Vibration-based damage precise localization in 3D ... - CiteSeerX

Preprint submitted to Structural Health Monitoring 17 December 2014

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Vibration-based damage precise localization in 3D structures: Single vs multiple response measurements Christos S Sakaris, John S Sakellariou and Spilios D Fassois

Abstract The goal of this study is the vibration based damage precise localization on 3D structures through the vector version of an advanced Functional Model Based Method. This version, which constitutes an extension of the original scalar method, is equipped with a Functional Model form that may simultaneously exploit multiple structural responses and incorporates the damage coordinates within its parameters through an appropriate operating parameter vector. Based on this model form, precise estimation of the damage coordinates is achieved within a nonlinear optimization framework with constraints representing the 3D structural topology, and corresponding damage confidence intervals are constructed. The effectiveness of the method is assessed through damage precise localization for numerous damage scenarios in a 3D truss structure, as well as through detailed comparisons with the previous, scalar method, demonstrating significant reduction in the localization error. Keywords Vibration based Structural Health Monitoring, statistical time series methods, damage precise localization, multiple signals, vector functional models

Introduction Multivariate models that use the information from more than one vibration signals for the modelling of the structural dynamics have been widely used in vibration based damage detection and localization methods [1],[2, pp. 45-224],[3–8],[9, pp. 361-375], [10, 11]. More specifically, these methods may be classified in two main categories [1],[2, p. 51], [9, pp. 9-10]: (i) those which are founded on analytical, physics-based, models of the structure and, (ii) those that utilize structural models developed exclusively through measurements from the structure, known also as data-based methods. Stochastic Mechanical Systems & Automation (SMSA) Laboratory, Department of Mechanical & Aeronautical Engineering, University of Patras, Patras, Greece Corresponding author: Spilios D Fassois, Stochastic Mechanical Systems & Automation (SMSA) Laboratory, Department of Mechanical & Aeronautical Engineering, University of Patras, GR 26504, Patras, Greece. Email: [email protected] Url: http://www.smsa.upatras.gr

Structural Health Monitoring Vol. 14(3), pp. 300-314, 2015. http://dx.doi.org/10.1177/1475921714568407

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Finite Element Model (FEM) updating based methods belong to the first category of methods and are commonly used for damage detection and precise localization. Damage precise localization on a continuous 3D structural topology is herein defined as the determination of the damage precise coordinates and is different from approximate localization, where the damage is roughly localized as belonging to a specific number of pre–selected regions (usually structural elements). FEM and, broader, analytical model based methods may potentially achieve damage precise localization but in practice such a task becomes cumbersome. More especially, for structures with relatively complicated geometries including several elements, bolted joints and other components that lead to a complex dynamic behavior, analytical modelling may reveal considerable discrepancies from reality and their tuning/training necessitates several prerequisites. For instance a FEM updating based method may lead to damage precise localization if the structure is modelled through a detailed, large-scale, model, the damage is properly incorporated into the FEM, and the updating parameters (such as the elements cross sectional area, the moment of inertia, the Young modulus and so on) are quite sensitive to the selected response(s) [12]. However these increase significantly the difficulty in real applications and require information from a high number of sensors [13–15] which may be impracticable is some cases. On the other hand less detailed, smaller-scale, models such as Neural Networks, state space models and Vector AutoRegressive with eXogenous excitation (VARX) type models are commonly used by data-based methods including the statistical time series and pattern recognition type methods [6, 16–23]. The performance of these methods is greatly affected by the selected model structure, based on which damage detection and localization is achieved; in most cases only approximate localization is performed. In such an approach damage is roughly localized as belonging to a single out of a specific number of preselected regions, thus reflecting a damage classification rather than damage localization problem [6, 16–24]. Furthermore, this treatment is much simpler than damage precise localization where the estimation of the precise damage coordinates on the investigated structural topology is sought. The problem of damage precise localization has been initially investigated within a statistical time series framework through a Functional Model Based Method (FMBM) using scalar (single response) Functional Models (FMs) [25] accounting for 1D structural elements (also see [26] for magnitude estimation), while in a recent preliminary study Vector dependent Functionally PooledARX (VFP-ARX) models have been employed for damage precise localization in 2D and 3D structural elements [27]. This, most recent, type of advanced (scalar) model structure offers a global and compact representation of the complete continuous topology of a 3D structure under damage at different locations. Although the performance of the FMBM equipped with scalar VFPARX models has been found to be particularly promising, in certain cases the localization error is significant [27]. Thus the goal of the present study is twofold: (i) the extension of the advanced Functional Model Based Method (FMBM) to its vector form in order to exploit multiple structural responses for achieving higher precision (compared to the scalar form) in damage localization upon complex structural topologies under realistic operating conditions of random natural excitation of low and limited bandwidth, and, (ii) its comparison with the previously developed scalar version. This study includes all details about the vector version of the FMBM, as well as an exhaustive experimental assessment and comparison with the scalar method, while a shorter version is recently presented in a conference paper [28]. The method’s extension necessitates the development of a suitable optimization framework within which the topology of 3D structures consisting of 1D elements is described through analyti-

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Figure 1. (a) The truss structure and the experimental setup: The force excitation (Point X) and the vibration acceleration measurement positions (Points Y1 & Y2); (b) Schematic diagram of the truss with the considered 3D Cartesian coordinate system with origin at node 8. The 16 nodes are indicated by circles (dimensions in cm); (c,d,e) The electromechanical exciter, the impedance head, and the lightweight accelerometers; (f) Zoom of an indicative node with bolts (node 13).

cal geometrical functions as well as multivariate FMs, that is VFP-Vector (V) ARX models. These models constitute extensions of the previously used, simpler (scalar), VFP-ARX models [27], and are capable of representing the structural dynamics as viewed from multiple measurement positions, while also accounting for their interrelations. It is worth noting that a simpler form (for 1D structures) of multivariate FMs has been used for structural identification [29] and damage detection under different temperatures [29, 30]. The vector version of the FMBM is assessed via laboratory experiments with a 3D aluminum truss structure (see Figure 1) that consists of 16 nodes jointed together with 74 bolts. Numerous damage scenarios are employed, each corresponding to the loosening of a single bolt at a different location, and damages shall be detected and precisely localized in terms of their Cartesian coordinates.

The structure, the damages and the experimental set-up The structure The truss structure is suspended in a free-free mode through a set of elastic cords and hooks from long rigid beams sustained by two heavy-type stands. It consists of 26 rods with rectangular cross

4 Table 1. Experimental details.

Structural State Healthy Damaged

Description Number of experiments no damage 10 (2 in the baseline phase) loosening of a bolt 390 (20 in the baseline phase) at a single location Sampling frequency: fs = 128 Hz, Signal Bandwidth: [3 − 59] Hz Signal length in samples (s): N = 9758 samples (76.23 s) Bandpass filter: Chebyshev Type II, 12th order (Matlab function: filter.m) sections (1.5 × 1.5 cm) jointed together via steel elbow plates and 74 bolts (see Figures 1(a), (b)). All parts are constructed from standard aluminum with the overall dimensions being 140 × 80 × 70 cm. A schematic diagram and photos of the truss and its suspension are shown in Figure 1. More details about the truss structure are provided in [27] where the scalar FMBM is used for damage precise localization based on a single acceleration response.

The damages Each considered damage corresponds to the complete loosening of a single bolt at a node (see for instance node 13 in Figures 1(b), (f)) with the total number of investigated damages being 390. The bolts are “on-off” type, which means that there is no intermediate level of tightening and they are distributed at 16 nodes (Figure 1(b)) with each node having certain coordinates defined based on the Cartesian origin (x = 0, y = 0, z = 0) taken at node 8 (see Figure 1(b)). Thus, a damage is designated as Fx,y,z , with x, y, z the coordinates along the corresponding Cartesian axes. It is noted that due to the small dimensions of their cross section, each rod of the truss is assumed to have negligible thickness.

The experimental set-up Damage detection and localization are based on a single excitation and two vibration acceleration response signals. The excitation is a zero-mean random Gaussian force applied horizontally along the y direction at Point X (Figures 1(a), (c)) via an electromechanical shaker equipped with a stinger, and measured via an impedance head (PCB 288D01, sensitivity 98.41 mV/lb; see Figure 1(c)). The acceleration response signals are measured via two lightweight accelerometers (PCB ICP 352C22, 0.5 gr, frequency range 0.003 - 10 kHz, sensitivity ∼ 1.052 mV/m/s2 ) along y direction at points Y1 and Y2 (see Figures 1(a), (d), (e)). The excitation-response points are selected so as the resonant frequencies being distinct in the Frequency Response Function (FRF) as well as sensitive to the effects of the known damage states which are used for the method’s training as described in the following. 10 experiments are carried out, initially with the healthy structure and subsequently 390 experiments with the structure under damage at different locations. Two experiments from the healthy state of the structure and 20 experiments with the damaged structure are used in the “training” (baseline) phase of the damage detection and localization methodology. The remaining excitation-response signals are solely used in the inspection phase (also see section “Experimental Results”). All signals are sample mean subtracted, scaled by the sample standard deviation, and filtered within the [3 − 59] Hz bandwidth. The experimental details are summarized in Table 1.

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The damage detection and localization methodology Damage detection is achieved based on the multivariate version of the Likelihood function based method [6, 10, 11]. Once a damage is detected the vector version of the Functional Model Based Method equipped with VFP-VARX models and an appropriate optimization framework is employed for damage precise localization. Both methods operate within the baseline and inspection phases as presented in the following subsections. As is common to all vibration methods, the boundary conditions and the excitation bandwidth are assumed to remain constant in the two phases. Furthermore, although the methodology is presented for the measurable force excitation case, the non–measurable case may be also accommodated via proper modifications.

Baseline phase Damage detection. A single experiment of nx-excitation ny-response signals from the healthy structure are used for the identification of a typical VARX model [31, pp. 91-93], [32] based on the successive fitting of VARX(na, nb) models (with na, nb designating the AR and X orders, respectively; herein na = nb = n) until a suitable model is selected (bold-face upper/lower case symbols designate matrix/column-vector quantities, respectively; T designates transposition): y[t] +

na X

Ai · y[t − i] =

i=1 i α1,1  . Ai =  .. i αny,1



nb X

B i · x[t − i] + e[t],

e[t] ∼ iid N 0, Σ)

(1a)

i=0

 i . . . α1,ny ..  .. , . .  i . . . αny,ny [ny×ny]

bi1,1  . B i =  .. biny,1 

 bi1,nx ..  .  i bny,nx [ny×nx]

... .. . ...

(1b)

T

with t = 1, . . . , N designating the normalized discrete time, x[t] = [x1 [t] . . . xnx [t]][nx×1] the nxT

excitation signals, y[t] = [y1 [t] . . . yny [t]][ny×1] the ny-response signals, Ai and B i the AR and X parameter matrices, respectively, e[t] = [e1 [t] . . . eny [t]]T[ny×1] the vector that includes the model residual (one-step-ahead prediction) with covariance matrix Σ = E{e[t]eT [t]} and E{e[t]e[r]} = 0 for r 6= t, where E{·} designates statistical expectation. iid stands for identically independently distributed and N (·, ·) designates Gaussian distribution with the indicated mean and variance. Model parameter estimation is based on Ordinary Least Squares (OLS) [31, pp. 203-207], model order selection is based on the Bayesian Information Criterion (BIC) [33, pp. 146–157], while model validation is accomplished based on the model residuals which for an accurate model shall be uncorrelated and uncorrelated with the excitation [33, pp. 157–174]. . The model parameter vector θ = vec([A A ... A .. B B ... B ]T ), with dimension [(na ny 2 o

1

2

na

0

1

nb

+(nb + 1)nx ny) × 1] (vec(·) operator transforms a matrix into a vector by stacking its columns) and the residual covariance matrix Σo which correspond to the healthy structure are initially estimated for their use within the statistical decision making framework of the Likelihood function based method [6, 10, 11] for damage detection. It is noted that one (as in the present study) or more experiments with the healthy structure are used in this phase for the determination of the method’s critical point [10, 11] based on which damage detection is achieved in the inspection

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phase. Damage localization. A VFP-VARX model capable of representing the investigated 3D structural topology under damage is identified in this phase of the methodology based on a number of experiments from different damage locations on the structure. A total number of M = M1 ×M2 ×M3 experiments are performed for a sample of potential damage locations that cover the considered 3D continuous structural topology, with each experiment being characterized by a specific damage location with coordinates xl , ym , zn on the Cartesian axes while the complete series of experiments covers [xmin , xmax ], [ymin , ymax ] and [zmin , zmax ] via the discretizations: xl ∈ {x1 , x2 , . . . , xM1 },

ym ∈ {y1 , y2 , . . . , yM2 },

zn ∈ {z1 , z2 , . . . , zM3 }

(2)

with each subscript designating the discretization index. Thus for a single damage located at a specific point on the 3D structural topology the following operating parameter vector k is defined: k = [xl ym zn ]T

⇐⇒ kl,m,n with l = 1, . . . , M1 ,

m = 1, . . . , M2 ,

n = 1, . . . , M3

(3)

and the complete set of baseline experiments yields a pool of M sets of nx-excitation and nyresponse signals, each of length N : xk [t] = [x1,k [t] . . . xnx,k [t]]T[nx×1] , with t = 1, . . . , N,

y k [t] = [y1,k [t] . . . yny,k [t]]T[ny×1]

xl ∈ {x1 , . . . , xM1 },

ym ∈ {y1 , . . . , yM2 },

(4) zn ∈ {z1 , . . . , zM3 }

It should be noted that in its present form, the damage localization method assumes a single type and single magnitude damages. This is in line with the structure employed, as damages correspond to complete bolt loosening (the only possible damage magnitude as the bolts are of the “on–off” type). Various types and magnitudes of single or multiple damages could be also accommodated via proper modifications at the expense of more complex VFP–VARX models (see [25, 26]). The availability of vibration signals from the healthy structure as well as from that under various damage types/locations/magnitudes is a general requirement for damage localization via data– based methods (which operate without a physics-based model of the structure in the inspection phase). For cases where such signals may not be available from the actual structure, they should be obtained either from a high fidelity finite element model or a good laboratory–scale prototypes. Thus based on the data set of equation (4), a mathematical representation of the continuous 3D structural topology under any potential damage location is obtained through the VFP-VARX (na, nb)p model of the form: y k [t] +

na X

Ai (k) · y k [t − i] =

i=1

p X j=1

B i (k) · xk [t − i] + ek [t]

(5a)

i=0


ek [t] ∼ iid N 0, Σe (k) Ai (k) =

nb X

Ai,j · Gj (k),

k ∈ R3

B i (k) =

(5b) p X j=1

B i,j · Gj (k)

(5c)

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i,j α1,1  =  ... i,j αny,1



Ai,j

 i,j . . . α1,ny  .. .. ,  . . i,j . . . αny,ny [ny×ny]

bi,j 1,1  =  ... bi,j ny,1 

B i,j

 bi,j 1,nx  ..  . i,j bny,nx [ny×nx]

... .. . ...

 E ekl,m,n [t] · ekq,r,h [t − τ ] = Σe (kl,m,n , kq,r,h ) · δ[τ ]

(5d)

(5e)

where ek [t] = [e1,k [t] . . . eny,k [t]]T[ny×1] is the model residual vector that is zero-mean, white (serially uncorrelated), with covariance matrix Σe (k) = E{ek [t]eTk [t]} of dimension [ny × ny]), potentially cross-correlated with its counterparts corresponding to different experiments and δ[τ ] the Kronecker delta (equal to unity for τ = 0 and equal to zero for τ 6= 0). The fully parametrized AR and X matrices Ai (k), B i (k) of equation (5c) have dimensions [ny × ny] and [ny × nx], respectively, and are expressed as explicit functions of the operating parameter vector k belonging to p-dimensional functional subspaces spanned by the (mutually independent) functions G1 (k), G2 (k), . . . , Gp (k). These form a functional basis that consists of polynomials of three variables (vector polynomials) obtained as tensor products from typical univariate polynomials such as Chebyshev, Legendre, Jacobi and so on [34, pp. 771-802]) (also see [27] for a detailed description of the functional basis). The matrices Ai,j , B i,j include the corresponding AR and X, coefficients of projection to the basis functions. The VFP–VARX (na, nb)p model of equations (5a)-(5e) may be written in linear regression form as follows: y k [t] = [I ny ⊗ ϕTk [t] ⊗ g T (k)] · θ + ek [t] = ΦTk [t] · θ + ek [t]

(6)

with I ny designating identity matrix with dimension [ny × ny], ϕk [t] = [−y k [t − 1] . . . − y k [t − . na] .. xk [t] . . . xk [t − nb]]T being the regression vector with dimension [(na ny + (nb + 1)nx) × 1], g(k) = [G1 (k) . . . Gp (k)]T[p×1] a vector containing the functional basis, θ the vector that includes the projection coefficients with dimension [(na ny 2 p + (nb + 1)nx ny p) × 1], and ⊗ the Kronecker product [31, p. 123]. Pooling together the VFP–VARX model of equation (6) for all operating parameter vectors k (k1,1,1 , k1,1,2 , . . . , kM1 ,M2 ,M3 ) considered in the experiments for a single value of t, leads to: 

y k1,1,1 [t] .. .





      = y kM ,M ,M [t] 1 | {z2 3 } [nyM ×1]

ΦTk [t] 1,1,1 .. .



      ΦTk [t] M1 ,M2 ,M3 | {z }



ek1,1,1 [t] .. .



      =⇒ y[t] = Φ[t]·θ +e[t] (7) ·θ + ekM ,M ,M [t] 1 {z2 3 } |

[nyM ×(na ny 2 p+(nb+1)nx ny p)]

[nyM ×1]

Then, following substitution of the data for t = 1, . . . , N yields: y =Φ·θ+e

(8)

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with:  y[1]  y[2]    y =  ..   .  y[N ]







,

 Φ[1]  Φ[2]    Φ =  ..   .  Φ[N ]

[N nyM ×1]

,

[N nyM ×(na ny 2 p+(nb+1)nx ny p)]

 e[1]  e[2]    e =  ..   .  e[N ]

[N nyM ×1]

The parameter vector θ may be thus estimated based on the Ordinary Least Squares estimator [27]: "N #−1 " N # X X  T −1  T  T T b= Φ Φ θ · Φ y = Φ [t] Φ[t] · Φ [t] y[t] (9) t=1

t=1

and the residual covariance matrix estimate is obtained as: N X b = 1 b T [t, θ] b b e (k, θ) Σ e [t, θ]e k N t=1 k

for k = k1,1,1 , k1,1,2 , . . . , kM1 ,M2 ,M3

(10)

The AR and X orders of the typical VARX(na, nb) model, as selected for the detection problem, are used for the VFP-VARX model as well. The determination of its functional subspace dimensionalities for a given basis function family is achieved via Genetic Algorithms (GAs) [25, 27] based on the minimization of the BIC [30]. Finally model validation is based on formal verification of the residual sequence uncorrelatedness (whiteness) hypothesis [33, pp. 157–174] corresponding to the M experiments used in model estimation.

Inspection phase Damage detection. A new experiment is conducted under the current, unknown, state of the structure and nx-excitation xu [t] and ny-response y u [t] signals are obtained. Under the null hypothesis, where the structure is still under its healthy state, the residual series generated by driving the current signals through the VARX model (with parameter vector θ o ) of the baseline phase should be iid Gaussian with zero mean and covariance matrix Σo . The decision making procedure is then based on the likelihood function that is determined for the current data, expecting it to be larger or equal to the critical point that is obtained in the baseline phase in order to accept the null hypothesis; otherwise a damage is detected (details in [6, 10, 11]). If the presence of a damage is detected, then damage localization is activated based on the FMBM as follows. Damage localization. The current signals are driven through the VFP-VARX(na, nb)p model of the baseline phase which is now re-parametrized in terms of the current damage coordinate vector k and the corresponding residual series eu [t, k] with covariance matrix Σeu yielding: y u [t] +

na X

Ai (k) · y u [t − i] =

i=1

nb X

B i (k) · xu [t − i] + eu [t, k]

(11)

i=0

Structural elements with negligible thickness (1D elements) may be expressed by a set of functions of an independent variable s [35, pp. 816-820]: x = x(s),

y = y(s),

z = z(s)

(12)

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with s ∈ S = [slb , sub ] ⊂ R defined by the structural topology. Thus the damage location k is a function of s, k(s) = [x(s) y(s) z(s)]T . The estimation of s is based on the Prediction Error principle (minimization of the Residual Sum of Squares, RSS), and is obtained through Nonlinear Least Squares (NLS) [31, pp. 327-329] over the considered structural topology:

sb = arg min RSS(s) = arg min s∈S

s∈S

N X be = 1 eu [t, k( sb )]eTu [t, k( sb )] eTu [t, k(s)]eu [t, k(s)], Σ u N t=1 t=1

N X

(13) with eu [t, k(s)] given by equation (11). The estimate sb may be shown to be asymptotically
(N → ∞) Gaussian distributed, with mean equal to the true µs and variance σs2 sb ∼ N (µs , σs2 ) coinciding with the Cramer-Rao lower bound (see Appendix 4): " σ bs2

=

N X

#−1

−1

T

b ·ξ [t, sb ] ξ[t, sb ]·Σ eu

,

ξ[t, sb ] =

t=1

  ∂eTu [t, k(s)] b T· I ny ⊗ϕ[t]⊗ ∂g(k(s)) i = − θ ∂s ∂s s=bs s=b s (14)

. b designating with ϕ[t] = [−y u [t − 1] . . . − y u [t − na] .. xu [t] . . . xu [t − nb]]T[(na ny+(nb+1)nx)×1] and θ the available (from the baseline phase) estimate of the projection coefficients vector. b = k( sb ) is a random variable with probability density Thus each damage coordinate in vector k function (pdf) that may be obtained from that of sb through the expression (hats are omitted for simplicity): −1 d x (x) for x (likewise for y, z) fx (x) = fs (x−1 (x)) · dx

(15)

where |·| designates absolute value and fs (s) is the normal pdf of sb provided that x (s) is one-to-one [36, pp. 213-214]. The mean and variance of x b, based on which the corresponding intervals are constructed, may be then obtained from typical expressions [36, pp. 119-125]. In the present study where each rod/element of the truss is considered as an 1D element (negligible thickness), it is modelled as a straight line connecting two nodes with coordinates b corresponding to equation (12) may be expressed {x0 y0 z0 } and {x1 y1 z1 }. The coordinates of k as: x = x0 + s (x1 − x0 )

and likewise for y, z

(16)

based on which estimates are presently constructed using the estimate of s. The random variable x b then follows Gaussian distribution with mean and variance: µx = x0 + µs (x1 − x0 ),

σx2 = (x1 − x0 )2 σs2

and the interval estimate of x then is (at the α risk level):   x b + Z α2 · σ bx , x b + Z1− α2 · σ bx

(17)

(18)

10 Table 2. Likelihood function damage detection based method details.

Signal length

Estimated model

No. of parameters

No. of outputs

N = 9758 samples VARX(62,62) 374 ny = 2 (76.23 s) Estimation method: Ordinary Least Squares (OLS); Matlab function: arx.m Samples Per Parameter (SPP) = 78.27; Condition number of inverted matrix = 3.27 × 1010 Critical Point = 3.255 × 104 VARX: Vector AutoRegressive with eXogenous excitation.

Table 3. Details on the VFP-VARX functional subspace dimensionality determination based on a Genetic Algorithm.

Population Elite Crossover size count fraction 100 20 0.7 Matlab function: ga.m

Maximum number of generations 1000

Function tolerance 10−4

Minimum BIC −227.6327

RSS/SSSa 0.0318 %

VFP-VARX: Vector dependent Functionally Pooled - Vector AutoRegressive with eXogenous excitation. BIC: Bayesian Information Criterion; RSS/SSS: Residual Sum of Squares / Series Sum of Squares. P PN PN a RSS/SSS : 1 T T b b k ( t=1 ek [t, θ]ek [t, θ])/( t=1 y k [t]y k [t]) × 100 %. M

bx the estiwith Z α2 , Z1− α2 designating the normal distribution’s respective critical points, and σ mated variance of x b obtained from that of s via equation (17). The obtained damage location estimate is accepted as valid after the estimated reparametrized model is successfully validated through typical statistical tests examining the hypothesis of residual b uncorrelatedness (whiteness). In the present study the well known Portmanteau Test [33, eu [t, k] pp. 169-171] is used.

Experimental results Baseline phase A conventional VARX(62,62) model (characterized by zero excitation delay, b0 6= 0) is estimated based on a single experiment of nx = 1 excitation – ny = 2 response signals from the healthy structure and the model residual covariance matrix Σo is obtained for use in the Likelihood function based method. An additional experiment with the healthy structure is used for the determination of the method’s critical point based on which damage detection is achieved in the inspection phase. The method’s details are shown in Table 2. The identification of a VFP-VARX model for the representation of the 3D truss structure under the considered damage type is based on M = 16 experiments of a single (nx = 1) excitation - two (ny = 2) response acceleration signals (also see subsection “The experimental set-up”). The AR and X orders of the estimated conventional VARX(62,62) model representing the healthy structure are also used for the VFP-VARX model. The model’s functional subspace is determined via a GA based procedure where an initial extended functional subspace consisting of 20 trivariate

11 Table 4. FMBM damage precise localization details.

Signal length

Estimated model

N = 2500 samples (19.53 s)

VFP-VARX(62,62)16

No. of experiments M = 16

No. of Projections coefficients

No. of outputs

5984

ny = 2

Baseline phase Estimation method : Ordinary Least Squares (OLS) Samples Per Parameter (SPP) = 20.05; Condition number of ΦT Φ= 3.18 × 1011 Functional basis consisting of 16 Trivariate Shifted Legendre polynomials : G = [G0,0,0 (x, y, z) G0,1,0 (x, y, z) G1,0,0 (x, y, z) G1,0,1 (x, y, z) G1,1,0 (x, y, z) G2,0,0 (x, y, z) G2,0,1 (x, y, z) G2,1,0 (x, y, z) G2,1,1 (x, y, z) G3,0,1 (x, y, z) G3,1,0 (x, y, z) G1,1,1 (x, y, z) G4,0,0 (x, y, z) G4,0,1 (x, y, z) G4,1,0 (x, y, z) G4,1,1 (x, y, z)] Inspection phase Estimation method: NLS estimator based on golden search & parabolic interpolation Tolerance of the objective function: 10−10 ; Tolerance of the estimated value: 10−10 Samples Per Parameter (SPP) = 5000; s ∈ [0, 1]; Matlab function: fminbnd.m Validation method: Portmanteau Test; Critical Point = 2.565 × 104 (number of lags = 28) VFP-VARX: Vector dependent Functionally Pooled - Vector AutoRegressive with eXogenous excitation. NLS: Nonlinear Least Squares.

Shifted Legendre polynomial basis functions is considered (also see [27, 28]), with the GA aiming at selecting the optimal functional basis subset based on BIC minimization (see details in Table 3). 16 trivariate Shifted Legendre are selected and the VFP-VARX(62, 62)16 model is then validated based on residual whiteness confirmed through the Portmanteau Test [33, pp. 169-171]. It is noted that further measurements from the structure under damage and especially from 4 locations (randomly selected) of those used in the VFP-VARX identification procedure are employed for the selection of an adjusted critical point for the Portmanteau Test. Thus, the complete identification procedure leads to a VFP-VARX(62,62)16 model (b0 6= 0) with functional subspace consisted of 16 trivariate Shifted Legendre polynomial basis functions (model identification details in Table 4).

Inspection phase Each set of nx = 1 excitation – ny = 2 response signals acquired from a current unknown structural state, is driven through the conventional VARX(62,62) model of the baseline phase and the obtained residual are used in the Likelihood function based method for damage detection. The results in Table 5 and Figure 2 demonstrate excellent damage detection as the method properly distinguishes the healthy from the damaged state of the structure without false alarms or missed damages. Once a damage is detected, its precise localization is achieved through the vector FMBM. The VFP-VARX(62,62)16 model is thus re-parametrized based on the current signals corresponding to each test case of the inspection phase (also see Table 1) and the NLS estimator in the left part of equation (13) (realized via golden search and parabolic interpolation [37, pp. 178–184]) leads to estimates of the unknown damage coordinates. Details on the method operation are provided in Table 4.

12 Table 5. Damage detection & precise localization results.

Damage Detection False Alarms 0/8 Type of FMBM Scalar Vector

Missed Damages 0/370

Damage Localization Model Validation Localization Error (sample mean ± std) 370/370 (100%) 15.1 ± 26.47 (cm) 370/370 (100%) 9.31 ± 9.28 (cm)

FMBM: Functional Model Based Method; std: standard deviation.

Figure 2. Damage detection results through the Likelihood based method (378 test cases). A damage is detected when the test statistic exceeds the critical point (dashed horizontal line).

Indicative damage localization results using the nonlinear optimization framework which searches for the damage coordinates over the truss structure through the minimization of the RSS with respect to variable s based on the scalar method (a VFP-ARX(84,84)16 is used; also see [27]) and the vector method, are shown in Figure 3. The color code shows the level of RSS at different locations on the structure, with the darkest value corresponding to the minimum point. Evidently, the vector method leads to more accurate damage localization results. Both test cases correspond to very small damages at different locations, which only slightly affect the structural dynamics. This is obvious through the indicative comparison between the Welch based Frequency Response Function (FRF) estimates [10, 11] for the healthy and damaged structures shown in Figure 4, which almost coincide. This figure also shows the excellent modelling of the damaged structural dynamics based on the VFP-VARX(62, 62)16 model which has an almost identical FRF with the corresponding Welch estimate. It should remarked that the present, vector version of the method, manages to successfully tackle certain “pathological” damage cases which were not successfully by the original, scalar, version [27]. This difficulty was associated with the very minor effects of damage on the single FRF considered in the scalar version, while its effective treatment by the present version is attributed to the fact that two response sensors (thus effectively two FRFs) are employed. This improved behaviour is nicely depicted in Figure 3(d). Point estimates of the damage coordinates, along with corresponding confidence intervals, for four indicative damage scenarios based on the scalar and vector versions of the FMBM are presented in Figure 5, where the improvement achieved by the vector version is evident. It is noted that the estimated models are successfully validated through the Portmanteau Test, as indicated in Table

13

Figure 3. Indicative damage localization results for two damage scenarios based on: (a) & (c) the scalar and, (b) & (d) the vector FMBM. The values of the RSS are color coded as on the right vertical scale and the darkest value indicates the minimum. The actual and estimated damage coordinates are shown above the plots (Actual damage location: —; Estimated damage location: – – ). FMBM: Functional Model Based Method; RSS: Residual Sum of Squares.

5. The higher accuracy in damage localization achieved by the vector FMBM (over its scalar counterpart) is also confirmed through the comparison of Figure 6 in terms of the Euclidean distance between the true and the estimated damage location for all considered damage scenarios. It is noted that certain “pathological” cases leading to errors higher than 85 cm by the scalar

14

Figure 4. Welch based FRF magnitude estimates for the healthy and a damage scenario. The VFPVARX(62, 62)16 model based FRF magnitude is also shown. (a) Point X-Point Y1; (b) Point X-Point Y2 (Matlab function: tfestimate.m, signal length 2500 samples, segment length 1020 samples, overlap 95%, Hamming window, frequency resolution 0.125 Hz). FRF: Frequency Response Function. VFP-VARX: Vector dependent Functionally Pooled - Vector AutoRegressive with eXogenous excitation.

method (not shown in the plot) are effectively treated by the vector FMBM, leading to errors in the range of 20-30 cm. The localization error sample mean and standard deviation for the scalar and vector versions of the FMBM are presented in Table 5, from which the improvement offered by the vector version is evident.

Conclusions An extension of the advanced Functional Model Based Method to a vector form achieving improved accuracy in the estimation of the exact damage coordinates on the topology of 3D structures was presented. The vector version of the method is founded on an appropriate nonlinear optimization framework which is equipped with vector Functional Models that may exploit multiple response measurements from the structure and also it may accurately represent the structural dynamics under damage at different locations through a single mathematical representation. The method’s as-

15

Figure 5. Indicative damage localization results and corresponding comparisons between the scalar and the vector version of the FMBM for four damage scenarios (•: true damage location; N: point estimate based on the vector FMBM; F: point estimate based on the scalar FMBM; |—|: confidence intervals at the 0.0001 risk level; the true and estimated damage location coordinates are numerically provided above each plot as well). FMBM: Functional Model Based Method.

sessment and its comparison with the previous scalar version were experimentally achieved through damage localization for numerous damages at different locations upon a 3D truss structure. The results demonstrated that the vector version of the FMBM based on even two sensors achieves damage precise localization for all considered damage scenarios, within a low and limited frequency bandwidth. It also significantly reduces the localization error compared with its previously

16

Figure 6. Damage localization results in terms of the Euclidean distance between the true damage location and its estimate via the scalar and vector versions of the FMBM (359 damage scenarios). VFP-ARX: Vector dependent Functionally Pooled - AutoRegressive with eXogenous excitation. VFP-VARX: Vector dependent Functionally Pooled - Vector AutoRegressive with eXogenous excitation. FMBM: Functional Model Based Method.

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Appendix 1 Conventions – Bold-face upper/lower case symbols designate matrix/column-vector quantities, respectively. – Matrix transposition is indicated by the superscript T . – A functional argument in parentheses designates function of a real variable; for instance P (x) is a function of the real variable x. – A functional argument in brackets designates function of an integer variable; for instance x[t] is a function of normalized discrete time (t = 1, 2, . . .). The conversion from discrete normalized time to analog time is based on (t − 1)Ts , with Ts designating the sampling period. b is an estimate – A hat designates estimator/estimate of the indicated quantity; for instance θ of θ. – The subscripts “o” and “u” designate quantities associated with the nominal (healthy) and current (unknown) state of the structure, respectively.

20

Appendix 2 Notation Ai (k), B i (k)

:

Ai , B i Ai,j , B i,j arg min E{·} ek [t] e[t] fx (x), fy (y), fz (z) fs fs (s) I ny Gj (k) k kl,m,n

: : : : : : : : : : : : :

L M na , n b nx, ny N N (· , ·) p s slb , sub vec xl , y m , z n xk [t]

: : : : : : : : : : : :

x[t] y k [t]

: :

y[t] Z(1− α2 ) k(s) α θo θ µs µx , µy , µz σs2 σx2 , σy2 , σz2

: : : : : : : : : :

AR, X parameter matrices at operating parameter vector k (VFP–VARX model) AR, X parameter matrices (VFP–VARX model) AR, X coefficients of projection (VARX model) minimizing argument statistical expectation residual vector at operating parameter vector k (VFP–VARX model) residual vector (VARX model) probability density function of coordinate x, y, z, respectively sampling frequency (Hz) probability density function of s identity matrix with dimension [ny × ny] basis functions (j = 1, 2, . . . , p ) operating parameter vector k = [xl ym zn ]T operating parameter vector values used for identification (l = 1, . . . , M1 , m = 1, . . . , M2 , n = 1, . . . , M3 , M = M1 × M2 × M3 ) likelihood function number of data records used for model identification AR, X model orders number of excitation, response signals, respectively data record length normal distribution with indicated mean vector and covariance matrix dimension of functional subspace independent variable lower and upper bound of variable s operator transforming a matrix into a vector by stacking its columns coordinates in 3D space (l = 1, . . . , M1 ,m = 1, . . . , M2 , n = 1, . . . , M3 ) excitation signal vector at operating parameter vector k (VFP–VARX model) excitation signal (VARX model) response signal vector at operating parameter vector k (VFP–VARX model) response signal (VARX model) normal distribution (1 − α2 ) critical point operating parameter vector corresponding to s risk level parameter vector (VARX model) projection coefficient vector (VFP–VARX model) mean of s mean of coordinate x, y, z, respectively variance of s variance of coordinate x, y, z, respectively

21 Σe (k)

:

⊗

:

residual covariance matrix at operating parameter vector k (VFP– VARX model) Kronecker product

: : :

convergence in distribution absolute value natural logarithm

d

−→ |·| ln Appendix 3 Abbreviations

BIC: Bayesian Information Criterion FEM: Finite Element Model FMBM: Functional Model Based Method FRF: Frequency Response Function FM: Functional Model GA: Genetic Algorithm iid: identically independently distributed LS: Least Squares NLS: Nonlinear Least Squares OLS: Ordinary Least Squares RSS: Residual Sum of Squares SPP: Sample Per Parameter SSS: Series Sum of Squares VARX: Vector AutoRegressive with eXogenous excitation (model) VFP-ARX: Vector dependent Functionally Pooled ARX (model) VFP-VARX: Vector dependent Functionally Pooled VARX (model) Appendix 4 Asymptotic distribution of sb The residuals eu [t, k] of equation (11), presently written as e[t, k(s)], are, for a valid model, independent and identically distributed. Assuming they are Gaussian distributed with zero mean and covariance matrix Σe , [∼ N (0, Σe )], the natural logarithm of their likelihood function may, in analogy to the conventional VARX model [38, p. 198] case, be written as: N

lnL(s, Σe ) = −

N 1X T nyN ln(2π) − ln det [Σe ] − e [t, k(s)] · Σ−1 e · e[t, k(s)] 2 2 2 t=1

(19)

with det[·] designating the determinant of the indicated matrix. As in the conventional case, the likelihood is maximized at the following point [38, p. 201-202]:

Σe =

N 1 X e[t, k(s)] · eT [t, k(s)] N t=1

(20)

22

The Maximum Likelihood estimate of Σe is given by the above form when the minimum value of s, which is obtained by the Nonlinear Least Squares of equation (13), is used. √ s −s) converges in distributions to the normal distribution with Thus, the random variable N (b zero mean and variance given by the Cramer-Rao lower bound [31, pp. 214-215]: √

d

N (b s − s) −→

N (0, σs2 )

with

σs2

   T −1 ∂ lnL(s, Σe ) ∂ lnL(s, Σe ) = E · ∂s ∂s

(21)

d

with −→ denoting convergence in distribution. The first derivative of equation (19) with respect to s is: N

1 X ∂ eT [t, k(s)] · Σ−1 ∂ lnL(s, Σe ) e · e[t, k(s)] =− ∂s 2 t=1 ∂s   N N X 1 X ∂ eT [t, k(s)] ∂ eT [t, k(s)] −1 T −1 =− · Σe + (Σe ) ·e[t, k(s)] = − · Σ−1 e · e[t, k(s)] (22) 2 t=1 ∂s ∂ s t=1 where: eT [t, k(s)] = (y u [t] − [I ny ⊗ ϕT [t] ⊗ g T (k(s))] · θ )T = y Tu [t] − ([I ny ⊗ ϕT [t] ⊗ g T (k(s))] · θ )T = y Tu [t] − θ T · [I ny ⊗ ϕT [t] ⊗ g T (k(s))]T = y Tu [t] − θ T · [I ny ⊗ ϕ[t] ⊗ g(k(s))] By defining: ξ[t, s] =

  ∂ eT [t, k(s)] ∂g(k(s)) = −θ T · I ny ⊗ ϕ[t] ⊗ ∂s ∂s

1,1 1,p 1,1 1,p na,1 na,p na,1 na,p .. 0,1 0,1 = −[α1,1 . . . α1,1 . . . α1,ny . . . α1,ny . . . α1,1 . . . α1,1 . . . α1,ny . . . α1,ny . b1,1 . . . b0,p 1,1 . . . b1,nx

.. 1,1 nb,1 nb,p nb,1 nb,p .. 1,p na,1 1,1 1,p . . . b0,p 1,nx . . . b1,1 . . . b1,1 . . . b1,nx . . . b1,nx . . . . . αny,1 . . . αny,1 . . . αny,ny . . . αny,ny . . . αny,1 . na,p nb,1 nb,p na,1 na,p . 0,1 0,1 0,p nb,1 nb,p . . . αny,1 . . . αny,ny . . . αny,ny .bny,1 . . . b0,p ny,1 . . . bny,nx . . . bny,nx . . . bny,1 . . . bny,1 . . . bny,nx . . . bny,nx ]   1 ... 0 T !  ∂G (k(s)) ∂G (k(s))  .. . . ..  p 1 ... ·  . . .  ⊗[−y u [t−1] . . . −y u [t−na] xu [t] . . . xu [t−nb] ]T ⊗ ∂s ∂s 0 ... 1 the substitution of equation (22) into (21) leads to: " N  T #−1 X
−1 σs2 = E ξ[t, s] · Σ−1 e · e[t, k(s)] · ξ[t, s] · Σe · e[t, k(s)] t=1

" =

N X t=1

#−1 T −1 ξ[t, s] · Σ−1 e · Σe · Σe · ξ [t, s]

=

"N X t=1

#−1 T ξ[t, s] · Σ−1 e · ξ [t, s]

(23)

23

For signals consisting of N data points, the variance of sb is estimated by replacing Σe with its estimate: "N #−1 X −1 b · ξ T [t, sb ] σ bs2 = (24) ξ[t, sb ] · Σ e t=1

with: ξ[t, sb ] =

  ∂ g(k(s)) ∂ eT [t, k(s)] T = −θ · I ⊗ ϕ[t] ⊗ . ny ∂s ∂ s s=bs s=b s