IUTAM Symposium on Statistical Energy Analysis (Solid Mechanics and Its Applications)

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In this section, the interpretation and effect of the frequency averaging in the time domain is discussed. Properties of convolutions can be used to evaluate time responses when the averaging width is constant. This is described and illustrated in the two following sections, in particular in the case of Gaussian averaging. The latter is called the characteristic function of the probability density function, p a. It has three main steps:. The inverse Fourier transforms of the discretized transfer functions were evaluated through an explicit trapezoidal integral.

The real Gaussian scaling functions affecting the impulse response when the inverse of the frequency average is taken are presented in h. Each of these operations has the option to be evaluated in various manners. For example, while the frequency average could be evaluated by using the modal decomposition used in the derivation in this paper, an efficient matrix alternative based on rational approximations and Krylov subspaces that has been proposed by Lecomte [ 31 ] allows by-passing any modal analysis and has been successfully applied by the author to large systems in this context of evaluating time responses as proposed here.

Similarly, the inverse Fourier transform could be evaluated by considering individual modes of a modal decomposition or numerically by sampling the average in the frequency domain. The evaluation through modal decomposition can also be applied to the precise approximate reduced models of [ 31 ].

Note finally that the current discussion provides a description of the eigenfunctions of the Gaussian averaging operator. This is detailed in appendix B in the electronic supplementary material. The increase in the error at the end of the two first time ranges, i.


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Both the evaluation of the average and inverse Fourier transform can be made much more precise, particularly by using the method of Lecomte [ 31 ]. The finite precision arithmetic remark corresponds to a more fundamental fact: If an engineer, who would for example want to design a structure that is only moderately affected by a shock or turbulence, is only interested in the maximum and minimum values of the impulse response during a given time period after some impact, as in a , he or she can thus define the maximum averaging width possible that provides accurate response in the smallest time range of interest.

The discussion of this section provides some explanation of the Gaussian frequency averaging process in that the smoothed out details and irregularities of the transfer functions correspond to details of the response at larger times.

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Nevertheless their being smoothed out, information about the details of the transfer functions does not disappear in the averaging process, at least when the second-order averages—averaged square and variance—are considered. The information about the total transient energy, including at larger times, is indeed preserved and retrievable. Before that, the case of variable averaging width is briefly discussed. The discussion of the previous section provides the further interpretation that their deterministic character can be understood as well defined—rather than blurred—transfer functions and consequently refined impulse responses at small times.

The alternative high-frequency focus corresponds instead to statistical , blurred transfer functions and a more and more imprecise notion of energy at larger and larger times. It is now stressed how the focus of the analysis can be qualitatively pushed into the direction of a more deterministic character, by pinpointing frequency regions. The frequency average response is by itself an approximation of the exact response, in the sense that the averaging process may allow to get rid of unimportant features of the response. As noted in the previous sections, this notion of unimportance is related to an accurate time response of the system at small times and a small frequency variance.

Steps for reducing the frequency variance may therefore lead to a focus on more important features of the system response. With this choice, the average reveals a number about 30 of resonances in the refined interval and globally reduces the variance there. Attention now turns to the imprecise or energy part of the responses that are assessed by the second-order averaged square and variance. Equivalence theorems of two kinds are first highlighted. Exact expressions of the energy in terms of the frequency moments are then presented and discussed. It is notably shown how the energy is partitioned in terms of the first and the second moment contributions.

The evaluation of the total impulse energy through averaged functions is based on theorems of equivalence of the energy in the three domains, namely the domains of time, frequency and frequency average. The first kind of equivalence between the frequency and time densities is provided by Plancherel's theorem below, whose proof is provided in appendix A in the electronic supplementary material for completeness.

Its application to time derivatives gives the expressions needed for the evaluation of the kinetic energy, e. The second kind of equivalence concerns the frequency integrals of functions of the responses and of their frequency average. The integrated quadratic term defined in equation 6. The integrals match similarly when the time derivatives of equation 6. The integrated term defined in equation 6. These and similar equivalence properties can be applied to evaluate the energy terms of a system, just from the knowledge of its frequency average first and second moments targeted in the proposed framework.

It is now demonstrated how this can be done. This expression can further be expanded in terms of the average and variance of the response by using equation 2. The integrated potential energy of a component of the system can similarly be expressed in terms of frequency averages. The integrated kinetic energy can also be expressed in either time or frequency domains. The integrand must first be modified. Starting from equation 3. This modal expression of the second moment can also be expressed in matrix form, by using expression 3.

In the typical particular case of the previous section, i. This can further be expanded in terms of the average and variance of the response, by using equation 6. Other energy terms can be similarly evaluated exactly by integrating frequency averaged quantities. The integral of the transient input power can also be evaluated in terms of the averaged expressions. For example, in the case of an impulse force, its frequency density is constant so that the total input energy is.

The averaged square and product of responses did not appear in the expressions of the energy terms in the previous sections. They are however an important feature of the response at higher mid or high frequencies and are therefore an integral part of the computational goals of the proposed framework. Without these notions, a pure energy analysis indeed loses information on the phase of the transfer functions that may possibly be critical for some applications. Although these terms should not be overlooked, they may of course be omitted in situations where they are negligible, or in asymptotic forms, in the same way as the variance may be omitted in the asymptotic case of a low-frequency , deterministic, analysis.

In this section, an interpretation of the second frequency moments is provided in the time domain. Similar properties exist here with regards to double or two-dimensional inverse Fourier transforms of the second-order averaging terms, that is of the covariance and averaged products of responses. These properties are now detailed, with a particular focus on the real Gaussian case. Starting from the definition of the inverse Fourier transform of equation 2.

Following the same derivation, for the case of the power term, one finds. In the real Gaussian case, p a. The variance derives directly from this expression. Each of the two expressions 7. While similar expressions would exist for other averaging distributions, the weighting function in the real Gaussian case, is remarkably also a Gaussian. In both cases, the outer product of the impulse responses is scaled by the Gaussian ridge function, r t A , t B , that becomes tighter and tighter for increasing values of a.

Illustration of the two-dimensional inverse Fourier transform of the second frequency moments as described in g. Different behaviours are visible for different averaging widths. Simplifications however occurred since irrelevant details have been smoothed out of the response.

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This corresponds to a low-frequency behaviour. This is the high-frequency behaviour. Observing both the covariance and averaged products provides a rational and quantitative tool with which to assess the actual behaviour of the dynamic system. As such, the quality of the existing methods to deal with the frequency methods can be analysed and compared by evaluating how well they preserve the actual frequency average, covariance and averaged product characteristics of the actual system.


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New methods can also be developed based on the actual important system features identified by the averaging process. Besides the asymptotic approximations, which provide evaluation of the averages that are accurate and economic in particular situations, the question of evaluating the averages efficiently and precisely in a general context is a relevant concern.

Consequently, the exact averages are approximated in terms of fast converging rational functions of the poles of the system. These rational functions may be expressed in matrix form and evaluated exactly through stable Krylov projection methods, using only solutions of a system with added damping.

Combining the projection subspaces at only a few interpolation frequencies can then provide a precise evaluation of the average function over a whole frequency range. This approach has been shown to be extremely efficient and able to provide the Gaussian averaged response in a whole frequency range, and with frequency varying averaging width, with only a few responses—as little as two—of the full system.

This has particularly proved valuable to obtain the response of the system through a single reduced model of the system that gives different low- to high-frequency focus through different frequency ranges. The same strategy of mixing rational expressions, interpolation, and model reduction is applicable to other weighting functions that may or may not be originally expressed in rational form. In terms of asymptotic approximations, different approaches can be used, for example with rational expansions being used at zero, intermediate or infinite frequency.

The question of the efficient approximation or representation of the covariance and averaged product is of essential importance for the fundamental understanding of the mid-frequency region. The consideration of the averaged product of responses additionally to the covariance is important in assuring this representation, as well as in preserving information on the phase, as previously mentioned.

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It is evident from the analysis and plots that it can, in theory, be retrieved from the covariance or energy information evaluated with a single averaging width over the whole frequency range. The question of optimality of the approximations of the responses as in [ 19 ] can be reframed and possibly extended in the proposed averaging framework. A solution framework for dynamic systems has been proposed that targets directly the frequency average of their responses and their higher, second frequency moments.

It describes the desired level of frequency precision or uncertainty of the solution at any frequency. Asymptotically zero values of the parameter correspond to a deterministic or low-frequency solution with zero variance, while larger values correspond to high-frequency solutions that can be understood in a more energetic or statistical sense.

The transitional mid-frequency region is smoothly supported and no distinction or boundaries between the regions is necessary. The frequency averaged power or energy of the system, which are statistics that are usually considered at high-frequency, are found as function of the frequency average and of the averaged square of the absolute value of the response.

It has been explicitly stressed that this fact considered in [ 38 ] for the case of a uniform distribution can be applied for a general weighting function. The general explicit analytical expressions are provided and demonstrated here for the case of a real Gaussian averaging function. Contrary to the case of many statistical energy analysis and high-frequency methods, the power or power average is not the only result of the analysis in the proposed framework.

Many existing low-, mid- and high-frequency approaches can actually be placed and examined in the proposed framework. For example, deterministic modes, models of low-frequency physical components, such as springs and masses, SEA systems, analytical waves or hybrid models can be integrated into a single model. The coupling between any of these components within the framework must be characterized in such a way that the average, averaged square and variance of the responses of the global system are accurately represented.

This may be seen as a generalization of the fact that the coupling of high-frequency or SEA components must be such that the transfer of energy in a whole SEA system is accurately represented and that the coupling between low-frequency or deterministic systems must be such that the deterministic transfer functions, i. Coupling conditions assuring matching of the averages and variance also offer a point of view for the coupling conditions between the deterministic and SEA components of hybrid models [ 46 , 47 ]: Low-, mid- and hybrid mid-frequency coupling conditions are also particular cases of the more general framework coupling conditions.

They are perfectly valid within the framework but only under certain conditions and for specific ranges of values of the parameter a. This is exactly in the same manner as the validity of a simplified model of a spring or a mass starts breaking down when the frequency increases and the wavelengths become closer to the dimensions of the actual physical spring or mass. Other topics of the literature that have not yet been covered here but would be worth studying in future works include the averaging in the location of force, measurement and interface areas, the treatment of higher moment such as the variance of the general energy terms, and the conjunction of frequency and statistical averages.

On the other hand, material more rarely covered in the literature, that has been studied and highlighted here includes the treatment of the averaged product of responses and the covariance terms and the interpretations of the averages in the time domain. An efficient approach based on the frequency averages to estimate time responses has notably been proposed. Besides the fact that many approximate or asymptotic methods can be integrated into a single analysis, the framework also provides the advantage of offering a natural environment in which they can be better understood and expanded.

The precision of various methods to approximate low-, mid- or high-frequency components can also be studied in reference to how they affect the precision of the average and variance of transfer functions. The author is grateful to have found inspiration, attentive and patient ears, encouragement, support and useful feedback in several of his colleagues, including Paul Barbone, Neil Ferguson, Robin Langley, Brian Mace, J. Brainstorming on the statistical interpretation of the energy with Sujit Sahu, Sue Lewis and Dave Woods has been useful to clarify the interpretation of the second moments.

Detailed comments from two reviewers significantly improved the general presentation of the paper. Part of this work was carried out with financial support provided by the Engineering and Physical Sciences Research Council under grant no. Related Video Shorts 0 Upload your video. Customer reviews There are no customer reviews yet. Share your thoughts with other customers. Write a customer review. Amazon Giveaway allows you to run promotional giveaways in order to create buzz, reward your audience, and attract new followers and customers.

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IUTAM Symposium on Statistical Energy Analysis

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