Kundoc.com_statistical-energy-analysis-of-offshore-structures.pdf

Statistical energy analysis of offshore structures A. J. Keane

Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK (Received October 1992, revised version accepted March 1993) This paper briefly introduces the concepts of statistical energy analysis (SEA) and then discusses the potential for using the approach in the design of offshore marine structures. Some inherent limitations in the method as currently applied, mainly in the field of aerospace structures, are indicated and suggestions made as to areas where further research could be expected to overcome some of them. Of these, the problem of dealing with fluid actions in an SEA framework is perhaps most deserving of effort by the marine engineering community. A highly idealized example is used to illustrate some of these points.

Keywords: statistical energy analysis, offshore marine structures

The role of structural analysis has been fundamental to the exploration and development of offshore oil fields. It hardly seems worth mentioning that is is now possible to design and build vast concrete and steel structures that will withstand the worst that nature can throw at them. However, it is inevitable that, as current problems are tackled and brought under control, higher standards are desired both by operators and regulators. Initially, survivability was the main design criterion, now this is considered alongside reduced through life cost, ease of maintenance, high standards of habitability, etc. One of the consequences of this shift in horizon is the need for new design and analysis tools to be brought on stream to deal with problems that have, until recently, not been seen as high priorities. One area that is becoming increasingly important in all maritime spheres is that of noise and vibration control. Noise standards are become increasingly stringent for accommodation spaces and structural vibration can lead to rapid fatigue failures, particularly in lightweight structures. This topic has received considerable attention for very many years in related fields and a number of standard palliative techniques such as vibration isolation mountings, acoustic claddings, etc., are routinely used in marine construction. Indeed, it is now not unusual for entire deckhouses on ships to be mounted on resilient supports. None the less, most such techniques are rather ad hoc in nature and it would clearly be desirable to be able to calculate the effects of various measures during the design process. It would be particularly useful to be able to assess the paths that vibrational energy takes when flowing around a structure and also the modes of energy transport. Armed with such information it would then be possible to take appropriate steps. The resilient mounting of a deckhouse is a good example of an approach based on knowledge of energy paths: obviously all the noise energy measured in a cabin

0141-0296/94/020145-13 © 1994 Butterworth-HeinemannLtd

that is part of a deckhouse must enter through the foundations (assuming that there is no machinery in the structure itself) and thus resilient mounts block this flow in an efficient manner. Unfortunately, most of the structures found in offshore installations are quite complex and, particularly at low frequencies (50-300 Hz, say) their acoustic behaviour cannot be divorced from their structural dynamics. These frequency ranges are not amenable to analysis using finite elements (FE) or standard acoustics methods. The FE method, for instance, depends heavily upon numerical procedures which demand large, fast computational facilities in order to deal with mathematical models representing very detailed idealizations of the physical structures. The computational demands increase with structural and material complexity, and with analysis frequency range. Even today, when computational methods are highly developed and optimized, it is not generally practicable to predict the detailed vibrational behaviour of such structures at frequencies beyond a few tens of hertz. It is, in principle, possible to extend analyses to higher frequencies, at the expense of rapidly increasing demands in terms of the size of the model, and consequent analysis time and cost. However, as frequencies increase, the results become increasingly dependent upon fine structural detail, including structural connections, which cannot be mathematically represented with sufficient accuracy. In addition, the high frequency vibrational behaviour of individual physical realizations of nominal structures are observed to differ, often greatly, because of the influence of fabrication tolerances. It is clear that the high costs of computational procedures based on deterministic models cannot be justified if they do not yield reliable results. Conversely, acoustics methods rely heavily on statistical assumptions that may well not be valid until rather high frequencies are encountered where many modes of the structure are excited. Clearly

Engng Struct. 1994, Volume 16, Number 2 145

Statistical energy analysis of offshore structures. A. J. Keane

they are of little use when attempting to predict the fatigue life of offshore structures. In such circumstances other methods may be more appropriate. One such method is statistical energy analysis (SEA) which has been developed to deal with problems where FE and other approaches are unworkable 1. This method, or more accurately, analysis philosophy dates from the early 1960s, when engineers sought new analytical tools for dealing with the problem of predicting the response of launcher and payload structures to rocket noise at launch. Statistical concepts and models of dynamic behaviour which had been exploited for many years in the analysis of sound fields were borrowed and adapted to structural systems. System parameters were expressed in probabilistic terms, and the objective of an analysis was seen to be the prediction of the ensemble-average behaviour of sets of grossly similar realizations of an archetypal system (such as the products of an industrial production line). System response to vibrational inputs was characterized by time-average vibration energy; energy flows between coupled subsystems were expressed in terms of energy transfer coefficients; and vibration distribution was determined from power balance equations. The use of SEA has not, however, always been well received 2. In particular, it is, as originally conceived, poorly suited to dealing with problems where the structure of interest cannot be broken down into subsystems whose interactions are in some sense weak. This situation is worsened if the individual subsystems do not have many, varied and lightly damped vibrational modes since the statistics of such structures become less well behaved. Moreover, being a statistical tool, methods are required for making confidence estimates for predicted results; such methods have not been widely studied although they are the subject of current research, albeit at an early stage 3. Another fundamental problem concerns the characterization of the couplings between subsystems, where energy transfer coefficients must be identified, either theoretically or experimentally. Nonetheless, a number of companies are bringing SEA-based computer packages to the market and SEA is routinely used by some aerospace agencies. Given this background, further work on SEA would still seem to be required before it gains wide acceptance as a practitioner's tool although its long term acceptance seems assured. This paper attempts to set out some of the areas where such effort is required and, more importantly, to discuss some of the limitations SEA is likely to suffer from due to the inherent assumptions of the approach.

Preliminaries Before plunging into the detail of SEA it is worth outlining two of the most important ideas underlying all work in this area. The first concerns the primary variables used to describe problems: these are long term averages of (kinetic) energy flows (or levels). Since these energy flows are time invariant and also because all dissipative flows are detailed explicitly, energy balance equations may be set up and solved to find the long term average energy levels in all parts of the system under investigation. These equations encompass details of the (known) forcing functions and system parameters and allow a designer to see how vibrational energy is distributed around a system. This information can then be

146

Engng Struct. 1994, Volume 16, Number 2

related to the various average motions, stress levels, etc. The formulation inevitably leads towards an analysis in the frequency domain and places limits on what the method can reveal (i.e., extreme transient behaviour cannot be predicted). One consequence is that SEA is of most use in dealing with situations involving forces that are characterized by stationary random processes, e.g., flow noise or complex machinery vibrations. Secondly, and perhaps more importantly, SEA is based on the concept of structures with random parameters in which responses are calculated as averages across ensembles of similar systems. This leads to the idea of random natural frequencies, i.e., the analysis requires that the number of natural frequencies occurring in a given frequency band be known, but not their exact positions. As a combination of these two ideas SEA is therefore concerned with the calculation of average response spectra, e.g., the spectra are those that would be found if very many similar, but not identical systems were examined and the individual spectra averaged frequency by frequency, across all the results. A good analogy is to liken the time and ensemble averaged vibrational energy of a structure to that of heat energy in series of connected bodies, i.e., the energy flows from 'hot' bodies to 'cold' ones and eventually an equilibrium condition is reached when the energy being injected into the system is balanced by that rejected by the various bodies. Here energy injection occurs because of various noise sources such as engines, compressors, fluid actions, etc. and rejection takes the form of dissipation via damping (eventually to heat).

Statistical energy analysis: theory As has been stated, in SEA studies it is the energy levels of the various subsystems that are calculated, given knowledge of the energy inputs and the coupling and loss characteristics of the system; the capacity of a vibrating subsystem to hold energy being measured by the number of resonant vibrational modes it possesses. This concept is illustrated in Figures 1-3 where a simplified offshore structure is shown and then conceptually broken down into interconnected subsystems with four external excitations. The various FIuN represent these inputs, the Fll, j, flows between the subsystems, and the Flwlss, E~ and N~, dissipations, energy levels and numbers of modes interacting within the subsystems, respectively. This is a very attractive framework since relatively few quantities are used to build models of potentially highly complex structures. Moreover, such models may be built at an early stage in design when few structural details are known. It turns out that this approach can be shown to be correct for certain limiting cases where the following assumptions hold: • Subsystems and coupling mechanisms are assumed linear • The damping is proportional, so that modal analysis generates uncoupled principal coordinates within each subsystem • Energy balances are time invariant, implying that the long term energy flow averages are constant • Conservative coupling mechanisms join the subsystems together

Statistical energy analysis of offshore structures: A. J. Keane

Figure

I

Outline sketch of simplified offshore structure

• The driving forces are statistically independent between each subsystem • The modal driving forces on each subsystem are incoherent and excite all modes to a similar extent (this is the so-called 'rain on the roof' assumption) • The probability density functions describing the subsystem natural frequencies are constant with respect to frequency in the ranges of interest (this implies that the probability of interacting modes occurring outside the range of interest is zero, i.e., nonresonant modes have no effect) • The ensemble mean of the square of all mode shapes at the coupling points are unity (this implies that the actual values of the mode shapes do not, on balance, affect the coupling behaviour and therefore that all modes tend to couple equally well) • The frequency ranges of interest contain many interacting modes (i.e. the mode counts within subsystems are large) • Interest centres on 'narrow' frequency ranges • The coupling between subsystems is 'weak', so that the modal behaviour of the uncoupled subsystems is not

greatly affected when the subsystems are brought together • The internal damping of the subsystems is 'light' Under such circumstances it is possible to show that the average energy parameters are related by the following matrix equation 4 {n,N} =

[c]{eMoo}

where {EMoo} is the vector of average modal energies, i.e., {EMoD} = {E/N}. Here [C] is the matrix of loss factors with

and C~,~ = -~oN~t/i, j j ¢ i q~ are the viscous damping loss factors and th, ~ are the so-called coupling loss factors. It should be clear from the foregoing list of assumptions that SEA is not guaranteed to work in all cases!

Engng Struct. 1994, Volume 16, Number 2

147

Statistical energy analysis of offshore structures: A. J. Keane

Deck E (R)

Deck D (R)

Derrick

Machinery

Decki C

~

Accomm.

DeckB

~ ~ .

Deck A

~/

Derrick Derric Base

D e c k E (L)

D e c k D (L)

_1 Figure 2

,!

J

I Legs

Subsystems chosen for rig model and their connections

Indeed, is it by no means obvious what all these assumptions mean for any particular case. It therefore comes as no surprise that most of the research effort that has been carried out on SEA since its initial formulation has been concerned with how far this very simple model may be stretched without breaking and in finding ways of calculating or measuring the coupling and dissipation parameters needed in the equations. Conversely, the computational demands of solving the equations, even for very many subsystems are trivial given modern computing facilities; indeed some commercial packages can be easily run on lap-top computers. In the following section the basic steps involved in applying SEA in a marine context are outlined. Some of the problems likely to be found are discussed and areas in need of further research effort highlighted. These are then illustrated by a brief example based on an offshore platform.

Statistical energy analysis: practice The steps involved in most practical SEA analyses may be given as (1) Choice of subsystems and calculation/measurement of modal densities

148

Cr2rane

Engng Struct. 1994, Volume 16, Number 2

(2) (3) (4) (5)

Evaluation of energy inputs Selection/measurement of damping loss factors Calculation/measurement of coupling loss factors Calculation of subsystem mean energy levels and other response variables (6) Estimation of reliability of results

These are discussed in turn in the following subsections.

Subsystems As has already been mentioned, SEA is couched in terms of energies flowing between coupled subsystems. Therefore, one of the principal decisions that must be made when applying SEA lies in the choice of subsystem boundaries. First, the user of SEA must decide upon either a high degree of discretization or a rather crude one. A crude SEA model of an offshore rig might, for example, consider a drill derrick to be a single SEA subsystem whereas a more sophisticated approach could consider each beam within the derrick structure to be an independent subsystem. However, since modal information is required for each subsystem the crude approach leads to difficulties in deciding on the number and nature of the vibrational modes that characterize the vibrational

Statistical energy analysis of offshore structures: A. J. Keane 1-[ 14DISS

E14,N14

I I-[13,14 I-I13DISS

II8/N

II1.4

I'I$DISS

F[TDISS

I-[4,5 ]~

I'I6DISS Er,N6

).

I-Itwtss ExI,Nll

5

I I-I2,3

I !-Ii,2 I'IIDIS S

E1,N1

L

I-II1/N

,9 I"I9/N

I i-I3.4

FIzmss E2,N2

behaviour of a possibly complex structure. Of course, FE methods could be used for this purpose, where instead of a detailed modal description the results may be used in the form of just the modal density with all other information ignored. Further, if previous structures of this kind have been studied before, the old data might be used, scaled in some suitable fashion. Ultimately, such simplifications must, of course, jeopardize the accuracy of the final results with the 'big block' approach being best used during the early stages of design while the more detailed approach awaits further design decisions. If a detailed model is being built the modal densities are usually readily available via calculation or from small-scale experimental work and this is one of the desirable characteristics of such models. Of course, one of the attractions of the SEA framework is the possibility of modelling certain areas of the structure in detail while others are handled in a more conceptual fashion. Having made a global decision on the basic level of detail to be incorporated the user must then study the nature of the couplings. The fact that all the couplings within a traditional SEA model must be conservative

I F111,12

Fl3otss E3,N3

YIso/ss Es,N5

Figure 3 SEA parameters for rig model

E12,N12

I-[4DtSs E4 ,N 4

Es,Ns ET,N7

II 12DISS

EI3,NI3 I I'14,13

I"I9DISS

E9,N9

,9

(

I I'I9.10

1-I1.11

II 100lSS

E10,N10

I'll ,3

Y

implies some fairly obvious restrictions here as does the desire for the couplings to be weak. Clearly, if a structure contains viscoelastic vibration absorbers these cannot be modelled as couplings, even though they may appear to be an obvious choice in this respect. Instead they must form separate subsystems with couplings at either end. Similarly, if two plates lying in the same plane are welded along a common edge, unless they have markedly different properties, such as thicknesses, or meet at a junction with another plate, they cannot easily be considered as lying in two separate subsystems. Such problems are much more commonly met when producing a detailed model than one where large pieces of sensibly independent structure are connected at a few key points. In following the detailed model path there are in addition, some rather more subtle distinctions that must be made if all the traditional SEA requirements are to be met; specifically the requirement that all modes lying in the frequency range of interest within a subsystem interact with those of the connected subsystems to roughly equal extent can cause problems as can the need for external forcing to excite all modes equally well. These

Engng Struct. 1994, Volume 16, Number 2 149

Statistical energy analysis of offshore structures. A. J. Keane arise because most basic subsystems are capable of supporting a variety of independent wave types (e.g., transverse and axial motions) and these types may not be equally well excited because of the nature of the couplings (joining a pair of flat plates at right angles will cause the transverse motions of one to couple mainly to the axial motions of the other and vice versa, for instance) or the forcing (an acoustic loading will tend to excite only the transverse modes in a plate). Consequently, it is good practice when building detailed models to consider each physical subsystem as consisting of a number of independent SEA subsystems with one wave type per subsystem. Even though initially independent these may well be coupled to each other through the actions of other subsystems and their connections to the subsystem of interest. Another problem of particular interest to the marine sector concerns the modelling of fluid actions. SEA is well suited to dealing with air loadings since acoustic spaces readily fulfil the requirement for many resonant modes. Moreover, SEA was originally thought of by people working in the field of acoustics who therefore incorporated such elements from the outset. Conversely, relatively little work has been caried out on water loaded structures, although there has been some 5'6. Since there may be significant energy dissipation within the water surrounding a marine structure and also since energy may travel by this 'wet' route to distant parts of the structure some account of the fluid actions is necessary, presumably as a separate subsystem. However, the term 'resonant mode' seems rather difficult to support in this context and clearly further theoretical work is required here. Most current practitioners in the marine sector seem to ignore the 'wet' energy path altogether, although they usually account for the added damping and mass effects. In summary, the choice of subsystem boundaries tends to lead to models with few (ten or so) complex subsystems or many (perhaps thousands) of rather detailed nonphysical subsystems. The recent trend in SEA research has been towards this latter approach because crude models tend to be insufficiently precise for describing the kinds of problems most workers are interested in. Moreover, crude models tend to be rather difficult to validate because they inherently contain so many assumptions; although it should be remembered that it is precisely just such models that SEA was originally conceived to deal with. Few models of either kind deal well with liquid-loaded structures. The best example of a detailed SEA model applied in the marine field currently seems to be that produced in the course of a major Japanese program looking at ship cabin noise in the late 1970s and early 1980s 7-9. However, even this study did not characterize all the wave types in each subsystem and, at least from the published data, it is not clear how satisfactory the final results were, although the authors were clearly quite pleased. Certainly, little attention was paid to energy flows through the surrounding water or internal tanks.

Energy inputs Having broken a structure of interest into a number of either (possibly both) simple or complex subsystems, SEA requires details (frequency spectra) to be supplied concerning the energy flowing into each subsystem as a

150

Engng Struct. 1994, Volume 16, Number 2

result of external forcing. Such information is notoriously difficult to calculate and the normal approach is to use either measured data or to assume unit flat white noise inputs, either at specified points or else incoherently and evenly spread across entire subsystems. The purpose of the white noise approach is to consider each subsystem that is likely to be subject to external forcing in turn, with all other inputs set to zero, so that a sensitivity analysis can be carried out. The results of this aproach allow major conduits of energy flow to be identified and possibly altered. In either case, if the structure is modelled at a detailed level, a noise source might need to be considered as exciting various wave types in adjacent SEA subsystems (i.e., the same physical subsystem) which would, of course, not satisfy the requirement of being uncorrelated. The same problem occurs if a piece of machinery situated in a machinery space is the noise source; here it will excite the structure via its foundations and also via airborne noise. These two sources may well meet up in the sides of the space and some approximation must be made, usually to ignore the correlation. The application of SEA to problems with correlated forcing, is in theory possible, but even the simplest cases result in enormous complications in the governing equations 1°. If ignoring correlations is thought to be a problem, the only simple alternative would seem to be to model all subsystems that are externally forced as possessing mutiple wave types, despite the earlier comments concerning differences in coupling behaviour and the requirement of equal susceptibility to external forcing. Certainly more research on this topic is required.

Damping lossfactors The preceding equations for SEA explicitly require the damping loss factors for all the subsystems in the model. Such factors may well vary with frequency, they will almost certainly vary with mode type. The only acceptable source for such information seems to be experimental, although fortunately they seem to be rather insensitive to the actual structural details of any particular class of subsystem, typical values lying in the range 0.0005-0.01 for most welded steel structures. Fortunately, it is primarily their relative values that are of importance in an SEA model since all the energy injected will be dissipated within the system whatever the absolute levels of damping. One topic needing further research in this area concerns the effects of liquid loading and subsequent 'added damping'. It is not obvious for instance, how a partially filled fuel tank lying within a complex structure should be modelled, given the ability of energy to be stored and dissipated within the liquid as well as having the ability to flow through it. Moreover, fluid damping tends not to lead to uncoupled modal responses and this leads to further sources of error in the analysis. The assessment of fluid damping may, however, be amenable to the methods used in computational fluid dynamics studies. Another topic concerns the adoption of 'modern' materials such as fibre-reinforced plastics. These tend to have much higher intrinsic damping and this can be very beneficial when attempting to reduce noise transmission. However, such damping also tends to reduce the resonant behaviour that underpins SEA. This does not render SEA unworkable, indeed it tends to reduce the variances found between structures, but it

Statistical energy analysis of offshore structures: A. J. Keane does require some modifications to the approach. These have not been widely studied although some work has been carried out on this topic 11.

Couplino loss factors The area in which perhaps the greatest research effort has been expended during the evolution of SEA is that of determining coupling loss factors. This is perhaps not surprising as it is the concept of coupling losses that forms the basis of SEA. These factors are very difficult to measure directly since, for mechanical subsystems, both displacement and velocity must be simultaneously measured over the entire coupling. This is obviously very difficult for line or surface junctions but by no means easy even for point couplings unless a mechanical power flow element is inserted between the subsystems. Most experimental techniques instead rely on assuming the main SEA equations are correct and then working back to the coupling loss factors having measured the energy inputs, resulting subsystem energy levels, various damping values and mode counts. Of course, such methods are enormously prone to errors, especially if the basic system topology leads to an ill conditioned problem (i.e., attempts are made to infer coupling loss factors where the energy flows in the relevant coupling are rather small compared to flows in the other couplings). Moreover, since SEA is fundamentally concerned with the variability found between the behaviour of nominally identical subsystems such measurements should be carried out and averaged over a large number of realizations of the structure. This is hardly ever done and frequency averaging is usually used instead. Such averaging is, of course, prone to any peculiarities of the particular specimen under test. Consequently, most effort has been directed towards theoretical evaluations. In the aerospace field such work continues apace with the behaviour of quite complex couplings being studied (e.g., flat plates coupled to cylindrical shells, junctions that incorporate offset beams, etc.,). These are mostly based on wave analysis of pairs of joined semi-infinite subsystems. Even studies based on junctions simplified in this way are still far from straightforward; they are certainly best carried out only by those with considerable analytical experience in this area. However, a cursory examination of this field reveals that each new type of coupling seems to have been considered in isolation, with the long term intention seeming to be the construction of a complete library of coupling loss factors, suitable for dealing with all problems. The main difficulty with such an approach is that as the number of classes of subsystems increases the number of possible coupling types rises on a factorial basis! By comparison, one of the major strengths of FE methods is that almost all elements can be coupled to almost all others using the lowest common denominator of the joint (node) displacements (and rotations). This author, for one, feels that attempts to study all possible coupling types are rather misguided and that an approach similar to that used in FE is more realistic. Instead of considering all possible permutations of junctions, basic coupling elements should be used to link the motions at discrete points within the subsystems. For example, two plates might be thought of as being coupled by a line of linear springs connecting points on the

subsystems in the way that joints (nodes) are used in FE models. This decouples the behaviour of the subsystems requiring only that the point response functions of each class of subsystem be found in isolation (i.e., the Green functions) 12. This approach also has the advantage that the behaviour of individual subsystems can be given either in terms of algebraic Green functions or of modal information, which, in turn, may be algebraic, the results of FE analyses, based on measurements or specified by reference to various probability density functions. This ability to mix and match the desired subsystems and couplings without a fundamental re-analysis of the mechanics of every junction seems to offer one systematic way forward although, perhaps inevitably, it involves rather greater computational effort than approaches based on libraries of wave analysis derived coupling loss factors. Before leaving the topic of coupling loss factors it is perhaps worth noting that such factors are also required to link the behaviour of different wave classes within a single piece of structure if these are handled as separate SEA subsystems and the wave types are not completely independent (such as in a large stiffened panel, unless every bay and stiffener is modelled as an individual subsystem, which can lead to an explosion in the number of subsystems). Such pieces of structure may well be best handled as equivalent elements where the detailed modal behaviour is obtained via other methods such as traditional FE calculations or perhaps even analytically.

Subsystem mean energy levels Having gathered together all of the data required to form an SEA model it is then a relatively trivial task to invert the matrix of loss factors and to derive the subsystem mean energy levels for any given forcing model. These may then be linked to mean stress or motion levels but are usually used directly in attempts to understand the nature of the energy flows around the structure. Often this leads to a parametric survey of some kind where various design changes are made and their relative effects studied. However, all this depends on whether the results can be believed or not, given the initial assumptions and the fact that only average results are available.

Reliability of results Leaving aside the very many assumptions implicit within SEA it is, as the name makes clear, a statistical tool, that is, it gives results in terms of mean or average values. These averages should be taken across ensembles to maintain the true spirit of SEA but may be frequency or even spatial averages, sometimes a combination of these. Of course, any given realization of a structure, or its response at some specified frequency or location must be expected to deviate from the average. During its early development relatively little attention was paid to such deviations but this is obviously not acceptable for practising designers seeking absolute values for stress levels, noise intensities, etc., or even those using probabilistic design criteria where the probabilities of certain levels being exceeded are needed. Confidence bounds or some other higher-order statistics must be presented alongside the mean value predicted by SEA for the mean to be of much use. The formulation of such bounds is by no

Engng Struct. 1994, Volume 16, Number 2

151

Statistical energy analysis of offshore structures: A. J. Keane means easy and this remains an important topic of research. Recent work in this field 3 has shown that, even where the variation in the physical properties of a system are essentially Gaussian in nature, the resulting energy level statistics are often very far from Gaussian. In a number of cases it would appear that a log Gaussian description more nearly fits the results, sometimes extremely well. This is perhaps not so surprising since almost all practical measurements of vibrations are plotted on logarithmic scales. Results that fit log Gaussian descriptions point to the use of geometric rather than arithmetic averaging as being perhaps the most appropriate form for energy studies of structures, a point that has been made elsewhere 13. Unfortunately, the adoption of geometric averaging leads to very great complications in the theoretical analysis of SEA and prevents the simple equations presented earlier from being adopted. However, if this is the best way of making reliable predictions of confidence bands no doubt such difficulties will be overcome in due course. Conversely, if the traditional and simple arithmetic mean-based SEA approach is to be retained it would be useful to know how far the model could be reasonably expected to stretch before breaking, i.e., to what class of problems should it be restricted if extreme variations from the mean are to be avoided. Perhaps it should only be used when trying to establish the likely effects of changes to a design when it is only trends that are being sought. Certainly this was one of the original purposes of SEA in its early days. Another aspect of the prediction of confidence bands concerns identifying which characteristics of a problem control these bandwidths. If these were known it might be possible to predict variances more easily, or, eventually, designs might be produced that showed reduced susceptibility to variations in the responses of nominally similar structures. In this context it is clear that certain topological arrangements of couplings between subsystems will give rise to greater variations, e.g., the behaviour of a subsystem at one end of a long chain of random subsystems will show very wide variations to forcing at the other end whereas structures where all subsystems are mutually connected to all others will not. These two cases represent extremes, but it would clearly be useful if, say, the number of couplings on the shortest path between two subsystems could be related to the variance in the resulting energy levels. This, of course, would further depend on the relative strengths of the couplings with more strongly coupled paths tending to dominate more direct but more weakly coupled ones. A related topic concerns the extensive geometric repetition found in most modern steelwork. Such repetition is, of course, seen as highly desirable from a fabrication standpoint but it does tend to lead to some adverse characteristics in the subsequent noise performance. It is fairly obvious that if a piece of structure allows the propagation of a particular band of frequencies (perhaps because the frame spacing aligns with the wavelengths) then a structure built of such elements will have very poor characteristics if excited at this frequency. Conversely, using a mixture of different geometries might well ease this highly tuned susceptibility. Why then, not design structures with random stiffener spacings! Although at first thought this may seem bizarre modern computer controlled welding systems are easily capable of handling such structures as are modern CAD pack-

152

Engng Struct. 1994, Volume 16, Number 2

ages. They can also be easily fabricated out of librereinforced materials. Clearly, considerable further work is required on such problems and the author is engaged on a long term programme of research in this area~

A brief example To place the preceding discussion in perspective a very brief example will be given, based on the idealized offshore structure and equivalent SEA model of Figures 1-3. For the purposes of example the aim of the study is taken to be the assessment of the effects of machinery and wave noise in the accommodation block sited at the lefthand end of the rig. The range of frequencies of interest is chosen as 50 150 Hz and average energy levels are required based on modes interacting within 50 Hz bands. This bandwidth ensures that no subsystem has too few interacting modes while being narrow enough to allow all parameters to be considered invariant with frequency within the bands. Here, it is assumed that the designer is engaged on a concept study where few structural details are known but the overall format is fixed. Consequently, a big block approach is adopted and the structure broken into the 14 subsystems shown in Figures 2 and 3 and detailed in Table l. The choice of couplings between these subsystems is mostly self-evident. Note, however, that the model assumes that the legs pass through the lower decks rather than that the leg structures between the decks form separate subsystems. Also, much smallscale structure has been ignored at this early stage of design. The derrick and crane have been included not because of interest in their vibrations but because their masses and vibrations will have significant effects on the supporting structure. Lastly, lumping all of the support structure into one subsystem neatly avoids having to quantify the effects of energy flowing between the legs, etc., via the surrounding sea. Of course, one would expect subsequent iterations to use more refined modelling. Having chosen a frequency range and subsystems the designer must next quantify the modal densities of each subsystem. These will most probably be derived via FE studies of each subsystem backed up by measurements taken on real pieces of structure plus analytical estimates. For example, here the designer might know, from previous FE calculations, that a drill derrick typically had 20

Table 1 Notional 50 Hz bandwidth mode counts and damping loss factors at 100 Hz for example rig

Subsystems Name

No.

Mode counts N,

Damping loss factors q~

Legs Deck A Deck B Deck C Accomm. Deck D(L) Deck E(L) Machinery Crane Jib Derrick base Derrick Deck D(R) Deck E(R)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

35 100 100 100 250 30 30 15 12 25 5 18 25 25

0.2 0.01 0.01 0.01 0.05 0.01 0.01 0.02 0.02 0.003 0.02 0.003 0.01 0.01

Statistical energy analysis of offshore structures: A. J. Keane global modes of vibration in the frequency range of interest and that the modes associated with individual beams of the derrick were well above this range. He would then assume that unless the proposed design differed greatly this would be an adequate basis for scaling, probably on mass. Moreover, as noise paths to the accommodation block from machinery etc. are of interest it would seem sensible to model the derrick in a crude fashion since its detailed behaviour would seem unlikely to be of great importance. This part of the exercise is likely to consume a considerable amount of effort, particularly if data from previous analyses is not available. Table I sets out the values to be used here, although it must be emphasized that these are entirely notional and simply for the purposes of illustration. The major noise sources of interest for this study are taken to be the various items of machinery associated with the machinery block (generators, etc.), the derrick and perhaps the crane, plus those due to wave impacts on the legs and support structure, i.e., subsystems 8, 11, 9 and 1, respectively. The designer finds these difficult to quantify but places them in an order of priority based on previous experience and measurement as (1), derrick operations; (2), machinery block noise; (3), wave loads; and (4), crane operations. His choices being biased not only by absolute noise levels but also by likelihood of occurrence. Thus, his aim becomes one of predicting how unit input of energy at each frequency from each of these sources will affect the accommodation, leaving as a separate task decisions concerning their levels, frequency spectra and likelihood of occurrence. The choice of damping levels for the above water structure is based on decay tests on previous structures, which is acceptable as long as novel materials or fabrication methods will not be employed on the new design. The fluid damping on the legs would be based on computational fluid dynamics predictions or empirical expressions. Intermodal coupling via the fluid actions on the leg structure is ignored. This results in the 14 damping loss factors of Table 1 and again it is noted that the numbers chosen here do not reflect serious analysis but are for the purposes of example only. For the rig structure being studied the couplings will all be elastic although highly complex in character given the nature of the subsystems. Hopefully, they will be Table 3

relatively weak since the whole purpose of using big blocks is to break the structure into components that do not greatly affect each other. The quantification of these constants represents the most significant amount of effort in the SEA analysis. In due course, commercial SEA packages should enable this task to become more straightforward, although unusual subsystems or couplings will always lead to some difficulties. Hopefully, FE models, constructed for the purposes of static strength calculations, will become useful in this context as well. Table 2 lists the figures to be used in the example and again it is stressed these are purely notional and should not be taken as typical. Having quantified all the data needed for the main SEA equations, the matrix of loss factors can be inverted and the results studied for the various excitation forces of interest. Table 3 shows the relative mean energy levels of all 14 subsystems for each source of excitation at a frequency of 100 Hz. Notice that these figures are per unit of frequency and to gain the total energy present in the 50 Hz band they must be multiplied by 50 × 2n. The

Table 2

Notional coupling loss factors for example rig (7i, i =

7i, jNilNj) Coupling loss factors 71,2 71.3 71.4 71, 11 ~2,3 72, 5

72,9 ~3,4

73,5 73,9 74.5 74.e 74,9 r/4,11 r/4,13 ~5.6 7s,7

79.10 711.12 713.14

0.183E-02 0.148E-02 0.940E-03 0.196E-02 0.474E-03 0.474E-04 0.621 E-03 0.288E-03 0.579E-04 0.759E-03 0.307E-04 0.822E-04 0.403E-03 0.320E-03 0.192E-04 0.467E-03 0.467E-03 0.526E-02 0.259E-02 0.268E-03

Legs to Deck A Legs to Deck B Legs to Deck C Legs to Derrick base Deck A to Deck B Deck A to Accomm. Deck A to Crane Deck B to Deck C Deck B to Accomm. Deck B to Crane Deck C to Accomm. Deck C to Machinery Deck C to Crane Deck C to Derrick base Deck C to Deck D(R) Accomm. to Deck D(L) Accomm. to Deck E(L) Crane to Jib Derrick base to Derrick Deck D(R) to Deck E(R)

Resulting average energy levels (KJs x 1012) at 100 Hz for example rig for unit white noise inputs at four points of excitation Subsystems

Energy levels E~

Name

No.

For unit (legs)

Legs Deck A Deck B Deck C Accomm. Deck D(L) Deck E(L) Machinery Crane Jib Derrick base Derrick Deck D(R) Deck E(R)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

9.363 1.896 1.651 1.61 6 0.0047 0.0002 0.0002 0.0066 0.1978 0.3466 0.9719 0.8401 0.0031 0.0001

1"11/N

~91N

For unit ]']8/N (machinery)

For unit (Crane)

0.0332 0.1987 0.5390 11.15 0.0077 0.0004 0.0004 198.8 0.3226 0.5651 0.1873 0.1619 0.0214 0.0006

0.5935 72.06 87.64 47.10 0.2002 0.0094 0.0094 0.1936 130.9 229.4 0.8369 0.7233 0.0906 0.0024

For unit ]-[11/N (derrick base) 1.21 5 0.4387 0.7492 11.39 0.0084 0.0004 0.0004 0.0468 0.3487 0.6108 17.40 15.04 0.021 9 0.0006

Engng Struct. 1994, Volume 16, Number 2

153

Statistical energy analysis of offshore structures: A. J. Keane

distribution of energy around the structure is as anticipated given the geometry and parameters used. Note that the crane jib has higher energy levels than its base because of its relatively greater modal density and lighter damping. Also, the machinery block is much more easily excited by external forces than the crane and derrick base with the rig legs being least easily excited. This is a consequence of the relative modal densities of these structures and is, of course, related to their differing masses. Turning to the accommodation block, it can be seen that energy injected into the crane most easily travels to the block with that from the legs passing least well. However, it should not be concluded that crane noise will cause the greatest problems because excitation of the crane would probably be the lowest level source and it would not be in constant operation. It is perhaps most likely that energy originating in the derrick base will be the worst problem, since the source is likely to be quite large and is also well coupled to the accommodation (Figure 4 illustrates the magnitudes and directions of the various energy flows for this case). The machinery block is also likely to be worth considering in more detail because it is well coupled to the accommodation and, in addition, is likely to operate on a continuous basis.

Finally, it should be noted that there is no simple way of quantifying the likely variances that would be found in practice using this simple SEA model. Although unlikely, it might turn out that a typical rig corresponding to this model showed energy levels in the accommodation block that were 10 15 dB, different from those given here, even if the various parameters and assumptions used in the SEA model were correct. To gain some insight into the sensitivity of this model to changes, it is a simple matter to vary the various parameters in Tables 1 and 2; indeed it would not be difficult to minimize the energy level in the accommodation block by employing an optimization package to modify the parameters within sensible bounds. Certainly, one major use of a model such as the one developed here would be to find out what effects changes in damping and coupling levels have so that the most cost effective solutions could be found. This could even be extended to studies where whole elements are resited and the topology of the resulting SEA model changed, i.e., the machinery might be placed on a different deck to see if this helped. By way of illustration, assume that the designer wishes to reduce the noise levels in the accommodation arising from noise injection at the derrick base. Three

D e c k E (R) /X

D e c k D (R)

I I I I

Machinery

Deck C

Accomm.

DeckB

Derrick

,

~

f _ _

t

_

_

Derrick

/ DeckE(L)

I~ " ,

t

Crane

//f

D e c k D (L) ~ " "

/

Deck A

t Figure 4

154

Relative energy flows for power injection to derrick base

Engng Struct. 1994, Volume 16, Number 2

t

Legs

/

Jib

Statistical energy analysis of offshore structures: A. J. Keane

ways of doing this are to be considered: (1) (2)

(3)

Increase the damping within the accommodation block by 50% Increase the damping within the derrick base by

50% (3)

Reduce the structural connections between the accommodation block and the rest of the rig by removing connections between deck C and the block, leaving decks A and B plus the leg structure to support it (i.e., set r/4, s and ~/s,4 to zero).

Tables 4-6 show the results of these three changes, respectively. Comparison with the original results of Table 3 allows a number of observations to be made

(1)

Increasing the damping within the accommodation block simply reduces the vibration levels pro rata for all noise sources (2) Increasing the damping within the derrick base only reduces vibrations due to noise originating from that area

Decoupling the accommodation block from deck C decreases the vibration levels in the block from all sources, being most effective at reducing that originating from the machinery block and derrick base and least from that coming from the legs, where it is slightly less effective than increasing the accommodation internal damping.

It would be reasonable to expect that these trends would be relatively impervious to statistical variations between individual realizations, although they would, of course, depend on the accuracy of the various parameters and the degree to which the underlying assumptions detailed in earlier sections were met. Although highly idealized and simplistic, perhaps this brief example has set the analysis process in context. No doubt, it raises many questions as to the reliability and accuracy of SEA, the methods used to arrive at the parameter values needed and the interpretation of the results. Hopefully, it also demonstrates the considerable insight that can be gained into the way vibrational energy flows around a structure using SEA.

Table 4 Resulting average energy levels (KJs x 1012) at 100 Hz for example rig for unit white noise inputs at four points of excitation. 50% increased accommodation block damping Subsystems

Energy levels E.

Name

No.

For unit (legs)

Legs Deck A Deck B Deck C Accomm. Deck D(L) Deck E(L) Machinery Crane Jib Derrick base Derrick Deck D(R) Deck E(R)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

9.363 1.896 1.651 1.61 6 0.0032 0.0001 0.0001 0.0066 0.1978 0.3466 0.9719 0.8401 0.0031 0.0001

rl]/N

For unit 11slN (machinery)

For unit l-Igm (crane)

For unit I-Ill m (derrick base)

0.0332 0.1987 0.5389 11.15 0.0051 0.0002 0.0002 198.8 0.3226 0.5651 0.1873 0.1619 0.0214 0.0006

0.5935 72.06 87.64 47.10 0.1331 0.0062 0.0062 0.1936 130.9 229.4 0.8369 0.7233 0.0906 0.0024

1.215 0.4387 0.7492 11.39 0.0056 0.0003 0.0003 0.0468 0.3487 0.6108 17.40 15.04 0.0219 0.0006

Table 5 Resulting average energy levels (KJs x 1012) at 100 Hz for example rig for unit white noise inputs at four points of excitation, 50% increased derrick base damping Subsystems

Energy levels Ei

Name

No.

For unit f i l m (legs)

For unit ]'I81N (machinery)

For unit (crane)

l'I91N

Legs Deck A Deck B Deck C Accomm. Deck D(L) Deck E(L) Machinery Crane Jib Derrick base Derrick Deck D(R) Deck E(R)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

9.362 1.896 1.651 1.609 0.0047 0.0002 0.0002 0.0066 0.1976 0.3462 0.6410 0.5540 0.0031 0.0001

0.0287 0.1971 0.5362 11.11 0.0077 0.0004 0.0004 198.8 0.3213 0.5628 0.1228 0.1061 0.0214 0.0006

0.5734 72.05 87.62 46.91 0.2000 0.0094 0.0094 0.1928 130.9 229.4 0.5486 0.4742 0.0902 0.0024

For unit 11111N (derrick base) 0.7964 0.2876 0.4911 7.469 0.0055 0.0003 0.0003 0.0307 0.2286 0.4004 11.41 9.862 0.0144 0.0004

E n g n g Struct. 1 9 9 4 , V o l u m e 16, N u m b e r 2

155

Statistical energy analysis of offshore structures. A. J. Keane Table 6 Resulting average energy levels (KJs × 1012) at 100 Hz for example rig for unit white noise inputs at four points of excitation. Decoupled deck C and accommodation block Subsystems

Energy levels E~

Name

No.

For unit r i l l N (legs)

For unit r[81N (machinery)

For unit rIgl N (crane)

For unit FI~I~N (derrick base)

Legs Deck A Deck B Deck C Accomm. Deck D(L) Deck E(L) Machinery Crane Jib Derrick base Derrick Deck D(R) Deck E(R)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

9.340 1.888 1.637 1.402 0.0037 0.0002 0.0002 0.0058 0.1913 0.3350 0.9661 0.8350 0.0027 0.0001

0.0332 0.1987 0.5389 11.1 5 0.0008 0.0000 0,0000 198.8 0.3226 0.5651 0.1873 0.1 61 9 0.0214 0.0006

0.5935 72.06 87.64 47.10 0.1710 0.0080 0.0080 0.1936 130.9 229.4 0.8369 0.7233 0.0906 0.0024

1.21 5 0.4387 0.7492 11.39 0.0013 0.0001 0.0001 0.0468 0.3487 0.6108 1 7.40 15.04 0.021 9 0.0006

Future prospects Before drawing some conclusions it would seem to be worth considering the kind of progress that would be desirable in SEA over the next five to ten years and also what might realistically be achieved. To begin with, five broad areas have been identified in the preceding sections as needing more effort (1)

(1) (3)

(4)

(5)

The modelling of internal and external liquid (water, fuel, etc.,) allowing for energy flows through the liquid The modelling of small groups of subsystems with coherent external forcing The study of the effects of inherently heavily damped structures such as those made from 'modern' materials The development of a more systematic means of dealing with coupling loss factors so that enormous libraries of factors are not needed The ability to quantify the variance likely to be found around the mean values predicted by SEA

Of these, the second is probably least important although there are no doubt circumstances where ignoring the coherence in external forcing may lead to significant errors. Certainly, there seems to be no effort being expended in this direction at the moment. The fourth and fifth topics are the subject of much current interest and good progress can be expected in the near future. However, many workers are still committed to producing libraries of coupling loss factors and they seem most interested in aerospace structures. Therefore, unless the point coupling approach mentioned earlier becomes more widely accepted, some effort may be needed in studying the kinds of coupling more commonly found in marine structures. Finally, there seems to be very little effort going into the study of liquid actions or the effects of fibre-reinforced materials on SEA. The first of these is one area which is particularly important to the marine industry and where it is largely on its own. Conversely, modern materials should be of wider interest and can be expected to be studied as and when they are more commonly used in structural design.

156

E n g n g S t r u c t . 1 9 9 4 , V o l u m e 16, N u m b e r 2

On a more general level, two points can be made. First, there appear to be few guidelines as to when SEA can be expected to work. There seems to be little information on what the minimum modal densities and maximum damping factors can be for the underlying assumptions to be valid. Also, on how weakly subsystems must be coupled before reliable results can be guaranteed. Widely agreed guidelines on these aspects would be welcome. Secondly, most designers prefer not to have a great number of tools that need mastering when dealing with essentially related problems. Moreover, data collation and entry into analysis software should be minimized. This leads one to wish that SEA and FE could be combined within a single package working from a common design description. Perhaps this would be a crude model during the initial stages and SEA would be used at first to study noise performance. The design could then be fleshed out and traditional FE calculations run, both for assessing static strength and also as an input to low frequency dynamics work. The more refined FE model could then be used to carry out a more sophisticated SEA analysis, perhaps using some of the FE results to give details of mode counts, etc., for some subsystems within the model. Indeed, since both FE and SEA involve a number of matrix calculations the same low level routines might be used for both types of analysis. Although such an amalgamation seems obvious it will not come about until users of existing FE systems start to ask for built-in SEA tools and this will presumably not happen until SEA becomes more widely accepted. It can, however, be seen as a long term aim.

Conclusions This paper has taken a brief but general view of SEA and considered where this field is going from a marine viewpoint. Although inevitably biased by the author's own opinions it has attempted to highlight a few key areas in SEA research that seem necessary to bring wider acceptance to the technique. It has also considered how SEA might be fitted into existing analysis tools and noted that novel designs might be produced that exploit such

Statistical energy analysis of offshore structures: A. J. Keane capabilities. O f the areas m o s t requiring further research the aspect of energy flows t h r o u g h external a n d contained liquids seems to be the one that is m o s t deserving of s u p p o r t by the m a r i t i m e c o m m u n i t y , since workers in other fields are little affected by such problems. O n a more general front, m a r i n e designers c a n expect reasonable progress to be m a d e in q u a n t i f y i n g the variances in SEA estimates a n d also in h a n d l i n g a n ever wider range of s t r u c t u r a l connections. It is felt that this latter aspect is one where a fresh a p p r o a c h m a y be useful a n d one such m e t h o d has been o u t l i n e d here.

References 1 Lyon, R. H. Statistical Energy Analysis of Dynamical Systems: Theory and Applications, MIT Press, 1975 2 Fahy, F. J. 'Statistical energy analysis--a critical review,' Shock Vibr. Digest, 1974, 6, 14-33 3 Keane A. J. and Manohar, C. S. 'Power flow variability in a pair of coupled stochastic rods', J. Sound Vib. 1993 164 (2)

4 Langley, R. S. 'A general derivation of the statistical energy analysis equations for coupled dynamic systems', J. Sound Vib. 1989, 135 (3), 499-508 5 Gulizia, C. and Price, A. J. 'Power flow between strongly coupled oscillators', J. Acoustic Soc Amer. 1977, 61 (6), 1511-1515 6 Hattori, K. Nakamachi, K. and Sanada, M. 'Prediction of underwater sound radiated from a ship's hull by using statistical energy analysis', Proceedings of Inter-Noise 85, Munich, pp 645--648 7 Suhara, J. 'Analysis and prediction of shipboard noise', Proceedings of the International Symposium on Practical Design in Shipbuilding (PRADS), Tokyo, pp 189-196 8 Irie, Y. and Takagi, S. 'Structure borne noise transmission in a steel structure like a ship', Proceedings of Inter-Noise 78, San Francisco, pp 789-794 9 Irie, Y. and Nakamura, T. 'Prediction of structure borne sound transmission using statistical energy analysis', Bull. Marine Engng Soc. Japan, 1985, 13 (2), 60 72 10 Davies, H. G. 'Exact solutions for the response of some coupled multimodal systems', J. Acoust. Soc. Am. 1972, 51 (1), 387-392 11 Keane, A. J. and Price, W. G. 'A note on the power flowing between two conservatively coupled multi-modal sub-systems', J. Sound Vib., 144 (2), 185-196 12 Keane, A. J. 'Energy flows between arbitrary configurations of conservatively coupled multi-modal elastic subsystems'. Proc. R. Soc. Lond., 1992, A436, 537-568 13 Hodges, C. H. 'Confinement of vibration by one-dimensional disorder: theory of ensemble averaging', TOP/14/81/1, Topexpress Ltd, Cambridge, UK, 1981

Engng Struct. 1994, Volume 16, Number 2

157