Application of Etreme Value Statistics for Structural Health Monitoring Hoon Sohn Weapon Response Group Engineering Sciences and Applications Division Los Alamos National Laboratory Los Alamos, New Meico, USA. Presented at Uncertainty Quantification Working Group May 1st, 2003
Four Goals for Structural Health Monitoring Step 1: Damage Identification Step 2: Damage Localization Step 3: Damage Quantification Step 4: Damage Prognosis 2
Structural Health Monitoring Is this bridge damaged? Perform pattern comparison. 3
Five Steps for Structural Health Monitoring Step 1: Operational Evaluation Step 2: Data Acquisition Step 3: Data Normalization Step 4: Feature Etraction Step 5: Statistical Inference 4
Environmental Variation E-1 E-2 E-3 E-4 Freq. Hz 2.55 2.5 2.45 2.4 2.35 Dam0 Dam1 Dam2 Change of the first mode frequency w.r.t. damage cases Dam3 2.3 Dam0 Dam1 Dam2 Dam3 Dam4 5
Eample of Debris in Epansion Joint of the Alamosa Canyon Bridge This debris can effect the boundary conditions of the structure and its response to environmental changes 6
Statistical Pattern Recognition Paradigm for SHM After Sep. 11, 2001 Before Sep. 11, 2001 7
Data Normalization Eample [Sohn et al. 2002] m8 m1 Shaker Bumper List of time series employed in this study Case Description Input level Data # per input Total data # 0 No bumper 3, 4, 5, 6, 7 Volts 15 sets 75 sets 1 Bumper between m1-m2 3, 4, 5, 6, 7 Volts 5 sets 25 sets 2 Bumper between m5-m6 3, 4, 5, 6, 7 Volts 5 sets 25 sets 3 Bumper between m7-m8 4, 5, 6, 7 Volts 5 sets 20 sets 8
9 Outline of Damage Diagnosis using AR-ARX, Auto-Associative Network and Hypothesis Test 1. COLLECT BASELINE DATA Time series from various operation & environmental conditions t 2. FEATURE EXTRACTION = = = b j i a i i j t e i t t t 1 1 β α ε 4. FEATURE EXTRACTION = = = b j i a i i y j t e i t t t 1 1 ˆ ˆ β α ε 3. DATA NORMALIZATION 6. DECISION MAKING 1 2 3 4 5 6 7 8 0 0.5 1 1.5 2 2.5 3 3.5 Bumper between m1 and m2 DOF σε y /σε 5. STATISTICAL INFERENCE : : 1 y y o H H ε σ ε σ ε σ ε σ
Auto-Associative Neural Network for Data Normalization Mapping function G De-mapping function H Put α and β N N Predict and Predict α and β α i β i N Put ecitation level Predict ecitation level N N V L Vˆ N N N L L L αˆi βˆi N N L N N Mapping layer Bottleneck layer De-mapping layer Output layer 10
Damage Localization 1.5 No No bumper Bumper 3.5 Bumper between m1 m1 and and m2 m2 3.03 σε y /σε σ ε / σ ε y 1.01 0.5 σε y /σε 2.5 2.02 1.5 1.01 0.5 0.00 1 2 3 4 5 6 7 8 DOF 0.00 1 2 3 4 5 6 7 8 DOF 3.03 Bumper between m5 and and m6 m6 2.5 Bumper between m7 m7 and and m8 m8 2.5 2.02 σε y /σε σ ε / σ ε y 2.02 1.5 1.01 σε y /σε 1.5 1.01 0.5 0.5 0.00 1 2 3 4 5 6 7 8 DOF 0.00 1 2 3 4 5 6 7 8 DOF 11
Real World Application [Sohn et al. 2001] a Surface-Effect Fast Patrol Boat A,B Courtesy of Naval Research Laboratory H,I K J C,D G E,F I B D F b Fiber optic strain gauges K J A G E H C 12
Raw Dynamic Strain Time Series Data Low Speed High Speed Strain Strain Strain Strain Strain Strain 1000 1000 500 500 0-500 0 500-1000 Data Point # = 26980 Time Period = 601.17 sec t = 0.02228 sec 1000 0 100 200 300 400 500 600 700 Signal 2 2 1000 1000 500 500 0-500 0 500-1000 1000 0 100 200 300 400 500 600 700 Signal 3 3 1000 1000 500 500 0-500 0 500-1000 Signal 1 1 1000 0 100 200 300 400 500 600 700 Time Sec Time sec Structural Condition I undamaged Structural Condition II damaged 13
Damage Classification using σ ε σ ε. y 3.5 3.03 Signal 1 Signal 2 Signal 3 From from first half half Signal 3 From from second half half 2.5 σ y σε y /σε ε / σ ε 2.0 2 H=1.85 h=1.85 1.5 1.0 1 0.5 0 2 4 6 8 10 12 14 16 18 20 Test Number 14
A Moment Resisting Frame Structure Model Damage 2 Damage 1 15
Establishment of Decision Boundaries Unsupervised Learning: Use only the baseline data for training Undamaged Training Undamaged Testing Damaged Testing # of Epected outliers = 1% of 1000 points = 10 outliers From Normal # of outliers: 48 16
Normality Assumption of Data Let s look at the baseline prediction errors to see whether they have a normal distribution or not. A normal probability plot graphically assesses whether the data come from a normal distribution. If the data are normal, the plot will be linear. Otherwise, there would be curvature in the plot. The central population of data seems to fit to the normal distribution well, but the tails do not. Probability CDF 0.999 0.500 0.001 Normal Probability Plot -1.0 0.0 1.0 Prediction Errors 17
Why Etreme Value Statistics? In general, the distribution type of the parent data is unknown, and there are infinite numbers of candidate distributions. There are only three types of distributions for etreme the etreme maimum or minimum values regardless the distribution type of the parent data [Fisher and Tippett, 1928]. That means, the model selection for the etreme values becomes much easier, because there are only three models to choose.gumbel, Weibull, Frechet distributions 18
19 Feasible Cumulative Density Functions for Maima From Castillo [1988]: = otherwise ep if 1 β δ λ λ F Weibull: = otherwise 0 if ep λ λ δ β F Frechet: Gumbel 0, ep ep > < < = δ δ λ F
Fitting to Gumbel Distribution 0.999 Normal Probability Plot 1.0 Gumbel s CDF Probability 0.500 Probability 0.5 Points used Fitted CDF Empirical CDF Points used Fitted CDF 0.001-1.0 0.0 1.0 Data 0.0 0.2 0.3 0.4 0.5 Order statistic maima Divide the original times series with 8192 data points into 128 time series with 64 points. Compute the maimum value from each block and fit the 128 maima to a Gumbel distribution. 20
Establishment of Decision Boundaries Undamaged Training Undamaged Testing Damaged Testing From EVS # of outliers: 9 # of Epected outliers = 1% of 1000 points = 10 outliers From Normal # of outliers: 48 21
Detection of Rattling Container Frictionless Roller In-Ais Accelerometer Spring Internal component Off-Ais Accelerometer Bumper Base Ecitation at 18 Hz 22
Acceleration Response Rattling Rattling 23
Discontinuity Detection via Holder Eponent Definition: The Holder Eponent is a measure of the regularity of the signal. The regularity of the signal is the number of continuous derivatives that the signal possesses. Objective: Identify discontinuity in signals that can be caused by certain types of damage. Application: Eamples of damage that might induce discontinuity into the dynamic response signal include: Opening and closing of cracks A loose joint that is allowing contact rattle to occur 24
Holder Eponent Analysis Time Signal Wavelet Transform 1 t u Wf u s = *, f t ψ dt s s Translation Scalogram Dilation 25
Holder Eponent Analysis Scalogram Slop = Holder Eponent 26
Scalogram from Wavelet Transform 27
Holder Eponent Analysis 1 Find the ma drop 2 Threshold = C ma drop 3 Discontinuity > Threshold 28
Summary Cast structural health monitoring problems in the framework of statistical pattern recognition. Developed various signal-based damage detection algorithms. Embed damage detection algorithms into on-board microprocessors. Address data normalization issue eplicitly. Decision making is based on rigorous statistical modeling. Provide a suite of data interrogation algorithms for structural health monitoring in the format of GUI software called DIAMOND II patent pending. 29