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Research Papers

Vibro-Acoustic Response of Engineering Structures With Mixed Type of Probabilistic and Nonprobabilistic Uncertainty Models

[+] Author and Article Information
Alice Cicirello

Department of Engineering,
University of Cambridge,
Trumpington Street, Cambridge CB2 1PZ, UK
e-mail: ac685@cam.ac.uk

Robin S. Langley

Department of Engineering,
University of Cambridge,
Trumpington Street, Cambridge CB2 1PZ, UK
e-mail: rsl21@cam.ac.uk

Manuscript received October 25, 2014; final manuscript received February 4, 2015; published online October 2, 2015. Assoc. Editor: Alba Sofi.

ASME J. Risk Uncertainty Part B 1(4), 041001 (Oct 02, 2015) (13 pages) Paper No: RISK-14-1078; doi: 10.1115/1.4030470 History: Received October 25, 2014; Accepted April 28, 2015; Online October 02, 2015

The response of engineering structures is often sensitive to uncertainty in the system properties. The information on the uncertain parameters is frequently incomplete, limited to experts’ opinion, based on previous knowledge, or a combination of those. There have been recent advances in building mathematical models of structures, which combine nonparametric probabilistic and parametric (probabilistic or nonprobabilistic) models of uncertainty. Here, several strategies for establishing the response of random systems whose uncertainties are described by parametric probabilistic and nonprobabilistic approaches in combination with a nonparametric probabilistic approach are presented. The proposed strategies are illustrated by analyzing a built-up plate structure.

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References

Figures

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Fig. 1

Velocity response surface and contour obtained at the first mass at 10 Hz

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Fig. 2

Generic random system with interval ([bL,bU]), probabilistic parametric (p(b^)), and nonparametric (p(ωn,φn)) uncertainty descriptions

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Fig. 3

Conversion of a PDF description of an uncertain parameter b^ into an interval description bint*

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Fig. 4

Conversion of an interval description of an uncertain parameter bint=[bL;bU] into a PDF description b^*

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Fig. 7

Built-up plate system under investigation

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Fig. 8

Bounds on the nonparametric average response over the parametric ensemble considering only the extreme values of the interval variables. Top: vibrational energy of the plates (left: plate 1, right: plate 2)—units dB ref. 1 J. Bottom: auto-spectra of the mass displacement (left: mass 1, right: mass 2)—units dB ref 1 m. Upper bounds: continuous line; lower bounds: dashed lines; thick lines: hybrid FE/SEA + interval results; thin lines: benchmark model.

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Fig. 9

Method A (hybrid FE/SEA + interval) compared to results yielded considering just the bounds of the interval variables. Bounds on the nonparametric average response over the parametric ensemble. Top: vibrational energy of the plates (left: plate 1, right: plate 2)—units dB ref. 1 J. Bottom: auto-spectra of the mass displacement (left: mass 1, right: mass 2)—units dB ref 1 m. Upper bounds: continuous line; lower bounds: dashed lines; thick gray lines: Method A results considering 50 subintervals within each interval variable; thin black lines: results considering only extreme points of the interval variable.

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Fig. 10

Method A for 900, 2500, and 22,500 sets of springs. Bounds on the nonparametric average response over the parametric ensemble. Top: vibrational energy of the plates (left: plate 1, right: plate 2)—units dB ref. 1 J. Bottom: auto-spectra of the mass displacement (left: mass 1, right: mass 2)—units dB ref 1 m. Upper bounds: continuous line; lower bounds: dashed lines; 22,500 sets of springs: thick black lines; 2500 sets of springs: thick gray lines; 900 sets of springs: thin black lines.

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Fig. 11

Method A for 22,500 sets of springs compared to the hybrid FE/SEA + random sampling intervals for 900 and 22,500 sets of springs. Bounds on the nonparametric average response over the parametric ensemble. Top: vibrational energy of the plates (left: plate 1, right: plate 2)—units dB ref. 1 J. Bottom: auto-spectra of the mass displacement (left: mass 1, right: mass 2)—units dB ref 1 m. Upper bounds: continuous line; lower bounds: dashed lines; Method A 22,500 sets of springs: thick black lines. Hybrid FE/SEA + random sampling intervals: (1) 22,500 sets of springs: thin black lines and (2) 900 sets of springs: thin gray lines.

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Fig. 12

Unconditional mean (left) and relative variance (right) of the vibrational energy (top—plate 1: solid lines, plate 2: dashed lines) and of the auto-spectra of the mass displacement (bottom—mass 1: solid lines, mass 2: dashed lines). Units: dB ref. 1 J for average energy and dB ref. 1 m for average auto-spectra. The cloud of gray curves represent each member of the ensembles of the responses calculated with the benchmark MCS.

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Fig. 13

99% Confidence intervals calculated with Method B (thick lines) versus Benchmark Monte Carlo results (thin lines). Top: vibrational energy of the plates (left: plate 1, right: plate 2). Bottom: auto-spectra of the mass displacement (left: mass 1, right: mass 2). Units: dB ref. 1 J for average energy and dB ref. 1 m for average auto-spectra.

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Fig. 14

Method C—Bounds on the unconditional average response considering only the extreme values of the interval variables. Top: vibrational energy of the plates (left: plate 1, right: plate 2)—units: dB ref. 1 J. Bottom: auto-spectra of the mass displacement (left: mass 1, right: mass 2)—units dB ref. 1 m. Upper bounds: continuous line; lower bounds: dashed lines; Method C results: thick black lines; Benchmark Monte Carlo results: fluctuating light curves.

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Fig. 15

Method C—Bounds on the unconditional average response compared to bounds obtained considering just the extreme value of the interval variable bounds. Top: vibrational energy of the plates (left: plate 1, right: plate 2)—units: dB ref. 1 J. Bottom: auto-spectra of the mass displacement (left: mass 1, right: mass 2)—units dB ref. 1 m. Upper bounds: continuous line; lower bounds: dashed lines; Method C results: thick gray lines; Hybrid FE/SEA + random sampling results: fluctuating light black curves; thick black lines: results considering only extreme points of the interval variable.

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Fig. 16

The statistics of the bounds on the nonparametric ensemble average values computed over the probabilistic parameters. Top: vibrational energy of the plates (left: plate 1, right: plate 2)—units: dB ref. 1 J. Bottom: auto-spectra of the mass displacement (left: mass 1, right: mass 2)—units dB ref. 1 m. Upper bounds: continuous line; lower bounds: dashed lines; Method D results: thick black lines; Hybrid FE/SEA + random sampling: fluctuating light curves.

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Fig. 17

Method C (gray curves) versus Method D (black curves). Top: vibrational energy of the plates (left: plate 1, right: plate 2). Bottom: auto-spectra of the mass displacement (left: mass 1, right: mass 2). Units: dB ref. 1 J for energy and dB ref. 1 m for auto-spectra.

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Fig. 18

Comparison between Methods A (black lines) and Method C (gray continuous line)—bounds on average results. Top: vibrational energy of the plates (left: plate 1, right: plate 2). Bottom: auto-spectra of the mass displacement (left: mass 1, right: mass 2). Units: dB ref. 1 J for average energy and dB ref. 1 m for average auto-spectra.

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Fig. 19

Comparison between Methods B (black thin lines) and Method C (gray continuous line) 99% confidence bounds results. Top: vibrational energy of the plates (left: plate 1, right: plate 2). Bottom: auto-spectra of the mass displacement (left: mass 1, right: mass 2). Units: dB ref. 1 J for average energy and dB ref. 1 m for average auto-spectra.

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