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TECHNICAL PAPERS

Improving Excitations for Active Sensing in Structural Health Monitoring via Evolutionary Algorithms

[+] Author and Article Information
Colin C. Olson, M. D. Todd

Department of Structural Engineering, University of California, San Diego, La Jolla, CA 92093-0085

Keith Worden

Department of Mechanical Engineering, Sheffield University, Sheffield S1 3JD, UK

Charles Farrar

Los Alamos National Laboratory, Engineering Institute, MS T-001, Los Alamos, NM 87545

J. Vib. Acoust 129(6), 784-802 (Mar 14, 2007) (19 pages) doi:10.1115/1.2748478 History: Received November 29, 2006; Revised March 14, 2007

Active excitation is an emerging area of study within the field of structural health monitoring whereby prescribed inputs are used to excite the structure so that damage-sensitive features may be extracted from the structural response. This work demonstrates that the parameters of a system of ordinary differential equations may be adjusted via an evolutionary algorithm to produce excitations that improve the sensitivity and robustness to extraneous noise of state-space based damage detection features extracted from the structural response to such excitations. A simple computational model is used to show that significant gains in damage detection and quantification may be obtained from the response of a spring-mass system to improved excitations generated by three separate representative ordinary differential equation systems. Observed differences in performance between the excitations produced by the three systems cannot be explained solely by considering the frequency characteristics of the excitations. This work demonstrates that the particular dynamic evolution of the excitation applied to the structure can be as important as the frequency characteristics of said excitation if improved damage detection is desired. In addition, the implied existence of a globally optimum excitation (in the sense of improved damage assessment) for the model system is explored.

Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

2-DOF spring mass system: m=0.0001, c=0.001, k1=k2=1 in the undamaged case, k1=0.97*k2 in the damaged case.

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Figure 2

Schematic of the overall excitation optimization technique

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Figure 3

Maximum population fitness versus generation for the variable-ε optimization of the Lorenz system. The variation in fitness for a given solution arises because fitness is recalculated each generation and prediction error is a stochastic feature.

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Figure 4

Histograms of prediction error for the baseline-baseline (xb1 versus xb2) comparison (left-most distribution in both plots) and baseline-damaged (xb1 versus xc) comparison for the optimized (a) and unoptimized chaotic (b) Lorenz system. Parameters used for the optimized excitation are provided in Table 1 and parameters for the chaotic Lorenz are typical ones given in Ref. 34. Fitness of optimized excitation is 0.0013 versus the unoptimized fitness of 0.00007. The abscissa is composed of 500 bins for each plot and 10,000 prediction error values are plotted for each distribution. ε=1 and SNR=10dB for this comparison.

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Figure 5

2D projection of the undamaged (a) and 3% damaged (b) SVD embeddings of the first mass time series for the spring system forced by the optimized Lorenz excitation. Parameters used for the optimized excitation are provided in Table 1. The corresponding undamaged (open circles) and damaged first mass time series are shown in (c). Histograms of the prediction error for the baseline-baseline comparison and baseline-damaged comparison are plotted in (d) with 10,000 PE values for each distribution. Fitness=0.0011, histograms are plotted with 500 bins for the abscissa. SNR=10dB in both cases and ε=1 for this optimization.

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Figure 6

2D projection of the undamaged (a) and 3% damaged (b) SVD embeddings of the first mass time series for the spring system forced by the optimized Sprott excitation. Parameters used for the optimized excitation are provided in Table 1. The corresponding undamaged (open circles) and damaged first mass time series are shown in (c). Histograms of the prediction error for the baseline-baseline comparison and baseline-damaged comparison are plotted in (d) with 10,000 PE values for each distribution. Fitness=0.0015 and histograms are plotted with 500 bins for the abscissa. SNR=10dB in both cases and ε=1 for this optimization.

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Figure 7

2D projection of the undamaged (a) and 3% damaged (b) SVD embeddings of the first mass time series for the spring system forced by the optimized Rossler excitation. Parameters used for the optimized excitation are provided in Table 1. The corresponding undamaged (open circles) and damaged first mass time series are shown in (c). Histograms of the prediction error for the baseline-baseline comparison and baseline-damaged comparison are plotted in (d) with 10,000 PE values for each distribution. Fitness=0.0033 and histograms are plotted with 500 bins for the abscissa. SNR=10dB in both cases and ε=1 for this optimization.

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Figure 8

2D projection of the undamaged (a) and 3% damaged (b) SVD embeddings of the first mass time series for the spring system forced by the optimized Rossler excitation. Parameters used for the optimized excitation are provided in Table 2. The corresponding undamaged (open circles) and damaged first mass time series are shown in (c). Histograms of the prediction error for the baseline-baseline comparison and baseline-damaged comparison are plotted in (d) with 10,000 PE values for each distribution. Fitness=0.002 and histograms are plotted with 500 bins for the abscissa. SNR=10dB in both cases. ε was allowed to vary for this optimization.

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Figure 9

2D projection of the undamaged (a) and 3% damaged (b) SVD embeddings of the first mass time series for the spring system forced by the optimized Lorenz excitation. Parameters used for the optimized excitation are provided in Table 2. The corresponding undamaged (open circles) and damaged first mass time series are shown in (c). Histograms of the prediction error for the baseline-baseline comparison and baseline-damaged comparison are plotted in (d) with 10,000 PE values for each distribution. Fitness=0.0137 and histograms are plotted with 500 bins for the abscissa. SNR=10dB in both cases. ε was allowed to vary for this optimization.

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Figure 10

2D projection of the undamaged (a) and 3% damaged (b) SVD embeddings of the first mass time series for the spring system forced by the optimized Sprott excitation. Parameters used for the optimized excitation are provided in Table 2. The corresponding undamaged (open circles) and damaged first mass time series are shown in (c). Histograms of the prediction error for the baseline-baseline comparison and baseline-damaged comparison are plotted in (d) with 10,000 PE values for each distribution. Fitness=0.123 and histograms are plotted with 500 bins for the abscissa. SNR=10dB in both cases. ε was allowed to vary for this optimization.

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Figure 11

Fitness versus SNR for the Rossler (dashed), Sprott (solid), and Lorenz (dotted) excitations from the constant-ε optimization. The mean and standard deviation of ten separate comparisons (excitation held constant with a new instance of additive noise) made at each noise level are shown. The lowest SNR value plotted for each excitation represents the final point where a decision between the undamaged and damaged case could be made with 100% confidence (finite data) for all ten comparisons. Lowest SNR points shown on the plots are accurate to approximately ±0.5dB.

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Figure 12

Fitness versus percent stiffness reduction of the first spring for various signal-to-noise ratios: SNR=20dB, SNR=15dB (dashed), SNR=10dB (dotted), SNR=5dB (solid). System is forced with best discovered output from the Sprott system optimized with 3% damage and ε allowed to vary. The mean and standard deviation of ten separate comparisons (excitation held constant, new instance of additive noise) made at each damage level are shown. The lowest damage percent value plotted for each excitation represents the final point where a decision between the undamaged and damaged case could be made with 100% confidence (finite data) for all ten comparisons. Lowest damage values shown on the plot are accurate to approximately ±0.025%.

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Figure 13

Fitness versus percent stiffness reduction of the first spring for various signal-to-noise ratios: SNR=20dB, SNR=15dB (dashed), SNR=10dB (dotted), SNR=5dB (solid). System is forced with best discovered output from the Lorenz system optimized with 3% damage and ε allowed to vary. The mean and standard deviation of ten separate comparisons (excitation held constant, new instance of additive noise) made at each damage level are shown. The lowest damage percent value plotted for each excitation represents the final point where a decision between the undamaged and damaged case could be made with 100% confidence (finite data) for all ten comparisons. Lowest damage values shown on the plot are accurate to approximately ±0.025%.

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Figure 14

Fitness versus percent stiffness reduction of the first spring for various signal-to-noise ratios: SNR=20dB, SNR=15dB (dashed), SNR=10dB (dotted), SNR=5dB (solid). System is forced with best discovered output from the Rossler system optimized with 3% damage and ε allowed to vary. The mean and standard deviation of ten separate comparisons (excitation held constant, new instance of additive noise) made at each damage level are shown. The lowest damage percent value plotted for each excitation represents the final point where a decision between the undamaged and damaged case could be made with 100% confidence (finite data) for all ten comparisons. Lowest damage values shown on the plot are accurate to approximately ±0.025%.

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Figure 15

Lowest damage percentage versus SNR for the Rossler, Lorenz, and Sprott excitations derived from variable-ε optimization. Ten separate comparisons (excitation held constant, new instance of additive noise) were made at each damage level and the lowest damage percent value plotted for each excitation represents the final point where a decision between the undamaged and damaged case could be made with 100% confidence (finite data) for all ten comparisons. Lowest damage values shown on the plot are accurate to approximately ±0.025%.

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Figure 16

Periodogram estimates of power spectral density (PSD) for the Lorenz (open circles), Rossler (dots), and Sprott excitations found from the constant-ε optimizations. Estimated with Kaiser–Bessel window, β=7.85 and no overlaps. The first and second natural frequencies are 9.84Hz and 25.75Hz for the undamaged system; 9.73Hz and 25.65Hz for the damaged system.

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Figure 17

Periodogram estimates of PSD for the Lorenz (open circles), Rossler (dots), and Sprott excitations found from the variable-ε optimizations. Estimated with Kaiser-Bessel window, β=7.85 and no overlaps. The first and second natural frequencies are 9.84Hz and 25.75Hz for the undamaged system; 9.73Hz and 25.65Hz for the damaged system.

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Figure 18

Fitness versus SNR for the Rossler (dashed), Sprott (solid), and Lorenz (dotted) excitations and their respective surrogates (same line style, lower fitness) from the constant-ε optimization. The mean and standard deviation of ten separate comparisons (excitation held constant, new instance of additive noise) made at each noise level are shown. The lowest SNR value plotted for each excitation represents the final point where a decision between the undamaged and damaged case could be made with 100% confidence (finite data) for all ten comparisons. Lowest SNR points shown on the plots are accurate to approximately ±0.5dB.

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Figure 19

Fitness versus SNR for the Rossler (dashed), Sprott (solid), and Lorenz (dotted) excitations and their respective surrogates (same line style, lower fitness) from the variable-ε optimization. The mean and standard deviation of ten separate comparisons (excitation held constant, new instance of additive noise) made at each noise level are shown. The lowest SNR value plotted for each excitation represents the final point where a decision between the undamaged and damaged case could be made with 100% confidence (finite data) for all ten comparisons. Lowest SNR points shown on the plots are accurate to approximately ±0.5dB.

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Figure 20

Attractor projections for the improved Lorenz (top row) and Sprott excitations from the variable-ε optimization

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Figure 21

Periodogram estimates of PSD for the Lorenz (open circles) and Sprott excitations. Estimated with Kaiser–Bessel window, β=7.85, and no overlaps. The first and second natural frequencies are 9.84Hz and 25.75Hz for the undamaged system; 9.73Hz and 25.65Hz for the damaged system.

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Figure 22

Optimized excitation attractors from the constrained-ε optimization (top row) and the variable-ε optimization. The Rossler, Lorenz, and Sprott excitations are shown from left to right.

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