Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 49, 3, pp. 727-756, Warsaw 2011 SCALAR AND VECTOR TIME SERIES METHODS FOR VIBRATION BASED DAMAGE DIAGNOSIS IN A SCALE AIRCRAFT SKELETON STRUCTURE Fotis P. Kopsaftopoulos Spilios D. Fassois University of Patras, Department of Mechanical and Aeronautical Engineering, Patras, Greece e-mail: fkopsaf@mech.upatras.gr; fassois@mech.upatras.gr A comparative assessment of several scalar and vector statistical time seriesmethods for vibration based Structural HealthMonitoring (SHM) is presented via their application to a laboratory scale aircraft skeleton structure in which different damage scenarios correspond to the loose- ning of different bolts. A concise overview of scalar and vectormethods, that is methods using scalar or vector signals, statistics, and correspon- ding models, is presented. The methods are further classified as non- parametric or parametric and response-only or excitation-response. The effectiveness of the methods for both damage detection and identifica- tion is assessed via various test cases corresponding to different damage scenarios. The results of the study reveal various facets of the methods and confirm the global damage diagnosis capability and the effectiveness of both scalar and vector statistical time series methods for SHM. Key words: Structural Health Monitoring (SHM), damage detection, damage identification Acronyms ARMA –AutoRegressiveMoving Average, ARX –AutoRegressive with eXo- genous excitation, BIC – Bayesian Information Criterion, FRF – Frequency Response Function, iid – identically independently distributed, LS – Least Squares, PE – Prediction Error, PSD – Power Spectral Density, RSS – Resi- dual Sum of Squares, SHM – Structural Health Monitoring, SPP – Samples Per Parameter, SPRT – Sequential Probability Ratio Test, SSS – Signal Sum of Squares. 728 F.P. Kopsaftopoulos, S.D. Fassois 1. Introduction Statistical time seriesmethods for damage detection and identification (locali- zation), collectively referred to as damage diagnosis, utilize randomexcitation and/or vibration response signals (time series) along with statistical model building and decision making tools for inferring the health state of a struc- ture (Structural Health Monitoring – SHM). They offer a number of advan- tages, including no requirement for physics based or finite element models, no requirement for complete modal models, effective treatment of uncerta- inties, and statistical decision making with specified performance characteri- stics (Fassois and Sakellariou, 2007, 2009). Thesemethods form an important, rapidly evolving category within the broader family of vibration based me- thods (Doebling et al., 1998; Sakellariou andFassois, 2008;Kopsaftopoulos and Fassois, 2010). Statistical time series methods for SHM are based on scalar or vector random (stochastic) vibration signals under healthy and potentially dama- ged structural states, identification of suitable (parametric or non–parametric) time series models describing the dynamics under each structural state, and extractionof a statistical characteristic quantity Qo characterizing the structu- ral state in each case (baseline phase). Damage diagnosis is then accomplished via statistical decision making consisting of comparing, in a statistical sense, the current characteristic quantity Qu with that of each potential state as de- termined in the baseline phase (inspection phase). For an extended overview of the basic principles and the main statistical time series methods for SHM, the interested reader is referred to Fassois and Sakellariou (2007, 2009). An experimental assessment of severalmethods is provided inKopsaftopoulos and Fassois (2010). Non-parametric time seriesmethods are those based on corresponding sca- lar or vector non-parametric time series representations such as spectral es- timates (Power Spectral Density, Frequency Response Function) (Fassois and Sakellariou, 2007, 2009), and have received limited attention in the literatu- re (Sakellariou et al., 2001; Liberatore and Carman, 2004; Hwang and Kim, 2004; Rizos et al., 2008; Kopsaftopoulos and Fassois, 2010). On the other hand, parametric time seriesmethods are those based on corresponding scalar or vector parametric time series representations such as the AutoRegressi- ve Moving Average (ARMA) models (Fassois and Sakellariou, 2007, 2009). This latter category has attracted significant attention recently, and their principles have been used in a number of studies (Sohn and Farrar, 2001; Sohn et al., 2001, 2003; Basseville et al., 2004; Sakellariou and Fassois, 2006; Scalar and vector time series methods for vibration... 729 Nair et al., 2006; Mattson and Pandit, 2006; Zheng and Mita, 2007; Car- den and Brownjohn, 2008; Gao and Lu, 2009; Kopsaftopoulos and Fassois, 2010). The goal of the present study is the comparative assessment of several scalar (univariate) and vector (multivariate) statistical time series methods for SHM via their application to an aircraft scale skeleton structure in which different damage scenarios correspond to the loosening of different bolts. The methods are further classified as non-parametric or parametric and response- only or excitation-response. Preliminary results by various methods may be found in our recent papers (Kopsaftopoulos et al., 2010; Kopsaftopoulos and Fassois, 2011). It should be noted that the structure has been used in the past for thedevelopment of novel scalar (univariate)methods for precise damage lo- calization andmagnitude (size) estimationusingdifferent (simulated)damages consisting of small masses attached to the structure (Sakellariou and Fassois, 2008; Kopsaftopoulos and Fassois, 2007). Due to their ability to address the precise localization and magnitude estimation problems, these methods are generally more complex. As the focus of the present study is more on damage detection and identification (the latter in the sense of estimating the damage scenario from a given pool of potential scenarios), only simpler methods are utilized. More specifically, four scalar methods, namely the Power Spectral Density (PSD), Frequency Response Function (FRF), model residual variance, and Sequential Probability Ratio Test (SPRT) basedmethod are employed, along with two vector methods, namely the model parameter based and residual likelihood function based method. A number of test cases (experiments) are considered, each one corresponding to a specific damage scenario (loosening of one or more bolts connecting various structural elements). Themain issues addressed in the study are: (a) Assessment of the methods in terms of their damage detection capability under various damage scenarios and different vibration measurement locations (classified as either “local” or “remote” to damage location). (b) Comparison of the performance characteristics of scalar and vector sta- tistical time series methods with respect to effective damage diagnosis: false alarm,missed damage and damagemisclassification rates are inve- stigated. (c) Assessment of the ability of the methods to accurately identify (classify) the damage type through “local” or “remote” sensors. 730 F.P. Kopsaftopoulos, S.D. Fassois 2. The structure and the experimental set-up 2.1. The structure The scale aircraft skeleton structure used in the experiments was designed byONERA (France) in conjunctionwith the Structures andMaterials Action Group SM-AG19 of the Group for Aeronautical Research and Technology in Europe (GARTEUR) (Degener and Hermes, 1996; Balmes andWright, 1997) and manufactured at the University of Patras (Fig.1). It represents a typical aircraft skeleton design and consists of six solid beams with rectangular cross sections representing the fuselage (1500× 150× 50mm), the wing (2000× ×100× 10mm), the horizontal (300× 100× 10mm) and vertical stabilizers (400×100×10mm) and the right and left wing-tips (400×100×10mm). All parts aremade of standard aluminumand are jointed together via steel plates and bolts. The total mass of the structure is approximately 50kg. 2.2. The damage scenarios and the experiments Damage detection and identification is based on vibration testing of the structure, which is suspended through a set of bungee cords and hooks from a long rigid beam sustained by two heavy-type stands (Fig.1). The suspension is designed in a way as to exhibit a pendulum rigid body mode below the frequency range of interest, as the boundary conditions are free-free. Fig. 1. The scale aircraft skeleton structure and the experimental set-up: The force excitation (Point X), vibrationmeasurement locations (Points Y1-Y4), and bolts connecting various elements of the structure The excitation is broadband random stationary Gaussian force applied vertically at the right wing-tip (Point X, Fig.1) through an electromechanical shaker (MBDynamicsModal 50A,max load 225N). The actual force exerted Scalar and vector time series methods for vibration... 731 on the structure is measured via an impedance head (PCBM288D01, sensiti- vity 98.41mV/lb),while the resultingvertical acceleration responses atPoints Y1,Y2, Y3 andY4 (Fig.1) aremeasured via lightweight accelerometers (PCB 352A10 miniature ICP accelerometers, 0.7g, frequency range 0.003-10kHz, sensitivity ∼ 1.052mV/m/s2). The force and acceleration signals are driven through a conditioning charge amplifier (PCB 481A02) into the data acquisi- tion system based on two SigLab 20-42 measurement modules (each module featuring four 20-bit simultaneously sampled A/D channels, two 16-bit D/A channels, and analog anti-aliasing filters). The damage scenarios considered correspond to the loosening of various bolts at different joints of the structure (Fig.1). Six distinct scenarios (types) are considered and summarized in Table 1. The assessment of the presen- ted statistical time series methods with respect to the damage detection and identification subproblems is based on 60 experiments for the healthy and 40 experiments for each considered damage state of the structure (damage ty- pesA,B,. . . ,F – see Table 1) – each experiment corresponding to a single test case. Moreover, four vibration measurement locations (Fig.1, Points Y1-Y4) are employed in order to determine the ability of the considered methods in treating damage diagnosis using single or multiple vibration response signals. The frequency range of interest is selected as 4-200Hz with the lower limit set in order to avoid instrument dynamics and rigid bodymodes. Each signal is digitized at fs = 512Hz and is subsequently sample mean corrected and normalized by its sample standard deviation (Table 1). Table 1.The damage scenarios and experimental details Structural state No of inspection Description experiments (test cases) Healthy – 60 Damage A loosening of bolts A1, A4, Z1, Z2 40 Damage B loosening of bolts D1, D2, D3 40 Damage C loosening of bolts K1 40 Damage D loosening of bolts D2, D3 40 Damage E loosening of bolts D3 40 Damage F loosening of bolts K1, K2 40 Sampling frequency: fs =512Hz, Signal bandwidth: 4-200Hz Signal length in samples (s): Non-parametric methods: N =46080 (90s) Parametric methods: N =15000 (29s) 732 F.P. Kopsaftopoulos, S.D. Fassois A single healthy data set is used for establishing the baseline (reference) set, while 60 healthy and 240 damage sets (six damage types with 40 expe- riments each) are used as inspection data sets. For damage identification, a single data set for each damage structural state (damage types A,B,. . . ,F) is used for establishing the baseline (reference) set, while the same 240 sets are considered as inspection data sets (each corresponding to the test case in which the actual structural state is considered unknown). The time series models are estimated and the corresponding estimates of the characteristic quantity Q are extracted (Q̂A,Q̂B, . . . , Q̂F in the baseline phase; Q̂u in the inspection phase). Damage identification is presently based on successive bi- nary hypothesis tests – as opposed to multiple hypothesis tests – and should be thus considered as preliminary (Fassois and Sakellariou, 2009). 3. Structural dynamics of the healthy structure 3.1. Non-parametric identification Non-parametric identification of the structural dynamics is based on N = = 46080 (≈ 90s) sample-long excitation-response signals obtained from four vibration measurement locations on the structure (see Fig.1). An L= 2048 sample-long Hamming data windowwith zero overlap is used (number of seg- ments K =22) for PSD (MATLAB function pwelch.m) and FRF (MATLAB function tfestimate.m) Welch based estimation (see Table 2). Table 2.Non-parametric estimation details Data length N =46080 samples (≈ 90s) Method Welch Segment length L=2048 samples Non-overlapping segments K =22 segments Window type Hamming Frequency resolution ∆f =0.355Hz The obtainedFRFmagnitude estimates for the healthy anddamage states of the structure for the Point X - Point Y2 transfer function are depicted in Fig.2. As it may be observed, the FRFmagnitude curves are quite similar in the 4-60Hz range; notice that this range includes the first five modes of the structure. Significant differences between the healthy and damage type A, C and F magnitude curves are observed in the range of 60-150Hz, where the Scalar and vector time series methods for vibration... 733 next fourmodes are included. Finally, in the range of 150-200Hz another two modes are present and discrepancies aremore evident for damage types A, B, C and F. Notice that the FRF magnitude curves for damage types D and E are very similar to those of the healthy structure. Fig. 2. Non-parametricWelch-based Frequency Response Function (FRF) magnitude estimates for the healthy and damaged structural states (Point X - Point Y2 transfer function) 3.2. Parametric identification Parametric identification of the structural dynamics is based on N = =15000 (≈ 29s) sample-long excitation and vibration response signals used in theestimation ofVectorAutoRegressivewith eXogenous excitation (VARX) models (MATLAB function arx.m). Themodeling strategy consists of the suc- cessive fitting of VARX(na,nb) models (with na, nb designating the AR and Xorders, respectively; na=nb=n is currently used) until a candidatemodel is selected. Model parameter estimation is achieved by minimizing a quadra- tic Prediction Error (PE) criterion (trace of the residual covariance matrix) leading to a Least Squares (LS) estimator (Fassois, 2001; Ljung, 1999, p.206). Model order selection, which is crucial for successful identification, may be based on a combination of tools including the Bayesian Information Criterion (BIC) (Fig.3), which is a statistical criterion that penalizes model comple- xity (order) as a counteraction to a decreasing model fit criterion (Fassois, 2001; Ljung, 1999, pp.505-507) and the use of “stabilization diagrams” which depict the estimated modal parameters (usually frequencies) as a function 734 F.P. Kopsaftopoulos, S.D. Fassois of the increasing model order (Fassois, 2001). BIC minimization is achieved for the model order n = 80 (Fig.3), thus a 4-variate VARX(80,80) model is selected as adequate for the residual variance, model parameter, and likeliho- od function based methods. The identified VARX(80,80) representation has d=1604 parameters, yielding a Sample Per Parameter (SPP) ratio equal to 37.4 (N× (no of outputs)/d). Fig. 3. Bayesian Information Criterion (BIC) for VARX(n,n) type parametric models in the healthy case It should be noted that the complete 4-variate VARX(80,80) model is em- ployed in conjunction with vector methods in Section 5. Yet, scalar parts of this model corresponding to excitation – single response are used in conjunc- tion with scalar methods in Section 4. This is presently done for purposes of simplicity and it is facilitated by the fact that only a single (scalar) excitation is present. 4. Scalar time series methods and their application Scalar statistical time series methods for SHM employ scalar (univariate) mo- dels and corresponding statistics. In this Section, two non-parametric scalar methods, namely a Power Spectral Density (PSD) based method and a Fre- quency Response Function (FRF) based method, and two parametric scalar methods, namely a residual variance basedmethod andSequential Probability RatioTest (SPRT)basedmethod, arebrieflypresentedand thenapplied to the scale aircraft skeleton structure. Theirmain characteristics are summarized in Table 3. 4.1. The Power Spectral Density (PSD) based method Damage detection and identification is in this case tackled via characteristic changes in the Power Spectral Density (PSD) S(ω) of themeasured vibration S c a l a r a n d v e c t o r t im e s e r ie s m e t h o d s f o r v ib r a t io n ... 7 3 5 Table 3. Characteristics of the employed statistical time series methods for SHM Method Principle Test Statistic Type PSD based Su(ω) ? =So(ω) F = Ŝo(ω)/Ŝu(ω)∼F(2K,2K) scalar FRF based δ|H(jω)|= |Ho(jω)|− |Hu(jω)| ? =0 Z = |δ|Ĥ(jω)||/ √ 2σ̂2H(ω)∼N(0,1) scalar Residual variance σ2ou ? ¬σ2oo F = σ̂2ou/σ̂2oo ∼F(N,N −d) scalar SPRT based σou ? ¬σo or σou ? ­σ1 L(n)=n log σoσ1 + σ2 1 −σ2 o 2σ2 o σ2 1 ∑n t=1e 2[t] scalar Model parameter δθ=θo−θu ? =0 χ2θ = δθ̂ ⊤ (2P̂θ) −1δθ̂∼χ2(d) vector Residual likelihood θo ? =θu ∑N t=1(e ⊤ ou[t,θo] ·Σo ·eou[t,θo])¬ l vector Explanation of Symbols: S(ω): Power Spectral Density (PSD) function; |H(jω)|: Frequency Response Function (FRF)magnitude σ2 H (ω)= var[|Ĥo(jω)|]; θ: model parameter vector; d: parameter vector dimensionality; Pθ: covariance of θ σ2 oo : variance of residual signal obtained by driving the healthy structure signals through the healthy model σ2 ou : variance of residual signal obtained by driving the current structure signals through the healthymodel eou[t,θo]: vector residual sequence obtained by driving the current structure signals through the healthymodel σo,σ1: user defined values for the residual standard deviation under healthy and damage states, respectively e: k-variate residual sequence; Σ: residual covariancematrix; l: user defined threshold The subscripts “o” and “u” designate the healthy and current (unknown) structural states, respectively 736 F.P. Kopsaftopoulos, S.D. Fassois response signals (non-parametricmethod) as the excitation is assumed unava- ilable (response-only method). Thus, the characteristic quantity is Q= S(ω) (ω designates frequency – see Table 3). Damage detection is based on confir- mation of statistically significant deviations (from thenominal/healthy) in the current structure’s PSD function at some frequency (Fassois and Sakellariou, 2007, 2009). Damage identificationmay be achieved by performing hypothesis testing similar to the above separately for damages of each potential type. It should be noted that response signal scaling is important in order to properly account for potentially different excitation levels. Application Results. Typical non-parametric damage detection results using the vibration measurement location of Point Y1 are presented in Fig.4. Evidently, correct detection at the α=10−4 risk level is obtained in each case as the test statistic is shownnot to exceed the critical point (dashed horizontal Fig. 4. PSD basedmethod: Representative damage detection results (sensor Y1) at the α=10−4 risk level. The actual structural state is shown inside each plot lines) in the healthy test case, while it exceeds it in each damage test case. Observe that damage types A, B and C (see Fig.1 and Table 1) are more easily detectable (note the logarithmic scale on the vertical axis of Fig.4), while damage types D and E are harder to detect. This is in agreement with Scalar and vector time series methods for vibration... 737 the remarks made in Subsection 3.1. Furthermore, notice that the frequency bandwidth of [150-170]Hz is more sensitive to damage. This is also in agre- ement with the remarks made in Subsection 3.1 and seems to be due to the fact that the two natural frequencies in this bandwidth are more sensitive to the considered damage scenarios (see Fig.2). Representative damage identification results at the α = 10−4 risk level and using (as an example) the vibration measurement location at Point Y3 are presented in Fig.5, with the actual damage being of type A. The test statistic does not exceed the critical point when the Damage A hypothesis is considered, while it exceeds it in all remaining cases. This correctly identifies damage type A as the current underlying damage. Fig. 5. PSD basedmethod: Representative damage identification results (sensor Y3) at the α=10−4 risk level with the actual damage being of type A. Each considered damage hypothesis is shown inside each plot Summary damage detection and identification results for each vibration measurement location (Fig.1) are presented in Table 4. The PSD based me- thod achieves accurate damage detection as no false alarms are exhibited, while the number of missed damage cases is zero for all considered damaged structural states. Themethod is also capable of identifying the actual damage type, as zero damagemisclassification errors are reported for damage typesA, 738 F.P. Kopsaftopoulos, S.D. Fassois C,D and F, while it exhibits somemisclassification errors for damage type E. Themisclassification problem is more intense for damage type B when either the Y3 or the Y4 vibration measurement location is used (Table 4). Table 4. Scalar methods: damage detection and identification summary re- sults Damage Detection Method False Missed damage alarms A B C D E F PSD based 0/0/0/0 0/0/0/0 0/0/0/0 0/0/0/0 0/0/0/0 0/0/0/0 0/0/0/0 FRF based 1/0/0/35 0/0/0/0 0/0/0/0 0/0/0/0 0/0/1/0 0/1/0/0 0/0/0/0 Res. varian.† 0/0/0/0 0/0/0/0 0/0/0/0 0/0/0/0 0/0/0/0 0/0/0/0 0/0/0/0 SPRT based 0/0/0/0 0/0/0/0 0/0/0/0 0/0/0/0 0/0/0/0 0/0/0/0 0/0/0/0 False alarms for response points Y1/Y2/Y3/Y4 out of 60 test cases per point. Missed damages for response points Y1/Y2/Y3/Y4 out of 40 test cases per point; †adjusted α. Damage Identification Method Damagemisclassification A B C D E F PSD based 0/0/0/0 0/0/21/21 0/0/0/0 0/0/0/0 0/0/1/2 0/0/0/0 FRF based 0/0/0/0 10/4/7/8 6/10/2/0 5/22/9/8 2/9/5/2 0/3/1/0 Res. variance† 0/0/0/0 0/0/0/0 0/0/0/0 0/0/0/0 0/0/0/0 0/0/0/0 SPRT based 0/0/0/0 0/0/0/0 0/0/0/0 0/0/0/0 0/0/0/0 0/0/0/0 Damage misclassification for response points Y1/Y2/Y3/Y4 out of 40 test cases per point; †adjusted α. 4.2. The Frequency Response Function (FRF) based method Thismethod is similar to the previous, but requires the availability of both the excitation and response signals (excitation-response method) and uses the FRFmagnitude |H(jω)| (with j= √ −1) as its characteristic quantity (non- parametric method), thus Q = |H(jω)| (see Table 3). The main idea is the comparison of the FRF magnitude Hu(jω) of the current state of the struc- ture to that of the healthy structure |Ho(jω)|. Damage detection is based on confirmation of statistically significant deviations (from the nominal/healthy) in the current FRF of the structure at one or more frequencies through a hy- pothesis testing problem (for each ω) (Fassois and Sakellariou, 2007, 2009). Damage identification may be achieved by performing a hypothesis testing similar to the above separately for damages of each potential type. Scalar and vector time series methods for vibration... 739 Application Results. Figure 6 presents typical non-parametric damage detection results via the FRF basedmethod using the vibrationmeasurement location of Point Y4. Evidently, correct detection at the α=10−6 risk level is achieved in each case, as the test statistic is shown not to exceed the critical points (dashed horizontal lines) in the healthy case, while it exceeds them in all damage cases. Again, damage types A, B andC aremore easily detectable (hence more severe), while damage types D and E are harder to detect. Fig. 6. FRF basedmethod: Representative damage detection results (sensor Y4) at the α=10−6 risk level. The actual structural state is shown inside each plot Representative damage identification results at the α = 10−6 risk level using (as an example) the vibration measurement location of Point Y2 are presented in Fig.7, with the actual damage being of type E. The test statistic doesnot exceed the critical pointwhentheDamageEhypothesis is considered, while it exceeds it in all other cases. This correctly identifies damage type E as the current underlying damage. Summary damage detection and identification results for each vibration measurement location (Fig.1) are presented in Table 4. The FRF based me- thod achieves effective damage detection as no false alarms ormissed damages are reported (Table 4). On the other hand, the method exhibits decreased accuracy in damage identification as significant numbers of damagemisclassi- fication errors are reported for damage types B and D (Table 4). 740 F.P. Kopsaftopoulos, S.D. Fassois Fig. 7. FRF basedmethod: Representative damage identification results (sensor Y2) at the α=10−6 risk level with the actual damage being of type E. Each considered damage hypothesis is shown inside each plot 4.3. The residual variance based method In thismethod (excitation-response case) the characteristic quantity is the residual variance. The main idea is based on the fact that the model (para- metric method)matching the current state of the structure should generate a residual sequence characterized byminimal variance (Fassois and Sakellariou, 2007, 2009). Damage detection is based on the fact that the residual series obtained by driving the current signal(s) through themodel corresponding to the nominal (healthy) structure has a variance that is minimal if and only if the current structure is healthy (Fassois and Sakellariou, 2007, 2009). This method uses classical tests on the residuals and offers simplicity as there is no need for model estimation in the inspection phase. The main characteristics of the method are shown in Table 3. Application Results. The residual variance based method employs an excitation – single response submodel obtained from the complete 4-variate VARX(80,80)models identified in the baseline phase, aswell as the correspon- ding residual series obtained by driving the current (structure in an unknown Scalar and vector time series methods for vibration... 741 state) excitation and single response signals through the same submodel (in- spection phase).Damage detection and identification is achieved via statistical comparison of the two residual variances. Representativedamagedetection and identification results obtainedvia the residual variance based method (when the vibration measurement location of PointY2 is used) are presented inFigs.8 and9, respectively. Evidently, correct detection (Fig.8) is obtained in each considered case as the test statistic is shown not to exceed the critical point in the healthy case, while it exceeds it in each damage test case. Moreover, Fig.9 demonstrates the ability of the method to correctly identify theactual damage type– in this case thevibration measurement location of Point Y3 is used. Fig. 8. Residual variance basedmethod: Representative damage detection results (sensor Y2; healthy – 60 experiments; damaged – 200 experiments). A damage is detected if the test statistic exceeds the critical point (dashed horizontal line) Summary damage detection and identification results for each vibration measurement location (Fig.1) are presented in Table 4. The method achie- ves effective damage detection and identification as no false alarms, missed damage, or damage misclassification errors are observed. 4.4. The Sequential Probability Ratio Test (SPRT) based method Thismethodemploys theSequentialProbabilityRatioTest (SPRT)(Wald, 2004; Ghosh and Sen, 1991) in order to detect a change in the standard devia- 742 F.P. Kopsaftopoulos, S.D. Fassois Fig. 9. Residual variance basedmethod: Representative damage identification results (sensor Y3; 240 experiments) with the actual damage being of type D. A damage is identified as type D if the test statistic is below the critical point (dashed horizontal line) tion σ of the model scalar residual sequence (e[t] ∼ N(0,σ2), t = 1, . . . ,N) (parametric method). An SPRT of strength (α,β), with α, β designating the type I (false alarm) and II (missed damage) error probabilities, respectively, is used for the following hypothesis testing problem: Ho : σou ¬σo (null hypothesis – healthy structure) H1 : σou ­σ1 (alternative hypothesis – damaged structure) (4.1) with σoudesignating the standarddeviationof a scalar residual signal obtained by driving the current excitation and response signals through the healthy structuralmodel, and σo, σ1 user defined values. The basis of the SPRT is the logarithm of the likelihood ratio function based on n samples L(n)= log f(e[1], . . . ,e[n]|H1) f(e[1], . . . ,e[n]|Ho) = n∑ t=1 log f(e[t]|H1) f(e[t]|Ho) (4.2) =n log σo σ1 + σ21 −σ2o 2σ2oσ 2 1 n∑ t=1 e2[t] with L(n) designating the decision parameter of the method and f(e[t]|Hi) Scalar and vector time series methods for vibration... 743 the probability density function (normal distribution) of the residual sequence under hypothesis Hi (i=0,1). The following test is then constructed at the (α,β) risk levels: L(n)¬B =⇒ Ho is accepted (healthy structure) L(n)­A =⇒ H1 is accepted (damaged structure) B