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A frequency averaging framework for the solution of complex dynamic systems.

Lecomte C - Proc. Math. Phys. Eng. Sci. (2014)

Bottom Line: A frequency averaging framework is proposed for the solution of complex linear dynamic systems.It is remarkable that, while the mid-frequency region is usually very challenging, a smooth transition from low- through mid- and high-frequency ranges is possible and all ranges can now be considered in a single framework.An interpretation of the frequency averaging in the time domain is presented and it is explained that the average may be evaluated very efficiently in terms of system solutions.

View Article: PubMed Central - PubMed

Affiliation: Associate Member, Southampton Statistical Sciences Research Institute , University of Southampton , Southampton, UK.

ABSTRACT
A frequency averaging framework is proposed for the solution of complex linear dynamic systems. It is remarkable that, while the mid-frequency region is usually very challenging, a smooth transition from low- through mid- and high-frequency ranges is possible and all ranges can now be considered in a single framework. An interpretation of the frequency averaging in the time domain is presented and it is explained that the average may be evaluated very efficiently in terms of system solutions.

No MeSH data available.


Comparison of the average, variance and impulse response functions for a constant, original, averaging width a=0.1 rad Hz and an averaging width that has been refined in the interval (0.1,1.1) rad Hz. The approximate and exact impulse responses are, respectively, the inverse Fourier transforms of the exact transfer function and that of its average. Both were evaluated through an explicit trapezoidal integral as for the results of figure 5.
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RSPA20130743F6: Comparison of the average, variance and impulse response functions for a constant, original, averaging width a=0.1 rad Hz and an averaging width that has been refined in the interval (0.1,1.1) rad Hz. The approximate and exact impulse responses are, respectively, the inverse Fourier transforms of the exact transfer function and that of its average. Both were evaluated through an explicit trapezoidal integral as for the results of figure 5.

Mentions: Alternative approaches. An apparently evident alternative strategy to model the response of the system would be to consider the modal components of the system that contribute most to the response. Looking at the modal magnitudes of found from modal expression (3.5), as such a criteria, one can truncate the modal representations at a number of pairs of conjugate modes (since the response in time is real, each ωj comes in pair with an eigenvalue with opposite real part, −ℜ(ωj)+iℑ(ωj)). The magnitude of the impulse responses are presented in figure 3a for an increasing number of 1, 2, 5, 10 modal pairs whose reference eigenvalues, sorted by decreasing modal magnitude are ωj={0.5634+i0.000079, 0.5173+i0.000067, 6.2552+i0.009782, 0.6022+i0.000091, 0.4823+i0.000058, 0.7754+0.00015i, 0.6191+0.000096i, 0.3755+0.000035i, 0.7165+0.00013i, 0.8323+0.000024i, …}. While the predictions obtained through modal truncation are roughly similar to the actual impulse response, they are not particularly precise. For example, the time and value of the first local maximum, , of the impulse response is estimated as (0.3350[s],0.0805), when 10 pairs of modes are used. This is a 13 and 40% relative error. Another issue is that it is difficult to assess the quality of these approximations if information on the other modes is not used. Other approaches have been proposed over the last one or two decades with the objective of reducing model dimensions such that the reduced model preserves information of the original system, in particular in the time domain. Among the several options that have notably been proposed and explored by Barbone et al. [19], it is worth noting a connection of the current work with the smooth (mass) modal density function used to model a high-density discrete spectrum. Although starting from a different point of view, the present frequency averaging framework can indeed also be seen as offering smoothed frequency functions. Further comparison between the two points of view might therefore bring additional insight. Further efforts have also been pursued to identify and extract the most important modes of complicated subsystems, also with a focus on the quality of the time responses of coupled systems. The impulse response predictions obtained through the OMR algorithm of [45, Box 1, page 1669]1 are presented in figure 3b for reduced models of the attachment of dimensions 1 to 50. While such approach is better targeted than other modal truncation methods to the evaluation of impulse responses, its accuracy is still relatively limited for the amount of information preserved. Note that the possibly less precise predictions compared with those obtained via the previous modal truncation method may be explained by the fact that OMR operates without information on the actual system to which the complicated ensemble of attachments is connected. The predictions of both sets of impulse responses of figure 3 should be compared to the results presented in figures 4–6. It is seen that an accurate estimation of the average can be achieved with only a few equivalent modes and that it can provide extremely accurate time domain predictions. A practical implicit approach for evaluating such equivalent modes is through the rational Krylov projection method proposed in [31].Figure 3.


A frequency averaging framework for the solution of complex dynamic systems.

Lecomte C - Proc. Math. Phys. Eng. Sci. (2014)

Comparison of the average, variance and impulse response functions for a constant, original, averaging width a=0.1 rad Hz and an averaging width that has been refined in the interval (0.1,1.1) rad Hz. The approximate and exact impulse responses are, respectively, the inverse Fourier transforms of the exact transfer function and that of its average. Both were evaluated through an explicit trapezoidal integral as for the results of figure 5.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4042715&req=5

RSPA20130743F6: Comparison of the average, variance and impulse response functions for a constant, original, averaging width a=0.1 rad Hz and an averaging width that has been refined in the interval (0.1,1.1) rad Hz. The approximate and exact impulse responses are, respectively, the inverse Fourier transforms of the exact transfer function and that of its average. Both were evaluated through an explicit trapezoidal integral as for the results of figure 5.
Mentions: Alternative approaches. An apparently evident alternative strategy to model the response of the system would be to consider the modal components of the system that contribute most to the response. Looking at the modal magnitudes of found from modal expression (3.5), as such a criteria, one can truncate the modal representations at a number of pairs of conjugate modes (since the response in time is real, each ωj comes in pair with an eigenvalue with opposite real part, −ℜ(ωj)+iℑ(ωj)). The magnitude of the impulse responses are presented in figure 3a for an increasing number of 1, 2, 5, 10 modal pairs whose reference eigenvalues, sorted by decreasing modal magnitude are ωj={0.5634+i0.000079, 0.5173+i0.000067, 6.2552+i0.009782, 0.6022+i0.000091, 0.4823+i0.000058, 0.7754+0.00015i, 0.6191+0.000096i, 0.3755+0.000035i, 0.7165+0.00013i, 0.8323+0.000024i, …}. While the predictions obtained through modal truncation are roughly similar to the actual impulse response, they are not particularly precise. For example, the time and value of the first local maximum, , of the impulse response is estimated as (0.3350[s],0.0805), when 10 pairs of modes are used. This is a 13 and 40% relative error. Another issue is that it is difficult to assess the quality of these approximations if information on the other modes is not used. Other approaches have been proposed over the last one or two decades with the objective of reducing model dimensions such that the reduced model preserves information of the original system, in particular in the time domain. Among the several options that have notably been proposed and explored by Barbone et al. [19], it is worth noting a connection of the current work with the smooth (mass) modal density function used to model a high-density discrete spectrum. Although starting from a different point of view, the present frequency averaging framework can indeed also be seen as offering smoothed frequency functions. Further comparison between the two points of view might therefore bring additional insight. Further efforts have also been pursued to identify and extract the most important modes of complicated subsystems, also with a focus on the quality of the time responses of coupled systems. The impulse response predictions obtained through the OMR algorithm of [45, Box 1, page 1669]1 are presented in figure 3b for reduced models of the attachment of dimensions 1 to 50. While such approach is better targeted than other modal truncation methods to the evaluation of impulse responses, its accuracy is still relatively limited for the amount of information preserved. Note that the possibly less precise predictions compared with those obtained via the previous modal truncation method may be explained by the fact that OMR operates without information on the actual system to which the complicated ensemble of attachments is connected. The predictions of both sets of impulse responses of figure 3 should be compared to the results presented in figures 4–6. It is seen that an accurate estimation of the average can be achieved with only a few equivalent modes and that it can provide extremely accurate time domain predictions. A practical implicit approach for evaluating such equivalent modes is through the rational Krylov projection method proposed in [31].Figure 3.

Bottom Line: A frequency averaging framework is proposed for the solution of complex linear dynamic systems.It is remarkable that, while the mid-frequency region is usually very challenging, a smooth transition from low- through mid- and high-frequency ranges is possible and all ranges can now be considered in a single framework.An interpretation of the frequency averaging in the time domain is presented and it is explained that the average may be evaluated very efficiently in terms of system solutions.

View Article: PubMed Central - PubMed

Affiliation: Associate Member, Southampton Statistical Sciences Research Institute , University of Southampton , Southampton, UK.

ABSTRACT
A frequency averaging framework is proposed for the solution of complex linear dynamic systems. It is remarkable that, while the mid-frequency region is usually very challenging, a smooth transition from low- through mid- and high-frequency ranges is possible and all ranges can now be considered in a single framework. An interpretation of the frequency averaging in the time domain is presented and it is explained that the average may be evaluated very efficiently in terms of system solutions.

No MeSH data available.