ANÁLISIS GLOBAL DE LA GESTIÓN RECAUDATORIA

Fundamental principles of SEA

The inadequacy of element-based models to capture the high-frequency physics of dynamical problems has led to the development of a number of predominantly non- deterministic methods. In this section, statistical energy analysis is briefly introduced as the dominant approach adopted for solutions in the high-frequency domain. Some key developments and modern implementations, expanding the SEA framework to lower frequencies and wider range of problems, are then discussed. For completeness, newer schemes targeted at covering the problematic mid-frequency range are subsequently mentioned. Subsystem 1 Subsystem 2 Subsystem 3

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Figure 2.6: A schematic representation of the energy exchange between three SEA subsystems

The initial derivation of the SEA principles can be traced to the early 60s, motivated by the demand to analyse complex aerospace structures subjected to distributed ran- dom excitations. The observations made suggested an analogy between the laws of thermodynamics and vibroacoustic systems. That is, under appropriate conditions, the exchange of power between two coupled oscillators with one degree of freedom is

proportional to their energy difference. The governing principles were established by Lyon and DeJong [149], who are the original contributors to the creation of the SEA method. This classic book serves as a comprehensive resource on SEA, containing the relevant theoretical descriptions and also reflecting practical knowledge of its applica- bility gained since its inception.

Extension of the basic idea to complex systems involves a decomposition into several coupled substructures and performing a power balance between them, as depicted on Figure 2.6. The following annotation is defined: Pi,in and Pi,diss represent the power

injected to and dissipated by the i-th subsystem, respectively. Pij is the power ex-

changed between subsystems i and j. Writing the equilibrium equation is facilitated by expressing the various powers in terms of the time-averaged total energies Ei and

modal densities ni of the considered subsystems. To that end, several constants, typ-

ically referred to as the ’SEA parameters’ can be identified. Firstly, the damping loss factor (DLF) ηi is given by:

Pi,diss = ωηiEi (2.64)

Physically, ηi characterises the combined dissipative effect of structural damping mech-

anisms and acoustic radiation into the surrounding fluid, if such is present. In addition, under the assumption of proportionality between the modal energy difference and ex- changed power:

Pij = ω (ηijEi− ηjiEj) (2.65)

where ηij is known as the coupling loss factor (CLF). The reciprocal relationship linking

ηij and ηji is

niηij = njηji (2.66)

Combining equations (2.64)-(2.66), the SEA equation for k coupled subsystems can be written in matrix form:

      η1+Pkj6=1η1j −η21 . . . −ηk1 −η₁₂ η2+Pkj6=2η2j . . . −ηk2 .. . ... . .. ... −η_1k −η_2k . . . ηk+ Pk j6=kηkj            E1 E2 .. . Ek      =      P1,inj/ω P2,inj/ω .. . Pk,inj/ω      (2.67)

Due to the reciprocity relationship, provided ni = nj, the coefficient matrix on the

LHS of (2.67) is symmetric. Moreover, it is inherently much smaller than FEM or BEM derived descriptions of the global problem. Thus SEA is appealing precisely because it only needs top-level ’discretisation’, while simultaneously benefiting from the intrinsic parametric uncertainty of modes at high frequencies.

Estimation of SEA parameters and scope of validity

In contrast to the stated pros, the construction of (2.67) implies the knowledge of all modal densities, DLFs and CLFs. Obtaining them is usually not a trivial task and represents the main obstacle to the practical application of SEA.

The modal density is the estimated number of modes per frequency band. Its presence in the law of reciprocity means miscalculation of the ni terms gives rise to a notable

disparity between the real and estimated coupling loss factors. An analytical compu- tation approach for the modal densities involves considering the natural frequencies of the subsystems in a simply supported state, and is generally assumed to be reliable at high frequency. Examples for composite structures can be found in [150].

In general, typical wave-context approaches employed in the evaluation of the SEA coefficients rely on analytical solutions for (semi-)infinite systems with high modal overlap, and are therefore limited to relatively simple problems. Note that a ’wave approach’ to SEA approximately means viewing a complex system as an ensemble of propagating wavetypes, as opposed to a modal description. In addition, ’overlap’ refers to the diffuseness of the vibrational field of each subsystem (spatial modal overlap) and the number of modes that exist within the frequency band of application (spectral modal overlap).

In complex systems, either FE modelling (e.g. [151], [152]) or experimental measure- ment data for similar problems [153] can be used to find the SEA parameters. In both situations, the power injected method may be used. It involves applying a rain on the roof excitation to the system - essentially a set of arbitrary forces the combination of which reproduces white noise. The importance of this method is seen primarily in the frequency domain, where it is more accurate than wave approaches. More details on techniques applicable to finding CLFs and DLFs are available in [154].

Apart from the intricacy of extracting the SEA parameter values for complex systems, the practical usability of the method is hindered by a set of underlying hypotheses that have to be satisfied:

• The studied system is not in a transient regime. Excitations are random, station- ary and uncorrelated in a statistical sense.

• Each of the subsystems the model is partitioned into is only weakly dissipating. The coupling between any pair of them is weak, or stated equivalently, no global modes exist.

• Coupling between subsystem pairs is carried out by mass, stiffness and gyroscopic effects, therefore damping is small and losses in the coupling are ignored.

• The subsystems are reverberant, thus modal overlap is high, guaranteeing the correctness of a statistical representation.

• The behaviour of several grouped modes is amenable to description as an averaged mode. Additionally, the mean total energy in a frequency band is only affected by the constituent modes in that band, which also contribute to the entirety of the energy transmission to other subsystems.

In summary, the preceding requirements mostly translate into restrictions on the lower bound of the applicable frequency and the permitted characteristics of junctions be- tween subsystems.