Standard Statistical Energy Analysis Research Articles

Solving the stationary Liouville equation via a boundary element method

Energy distributions of linear wave fields are, in the high frequency limit, often approximated in terms of flow or transport equations in phase space. Common techniques for solving the flow equations in both time dependent and stationary problems are ray tracing and level set methods. In the context of predicting the vibro-acoustic response of complex engineering structures, related methods such as Statistical Energy Analysis or variants thereof have found widespread applications. We present a new method for solving the transport equations for complex multi-component structures based on a boundary element formulation of the stationary Liouville equation. The method is an improved version of the Dynamical Energy Analysis technique introduced recently by the authors. It interpolates between standard statistical energy analysis and full ray tracing, containing both of these methods as limiting cases. We demonstrate that the method can be used to efficiently deal with complex large scale problems giving good approximations of the energy distribution when compared to exact solutions of the underlying wave equation.

Boundary element dynamical energy analysis: A versatile method for solving two or three dimensional wave problems in the high frequency limit

Dynamical energy analysis was recently introduced as a new method for determining the distribution of mechanical and acoustic wave energy in complex built up structures. The technique interpolates between standard statistical energy analysis and full ray tracing, containing both of these methods as limiting cases. As such the applicability of the method is wide ranging and additionally includes the numerical modelling of problems in optics and more generally of linear wave problems in electromagnetics. In this work we consider a new approach to the method with enhanced versatility, enabling three-dimensional problems to be handled in a straightforward manner. The main challenge is the high dimensionality of the problem: we determine the wave energy density both as a function of the spatial coordinate and momentum (or direction) space. The momentum variables are expressed in separable (polar) coordinates facilitating the use of products of univariate basis expansions. However this is not the case for the spatial argument and so we propose to make use of automated mesh generating routines to both localise the approximation, allowing quadrature costs to be kept moderate, and give versatility in the code for different geometric configurations.

Wave Intensity Distributions in Complex Structures

The vibro-acoustic response of mechanical structures can in general be well approximated in terms of linear wave equations. Standard numerical solution methods comprise the nite or boundary element method in the low frequency regime and statistical energy analysis in the high-frequency limit. Major computational challenges are posed by the so-called mid-frequency problem that is, composite structures where the local wavelength may vary by orders of magnitude across the components. Recently, a new approach towards determining the distribution of mechanical and acoustic wave energy in complex built-up structures improving on standard statistical energy analysis has been proposed. The technique interpolates between statistical energy analysis and ray tracing containing both these methods as limiting cases. The method has its origin in studying solutions of wave equation with an underlying chaotic ray-dynamics often referred to as wave chaos. Within the new theory dynamical energy analysis statistical energy analysis is identi ed as a low resolution ray tracing algorithm and typical statistical energy analysis assumptions can be quanti ed in terms of the properties of the ray dynamics. We have furthermore developed a hybrid statistical energy analysis/ nite element method based on random wave model assumptions for the short-wavelength components. This makes it possible to tackle mid-frequency problems under certain constraints on the geometry of the structure. Dynamical energy analysis and statistical energy analysis/ nite element method calculations for a range of multi-component model systems will be presented. The results are compared with both statistical energy analysis results and nite element method as well as boundary element method calculations. Dynamical energy analysis emerges as a numerically e cient method for calculating mean wave intensities with a high degree of spatial resolution and capturing long range correlations in the ray dynamics.

Dynamical energy analysis for built-up acoustic systems at high frequencies

Standard methods for describing the intensity distribution of mechanical and acoustic wave fields in the high frequency asymptotic limit are often based on flow transport equations. Common techniques are statistical energy analysis, employed mostly in the context of vibro-acoustics, and ray tracing, a popular tool in architectural acoustics. Dynamical energy analysis makes it possible to interpolate between standard statistical energy analysis and full ray tracing, containing both of these methods as limiting cases. In this work a version of dynamical energy analysis based on a Chebyshev basis expansion of the Perron-Frobenius operator governing the ray dynamics is introduced. It is shown that the technique can efficiently deal with multi-component systems overcoming typical geometrical limitations present in statistical energy analysis. Results are compared with state-of-the-art hp-adaptive discontinuous Galerkin finite element simulations.

Dynamical energy analysis—Determining wave energy distributions in vibro-acoustical structures in the high-frequency regime

We propose a new approach towards determining the distribution of mechanical and acoustic wave energy in complex built-up structures. The technique interpolates between standard statistical energy analysis (SEA) and full ray tracing containing both these methods as limiting cases. By writing the flow of ray trajectories in terms of linear phase space operators, it is suggested to reformulate ray-tracing algorithms in terms of boundary operators containing only short ray segments. SEA can now be identified as a low resolution ray-tracing algorithm and typical SEA assumptions can be quantified in terms of the properties of the ray dynamics. The new technique presented here enhances the range of applicability of standard SEA considerably by systematically incorporating dynamical correlations wherever necessary. Some of the inefficiencies inherent in typical ray-tracing methods can be avoided using only a limited amount of the geometrical ray information. The new dynamical theory— dynamical energy analysis (DEA)—thus provides a universal approach towards determining wave energy distributions in complex structures in the high-frequency limit.

A scaling procedure for the response of an isolated system with high modal overlap factor

The paper deals with a numerical approach that reduces some physical sizes of the solution domain to compute the dynamic response of an isolated system: it has been named Asymptotical Scaled Modal Analysis (ASMA). The proposed numerical procedure alters the input data needed to obtain the classic modal responses to increase the frequency band of validity of the discrete or continuous coordinates model through the definition of a proper scaling coefficient. It is demonstrated that the computational cost remains acceptable while the frequency range of analysis increases. Moreover, with reference to the flexural vibrations of a rectangular plate, the paper discusses the ASMA vs. the statistical energy analysis and the energy distribution approach. Some insights are also given about the limits of the scaling coefficient. Finally it is shown that the linear dynamic response, predicted with the scaling procedure, has the same quality and characteristics of the statistical energy analysis, but it can be useful when the system cannot be solved appropriately by the standard Statistical Energy Analysis (SEA).

Variance of energy and energy density in statistical energy analysis of complex systems

Standard statistical energy analysis (SEA) predicts the mean energy level in each subsystem of a complex system at high frequencies. The resulting energy is to be seen as averaged over an ensemble of similar systems (like nominally identical products coming off an assemble line). This prediction of the mean has been recently extended to the ensemble variance of the energy. The variance gives indications of how far from the ensemble mean can be the response of one particular member of the ensemble. Using few additional assumptions, the variance can be used to predict the confidence intervals around the SEA mean prediction, and can be extended to the variance of the response at a point (i.e., the energy density). Some numerical validations for several types of vibro-acoustic systems are presented.

Phase determination of reverberant structural-acoustic systems using pole and zero distribution

This study investigates the phase behavior of a simple coupled structural acoustic system with high modal overlap. It proposes that phase is simply accumulated across the components making up the structure. Classical dynamic solutions breakdown with high modal overlap and complex geometries, and standard Statistical Energy Analysis (SEA) solutions work well for high modal overlap but give no phase information. Many applications such as active control predictions and time-domain signal analysis require phase information. Results of the investigation show that the amount of damping in the system is very important in the estimation of reverberant phase. The amount of energy lost due to coupling with adjacent subsystems was also demonstrated to be important for both acoustic and structural subsystems. The reverberant phase of an acoustic subsystem was found to be affected more by the inclusion of coupling loss to an adjacent structure than the reverberant phase of a structure attached to an acoustic subsystem. The phase of the overall system is calculated by simply accumulating the phase of each of the subsystems.

Sound transmission through parallel plates coupled along a line

This paper presents a theoretical model to predict bending wave transmission through double walls where the two leaves are coupled by a line joint. The model assumes that the joint comprises four semi-infinite plates connected at their edges by an inertia-less rigid beam and allows the structural transmission loss between the plates to be computed. These values can then be used in a standard statistical energy analysis model to predict either the structural transmission from one plate to another or airborne transmission through the wall. The predicted results show good agreement with measurements performed both on a cavity masonry wall and a dry-lined wall tested in a transmission suite.

Joint effects in the mid‐frequency vibration of connected plates.

Structure‐borne noise in coupled systems has been shown to be highly sensitive to joint properties. In this presentation, the effects of joint behavior is examined for coupled plates joined in the canonical L‐shaped configuration. The effects of the variation of the joint properties with respect to the coordinate axis along the two‐plate junction is of interest. Mid‐frequency loads are used, where the modal density is not sufficiently high for standard statistical energy analysis. The problem is analyzed using several methods. A mobility approach [J. M. Cuschieri, J. Acoust. Soc. Am. 87, 1159–1165 (1990)] is used as a benchmark, which, for the thin‐plate model, is exact except for mode truncation. The problem is then examined in the perspective of the following two approximate methods: a spatial and temporal averaging method using energy and intensity as the primary field variables [O. M. Bouthier and R. J. Bernhard, AIAA J. 30, 616–623 (1992)] and an asymptotic method, based on Skudrzyk’s Mean Value Theory [T. Igusa and Y. Tang, AIAA J. 30, 2520–2525 (1992)]. It is found that the effects of the variations in the joint properties can be interpreted using wave vector decompositions.

On the vibrational conductivity approach to high frequency dynamics for two-dimensional structural components

Abstract The vibrational conductivity approach to high frequency dynamics is applied here to the case of two-dimensional structural components: with this approach an equation which is analogous to the steady state heat conduction equation is used to model the spatial variation of the energy density of the structure. It is shown that the standard derivation of the method is incomplete, in the sense that the steady state heat conduction equation admits a class of solution which is not encompassed by the assumed form of response, and a revised derivation is presented. It is further shown that the associated boundary conditions for coupled structural components may be expressed in terms of the coupling loss factors which are used in statistical energy analysis. A variational principle for the differential equation and the associated boundary conditions is then derived which permits the application of the Rayleigh-Ritz technique to coupled problems, and it is shown that a one-term Rayleigh-Ritz solution leads to standard statistical energy analysis. The method is applied to a single plate and to two coupled plates, and it is found that the technique is unable to provide an accurate estimate of the direct field due to point loading. This difficulty is traced to the fact that the assumptions employed in the derivation become self-contradictory for this case. Finally, the usefulness or otherwise of the approach as an engineering tool is discussed.

Thermodynamic modelling of interconnected systems Part I: conservative coupling

In this paper we extend results of Wyatt, Siebert and Tan by deriving an energy flow model in terms of thermodynamic energy rather than stored energy as in the standard Statistical Energy Analysis (SEA) approach to energy flow modelling. This modified energy flow model shows that energy flows according to thermodynamic energy and that this property holds for an arbitrary number of nonidentical subsystems independently of the strength of the coupling. These results are compared with SEA energy flow predictions by means of illustrative examples. In particular, it is shown that for multiple coupled oscillators, SEA energy flow predictions based upon blocked energy may be erroneous, while predictions based upon thermodynamic energy are correct. In particular, the thermodynamic energy flow model correctly predicts zero net energy flow in the case of equal temperature subsystems.

A derivation of the coupling loss factors used in statistical energy analysis

The conditions under which an exact theory concerning the energy/power flow relationships for a complex dynamic system may be reduced to the standard Statistical Energy Analysis (SEA) equations are considered. A definition of weak coupling is introduced which encompasses a number of previous definitions, and it is shown that the presence of weak coupling does not guarantee the validity of the standard SEA assumption that the coupling loss factor (CLF) between two subsystems is zero unless the subsystems are directly connected. The standard assumption is likely to be reasonable for systems which are both weakly coupled and reverberant. Attention is then turned to the CLF between two connected subsystems and a general result is obtained which involves a space and frequency averaged Green function. It is shown that under the appropriate assumptions a number of existing results may be derived from this expression, including those produced by both the wave and the modal approaches to SEA. The exact theory may be used as a basis for developing improved models of the system dynamics.