Stationary or Non-Stationary Random Excitation for Vibration-Based Structural Damage Detection? An exploratory study Andriana S. GEORGANTOPOULOU & Spilios D. FASSOIS Stochastic Mechanical Systems & Automation (SMSA) Laboratory Department of Mechanical & Aeronautical Engineering University of Patras, GR 26504 Patras, Greece ageorgant@mech.upatras.gr, fassois@mech.upatras.gr http://www.smsa.upatras.gr 6th International Symposium on NDT in Aerospace Madrid, Spain, 12-14th November 2014 SMSA Lab - University of Patras Damage Detectability Madrid, Spain, November 2014
Outline Talk Outline 1. Introduction 2. The structure & the damage scenario 3. The non-stationary random excitation signal 4. Assessing damage detectability 5. Damage detectability: non-stationary vs stationary excitation 6. Conclusions SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 2/23
Introduction 1. Introduction Random vibration based damage detection is popular as (Fan & Qiao 2010; Sakellariou & Fassois 2007) random vibration: May be easily induced in a controlled environment Is also naturally available Equipment for its measurement is widely available & inexpensive It may be lead to effective detection of damage at an early stage SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 3/23
Vibration-based SHM The UNDERLYING THESIS is that damage affects the dynamical characteristics! Early methods based on response range or similar characteristics are generally less sensitive and/or slower in responding Damage Dynamics Response SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 4/23
Introduction In a user-controlled excitation context, the excitation is typically chosen as a random white signal. Random white excitation Random vibration response t t ACF ρ σ 2 ACF ρ PSD S 0 τ PSD S 0 τ 0 ω 0 ω White signal equally excites all frequencies PSD reveals structural modes SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 5/23
Introduction The question posed and explored in this study: Can a more complex ( richer ) random excitation exhibiting non-stationary characteristics lead to improved damage detectability? Motivation Natural random excitation often is non-stationary. Is this a benefit? In a user-controlled environment should the excitation be chosen as nonstationary? Study Method For answering the question non-stationary random excitation is compared to stationary white excitation in terms of damage detectability. Monte Carlo experiments based on a damaged composite structure are used. SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 6/23
Local time τ (s) Introduction What is non-stationary excitation? Non-stationary excitation is characterized by Time-Varying (TV) characteristics: mean, variance, autocovariance (acf), power spectral density (psd) The special stationary case: Mean μ t = E x t = μ = const Variance σ t 2 = E x t μ 2 = σ 2 = const autocovariance (acf) γ t, t + τ = E x t μ x t + μ = γ[τ] (function of time lag τ) γ[τ] 0 τ power spectral density (psd) S(ω) =F {γ[τ]} (function of ω) S(ω) 0 ω s /2 ω The general non-stationary case: Mean μ t = E x t Variance σ 2 t = E x t μ[t] 2 autocovariance (acf) γ t, t + τ power spectral density (psd) S(ω 1, ω 2 ) =F {γ t, t + τ } A time-frequency distribution is preferred S(ω,t) Time t (s) SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 7/23
2. The structure and the damage scenario 2. The structure and the damage scenario Information on the Composite Beams Manufacturing Several layers of woven and unidirectional fabric Processing based on one shot Resin Transfer Molding Sampling frequency fs = 4 654.5 Hz Sampling bandwidth 3 2 327.5 Hz Signal length N = 112 000 samples (24.06 s) Exciter Vibration Controller Signal conditioner Accelerometers Force sensor LDS Model V406 LDS COMET USB COM-200 PCB F482A20 8 chanels PCB ICP 352C22, Piezotronics Inc. PCB 288D01 impedance head Beam Dimensions Length: 600 mm Width: 65 mm Height: 65 mm Thickness: 3 mm Square hollow cross section SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 8/23
2. The structure and the damage scenario Two Structural states: Healthy & Damaged (Damage applied via a pendulum type impact hammer impact 15J) The two structural states are represented by a distinct ARX simulation model each. Model selection ARX (53,53) SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 9/23
3. The non-stationary random excitation signal 3. The non-stationary random excitation signal The non-stationary excitation is designed as: Zero-mean Gaussian Three lightly damped modes Two anti-resonant modes Signal realization Stationary white noise Time-Varying Filter Non-Stationary random excitation SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 10/23
3. The non-stationary random excitation signal The TV Filter is synthesized from the selected modes and is of the form: (Poulimenos & Fassois 2006) nb nc x t + b i t x t i = w t + c i t w t i, w t ~ NID(0, σ w 2 ) i=1 i=1 x t b i t, c i t : force excitation signal : Time-dependent AR/MA parameters n a =6, n b =4 w t : non-stationary, zero-mean, uncorrelated (innovations) signal with variance Time-dependent ARMA (TARMA) model for the non-stationary excitation σ w 2 Typical excitation realization SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 11/23
3. The non-stationary random excitation signal 2D TV-PSD The theoretical "frozen" TARMA-based TV-PSD of the synthesized non-stationary force excitation 3D TV-PSD SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 12/23
4. Assessing damage detectability 4. Assessing damage detectability In an output-only context damage detection may be based on the structural vibration response & specifically its frozen TV-PSD. (healthy or damaged) Estimate random structural response model Damage Detection Steps Obtain the model-based frozen TV-PSD Obtain the distance of the current TV-PSD to its healthy counterpart: Structure d t = S ω, t S 0 ω, t 2ω < threshold healthy else Structure current healthy damaged SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 13/23
4. Assessing damage detectability The random structural response model Non-stationary Recursive AR (RAR) model (Poulimenos & Fassois 2006) y t + na a i i=1 vibration response signal [t] y t i = e t, Time-dependent AR parameters zero-mean, uncorrelated (innovations) signal with TV variance e[t]~nid(0, σ e 2 [t]) Estimation Recursive Least Squares (RLS) with forgetting factor λ (Ljung 1999) θ t = θ t 1 + L t [y t φ Τ (t)θ t 1 ] L t = P t 1 φ(t) λ t + φ Τ (t)p t 1 φ(t) P t = P t 1 P t 1 φ t φτ t P t 1 λ t + φ Τ t P t 1 φ t /λ t Model-based Frozen TV-PSD S F ω, t σ e 2 [t] = 1 + n a i=1 1 a i [t] e jωt si 2 ω: frequency Ts: sampling period j: the imaginary unit : complex magnitude SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 14/23
5. Damage detectability: non-stationary vs stationary excitation 5. Damage detectability: non-stationary vs stationary excitation Non-stationary excitation Model & λ selection: Selected model: RAR(68) with λ=0.0998 Model order search Estimation find best BIC, RSS/SSS for n a =40,...,80 Forgetting factor λ=0.92:0.001:0.999 Recursive Least Squares (MATLAB rarx.m), initial parameter vector 0, initial covariance vector 10 8 I Selected model λ RSS/SSS (%) BIC SPP* RAR(68) 0.0998 9.604 8.385 144.92 *Samples Per Parameter SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 15/23
5. Damage detectability: non-stationary vs stationary excitation 2D TV-PSD Healthy state RAR(68)-based frozen TV-PSD 2D TV-PSD Damaged state SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 16/23
5. Damage detectability: non-stationary vs stationary excitation Model selection Stationary excitation Selected model: AR(62) Healthy vs Damaged PSD Estimation Model Least squares (MATLAB arx.m) AR(62) RSS/SSS (%) 29.7945 BIC -0.3672 SPP* 161.29 *Samples Per Parameter SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 17/23
5. Damage detectability: non-stationary vs stationary excitation Damage detectability: non-stationary vs stationary excitation d t = d S o ω, t, S ω, t = S o ω, t S ω, t 2ω (t is dropped in the stationary case) d(t) for experiment 1 d(t) for experiment 2 SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 18/23
5. Damage detectability: non-stationary vs stationary excitation d(t) for 50 experiments SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 19/23
5. Damage detectability: non-stationary vs stationary excitation Monte Carlo experiments for 3 damage levels (50 exps per level) Impact 15J Impact 10J Impact 5J SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 20/23
6. 4. Conclusions 6. Conclusions Q1: Natural random excitation often is non-stationary. Is this a potential benefit? Answer: yes, it could be. Q2: In a user-controlled environment should the excitation be chosen as non-stationary? Answer: As demonstrated, the use of non-stationary excitation may improve damage detectability. This study has provided a first indication on the potential of non-stationary excitation for improved damage detectability. More concrete results require further work. SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 21/23
Thank you for your attention! Acknowledgements The support of this study by the European Commission (FP7 Project 605549 on Global In Flight Health Monitoring Platform for Composite Aerostructures Based on Advanced Vibration Based Methods VIBRATION) is gratefully acknowledged. Thanks to all partners for their contributions. For more information please visit www.smsa.upatras.gr SMSA Lab (University of Patras) Damage Detectability Madrid, Spain, November 2014 22/23