equations between modes in different subsystems. This circumvents the SEA requirement of modal energy equipartition and enables applying SmEdA to the cases of low modal overlap, locally excited subsystems and to deal with complex heterogeneous subsystems as well. Yet, widening the range of application of SEA is done at a price with large models because the number of modes per subsystem can become considerable when the frequency increases. Therefore, it would be worthwhile to have at one׳s disposal tools for a quick identification and ranking of the resonant and non-resonant paths involved in modal energy transmission between subsystems. It will be shown that previously developed graph theory algorithms for transmission path analysis (TPA) in SEA can be adapted to SmEdA and prove useful for that purpose. The case of airborne transmission between two cavities separated apart by homogeneous and ribbed plates will be first addressed to illustrate the potential of the graph approach. A more complex case representing transmission between non-contiguous cavities in a shipbuilding structure will be also presented.

Relative variance of the mean-squared pressure in multimode media: rehabilitating former approaches.

The commonly accepted model for the relative variance of transmission functions in room acoustics, derived by Weaver, aims at including the effects of correlation between eigenfrequencies. This model is based on an analytical expression of the relative variance derived by means of an approximated correlation function. The relevance of the approximation used for modeling such correlation is questioned here. Weaver's model was motivated by the fact that earlier models derived by Davy and Lyon assumed independent eigenfrequencies and led to an overestimation with respect to relative variances found in practice. It is shown here that this overestimation is due to an inadequate truncation of the modal expansion, and to an improper choice of the frequency range over which ensemble averages of the eigenfrequencies is defined. An alternative definition is proposed, settling the inconsistency; predicted relative variances are found to be in good agreement with experimental data. These results rehabilitate former approaches that were based on independence assumptions between eigenfrequencies. Some former studies showed that simpler correlation models could be used to predict the statistics of some field-related physical quantity at low modal overlap. The present work confirms that this is also the case when dealing with transmission functions.

An efficient probabilistic approach to vibro-acoustic analysis based on the Gaussian orthogonal ensemble.

Vibro-acoustic analysis of complex systems at higher frequencies faces two challenges: how to compute the response without using an excessive number of degrees of freedom (DOFs), and how to quantify the uncertainty of the response due to small spatial variations in geometry, material properties, and boundary conditions, which have a wave scattering effect? In this study, a general method of analysis is presented that provides an answer to both questions while overcoming most limitations of statistical energy analysis. The fundamental idea is to numerically compute an artificial ensemble of realizations for the components of the built-up system that are highly sensitive to small random wave scatterers. This can be efficiently performed because their eigenvalue spacings and mode shapes conform to Gaussian orthogonal ensemble spacings and Gaussian random fields, respectively. The DOFs of the overall system are therefore limited to those of the deterministic components and the interface DOFs of the random components. The method is extensively validated by application to plate structures. Good agreement between the predicted response probability distributions and the results of detailed parametric probabilistic models is obtained, also for cases of low modal overlap, single point loading, and strong subsystem coupling.

Non resonant transmission modelling with statistical modal energy distribution analysis

Statistical modal Energy distribution Analysis (SmEdA) can be used as an alternative to Statistical Energy Analysis for describing subsystems with low modal overlap. In its original form, SmEdA predicts the power flow exchanged between the resonant modes of different subsystems. In the case of sound transmission through a thin structure, it is well-known that the non resonant response of the structure plays a significant role in transmission below the critical frequency. In this paper, we present an extension of SmEdA that takes into account the contributions of the non resonant modes of a thin structure. The dual modal formulation (DMF) is used to describe the behaviour of two acoustic cavities separated by a thin structure, with prior knowledge of the modal basis of each subsystem. Condensation in the DMF equations is achieved on the amplitudes of the non resonant modes and a new coupling scheme between the resonant modes of the three subsystems is obtained after several simplifications. We show that the contribution of the non resonant panel mode results in coupling the cavity modes of stiffness type, characterised by the mode shapes of both the cavities and the structure. Comparisons with reference results demonstrate that the present approach can take into account the non resonant contributions of the structure in the evaluation of the transmission loss.

Extension of SmEdA to non-resonant transmission

Statistical modal Energy distribution Analysis (SmEdA) may be used as alternative to Statistical Energy Analysis for describing subsystems with low modal overlap. In its original form, SmEdA predicts the power flow exchanged between the resonant modes of different subsystems. In the case of the sound transmission through a thin light structure, it is well-known than the non-resonant response of the structure may have a significant role on the transmission below the critical frequency. In this paper, one presents an extension of SmEdA taking into account the contributions of the non resonant modes of the thin structure. The dual modal formulation (DMF) is used for describing the behavior of two acoustic cavities separated by a thin structure knowing their subsystem modes. A condensation in the DMF equations is achieved on the amplitudes of the non-resonant modes. Using some simplifications, a new coupling scheme between the resonant modes of the three subsystems is obtained. It shows direct couplings of the cavity modes through stiffness elements characterized by the modes shapes of the cavities and the structure, both. Comparisons with reference results show the ability and the interest of the present approach for representing the non-resonant contributions of the structure.

Why a Reverberation Chamber Works at Low Modal Overlap

The physical parameter well adapted to assess the degree of overmodedness of a reverberation chamber (RC) is the number MM of modes overlapping in a mode bandwidth. The lowest usable frequency of an RC often corresponds to a low modal overlap of one or two modes. Notwithstanding, in spite of this poor number of modes the RC still works. We show, using Monte Carlo simulation, that the number of modes must, in fact, not be restrained to MM and that the number of modes contributing to the field statistics can be, even at low modal overlap, somewhat larger than expected.

Extension of the statistical modal energy distribution analysis for estimating energy density in coupled subsystems

The present article deals with an extension of the Statistical modal Energy distribution Analysis (SmEdA) method to estimate kinetic and potential energy density in coupled subsystems. The SmEdA method uses the modal bases of uncoupled subsystems and focuses on the modal energies rather than the global energies of subsystems such as SEA (Statistical Energy Analysis). This method permits extending SEA to subsystems with low modal overlap or to localized excitations as it does not assume the existence of modal energy equipartition. We demonstrate that by using the modal energies of subsystems computed by SmEdA, it is possible to estimate energy distribution in subsystems. This approach has the same advantages of standard SEA, as it uses very short calculations to analyze damping effects. The estimation of energy distribution from SmEdA is applied to an academic case and an industrial example.

Ensemble statistics of active and reactive sound intensity in reverberation rooms

This paper examines fundamental statistical properties of the active and reactive sound intensity in reverberant enclosures driven with pure tones. The existing theory for sound intensity in a diffuse sound field, which is based on Waterhouse's random wave model and therefore limited to the region of high modal overlap, is extended to the region of low modal overlap by taking account of the random fluctuations of the sound power emitted by the source that generates the sound field. The validity of the extended model is confirmed by experimental and numerical results.

Space averaging of vibration level differences measured in lightweight building structures

This paper examines space averaging methods applied to structural level differences measured in lightweight building structures (two beam junction and single plate and beam junction). The methods studied include the averaging assuming a constant input force, the averaging based on a constant vibration level across the source element, and the arithmetic averaging of level differences. Tests indicate that a fairly steady input force can be easily produced when using an impulsive source such as a hammer; the constant input force method is then appropriate and can be used to space average level differences (the two beam structure showing a standard deviation of 0.1 dB from the averaging based on levels normalised to the force). On the other hand, acceleration levels can exhibit significant variations with position due to the low mode count and modal overlap of some lightweight elements such as timber beams. Variations in the vibration level across the source element can then be responsible for large inaccuracies in the level difference calculated from the constant level assumption, the two beams’ results showing errors greater than 5 dB at several 1/3 octave band frequencies. The latter averaging method is then inappropriate and should not be used.

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.

Experimental identification of viscous damping in linear vibration

This paper is concerned with the experimental evaluation of the performance of viscous damping identification methods in linear vibration theory. Both existing and some new methods proposed by the present authors [A.S. Phani, J. Woodhouse, Viscous damping identification in linear vibration, Journal of Sound and Vibration 303 (3–5) (2007) 475–500] are applied to experimental data measured on two test structures: a coupled three cantilever beam with moderate modal overlap and a free–free beam with low modal overlap. The performance of each method is quantified and compared based on three norms and the best methods are identified. The role of complex modes in damping identification from vibration measurements is critically assessed.

Low-Loss Surface-Mode Waveguides for Terahertz Si–SiGe Quantum Cascade Lasers

A new design of low-loss terahertz waveguide for Si-SiGe quantum-cascade lasers (QCLs) is presented. Periodic surface gratings are used to define waveguides without the requirement of cleaved or etched end facets. As Si cleaves along the (111) planes and not in the vertical direction for standard Si (100) substrates, this significantly aids the fabrication of waveguides. Losses down to 2 cm-1 with modal overlap of 0.4 can be achieved for shallow gratings with etched depths of only 0.56 mum for an active material layer thickness of 8 mum. Such low loss and high modal overlap is key to any Si-SiGe QCL being realized

Vibration transmission through a frame typical of timber-framed buildings

A study of vibration transmission through a frame typical of timber-framed buildings was undertaken in order to establish whether or not statistical energy analysis (SEA) could be used to model such structures. Examination of individual beam junctions showed that the normal assumption that junctions are rigid was not valid and that the junctions were better modelled as pinned junctions (sometimes also called hinged junctions) through which no moments are transmitted. It was also found that there were significant fluctuations in the response due to the low modal overlap factor of the beams. When appropriate corrections for both the junction transmission and the low mode count were taken into consideration, there was satisfactory agreement between the measured and predicted results. These theories were then used to model vibration transmission through a large two-storey timber frame and it was found that standard SEA models could be used to predict the mean response and that the fluctuations about that mean could also be estimated.

Variability of the coupling loss factor between two coupled plates

Within statistical energy analysis (SEA), the coupling loss factor (CLF) is a statistical quantity defined in terms of the average behaviour of an ensemble of similar systems. Thus, the ‘effective’ CLF for a given realization may differ from the ensemble average. The effective CLF of two coupled plates with low modal density and low modal overlap fluctuates significantly compared with the ensemble average CLF. Accordingly, the CLF is the main parameter expected to determine the confidence intervals in the SEA prediction. The aim of the present paper is to quantify the variability of the effective CLF for the case of two finite rectangular plates. Extensive parameter investigations have been performed by using the dynamic stiffness method (DSM) and a ‘numerical’ power injection method (PIM) using the SEA power balance equations. The effects of frequency bandwidth and the number of modes in a band were separated by using frequency averages at a series of constant bandwidths rather than one-third octave averages, while the modal overlap is made independent of frequency by setting the damping loss factor to be inversely proportional to frequency. Finally, an improved empirical model for the variability of the CLF is derived, from these numerical results, in terms of the modal overlap factor and the number of modes in a frequency band. This model can subsequently be used to evaluate the uncertainty of the CLF of such a system.

Statistical energy analysis: coupling loss factors, indirect coupling and system modes

The distribution of energy among the subsystems of a system can be found in terms of the modes of the system. If there are enough modes in the frequency band of interest then the system can be described by an SEA model. However, in general this is a “quasi-SEA” model, which involves both direct and indirect coupling loss factors, whose values depend on the modal overlap. This paper explores the conditions under which the indirect coupling loss factors are zero, so that the system is described by a “proper-SEA” model. It also investigates the dependence of the direct and indirect coupling loss factors on the modal properties of the system and on the modal overlap. In summary, the indirect coupling loss factors are zero, so that a proper-SEA model can be formed, either if all the system modes are local or in the weak coupling regime as the modal overlap becomes large. It is seen that in the low modal overlap limit the coupling loss factors are proportional to the damping loss factor and equipartition of energy only occurs if all modes are global. For higher modal overlap the SEA parameters depend on the detailed statistics of the modes and the situation is complicated. However, if all the modes are local the indirect coupling loss factors are all zero. In the high modal overlap limit the coupling loss factors asymptote to constants, with indirect coupling loss factors becoming zero. This behaviour occurs because of mode shape correlation at the boundary between two directly connected subsystems.

Vibration transmission between coupled plates using finite element methods and statistical energy analysis. Part 1: Comparison of measured and predicted data for masonry walls with and without apertures

In Part 1 of this paper a study is made to assess the prediction of vibration transmission between masonry walls using Finite Element Methods (FEM) and Statistical Energy Analysis (SEA) through a comparison of measured and predicted data. Masonry walls typically have low modal overlap and low modal density at low frequencies, hence FEM is used to predict the large fluctuations in the vibration level difference. The issue of uncertainty in the physical description of masonry walls is addressed by using Monte Carlo methods with FEM. Two types of masonry wall junction are studied, an L-junction and a T-junction. The importance of in-plane wave transmission was demonstrated for L-junctions with apertures and T-junctions. The agreement between FEM, SEA and measured data at high frequencies indicated that the conversion between bending and in-plane waves in the FEM models was correct.

A basic hybrid finite element formulation for mid-frequency analysis of beams connected at an arbitrary angle

When beams are connected at an arbitrary angle and subjected to an external excitation, both longitudinal and bending waves are generated in the system. Since longitudinal wavelengths are considerably longer than bending wavelengths in the mid-frequency region, the number of bending wavelengths in the beams is considerably larger than the number of longitudinal wavelengths. In this paper, plannar beams connected at arbitrary angles are considered. The energy finite element analysis (EFEA) is employed for modelling the bending behavior of the beams and the conventional finite element analysis (FEA) is utilized for modelling the longitudinal vibration in the beams. Thus, a basic hybrid FEA formulation is presented for mid-frequency analysis of systems that contain two types of energy. The bending vibration is associated with the long members in the system and the longitudinal vibration is associated with the short members. The long members are considered to have high modal overlap and to contain several wavelengths within their dimension, and uncertainty effects are present. The short members contain a small number of wavelengths, and exhibit a low modal overlap. Due to the low modal overlap the resonant frequencies are spaced far apart in the frequency domain, therefore the short members exhibit resonant or non-resonant behavior depending on the frequency of the excitation. In this work, the bending and the longitudinal vibration within the same beam member are treated as a long and as a short member, respectively. A hybrid joint formulation is developed between long and short members. Power reflection and transmission coefficients are derived for each joint. The distribution of the energy throughout the system demonstrates a strong dependency on the power transfer coefficients. Several systems are analyzed by the hybrid FEA and by analytical solutions, and good correlation between them is observed.

The quantification of structure-borne transmission paths by inverse methods. Part 1: Improved singular value rejection methods

Structure-borne sound from installed machinery is often transmitted into a receiver structure via many connection points and several co-ordinate directions at each of them. In order to quantify the contributions from the various connection points, the operational forces at the interfaces, or an equivalent set of forces at some other locations, should be determined. These forces may be combined with measured transfer functions to determine their contributions to the sound at the receiver locations. Inverse methods are becoming widely used, in which a matrix of measured accelerances is inverted at each frequency and used with operational acceleration data to find the forces. Due to poor conditioning of this matrix, however, the results can often be unreliable. In this paper, using both simulations and measurements, an assessment is made of the success and failure of various strategies for dealing with the problems of ill conditioning, in particular over-determination and singular value rejection. In each case the test structure is a rectangular plate, and a wide frequency range is covered to include regions of both low and high modal overlap. Critical for the rejection of singular values is a suitable threshold. It is established that previously used thresholds, based on estimates of error in either accelerances or operational responses, cannot be used universally. An alternative approach is developed in which the accelerance matrix is perturbed by a different amount for each sample of the operational responses. Based on this approach a more robust strategy is proposed which takes account simultaneously of the effect of errors in both the accelerances and operational responses.