Statistical Energy Analysis Model Research Articles

STATISTICAL ENERGY ANALYSIS OF MULTIPLE, NON-CONSERVATIVELY COUPLED SYSTEMS

In what follows, the analysis of Keane [1] for the energy flows around an arbitrary configuration of coupled multi-modal subsystems is extended to the case of non-conservative coupling. Here, the joints between any two subsystems are modelled by a spring anda damper, thus allowing for dissipation of energy to occur at the joint. The aim of this study is to give greater insight into the problem of energy flows through non-conservative couplings which has not been extensively discussed in the literature. Interest is focused on the effect of damping in the joints on the magnitudes of energy flows between, and energy levels in, each subsystem. The model derived is used to demonstrate the well known fact that selecting the correct level of damping in the joints surrounding a driven subsystem may cause a large percentage of the power input to the subsystem to be dissipated in these joints. This minimizes the overall power dissipated within the subsystems and thus the system total energy level. A Statistical Energy Analysis (SEA) model for non-conservatively coupled systems is then suggested, in which coupling damping loss factors are introduced into the main SEA energy balance equations to account for the energy dissipation in the joints. This model is shown to be exactly correct for the limiting case of weak coupling. Numerical examples which illustrate these various ideas are presented, with the use of parameters adopted in previous studies.

Flanking paths of a floor–ceiling assembly in wood frame constructions: Measurements and predictions using statistical energy analysis

This paper examines the net sound isolation of a floor–ceiling assembly and the supporting walls that might be found in wood frame apartment constructions. The specimen was constructed in the flanking transmission facilities of the National Research Council of Canada. Measured results show that the presence of two load-bearing walls introduce flanking paths that can reduce the net sound isolation of an FSTC 59 floor–ceiling assembly by 6 FSTC points. Various methods of modeling the structural transmission at the floor–wall intersection are presented. The models indicate that beams at the joint (i.e., joist header, head, and sole plates) introduce significant high-frequency joint attenuation due to rotational and shear inertia. Comparison between measured and predicted velocity level differences indicates that floor–wall intersection in standard wood frame constructions should be modeled as two corner joints [(1) upper party wall to floor decking, (2) floor decking to lower party wall] sharing a common plate (the floor decking). A statistical energy analysis model is used to show that the dominant flanking path is the floor decking to the load bearing wall below, while the wall–wall flanking path is much less important. The predictions are shown to be in good agreement with measured results. a)Present address: Shuller International, 10100 West Ute Ave., Littleton, CO 80127.

Sound transmission in buildings with elastic layers at joints

The influence of elastic layers (ELs) at junctions between walls and floor slabs on airborne and structure-borne sound transmission in buildings is investigated. For this aim, we carried out measurements on scale models with ELs at the joints. The results were compared with the predictions of the statistical energy analysis model. This model included the theoretically calculated transmission coefficients for junctions between elastically coupled plates. As a practical application, we analysed the performance of ELs in a model for a real building structure and for various positions of ELs. The work showed that the benefits of the use of ELs depend on the specific situation: the effect can be either positive or negative. In fact, the global effect of ELs is only a change in the distribution of the total energy. If less energy is transmitted into one direction, more energy will be transmitted into the other one.

INVESTIGATION OF DAMPING EFFECTS ON STATISTICAL ENERGY ANALYSIS OF COUPLED STRUCTURES

The effects of damping on energy sharing in coupled systems are investigated. The approach taken is to compute the forced response patterns of various idealised systems, and from these to calculate the parameters of Statistical Energy Analysis model for the systems using the matrix inversion approach [1]. It is shown that when SEA models are fitted by this procedure, the values of the coupling loss factors are significantly dependent on damping except when it is sufficiently high. For very lightly damped coupled systems, varying the damping causes the values of the coupling loss factor to vary in direct proportion to the internal loss factor. In the limit of zero damping, the coupling loss factors tend to zero. This is a view which contrasts strongly with ‘classical’ SEA, in which coupling loss factors are determined by the nature of the coupling between subsystems, independent of subsystem damping. One implication of the strong damping dependency is that equipartition of modal energy under low damping does not in general occur. This is contrary to the classical SEA prediction that equipartition of modal energy always occurs if the damping can be reduced to a sufficiently small value. It is demonstrated that the use of this classical assumption can lead to gross overestimates of subsystem energy ratios, especially in multi-subsystem structures.

The Effect of Design Changes on Sound Transmission through a Building

A statistical energy analysis model of a building was used to assess the effect of design changes on sound transmission. Systematic changes were made to the material properties (density, Young's modulus, Poisson's ratio and internal loss factor) and to the dimensions (thickness and room size). These changes resulted in a redistribution of the energy throughout the building causing the noise level to go up in some rooms and to go down in others. For each case examined it was found that the effect of several changes could be estimated from the sum of the individual changes. Thus a change of 20% in the density resulted in approximately double the change in DnTw that was obtained from a 10% change in density. The same additive effect was also found to apply if more than one variable was changed at the same time. Thus the change in DnTw resulting from a small change in Young's modulus for the floors and a small change in the density of the walls can be estimated from the sum of the two individual effects. Changes to the thickness and density of the walls and floors have the greatest effect on sound transmission whilst changes to Young's modulus and Poisson's ratio have a much smaller effect. Damping can also have a significant effect on transmission particularly far from the source.

Sound transmission through masonry cavity walls

A theoretical model of sound transmission across metal ties in a cavity wall is presented in which power flows across independent point contacts between the two leaves of the wall. From a knowledge of the mobilities of the walls and the tie stiffness, the power flow across the ties can be determined. The expression for the power flow is then used as part of a statistical energy analysis model which enables the performance of the entire wall to be computed. Transmission across the air was modelled as an equivalent tie. The stiffness of standard ties was measured as was the performance of a number of different walls with those ties. The measured results agreed well with the theoretical model, except when there were no metal ties.

Implementation of Acoustic Blankets in Energy Analysis Methods with Application to the Atlas Payload Fairing

Energy analysis methods are commonly used to predict the vibroacoustic response of a spacecraft inside a launch vehicle payload fairing during liftoff. The payload fairing wall is often lined with acoustic blankets which can be difficult to model. This paper shows the development of a practical working method for accurately representing acoustic blankets in a VAPEPS statistical energy analysis model. A clear scaling relationship between the acoustic absorption coefficient and the blanket thickness is established. The scaling procedure is demonstrated using two sets of data for different absorption materials. The absorption data are implemented in VAPEPS by converting them to frequency-dependent damping loss factors. A post-processing procedure is derived to account for the increase in transmission loss due to the addition of the blanket layer. The entire process is used to predict the noise reduction in a VAPEPS model of the Atlas 14-ft-diameter payload fairing. Comparisons to ground-test and flight data for the blanketed fairing are presented to validate the procedure.

Use of Statistical Energy Analysis for Rotating Machinery, Part III: Experimental Verification With Machine on Foundation

A Statistical Energy Analysis (SEA) model of a journal bearing on an I-beam foundation is developed to predict the response of the components of the journal bearing. The SEA model consists of 16 subsystems that represent wave components in both the in-plane and out-of-plane directions. The coupling loss factors are estimated using experimentally measured mobility data at the junction locations of the subsystems. The internal loss factors of each of the substructures are measured using both a power injection method and a time decay method, with each substructure disassembled from the rest of the bearing. The mean square velocities on the surface of the bearing components, estimated using the SEA model, are compared with those directly measured. The agreement between the SEA estimated results and the experimentally measured results is good.

Use of Statistical Energy Analysis for Rotating Machinery, Part I: Determination of Dissipation and Coupling Loss Factors Using Energy Ratios

A method for determining the dissipation and coupling loss factors of a fully assembled machinery structure is presented in this paper. The method is based on the experimental measurements of the total loss factors and the energy ratios between the subsystems of the machine structure when fully assembled. The presented approach, which will be referred to as the energy ratio method, is suitable for both conservative (no energy dissipation at the junction) and non-conservative coupling. Not only does this energy ratio method determine the coupling loss factors for directly coupled subsystems, but it also determines the coupling loss factors between subsystems which are not directly coupled. This type of coupling exists when the subsystems of the structure are small compared to the wavelength of vibration and all coupling interfaces are in the near field of each other. A series of experiments are carried out to demonstrate the use of this approach and to verify the accuracy of this method for predicting the energy distribution among a set of coupled subsystems. The determined coupling and dissipation loss factors are used in a Statistical Energy Analysis (SEA) model of a rotating machinery component and the estimated results for the response are compared to the directly measured results. As expected, the agreement between the SEA estimated results and the directly measured results is good.

Plate acceleration and sound transmission due to random acoustic andboundary-layer excitation

The 'APEPS statistical energy analysis model is used in the frequency range of 315-5000 Hz to predict the acceleration response and sound transmission of plates that are exposed to random acoustic or turbulent boundary-layer excitation. Good agreement was found with measured noise reduction data for aluminum, acrylic, and graphite-epoxy plates with length and width larger than the acoustic wavelength. The VAPEPS predicted acceleration response of a 12 X 8 X 0.040-in. plate, installed in a 1-in. thick rigid plate inside a wind tunnel, agreed well with measured data for a stationary (random acoustic) as well as for a convected (turbulent boundary layer) pressure field. It was found that lowest order plate modal frequencies increase most with free flow velocity U. The overall acceleration level as well as the transmitted sound pressure level due to turbulent boundary-layer excitation were close to a Us.' dependence. Excitation by a laminar boundary layer resulted in a U3.6 dependence. Measured noise reduction data for random acoustic excitation of the plate in the wind tunnel fell between the VAPEPS predicted curves for resonant and nonresonant sound transmission showing the limitations of statistical energy analysis to accurately predict sound transmission at low modal densities. The noise reduction of the plate, when backed by a shallow cavity and excited by a turbulent boundary layer, was predicted using a simplified theory based on the assumption of adiabatic compression of the fluid in the cavity. VAPEPS predicted plate acceleration response was used as input in the noise reduction prediction, and reasonable agreement was found with the measured noise reduction in the frequency range of 315-1000 Hz.

The importance of in-plane waves in sound transmission through buildings

The effect of including in-plane waves in a statistical energy analysis model of a building is examined. It is shown that the additional waves make little difference close to the source, but large differences can occur far from the source if they are omitted. It is found that there is close to equipartition of energy between longitudinal and transverse modes. This allows the model to be simplified by combining longitudinal and transverse subsystems into a single in-plane subsystem. Agreement between the measured and predicted response of a building was reasonable except where there was significant airborne sound transmission along corridors. There was some scatter of results but much of this can be explained by the effects of small random errors in the individual coupling loss factors which cannot be predicted.

Universal prediction schemes for estimating flow-induced industrial pipeline noise and vibration

This paper addresses the issue of quantifying the internal noise levels/wall pressure fluctuations and external radiated nose from uninsulated industrial gas pipelines. The separate issues of transmission loss characteristics of pipeline acoustic lagging (porous layers and cladding) and structure-borne sound associated with cladding supports are not dealt with at this stage. Semi-empirical prediction schemes based upon (i) estimated vibration levels and radiation ratios, (ii) semi-empirical transmission loss models, and (iii) statistical energy analysis models, are presented and compared. Relative levels are quantified into absolute levels via non-dimensional internal wall pressure spectra. The first scheme requires prior knowledge of the pipe wall vibration levels either from measured or predicted values. The second and third schemes are ‘stand-alone’ schemes which do not require any experimental data other than the non-dimensional internal wall pressure spectra.

Sound transmission paths through a statistical energy analysis model

The most common method of determining the performance of a system using statistical energy analysis is to solve a series of simultaneous equations. However, there can be many advantages in determining the performance on a path by path basis. The sound transmitted by a path is derived and the valid paths defined. The advantages of this form of analysis are discussed.

Statistical energy analysis modeling and testing of an acoustic guitar

Principals of statistical energy analysis (SEA) were applied to develop a mathematical expression for the time-averaged, space-averaged energy of vibration in specified frequency bands for the top plate of an acoustic guitar. The model predicted energy in terms of string parameters, guitar parameters, and plucking data. Primary assumptions were that the damping of the strings was negligible, that energy flowed from the strings to the resonance box, and that energy was dissipated in the resonance box. Power supplied to the strings was estimated from a classical model of the vibrating string. An experimental plucking apparatus was designed and instrumented with Bruel & Kjaer vibration analysis equipment. The results of the SEA-based computer program were compared to actual experimental data for a Yamaha FG335 acoustic guitar. The SEA model tended to overpredict the energy in the top plate in most frequency bands. Possible causes for the differences were identified and discussed.

Practical industrial method of increasing structural damping in machinery, I: Squeeze-film damping with air

There is frequently a need to reduce sound radiation due to resonant flexural motion of stiff machinery panels. This can be achieved by applying squeeze-film damping to the vibrating panel by attaching an auxiliary plate parallel to the surface, thereby trapping a thin layer of air. Relative vibration of the plates pumps this air at high velocities, resulting in energy loss due to the air viscosity. In this study the damping below the critical frequency of the “thick plate” with an “attached plate” and air layer has been investigated by using an impedance approach. This model is incorporated into a two element Statistical Energy Analysis (SEA) model to predict the damping well above the critical frequency of the thick plate. The agreement between the predicted and measured results is remarkably good. Below the critical frequency the damping is pumping controlled, while above critical the plate couplings are the controlling factor.

The sound transmission loss of a single panel measured with the two-microphone and the conventional method-comparison with the statistical energy analysis model

The sound transmission loss of a single metal panel obtained with the sound intensity technique and the conventional method have been compared with the theoretical model of Statistical Energy Analysis (SEA). This method of comparison with a theoretical model is useful in order to explain small systematic deviations between the results of both experimental methods. It happens that the difference between the so-called residual reactivity level and the measured reactivity gives a good estimate of the accuracy of the intensity measurement. Even in very simple cases this quantity can disturb the measuring results due to the complex sound field close to the sound radiating panel. By correcting the sound transmission loss results with the well known Waterhouse correction and by choosing an appropriate probe distance to the sound radiating construction, good agreement is obtained between the results of both experimental methods as well as with the SEA model.

The effect of random errors on a large statistical energy analysis model

The framework of analysis known as Statistical Energy Analysis has many important applications particularly in systems where detailed information is not available. As a result of the approximations made, to simplify the calculations, random error can be introduced into the SEA model. For large systems this gives rise to uncertainty in the energy levels. It is shown that the effect of these errors on the model depends on the “shape” of the model. A compact model dominated by short paths is less affected than a model controlled by long paths. In either case the ratio of the average error in the resultant energy level to the average error in the coupling loss factor decreases as the errors increase. This means that large models may be used with confidence even when based on data that is known to be approximate.

Parameter sensitivity of a T-junction sea model; the importance of the internal damping loss factors

For a typical building acoustics configuration, a T-junction of plates formed by a light weight wall placed on a heavy floor, a statistical energy analysis (SEA) model is presented. Only structural systems (i.e., no acoustic wavefields) are considered. Besides bending waves also in-plane waves, quasi-longitudinal and plane transverse waves, in particular, are included in the calculation. A parametric survey is conducted on the T-junction model—for one frequency (1000 Hz) only—in order to find the sensitivity of the SEA model to the inaccuracies of its parameters. It is shown that, when using reverberation time measurements of the plates to determine the internal damping loss factors, the worst case variations of the internal damping loss factors cause variations in the junction dampings of bending waves of about one order higher than any of the other parameters. Therefore, the conclusion is that in cases where the internal damping loss factors are large with respect to the coupling loss factors, it is necessary to obtain more accurate estimates for the internal damping loss factors than are found with simple reverberation time measurements of plates.

SEA equivalences between large platelike systems and their subparts

A procedure was developed to equate the properties of a large system, which is composed of a number of plates connected to each other in a variety of ways, with the properties of the smaller plates for use in Statistical Energy Analysis (SEA) modeling. Each small subsystem is a single plate, which can be characterized by a uniform thickness, an area, a material, and some internal losses. Each connection between systems is defined by a junction, with a length, a number of systems connected at that junction, and thicknesses of each system connected at that junction. Equivalents thicknesses, areas, loss factors, and junction lengths were determined for the larger composite system. The analysis was extended to the case where two composite systems are connected to each other.

Direct Measurement of Transmission Loss of Aircraft Structures Using the Acoustic Intensity Approach

A measurement technique is developed in order to obtain the sound transmission loss of an aircraft fuselage which obviates the need for the two-room transmission suite. The sound transmission paths were determined in tests on a light aircraft fuselage using a two-microphone acoustic intensity method for measuring the acoustic intensity transmitted to the interior when the fuselage was exposed to an external random incidence sound-field. The intensity transmitted through different sections of the fuselage can be estimated accurately using this new technique. Results of these tests show that the plexiglass window is the major transmission path in the high frequency range. In addition, the transmission losses through a single and a double layer window were predicted theoretically by using the Statistical Energy Analysis Model. Very good agreement is found between the predictions and the measurements.