Statistical energy analysis (SEA) is a method for predicting the transmission of

sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...

and

vibration Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic function, periodic, such as the motion of a pendulum ...

through complex structural acoustic systems. The method is particularly well suited for quick system level response predictions at the early design stage of a product, and for predicting responses at higher frequencies. In SEA a system is represented in terms of a number of coupled subsystems and a set of

linear equations In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...

are derived that describe the input, storage, transmission and dissipation of energy within each subsystem. The parameters in the SEA equations are typically obtained by making certain statistical assumptions about the local dynamic properties of each subsystem (similar to assumptions made in

room acoustics Room acoustics is a subfield of acoustics dealing with the behaviour of sound in enclosed or partially-enclosed spaces. The architectural details of a room influences the behaviour of sound waves within it, with the effects varying by frequency. ...

and

statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...

). These assumptions significantly simplify the analysis and make it possible to analyze the response of systems that are often too complex to analyze using other methods (such as

finite element The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat t ...

and boundary element methods).

History

The initial derivation of SEA arose from independent calculations made in 1959 by Richard Lyon and Preston Smith as part of work concerned with the development of methods for analyzing the response of large complex aerospace structures subjected to spatially distributed random loading. Lyon's calculation showed that under certain conditions, the flow of energy between two coupled oscillators is proportional to the difference in the oscillator energies (suggesting a thermal analogy exists in structural-acoustic systems). Smith's calculation showed that a structural mode and a diffuse reverberant sound field attain a state of 'equipartition of energy' as the damping of the mode is reduced (suggesting a state of thermal equilibrium can exist in structural-acoustic systems). The extension of the two oscillator results to more general systems is often referred to as the modal approach to SEA. While the modal approach provides physical insights into the mechanisms that govern energy flow it involves assumptions that have been the subject of considerable debate over many decades. The theory that combines deterministic finite element methods (FEM) and SEA was developed by Phil Shorter and Robin Langley and is called hybrid FEM/SEA theory. In recent years, alternative derivations of the SEA equations based on wave approaches have become available. Such derivations form the theoretical foundation behind a number of modern commercial SEA codes and provide a general framework for calculating the parameters in an SEA model. A number of methods also exist for post-processing FE models to obtain estimates of SEA parameters. Lyon mentioned the use of such methods in his initial SEA text book in 1975 but a number of alternative derivations have been presented over the years

Method

To solve a noise and vibration problem with SEA, the system is partitioned into a number of components (such as

plate Plate may refer to: Cooking * Plate (dishware), a broad, mainly flat vessel commonly used to serve food * Plates, tableware, dishes or dishware used for setting a table, serving food and dining * Plate, the content of such a plate (for example: ...

s, shells, beams and acoustic cavities) that are coupled together at various junctions. Each component can support a number of different propagating wavetypes (for example, the

bending In applied mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element. The structural element is assumed to ...

longitudinal Longitudinal is a geometric term of location which may refer to: * Longitude ** Line of longitude, also called a meridian * Longitudinal engine, an internal combustion engine in which the crankshaft is oriented along the long axis of the vehicle, ...

and

shear Shear may refer to: Textile production *Animal shearing, the collection of wool from various species **Sheep shearing *The removal of nap during wool cloth production Science and technology Engineering *Shear strength (soil), the shear strength ...

wavefields in a thin isotropic plate). From an SEA point of view, the reverberant field of each wavefield represents an orthogonal store of energy and so is represented as a separate energy degree of freedom in the SEA equations. The energy storage capacity of each reverberant field is described by a parameter termed the 'modal density', which depends on the average speed with which waves propagate energy through the subsystem (the average

group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space. For example, if a stone is thrown into the middl ...

), and the overall dimension of the subsystem. The transmission of energy between different wavefields at a given type of junction is described by parameters termed 'coupling loss factors'. Each coupling loss factor describes the input power to the direct field of a given receiving subsystem per unit energy in the reverberant field of a particular source subsystem. The coupling loss factors are typically calculated by considering the way in which waves are scattered at different types of junctions (for example, point, line and area junctions). Strictly, SEA predicts the average response of a population or ensemble of systems and so the coupling loss factors and modal densities represent ensemble average quantities. To simplify the calculation of the coupling loss factors it is often assumed that there is significant scattering within each subsystem (when viewed across an ensemble) so that direct field transmission between multiple connections to the same subsystem is negligible and reverberant transmission dominates. In practical terms, this means that SEA is often best suited for problems in which each subsystem is large compared with a wavelength (or from a modal point of view, each subsystem contains several modes in a given frequency band of interest). The SEA equations contain a relatively small number of degrees of freedom and so can be easily inverted to find the reverberant energy in each subsystem due to a given set of external input powers. The (ensemble average) sound pressure levels and vibration velocities within each subsystem can then be obtained by superimposing the direct and reverberant fields within each subsystem.

Applications

Over the past half century, SEA has found applications in virtually every industry for which noise and vibration are of concern. Typical applications include: * Interior noise prediction and sound package design in automotive, aircraft, rotorcraft and train applications * Interior and exterior radiated noise in marine applications * Prediction of dynamic environments in launch vehicles and spacecraft * Prediction of noise from consumer goods such as dishwashers, washing machines and refrigerators * Prediction of noise from generators and industrial chillers * Prediction of air-borne and structure-borne noise through buildings * Design of enclosures etc. Additional examples can be found in the proceedings of conferences such as INTERNOISE, NOISECON, EURONOISE, ICSV, NOVEM, SAE N&V.

Software implementations

Several commercial solutions for Statistical Energy Analysis are available:
Actran SEA Module
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Free Field Technologies, MSC Software

VA One SEA Module (previously AutoSEA)
from

ESI Group ESI Group provides virtual prototyping software that simulates a product's behavior during testing, manufacturing and real-life use. Engineers in a variety of industries use its software to evaluate the performance of proposed designs in the early ...

, France * SEAM, SEAM 3D from Cambridge Collaborative Inc. USA, since April 2019 under Altair Hyperworks.
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Dassault Systèmes SIMULIA
* GSSEA-Light from Gothenburg Sound AB, Sweden * SEA+ from InterAC, France distributed by LMS International Free solutions: * Statistical Energy Analysis Freeware, * SEAlab - open code in Matlab/Octave from Applied Acoustics, Chalmers, Sweden (open source) * pyva - python toolbox for vibroacoustic simulation, Germany (open source) Other implementations: *NOVASEA, Université de Sherbrooke, Canada

References

{{reflist, 35em Statistical mechanics Mechanical vibrations Acoustics