F.J. Fahy

Statistical energy analysis: a critical overview

For the benefit of the ‘enquirer within’, who may not be familiar with the background and concepts of SEA, this overview opens with a discussion of the rationale for the use of probabilistic energetic models for high-frequency vibration prediction, and introduces the postulate upon which conventional SEA is based. It compares and relates the modal and travelling wave approaches, discusses the strengths and weaknesses of SEA as currently practised and points out needs and directions for future research. Critical discussions of individual contributions to the development of the subject are presented only in as much as they treat specific matters of concept, principle or reliability. The roles of SEA in providing a framework for experimental investigations of the high-frequency dynamic behaviour of systems and in interpreting observations on operating systems, although equally important, are not substantially addressed. Nor are specific experimental techniques which involve considerations of transducers, spatial sampling, signal processing, error analysis and data interpretation, which require a critical review in their own right.

A study of uncertainty in applications of sea to coupled beam and plate systems, part I: Computational experiments

Abstract Computational studies of the effects of random geometric perturbations of coupled beams and plates on frequency-band average vibrational power flow, and on derived coupling loss factors, are presented. Complementary experimental results will be presented in Part II of the paper. The results indicate that the two principal parameters which control the variability of power flow, and the associated coupling loss factor, are the modal overlap factors of the uncoupled subsystems and the number of coupled modes of the total system which are resonant in the frequency band considered. It is confirmed that the commonly employed “travelling wave” estimate of coupling loss factor generally exceeds the actual value when the average modal overlap factor is much less than unity. Under conditions of low modal overlap the normalized variance of the power flow tends significantly to exceed that of the perturbed geometric parameter, while that of the associated coupling loss factor can be of the order of unity: both variances become small when the modal overlap factor exceeds unity. Under conditions of low modal overlap, normally distributed geometric variations generate highly non-normal distributions of power flow and coupling loss factor. It is found that it is necessary to have at least five modal resonance frequencies in a band for stable estimates of coupling loss factor to be obtained. A problem of potential concern to analysts of variance in SEA is that the relationship between the variance of “input” and “output” quantities is not, in general, linear, thereby limiting the general validity of conclusions reached by specific computational or experimental studies. A conclusion of practical significance for experimental evaluation of coupling loss factors is that value derived from tests on one physical sample of a class of system may be quite unrepresentative of the ensemble-average value when modal overlap factors are small.

A note on the interaction between a Helmholtz resonator and an acoustic mode of an enclosure

Abstract A single Helmholtz resonator is coupled to an enclosure and tuned to the natural frequency of one of its low order acoustic modes. The effect on the free, and forced, vibrations of the fluid in the enclosure is analyzed. The conditions necessary for the resonator to increase the damping of the two resultant modes, and to control the room response to excitation at frequencies within the range embracing both natural frequencies, are investigated. A simple design graph is presented.

Some applications of the reciprocity principle in experimental vibroacoustics

The principles of acoustic and vibroacoustic reciprocity are explained. Examples are then given of applications of acoustic reciprocity to the experimental analysis of sound radiation by various systems of interest to noise control engineers. The final part of the paper is devoted to a presentation of examples of the practical application of Lyamshev reciprocity to problems of identifying and quantifying sources of noise that operate in a variety of engineering systems.

VIBRATION OF CONTAINING STRUCTURES BY SOUND IN THE CONTAINED FLUID.

Abstract A system which consists of a rigid rectangular box with one simply-supported flexible wall is analysed by numerical and statistical methods for the internal acousto-structural mode coupling factors and the corresponding modal average radiation efficiency. A lower limiting frequency is found above which the subcritical radiation efficiency is equal to that of a baffled panel radiating into free-field conditions but below which the radiation efficiency falls below the free-field values. This limit is important practically in the cases of small box volumes and gases of high sound speed. At supercritical frequencies, a value of radiation efficiency half that of a freely radiating panel is found. Measurements have confirmed the existence of the lower limiting frequency. It has also been observed in the case of a cylinder. Corrections to the normal statistical energy response and radiation equations are presented which take into account the fact that it may not be assumed, purely on the basis of modal density ratio considerations, that many acoustic modes couple with an individual structural mode.

Response of a cylinder to random sound in the contained fluid

Coupled oscillator theory is used to estimate the response of a closed cylindrical shell to broad-band sound in the enclosed air. A frequency is determined above which the response is the same as that to an externally incident, diffuse sound field. It is termed the lower limiting frequency. Below this frequency, the response falls below that produced by external excitation. Two different cylinder end acoustic boundary conditions are investigated experimentally. It is found that the size of an opening in the cylinder end, in terms of an acoustic wavelength, governs the lower limiting frequency phenomenon. Observations are made about the relative merits of statistical and non-statistical estimates of multi-mode coupling under conditions of broad-band excitation.

Power flow between non-conservatively coupled oscillators†

The time-averaged power flow between two oscillators coupled by spring and damping elements, subjected to white noise force sources, is shown to be not simply proportional to the time-averaged oscillator energy difference, but also to the time-averaged energy of the individual oscillators. The proportionality constants are functions only of the oscillator and coupling parameters.

The use of acoustically tuned resonators to improve the sound transmission loss of double-panel partitions

Double-leaf partitions are often utilized in situations requiring low weight structures with high transmission loss, an example of current interest being the fuselage walls of propeller-driven aircraft. In this case, acoustic excitation is periodic and, if one of the frequencies of excitation lies in the region of the fundamental mass-air-mass frequency of the partition, insulation performance is considerably less than desired. The potential effectiveness of tuned Helmholtz resonators connected to the partition cavity is investigated as a method of improving transmission loss. This is demonstrated by a simple theoretical model and then experimentally verified. Results show that substantial improvements may be obtained at and around the mass-air-mass frequency for a total resonator volume 15% of the cavity volume.

Wave propagation in damped, stiffened structures characteristic of ship construction

Abstract Efforts to attenuate structure-borne sound generated in ship structures by propulsion and auxiliary machinery by the application of damping treatments to the hull and decks have not been conspicuously successful. An attempt is made to evaluate the potential effectiveness of damping hull-frame structures, as a means of controlling the vertical transmission of vibration, by the theoretical waveguide analysis of an idealised model of a section of typical ship structure which consists of two parallel beams (frames) attached to a flat plate (hull). Wavenumbers of free propagation parallel to the beams, together with proportions of energy flux carried in the plate and the beams, are calculated. Attenuations produced by adding ten percent hysteretic damping to the plate and the beams separately are evaluated and it is concluded that the short circuiting effect of the frames greatly reduces the effectiveness of plate damping near the cut-off frequencies of the waveguide modes. Application of damping to the beams is found not to adequately suppress this adverse behaviour. Plate motion is confined to pure flexure and beam motion is confined to bending and torsion.

Application of a vibro-acoustic reciprocity technique to the prediction of sound radiated by a motored IC engine

Abstract An internal-combustion (IC) engine is a complicated noise source. Theoretical prediction of its sound field is problematic. This paper describes a vibro-acoustic reciprocity technique for the experimental prediction of sound radiated by the engine. The basis of the technique is the evaluation of the surface volume velocity distribution of the source by nearfield intensity measurement and of the transfer functions between the volume velocity and the pressure at a receiver point by a reciprocal measurement technique. The principal advantage of this technique is that it avoids direct vibrational velocity measurement. This paper describes the method and a laboratory experiment which demonstrates its application.