LOS MENSAJES DE CORREO ELECTRÓNICO: EL “E-MAIL”.

Often when predicting structure-borne sound and vibration it is convenient to model an assembly in such a way that the FRFs of the individual sub-structures are ob- tained independently, then coupled together mathematically. This method will be referred to here as `dynamic sub-structuring' (DSS), although may be found in the literature under a number of names, including; sub-structure synthesis, structural synthesis, sub-structure coupling, among others. The DSS concept oers two distinct advantages over the direct measurement of FRFs. Firstly, it enables components to be interchanged with ease and their inuence upon the global dynamic behaviour of an assembly assessed quantitatively. Secondly, the integration of analytical and/or numerical data with experimentally determined FRFs becomes feasible. Both of these oer clear advantages, particularly within the eld of virtual acoustic proto- typing. The general concept of DSS may be formulated in any of the 4 domains one typically encounters in structural dynamics and vibration; physical (where sys- tems are characterised by their mass, stiness and damping matrices), state-space (where systems are described in terms of their state-variables), modal (where sys- tems are characterised by their eigenvalue/vector and modal damping matrices) or FRF (where systems are characterised by their frequency response function matri- ces). The application of a given domain formulation is generally dependent upon

Chapter 2. Literature 20 whether the work undertaken is of a theoretical or experimental nature. Outlined in Table 2.2 are the relative occurrences of domain types for theoretical and ex- perimental work. With the work of this Thesis concerned with the development of experimental methodologies, it is the FRF based formulations that are of particular interest here.

Physical State-space Modal FRF Theoretical Always Often Typical Unusual Experimental Never Unusual Typical Always

Table 2.2: Dynamic sub-structuring domains and their use in theoretical and experimental studies. Table adapted from [68].

Historically, the roots of DSS may be traced to the eld of domain decomposition, where the desire to analyse complex problems was addressed by rst considering the solutions to the simpler problems of its constituent components, and then determin- ing an interface solution [69]. Perhaps the earliest example of domain decomposition was the iterative process proposed by Schwarz [70] in 1890, whereby the existence of a solution to a domain consisting of a coupled circle and square was proven. Fast for- ward 70 years and the concepts of domain decomposition had begun to make their way into the eld of structural dynamics. These early DSS ideas, largely known as `component-mode synthesis', were mostly developed as reduction techniques and likely fuelled by the papers of Hurty [71, 72]. It wasn't until the 1980's however, with advancements in multi-channel data acquisition, that these methods became attrac- tive tools to the experimental structural dynamic community. Perhaps the rst step towards an FRF based DSS procedure was made by Crowley et al. [73], who pro- posed the structural modication method `SMURF' (structural modication using experimental frequency response function). However, it was not until a few years later, when Jetmundsen [74] formulated the now classic FRF based sub-structuring method, that experimental DSS began to gain popularity. Since Jetmundsen, a number of alternative approaches have been proposed. Although, with the physics of the problem remaining unchanged these simply go about applying compatibility and equilibrium conditions in a dierent manner. However, an essential requirement common to all DSS approaches is the independence of the sub-structure FRFs. That is, the FRFs of each sub-structure must be obtained in a transferable manner, an example being their free-interface mobilities.

Chapter 2. Literature 21 Generally speaking, a DSS approach may be considered a member of one of two families, depending on whether its formulation considers interface displacements or forces as unknowns. These are referred to as the primal and dual formulations, respectively [75].

Of the DSS methods available, the `classical impedance coupling' approach (also referred to as the basic impedance coupling process [76] or primal impedance for- mulation [75]) is arguably the most straightforward, both conceptually and in terms of its implementation. The method itself exists within the primal family and can be realised in a number of ways. It is well known that the enforcement of com- patibility and equilibrium between single DoF mechanical sub-structures results in a coupled structure whose impedance is equal to the sum of the individual sub- structure impedances. The extension of this concept to multi-DoF systems forms the basis of the classical impedance approach. Its implementation requires the free- interface mobility of each sub-structure to be measured and subsequently inverted. The resulting impedance matrices are then summed accordingly before being in- verted back to mobility form (usually required). It is this approach that is most often used in the assembly of nite element models. The classical approach requires all DoFs to be included in the inversion process, including those remote from the coupling interface. As such, ill-conditioning is a serious concern, particularly when dealing with large DoF assemblies.. Furthermore, matrix inversions are computa- tionally expensive procedures. The multiple inversions required at each frequency, along with the limited computational power and precision of early computers, meant that more computationally ecient algorithms were required.

Jetmundsen developed [77] and subsequently proposed [74] a generalized DSS method- ology that not only provided an ecient synthesis of the coupled assemblies dynamic behaviour from experimental data, but was particularly well-suited for combining experimental and numerical data. The proposed method required only a single matrix inversion, whilst employing graph theory to form the required connections. Unlike the classical approach the single inversion is preformed on a matrix contain- ing only the coupling DoFs, thus oering the potential for improved conditioning and eciency. The approach proposed by Jetmundsen has since been reformulated according to the dual domain decomposition method to form what is referred to as the Lagrange Multiplier Frequency Based Sub-structuring (LM FBS) method [78]. The LM FBS approach oers a number advantages over its predecessor; not only

Chapter 2. Literature 22 may it be expressed more simply, but only single coupling matrix is required to coupledN sub-structures, unlike Jetmundsen's approach which required N + 1.

2.3.1 Sub-structure Decoupling

In more recent years the DSS concept has been reversed, and instead used in the de- coupling of assemblies [7983]. Referred to here as dynamic sub-structure decoupling (DSSD), the procedure may be used to extract the independent passive property of a given sub-structure (i.e. the target sub-structure) from the measured properties of the coupled assembly and that of the remaining uncoupled sub-structure (i.e the residual sub-structure). Such a procedure oers the distinct advantage that the independent passive property of a given sub-structure may be determined without the need for free suspension, as would normally be required. This is of particular interest in cases where free suspension is simply not possible, for example; in cases where sub-structures are very large and heavy, and resilient mounting becomes im- practical (for example the decoupling of train carriages from their bogies [84]), or in cases where sub-structures are very small and lightweight, and resilient mounting does not suciently represent a free suspension.

Like DSS, the concept of DSSD is also of relevance to the eld of virtual acoustic prototyping. As previously stated, a fundamental requirement of a general VAP framework is that each sub-structure is characterised independently. This includes their passive properties. DSSD potentially oers an alternative means to characterise the passive properties of both source and receiver sub-structures.