This section focuses on the application of Statistical Energy Analysis (SEA) to built- up structures and specifically buildings. The development of the fundamental theory of SEA is given in section 2.2. SEA is a framework that allows the analysis of sound and vibration transmission between subsystems that form a larger system. A system representing a building is divided into subsystems, each representing a space or structural element that can store modal energy. Applying a steady-state power input into one or more of the subsystems, the steady-state sound pressure or vibration level can be determined for each subsystem.
The framework of analysis that later became known as SEA started as research by Lyon and Maidanik [15] into two linearly coupled oscillators, followed shortly after, by Scharton and Lyon [16]. Both papers looked at excitation of the oscillators by broadband random noise. From this work SEA was developed to evaluate time- averaged sound pressure and vibration levels in systems comprising many subsystems. Hodges and Woodhouse [17] and Lyon and DeJong [18] set out a number of assumptions that have to be satisfied in order that a group of subsystems may be studied using SEA. These assumptions concern the excitation forces, the modal response of the subsystems, and the coupling between the subsystems.
Crocker and Price [19] used SEA to predict the direct airborne sound transmission, across a separating wall. The SEA model consisted of three coupled subsystems, two rooms and an aluminium plate, and included resonant and non-resonant transmission. The thin aluminium plate had a high-modal density and therefore many modes below the critical frequency, which is not always typical of heavyweight concrete or masonry building elements. The results showed good agreement between measured and SEA predicted sound transmission loss. This work is expanded upon by Price
and Crocker [20] who look at the direct sound transmission through a double leaf wall using SEA. The model consists of five subsystems, two rooms, two plates and a cavity and as with the single leaf construction discussed previously both the resonant and non-resonant transmission is included. Combinations of plates with different densities were used, thereby altering the critical frequency (or frequencies) of the double leaf wall. It was shown that SEA could predict the transmission loss of all the double leaf walls to within 5 dB.
Early work concerning the the transmission of bending waves between coupled plates was done by Cremer and Heckl [21], and for bending and in-plane waves by Kihlman [22]. This produced angular-averaged transmission coefficients which allow calculation of structural coupling loss factors around the corners of X-, T- and L- junctions and across the straight section of X- and T- junctions. When considering structure-borne sound transmission using SEA a diffuse vibration field is assumed [23], therefore the angular-average transmission coefficient is relevant and is calculated by integrating angle dependent transmission coefficients over the angle of incidence.
Gibbs and Gilford [24] looked at SEA methods to assess sound transmission in buildings, concentrating on a T- junction and a pair of rooms that share a common line junction. A T- junction forms the key element that is repeated throughout large buildings, in order to simplify the SEA model of the T- junction only bending wave transmission across the junction is considered. The agreement between the SEA model and measurements of a quarter scale model show good agreement at mid-high frequencies, at low frequencies the individual modes of the plates dominate the response giving poor agreement. A pair of rooms coupled along a junction line was modelled using fourteen subsystems, the results show good agreement between the SEA predicted energies and measurements from the quarter scale model. Discrepancies between the measured and predicted energies are encountered at low frequencies, this is attributed to a breakdown in the power flow concepts which require sufficient modes in a frequency band in order to hold true.
A large amount of work on applying SEA to buildings is summarised by Craik [25], the work focussed on the determination of subsystem properties and coupling loss factors and the evaluation of SEA models comprising many subsystems. Craik [26] investigated the use of SEA to predict the sound pressure and vibration level due to direct and flanking sound transmission in a heavyweight building comprising many rooms and plates. The SEA coupling loss factors are validated using simplified ESEA. Comparing the measured and SEA predicted energy levels shows good agreement, with larger errors at low frequencies, presumably where the few local modes of each subsystem dominate the response. It is noted that some of the large errors observed were due to the omission of airborne flanking paths in the SEA model. Craik [27, 28] showed the significance of in-plane waves when dealing with structure-borne sound transmission is large buildings comprising X- and T- junctions, both of which are key in converting bending waves into in-plane waves, and visa versa, across the junction. By comparing SEA models with and without in-plane waves to measured sound pressure and vibration levels, it was shown that that the further away the receiver subsystem is from the source subsystem the larger role in- plane waves, and flanking transmission played in sound transmission. The papers by Crocker, Price and Craik [19, 20, 26-28] amongst others show the development of SEA successfully being used to predict sound pressure and vibration levels for direct sound transmission and flanking transmission. Moving from research that looks at direct sound transmission to those that include flanking sound transmission as well, the SEA models include more structural coupling than radiation coupling, as in large building structural coupling provides the flanking transmission with radiation coupling providing the final path to give a sound pressure level in a room.
The calculation of sound transmission and, airborne and impact sound insulation using SEA are summarised by Gerretsen [29, 30]. The work goes on to compare measured and predicted sound insulation values, the agreement for the frequency weighted sound reduction index is non-biased and within 5 dB. For a single separating wall in a transmission suite it is shown that SEA gives the same results for airborne sound insulation as the classical approach. As SEA considers all flanking
paths explicitly (as well as the direct path), given the parity between the two prediction methods it is assumed that the interaction between the flanking paths is negligible. While this may be the case for a confined system such as a transmission suite, this is not the case for system representing larger buildings where due to the number of flanking paths the flanking sound transmission is significant compared to the direct sound transmission, see Craik [26].
Hopkins [31, 32] looks at the effect of apertures in masonry walls on the vibration transmission across plates forming T- and L- junctions. The papers compare measured data to predicted data from SEA and Finite Element Methods (FEM) models, although this thesis is not concerned with FEM comparisons with SEA highlight certain attributes. Although SEA does not compare favourably to FEM in predicting the vibration level difference, SEA consistently underestimates the vibration level difference, SEA is not necessarily expected to perform well as there are low mode counts and modal overlap, Ns < 5 and M < 1, over most of the
frequency range considered. The key conclusions from this work which pertain to SEA are; that SEA cannot account for the change in the modal behavior of a plate due to the presence of a aperture, and without modification SEA cannot predict the variation due to the inherent uncertainty in the physical parameters of a test construction. Hopkins [23] systematically shows the effect of mode count, Ns, and
modal overlap, M, for airborne sound transmission as well as structure-borne sound transmission on heavyweight walls, highlighting where prediction models break down due to low mode counts and low modal overlap.
Current Standards for airborne and impact sound insulation [33, 34] use an SEA- based model to predict steady-state sound and vibration transmission. Like SEA only a steady-state structure-borne or airborne power input can be used. Although SEA can only predict steady-state sound pressure and vibration levels, it provides the basis of knowledge from which TSEA has been developed, and can be developed further.