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Journal of Vibration and Acoustics | Volume 135 | Issue 5 | research-article
Research Papers

Characterization of Dynamic Response of Structures With Uncertainty by Using Gaussian Processes

[+] Author and Article Information
J. Tang

e-mail: jtang@engr.uconn.edu
Department of Mechanical Engineering,
University of Connecticut,
Storrs, CT 06269

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Vibration and Acoustics. Manuscript received August 3, 2011; final manuscript received March 8, 2013; published online June 18, 2013. Assoc. Editor: Michael Brennan.

J. Vib. Acoust 135(5), 051006 (Jun 18, 2013) (13 pages) Paper No: VIB-11-1168; doi: 10.1115/1.4023998 History: Received August 03, 2011; Revised March 08, 2013
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Characterizing dynamic characteristics of structures with uncertainty is an important task that provides critical predictive information for structural design, assessment, and control. In practical applications, sampling is the fundamental approach to uncertainty analysis but has to be conducted under various constraints. To address the frequently encountered data scarcity issue, in the present paper Gaussian processes are employed to predict and quantify structural dynamic responses, especially responses under uncertainty. A self-contained description of Gaussian processes is presented within the Bayesian framework with implementation details, and then a series of case studies are carried out using a cyclically symmetric structure that is highly sensitive to uncertainties. Structural frequency responses are predicted with data sparsely sampled within the full frequency range. Based on the inferred credible intervals, a measure is defined to quantify the potential risk of response maxima. Gaussian process emulation is proposed for Monte Carlo uncertainty analysis to reduce data acquisition costs. It is shown that Gaussian processes can be an efficient data-based tool for analyzing structural dynamic responses in the presence of uncertainty. Meanwhile, some technical challenges in the implementation of Gaussian processes are discussed.
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Figures

Grahic Jump Location
Fig. 1

Geometry and meshing of the example bladed disk

+Geometry and meshing of the example bladed disk
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Geometry and meshing of the example bladed disk
Grahic Jump Location
Fig. 2

The seconds engine order frequency responses on 18 blade tips. Solid lines—tuned tip responses (identical for all blades); dotted lines—mistuned tip responses. (a) 2% mistuning SD; (b) 5% mistuning SD; (c) 8% mistuning SD; and (d) 12% mistuning SD.

+The seconds engine order frequency responses on 18 blade tips. Solid lines—tuned tip responses (identical for all blades); dotted lines—mistuned tip responses. (a) 2% mistuning SD; (b) 5% mistuning SD; (c) 8% mistuning SD; and (d) 12% mistuning SD.
View Large  |  Save Figure  |  Download Slide (.ppt)
The seconds engine order frequency responses on 18 blade tips. Solid lines—tuned tip responses (identical for all blades); dotted lines—mistuned tip responses. (a) 2% mistuning SD; (b) 5% mistuning SD; (c) 8% mistuning SD; and (d) 12% mistuning SD.
Grahic Jump Location
Fig. 3

The second engine order frequency responses on blade 2 tip by Monte Carlo sampling (5% mistuning SD). Gray lines—1000 sampled responses; black line—nominal responses; crosses—response peaks of individual runs.

+The second engine order frequency responses on blade 2 tip by Monte Carlo sampling (5% mistuning SD). Gray lines—1000 sampled responses; black line—nominal responses; crosses—response peaks of individual runs.
View Large  |  Save Figure  |  Download Slide (.ppt)
The second engine order frequency responses on blade 2 tip by Monte Carlo sampling (5% mistuning SD). Gray lines—1000 sampled responses; black line—nominal responses; crosses—response peaks of individual runs.
Grahic Jump Location
Fig. 5

Statistical characterization of frequency responses on blade 2 tip: (a) Monte Carlo sampling with 1000 runs; (b) Monte Carlo sampling with 50 runs; and (c) inferred envelope compared with true envelope. Gray line in (a) and (b)—individual sampled response curve; dotted line in (a) and (c)—upper envelope of 1000 sampled response curves; dotted line in (b)—upper envelope of 50 sampled response curves; solid line in (a)—nominal response curve; solid line in (c)—inferred upper envelope; plus sign - feature point for inference; shaded area in (c)—95% credible interval.

+Statistical characterization of frequency responses on blade 2 tip: (a) Monte Carlo sampling with 1000 runs; (b) Monte Carlo sampling with 50 runs; and (c) inferred envelope compared with true envelope. Gray line in (a) and (b)—individual sampled response curve; dotted line in (a) and (c)—upper envelope of 1000 sampled response curves; dotted line in (b)—upper envelope of 50 sampled response curves; solid line in (a)—nominal response curve; solid line in (c)—inferred upper envelope; plus sign - feature point for inference; shaded area in (c)—95% credible interval.
View Large  |  Save Figure  |  Download Slide (.ppt)
Statistical characterization of frequency responses on blade 2 tip: (a) Monte Carlo sampling with 1000 runs; (b) Monte Carlo sampling with 50 runs; and (c) inferred envelope compared with true envelope. Gray line in (a) and (b)—individual sampled response curve; dotted line in (a) and (c)—upper envelope of 1000 sampled response curves; dotted line in (b)—upper envelope of 50 sampled response curves; solid line in (a)—nominal response curve; solid line in (c)—inferred upper envelope; plus sign - feature point for inference; shaded area in (c)—95% credible interval.
Grahic Jump Location
Fig. 4

Predictions of the second engine order frequency responses at the tip of blade 2. Plus—training point; cross—validation point; solid line—inferred responses; dashed line—true responses; shaded area—95% credible interval. Plots (a), (d), and (g) are obtained with same nine training data but different initial hyperparameters; plots (b), (e), and (h) are obtained with same seven training data but different initial hyperparameters; and plots (c), (f), and (i) are obtained with different posterior hyperparameters φ∧={1.0,0.2106392,0}, {0.3,0.1051911,0}, and {0.07,0.09349025,0}, respectively.

+Predictions of the second engine order frequency responses at the tip of blade 2. Plus—training point; cross—validation point; solid line—inferred responses; dashed line—true responses; shaded area—95% credible interval. Plots (a), (d), and (g) are obtained with same nine training data but different initial hyperparameters; plots (b), (e), and (h) are obtained with same seven training data but different initial hyperparameters; and plots (c), (f), and (i) are obtained with different posterior hyperparameters φ∧={1.0,0.2106392,0}, {0.3,0.1051911,0}, and {0.07,0.09349025,0}, respectively.
View Large  |  Save Figure  |  Download Slide (.ppt)
Predictions of the second engine order frequency responses at the tip of blade 2. Plus—training point; cross—validation point; solid line—inferred responses; dashed line—true responses; shaded area—95% credible interval. Plots (a), (d), and (g) are obtained with same nine training data but different initial hyperparameters; plots (b), (e), and (h) are obtained with same seven training data but different initial hyperparameters; and plots (c), (f), and (i) are obtained with different posterior hyperparameters φ∧={1.0,0.2106392,0}, {0.3,0.1051911,0}, and {0.07,0.09349025,0}, respectively.
Grahic Jump Location
Fig. 6

Response peaks on blade 2 tip inferred with 21 training data (a) at all 1000 configuration points and (b) in a zoomed-in region. Cross—true peak value; plus—inferred peak value; circle—data point; shaded area—95% credible interval.

+Response peaks on blade 2 tip inferred with 21 training data (a) at all 1000 configuration points and (b) in a zoomed-in region. Cross—true peak value; plus—inferred peak value; circle—data point; shaded area—95% credible interval.
View Large  |  Save Figure  |  Download Slide (.ppt)
Response peaks on blade 2 tip inferred with 21 training data (a) at all 1000 configuration points and (b) in a zoomed-in region. Cross—true peak value; plus—inferred peak value; circle—data point; shaded area—95% credible interval.
Grahic Jump Location
Fig. 7

Response peaks on blade 2 tip inferred with 51 training data (a) at all 1000 configuration points and (b) in a zoomed-in region. Cross—true peak value; plus—inferred peak value; circle—data point; shaded area—95% credible interval.

+Response peaks on blade 2 tip inferred with 51 training data (a) at all 1000 configuration points and (b) in a zoomed-in region. Cross—true peak value; plus—inferred peak value; circle—data point; shaded area—95% credible interval.
View Large  |  Save Figure  |  Download Slide (.ppt)
Response peaks on blade 2 tip inferred with 51 training data (a) at all 1000 configuration points and (b) in a zoomed-in region. Cross—true peak value; plus—inferred peak value; circle—data point; shaded area—95% credible interval.
Grahic Jump Location
Fig. 8

Comparison of true and inferred response peaks on blade 2 tip (a) with 21 training data and (b) with 51 training data

+Comparison of true and inferred response peaks on blade 2 tip (a) with 21 training data and (b) with 51 training data
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Comparison of true and inferred response peaks on blade 2 tip (a) with 21 training data and (b) with 51 training data

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