This section reviews the literature relating to the concepts, main assumptions of and some limitations of SEA.
2.2.1.1 Concepts and assumptions
The origins of SEA concern the analysis of a linear system comprised of two ‘weakly’ coupled oscillators excited by independent broadband random noise [7]. It is found that power flow is proportional to the difference in energies of uncoupled resonators and it always flows from the resonator with higher energy to the one with lower energy. The analysis was extended to solve more complicated multi-modal subsystems under the assumption that the energy flow between two multi-modal subsystems is proportional to the difference in their modal energies. However, this statement can only be justified under the following assumptions [17]:
(1) ‘Weak’ or ‘light’ coupling between subsystems
In a modal approach, weak coupling can be considered to occur when the local modes of an uncoupled subsystem hardly change when it is coupled to other subsystems so that energy flow can be related to the local modal energies [19]. In terms of waves, weak coupling requires the wave field incident upon either side of the junction between two subsystems to be incoherent [20]. In the case of weak coupling, the energy flow is only dependent on the local properties of the subsystems, whilst if it is strong coupling, energy flow between subsystems is largely dependent on the global properties of the system where standard SEA formulation will no longer hold. Various criteria have been proposed to evaluate the validity of the condition of weak coupling and the applicability of SEA in previous studies. Langley [21, 22] proposed a definition of weak coupling where the difference between the Green function of a coupled subsystem and that of the uncoupled subsystem is sufficiently small. Fahy and James [23, 24] extended
Langley’s definition and proposed a practical method to determine the strength of coupling between two subsystems by using the time delay in the rise of kinetic energy of one subsystem when an impulsive excitation is injected to the other subsystem. Mace [25] gave a simple evaluation of coupling strength both from the modal analysis and the wave analysis. All of these methods try to ensure a well-conditioned energy response matrix in SEA for the matrix inversion.
(2) Equipartition of modal energy in subsystems
This assumption means that each mode of the subsystem carries equal amount of energy. The modal responses for subsystems are also assumed to be incoherent. To satisfy these assumptions requires a selection of similar mode groups to form subsystems often based on the similarity of geometries of the structures. It is often considered that subsystems with low damping tend to get close to the condition of equipartition of modal energy. The extreme situation of ‘true’ equipartition of modal energies can only be achieved when the subsystem damping is zero [7]. However, Yap and Woodhouse [26] indicated a contrary conclusion against the classical SEA prediction that subsystems with low damping didn't always yield equipartition of the modal energies and SEA could significantly overestimate the modal energies of those subsystems that are not physically connected to the source subsystem (for example, a chain of subsystems).
(3) Subsystem response to be dominated by the reverberant field
Under this assumption, the energy in a subsystem can be considered uniformly distributed. With highly damped subsystems, however, this assumption will not be true.
(4) Equal probability of natural frequencies occurring in the interested
frequency bands
This assumption means that each subsystem is a member of a population of systems that are generally physically similar, but different enough to have randomly distributed parameters [18].
Such excitation applies equal modal forces to all the subsystem modes and injects energy into the direct wave field equally at all points of the excited subsystem [27] so that the basic SEA assumption of equal partition of modal energies among subsystems is satisfied. Statistically independent excitations are described as over the plate surface with constant amplitude but with phase randomly distributed with location. Under this type of excitations, the energy response can be calculated by summing the energy response due to excitation applied to each point in the excited subsystem [28]. It can be realized using rain-on-the-roof excitation where the complex forces are delta-correlated, broadband excitations applied on the subsystem with magnitude at any location proportional to the local mass density and the phase follows a uniform probability in the range of (0, 2π). Ideal rain-on-the-roof can excite the local modes of the excited subsystem equally [28].
2.2.1.2 Limitations
Limitations of SEA result from the constraints of the assumptions that the development of SEA is based upon.
SEA can only give the estimate of the statistically averaged global response for a subsystem and cannot predict the distributions of the energy field. This may cause significant error or even the failure of SEA if the local response within a subsystem dominates the total response instead of the global response. In other words, the assumption of equipartition of modal energies is not satisfied. Therefore, additional procedures need to be taken in order to incorporate the large local response. This assumption of equipartition of modal energies in the modelling was removed by Maxit and Guyader [29] by incorporating the modal energy distribution in the SEA formulation. The modal information of subsystems needs to be calculated and for complex structures, this can be achieved using the Finite Element Method (FEM). This procedure is only applied to those subsystems for which equipartition doesn't occur. The rest of the subsystems are modelled using classical SEA theory.
Another issue that limits the application of SEA is in frequency bands where the subsystems have low mode count and low modal overlap (more discussions see
section 2.2.4). Craik et al. [30] discussed the application of SEA at low frequencies where low mode count and modal overlap occurs. Theoretical models by spatially averaging the mobilities of the subsystem were used to determine the coupling loss factors. It was stated that “modal properties of the receiving subsystem affect coupling between two subsystems”. Large fluctuations of coupling loss factors from the measurements were observed at low frequencies and they seemed to follow the same manner as the mobility of the receiving subsystem. The theoretical method was also able to give the upper and lower limit of coupling loss factor at low frequencies. Hopkins [31] investigated the application of SEA for different structural junctions with low modal overlap and mode count. It was shown that small variation of material properties can cause significant differences in the coupling parameters. Therefore, it is necessary to use numerical or experimental ensemble average to determine the coupling loss factors instead of one single deterministic analysis.
The assumption of weak coupling is also one of the concerns in the application of SEA as in many engineering structures the coupling between subsystems can be considered as ‘strong’. Although weak coupling has been considered as one of the basic assumptions in the derivation of SEA, Scharton and Lyon [32] showed that this assumption could actually be removed in SEA by redefining the subsystem ‘blocked’ energies. Mace and Rosenberg [33] related the coupling strength to the damping of the subsystems and it was indicated that when the coupling is strong (small damping), more information is needed for each subsystem than normal SEA in order to give accurate predictions (i.e. the coupling loss factor results can be sensitive to the shape of the subsystem).