Zoran Djinovic
email: zoran.djinovic@acmit.at

Michael Scheerer

Milos Tomic

Marijana Stojkovic

Martin Schueller

SIMULTANEOUS DAMAGE DETECTION AND DEFLECTION MEASUREMENT OF MORPHING WING STRUCTURES BY FIBER OPTIC SENSING SYSTEM

Keywords:

In this paper we present results of investigation of simultaneous damage detection and deflection measurement of morphing CFRP honeycomb structure by fiber optic sensing system developed in the frame of EU-FP7 project "Fiber Optic System for Deflection and Damage Detection (FOS3D)". The system is based on low-and highcoherence interferometry performed as "all-in-fiber" sensing configuration. Raw signals have been on-and off-line processed by "arctang" algorithm. Deflection data, expressed as phase angle change of the interferometer, have been simultaneously acquired with the reference data. A linear relationship between the fiber-optic sensing and reference technique is obtained. Slope of the linear fit line of 8 rad/mm denotes sensitivity of this sensor. Noise floor of about ±70µrad/√Hz determines the lowest measurable wing deflection of about 6µm. Damage events have been simulated by pencil break and hammer impact with-and without damages over the CFRP honeycomb structure. Raw fiber-optic interference and reference acoustic emission sensor signals have been simultaneously acquired. Figure of merit is given as probability of detection (POD) and localization of simulated impact event. All events can be detected and located with a POD of more than 98% within the localization error of around 15 mm.

INTRODUCTION

The morphing wing concept of aircrafts finds a deep inspiration in the flight capabilities of birds [START_REF] Costa Aleixo | Morphing Aircraft Structures; Design and Testing an Experimental UAV[END_REF]. It has to assure different flight manoeuvres in a wide range of environmental conditions just as current aircrafts. Every operational task has to be accomplished efficiently, safety and reliable [START_REF] Grady | Multi-Objective Optimization of a Three Cell Morphing Wing Substructure[END_REF]. For this purpose it is necessary to have complex mechanical structures and materials capable to actively change the aerodynamic shape of the wing [START_REF] Raither | Morphing Wing Structures with Controllable Twist Based on Adaptive Bending-Twist Coupling[END_REF]. Carbon-or glass-fiber reinforced plastics (CFRP, GFRP) have already found application in modern aircrafts and in morphing field as well. Honeycomb structures equipped with different actuators are suitable for the morphing purpose [START_REF] Bergamini | A Sandwich Beam with Electrostatically Tunable Bending Stiffness[END_REF]. In a real application it is necessary to have a closed loop between the actuators and sensors, which are responsible for providing feedback information about the real time position of the wings. During morphing action materials and structures suffer large loadings, which eventually could lead to the damage. Therefore, sensors have to assure on board structural health monitoring (SHM) of the structure in order to avoid any catastrophic scenarios.

In this paper we present results of investigation of simultaneous damage detection and deflection measurement of morphing CFRP structure by fiber optic sensing system developed in the frame of an EU-FP7 project named "Fiber Optic System for Deflection and Damage Detection (FOS3D)" [START_REF] Djinovic | Design and characterization of fiberoptic interferometric sensor for deflection and damage detection of morphing wing structures[END_REF]. The system is based on low-and high-coherence interferometry performed as "allin-fiber" sensing configuration. It is composed of four fiber-optic sensors, which are adhesively bonded onto the CFRP honeycomb structure. Interference signals were obtained by recombination of back reflected signals from the fiber tips of sensing and reference arms. Raw signals have been on-and off-line processed by "arctang" algorithm. Output result is voltage phase angle signal of every particular interferometer, which depends on the deflection magnitude. Deflection data have been acquired by reference technique using an inductive displacement sensor. A linear relationship between the fiber-optic sensing and reference techniques was obtained. Slope of the linear fit line of about 8 rad/mm denotes sensitivity of this sensor. Noise floor of about ±70µrad/√Hz determines the lowest measurable deflection of about 6µm.

Damage events have been simulated by pencil break and hammer impact with-and without damages over the CFRP structure. Four raw interference and four reference signals obtained by acoustic emission sensors have been simultaneously acquired by the conventional Acoustic Emission (AE) system. After signal processing of both kinds of AE signals damage detection procedure was accomplished. Figure of merit is given as probability of detection (POD) of simulated impact events. All events can be detected and located by the algorithm of zonal detection with a POD of more than 98% that is equal to the reference system. Nevertheless all damaging impacts can be detected and located with an average localization error of around 15 mm with both systems.

PRINCIPLE OF OPERATION

Basic principle of simultaneous wing flap deflection measurement and damage detection is an optical phase change occurred in the sensing coil of a fiber optic Michelson interferometer, Figure 1a. Fiber optic 3x3 single-mode (SM) coupler splits light emitted by source LS, a superluminescent diode (SLD) of 50µm of coherence length at 1300nm, into the three arms. One of the arms (so called inert arm) is terminated by cleaving of the fiber end at 8° in order to suppress back reflection of spurious light beam, which could be a source of intensity noise. The other two arms form the sensing and reference fiber optic coils. Light beam travels toward the fiber tips of both arms and reflects back from the fiber end surfaces and returned to the 3x3 coupler to recombine. Recombined light signals, carrying information of the optical path length difference (OPD) between the sensing and reference arms, hit the two receiving InGaAs photodiodes (PD1, PD2) generating voltage signals.

The sensing and reference coils are firmly fixed to the wing structure and sustain the static deformation originating from the wing deflection. Simultaneously, the coils sense the acoustical waves traveling through the wing structure, caused by an external impact or structure crack damage.

Sensing configuration, depicted in Figure 1a, is basically a classical two beam interferometer where irradiance at photodiodes is given by:

    12 12 11 2 1 2 1 2 cos 2 L k L I I I I I PD             (1)
where L 12 =L 1 -L 2 , L 1 and L 2 are length of sensing and reference coils, I 1 and I 2 are irradiances of beams reflected from the mirrored end of the sensing and reference fibers, k=2π/ is wave number of light and  11 is the degree of coherence of light.

When an external force acts on the fiber that carries an optical signal, it induces a change of the optical path length in the sensing arm. There are two basic mechanisms to produce the phase change: direct and indirect coupling of strains to the fiber and thermal effects. Hence, the phase change induced by the applied load is given by the expression (2) [START_REF] Kirkendall | Overview of High-Performance Fiber-Optic Sensing[END_REF]:

             dn L dL n L k L n d k d ) (  (2)
where: n is the refractive index of glass in the core region and L is the overall sensing length of the fiber. The term ndL reflects the physical change of the fiber length while the term Ldn reflects the change of the refraction index of the glass. a) b) Figure 1: a) Base sensing configuration: Michelson interferometer with 3x3 fiber optic coupler; "deformation"-perturbation caused by wing deflection, "acoustical wave"-produced by damage-like events; M1, M2-mirrors (fiber tips), PD1, PD2-photodiodes, LS-light source; b) Schematically presentation of placement of fiber-optic sensors over the morphing wing structure in the area to be monitored Figure 1b presents overall configuration of morphing wing structure equipped with an array of circular (or elliptical) shape fiber-optic sensors, which are connected to the 3x3 couplers (Figure 1a) and multiplexed in a common sensing system (see Figure 2) [START_REF] Scheerer | Fiber Optic System for Deflection and Damage Detection in Morphing Wing Structures[END_REF]. In the, so called, single sensing configuration, only sensing coil is adhesively bonded over the morphing area, while reference coil is suited in a proximity of the first one. In the push-pull configuration one coil is on the top and other on the bottom side of the wing. Such a configuration is applied in this work because of normalization of parasitic effect, e.g. temperature, typically occurred in the subjected CFRP structure. Maximal required wing deflection is ±5° with resolution of 0.1° and damage of about 20mm in diameter with a location accuracy of 5% in respect to the average sensor spacing.

Combination of low-and high-coherence radiation allows: 1) absolute determination of morphing wing position in respect of the initial (typically non-deformed) wing position (the role of the low-coherence source) and deflection measurement in the entire range of ±5° (the role of the high-coherence source). Damage detection is provided by acquisition, band pass filtration from 5 kHz to 500 kHz and processing of just one high-coherence interference signals. Determination of morphing wing deflection is performed by signal processing using "arctang" algorithm of lowfrequency (of about 100Hz) raw interference signals.

SYSTEM DESIGN

The concept of the system is schematically presented in Figure 2. Here is applied, so called, pushpull sensing configuration because one sensing coil is on the top of the subjected CFRP structure and other is underneath on the bottom side of the CFRP structure at the same coordinate in respect of the overall dimensions of the plate of 400x200x50mm. The plate has a specific triangular shape in longitudinal cross-section.

The entire setup is composed of four multiplexed fiber-optic sensors denoted by S1-S4. Both (sensing and reference) sensors are in the coil form of 20mm in diameter with just one turn, which are adhesively bonded onto CFRP structure. They are made by turning of single-mode optical fiber Corning e28+. Both coils are parts of the output arms of a 3x3 coupler which is used to build up two Michelson interferometers. It is of primary importance to say that both arms, including sensing coils, are short having of about 200mm in length. The length of the arm was precisely adjusted so the every interferometer was set to be in the equilibrium state when sensitivity of sensors was maximal. By this approach a substantial benefit in respect of better stability of interference working point and eventually higher SNR is obtained. Static phase of the arms are mutually shifted for about 120° that assures obtaining of quasi-quadrature interference signals.

EXPERIMENT

Figure 3 shows an overall view of the FOS3D sensing system and loading rig together with the subjected CFRP honeycomb structure. The FOS3D sensing system consists of: two-channel DC power supply, high-and low-coherence light sources (HCS, LCS), FOS3D16TIA-16 InGaAs photodiodes with accompanied transimpendance amplifier and four FOS3D DEF/DAM unit.

Top side of the CFRP structure, depicted in Figure 4a, is equipped with four fiber-optic sensors, four piezo-ceramic transducers (PZT) and four strain gauges. Bottom side of the structure is of the same sensors configuration and two inductive sensors touching the CFRP structure. Four input and eight output arms of the four 3x3 couplers were connected to the FOS3D sensing system. During deflection (from 0 to 10mm) and damage test campaigns, data acquisition of on-line calculated phase signal, raw high-coherence photodiode signals and accompanied reference signals from inductive sensors have been done by 16bit NI DAQ and by a commercial Acoustic Emission system. Figure 6a presents a linear relationship between the phase angle measured in fiber-optic sensor versus deflection of subjected CFRP structure. Out of this diagram we determined sensitivity of every particular sensor as a slope of the liner fitting line. Sensor sensitivity depends on its position on the CFRP structure. For example, sensor S1 demonstrates the largest sensitivity since at this place the CFRP structure and naturally optical fiber of sensor S1 are exposed to the highest strain. Other two sensors S2 and S3 show gradually smaller sensitivity, while sensor S4 shows the smallest sensitivity as it was found by the final test under the tensile test machine.

Minimal measurable deflection value by the FOS3D system is determined by finding the noise floor of the overall system. Figure 6b presents typical noise signal of fiber-optic sensors recorded by generation of minimal deflection of 35μm. The main noise source is about 10mV, which determines the minimal detectable phase angle signal of about 60 mrad, i.e. minimal detectable deflection of about 6-7μm. This is much better than required resolution of about 200µm. Figure 7 summarizes results of simultaneous deflection measurement and damage detection in case of cyclic loading from zero to 15mm of total deflection. CFRP structure was exposed to the hammer impact at around 240s of deflection experiment (denoted by ellipse) as can be seen out of Figure 7a. The hammer impact produced a sudden phase change that appeared as jump of voltage signal, depicted in the inset. Figure 7a shows voltage signal of fiber-optic sensors in dependence on deflection (stroke) recorded by tensile test machine. Figure 7b shows calibration diagram of phase angle change in radians versus strain which is obtained for the case of simultaneous deflection measurement and damage detection. Sensitivity of all fiber optic sensors was determined as a slope of linearly fitted line (yellow) and it is of about 0,1rad/με. The most sensitive parameter for damage detection is the AE energy and the average frequency. All pencil breaks, non-damaging impacts and damaging impacts produce hits with a large scatter band of parameters. Figure 8a illustrates three different clusters of hits: red cluster: first hits due to damaging impacts, yellow cluster: first hits due to non-damaging but higher energetic impacts, green cluster: hits from pencil breaks and reflections from the boundaries of the non-damaging and damaging impacts. Only the data from the red cluster serve as the basis for the detection of damaging impacts. They are obtained after data filtering for: amplitude > 95 dB, average frequency > 80 kHz and energy > 1 x 10 9 . All damaging impacts can be located with a 2-D localization algorithm using the threshold crossing for the determination of the arrival time (preferred localization algorithm) within an average localization error of less than 15 mm, shown in Figure 8b. All events can be detected and located with a POD of more than 98%. Due to the energy of the damaging impacts the quick S0 mode is strong enough to trigger all sensors which is required for the proper function of the mentioned localization algorithm.

CONCLUSION

Main characteristic of transportable FOS3D sensing system, calibration diagram of phase angle signal (rad) vs. deflection (mm), has been experimentally determined by numerous deflection campaigns of CFRP honeycomb structure. Overall sensitivity of all fiber-optic sensors of about 0,1rad/με was determined out of phase angle signal (rad) vs. strain (µε) calibration diagrams. It was also achieved very good linearity of all sensors in the entire deflection range from zero to 20mm. In respect of the initial requirements of resolution of 0.1° and deflection range of ±5°, required resolution of phase angle signal should be of about 3rad that corresponds with about 30με of measured strain. It was proved that FOS3D system can fulfil these requirements since minimum detectable phase angle signal is of about 120mrad. Finally, the main goal of the project was successfully proved by experimentally verification of capability of the FOS3D system for simultaneous deflection measurement and damage detection with POD of more than 98% and localization accuracy of about 15mm.

Figure 2 :

 2 Figure 2: Schematic presentation of multiplexed LCS_HCS sensing configuration composed of four fiberoptic sensors: S1-S4. Testing structure: CFRP honeycomb plate in size of 400x200x50 mm. HCS highcoherence source, LCS low-coherence source, photodiodes PD1, PD3, PD5, PD7,PD9, PD11, PD13, PD15 (HCS), PD2, PD4, PD6, PD8, PD10, PD12, PD14, PD16 (LCS), OI optical isolator, WDMC wavelength division multiplexing coupler

Figure 3 :FOS3DFigure 5 :

 35 Figure 3: Overall view of the entire FOS3D sensing system and loading rig, LCS low-coherence source, HCS high-coherence source Figure 4b illustrates the same CFRP structure exposed to the compression loading in a tensile test machine for final test. Here we induced cyclic loading from 0 to 15mm and from 0 to 20mm of deflection. Deflection was directly measured by stroke of the machine. Number of cyclic loading was three, five and ten. During this testing several signals have simultaneously been acquired: eight strain gauges for strain measurement, eight PZT for acoustic signal measurement, and all aforementioned fiber-optic signals.

  a) Calibration diagram of phase angle vs. deflection of all four sensors b) Minimal detectable phase angle (V) of fiber-optic sensor S1 (~6μm of deflection)

Figure 7 :

 7 Figure8apresents common results of damage detection produced by different types of impact: pencil breaks (small-non-damaging), non-damaging impacts with an impact hammer; damaging impacts with an impact hammer, damaging impact during morphing of the CFRP structure. All impacts have been analysed regarding detectability and localization using zonal and 2D localization algorithm, where the arrival time of the signals at the individual sensors were determined using first threshold crossing.

EWSHM 2014 -Nantes, France

ACKNOWLEDGMENT

The research leading to these results has received funding from the European Union`s Seventh Framework Programme (FP7/2007-2013) for the Clean Sky Joint Technology Initiative under grant agreement n_CSJU-GAM-SFWA-2008-001.


Gwendal Cumunel
email: gwendal.cumunel@enpc.fr

Thi-Thu-Ha Le

Nelly Point

Pierre Argoul

DETECTION OF STRUCTURAL MODIFICATIONS OF AN EULER-BERNOULLI BEAM THROUGH MODAL PARAMETERS VARIATION

Keywords: structural health monitoring, damage detection, localization of cracks, modal analysis, Euler-Bernoulli beam

This paper deals with a class of identification problems mainly concerning the localization of structural modifications in elastic beams with or without tension by dynamic analysis. Variations of tension, mass density and/or bending stiffness induce variations of the beam natural frequencies. For each natural frequency, a first order estimate of this modification is then established. It depends on the variations of tension, density and bending stiffness and also on the corresponding mode shape and curvature mode shape of the initial state. When the variations of density and flexural rigidity are localized along the beam (as for a notch in a beam or a crack in a cable), this estimate is used to propose a procedure permitting detection, localization and, eventually, quantification of these variations knowing the modifications of the first natural frequencies and the variation of tension. This procedure is applied successfully to experimental data found in literature concerning beams without tension and only bending stiffness modifications.

INTRODUCTION

The demand for enhanced performance and reliability of structures in terms of safety, noise and durability is ever increasing. A lot of research has been done for decades on structural health monitoring using a large number of different techniques (based on ultrasound [START_REF] Broda | Modelling of nonlinear crackwave interactions for damage detection based on ultrasounda review[END_REF], genetic algorithms [START_REF] Chou | Genetic algorithm in structural damage detection[END_REF], or vibration methods [START_REF] Salawu | Detection of structural damage through changes in frequency: a review[END_REF][START_REF] Doebling | A summary review of vibration-based damage identification methods[END_REF] for example) and applied in various fields like aeronautics [START_REF] Gobbato | A reliability-based framework for fatigue damage prognosis of composite aircraft structures[END_REF], industry [START_REF] Lei | A review on empirical mode decomposition in fault diagnosis of rotating machinery[END_REF] or civil engineering [START_REF] Sposito | A review of non-destructive techniques for the detection of creep damage in power plant steels[END_REF].

Beams are traditionally used to model buildings or civil engineering structural elements, but also smaller structures such as frames of trucks, automobiles or machines. Other mechanical systems contain beam structures and are designed and analyzed in a similar fashion. The transversal deflection of cables of suspension bridges can be modelled as that of a long beam with flexural rigidity varying with space under strong tension, with the bending moment at each point proportional to the local curvature. The question of localizing defects in a beam has been studied by many authors in the case of a beam without tension [START_REF] Pandey | Damage detection from changes in curvature mode shapes[END_REF][START_REF] Pandey | Damage detection in structures using changes in flexibility[END_REF][START_REF] Cao | Sensitivity of fundamental mode shape and static deflection for damage identification in cantilever beams[END_REF][START_REF] Jassim | A review on the vibration analysis for a damage occurrence of a cantilever beam[END_REF]. For example, Friswell [START_REF] Friswell | Crack modeling for structural health monitoring[END_REF] have compared the influence of different models of cracks in beam structures on the variation of the eigenfrequencies. To compare his modelizations with experimental data, he used the measures obtained in [START_REF] Rizos | Identification of crack location and magnitude in a cantilever beam from the vibration modes[END_REF], in which a crack location can be found from the measured amplitudes at two points of the structure vibrating at one of its natural modes, the respective vibration frequency and an analytical solution of the dynamic response. Many authors use a sensitivity method based on the eigenvalue equation but most of them with a discrete form, i.e. with mass and stiffness matrices [START_REF] Adhiakri | Rates of change of eigenvalues and eigenvectors in damped dynamic system[END_REF][START_REF] Esfandiari | Theoretical and experimental structural damage diagnosis method using natural frequencies through an improved sensitivity equation[END_REF]. In [START_REF] Adhiakri | Rates of change of eigenvalues and eigenvectors in damped dynamic system[END_REF], Adhiakri proposed a method based on rates of change of eigenvalues and eigenvectors of a damped linear discrete dynamic system with respect to the system parameters. Results show the efficiency of the proposed method and the importance of including the changes of the mode shapes through a frequency-based damage detection algorithm. To localize and evaluate the change of the bending stiffness, Morassi [START_REF] Morassi | Damage detection and generalized Fourier coefficients[END_REF] has also proposed a method based on generalized Fourier coefficients to calculate Eigenfrequency sensitivity to the stiffness variation caused by the damage.

In the articles cited above, the modification of mass is not taken into account. As tension is of interest for cables, in this article, we study the transverse deflection of an Euler-Bernoulli beam with tension. We establish a first order general estimate of the relative variation of each frequencies as a function of the simultaneous variations of different mechanical parameters of the beam such as tension, local density and local bending stiffness. Based on this estimation, a method to detect, localize and quantify a damage is proposed using only, at least, three frequencies of the undamaged and damaged beam, and the corresponding mode shapes (for a mass variation) and curvature mode shapes (for a stiffness variation) of the undamaged beam.

MODAL PARAMETERS VARIATIONS DUE TO MECHANICAL CHARACTERISTICS MODIFI-

CATIONS

Several assumptions are made: the beam material is homogeneous, isotropic, and linearly elastic, the tension (axial force) is constant (positive or null) along the beam and viscous damping is ignored. The corresponding partial differential equation governing the transverse deflection w(x,t)) of an Euler-Bernoulli beam under axial force is:

µ(x) ∂ 2 w(x,t) ∂t 2 -T ∂ 2 w(x,t) ∂ x 2 + ∂ 2 ∂ x 2 EI(x) ∂ 2 w(x,t) ∂ x 2 = p(x,t) (1)
where w(x,t) is the transverse deflection at point M(x) and time t, µ(x) the mass per unit length, T the tension force in the beam, EI(x) the flexural rigidity (or bending stiffness) and p(x,t) the excitation force per unit length. The particular case of the vibrating string is retrieved for EI = 0 (not studied here) and the case of the Euler-Bernoulli beam without tension is obtained for T = 0.

The normal modes φ n are functions which are square integrable on [0, L], as well as φ ′ n and φ ′′ n (i.e. φ n ∈ H 2 (0, L)). By using the method of separation of variables, it can be shown that the normal modes φ n of the beam verify the following equation:

(EI(x)φ ′′ n (x)) ′′ -T φ ′′ n (x) -λ n µ(x)φ n (x) = 0 (2)
for some boundary conditions. Multiplying this equation by any function u(x) belonging to H 2 (0, L), an integration by parts leads to the relation:

L 0 EI(x)φ ′′ n (x)u ′′ (x) + T φ ′ n (x)u ′ (x) -λ n µ(x)φ n (x)u(x) dx +C = 0 (3)
with C depending on the boundary conditions:

C = [ ∂ ∂ x (EI(x)φ ′′ n (x))u(x)] L 0 -[EI(x)φ ′′ n (x)u ′ (x)] L 0 -[T φ ′ n (x)u(x)] L 0 (4) If the beam is clamped in x = a then φ n (a) = φ ′ n (a) = 0, if it is simply supported then φ n (a) = φ ′′
n (a) = 0 and at a free end then φ ′′ n (a) = φ 3 n (a) = 0 (in such case the tension is equal to zero). For a beam verifying boundary conditions of the type listed above, the quantity C is equal to zero for any function u(x) belonging to the space H 2 (0, L) and verifying the same boundary conditions as the normal modes.

Hence for a beam with classical boundary conditions, the following relation is verified for any function u ∈ H 2 (0, L) verifying the boundary conditions:

L 0 EI(x)φ ′′ n (x)u ′′ (x) + T φ ′ n (x)u ′ (x) -λ n µ(x)φ n (x)u(x) dx = 0 (5)
We now consider changes of intrinsic characteristics of the beam due to local reduction or increase of flexural rigidity and weight in some area combined or not with a global variation of the tension. The modifications of the flexural rigidity and/or of the mass density will be qualified in the following by internal and that of the tension by external, respectively. We study the variation of the modal parameters due to these modifications which are assumed to be small in order to perform first order estimates. The modified mass and flexural rigidity are denoted by EI(x) and µ(x) with:

EI(x) = EI(x) + ∆EI(x) µ(x) = µ(x) + ∆µ(x) (6)
where the variation of the density and of the bending stiffness are denoted by ∆µ(x) and ∆EI(x) respectively. The tension is assumed to be constant along the beam. The variation of the tension is denoted by ∆T and the modified tension is equal to:

T = T + ∆T. (7)
These changes induce a variation of the mode shapes and of the eigenvalues:

φn (x) = φ n (x) + ∆φ n (x) λn = λ n + ∆λ n (8)
For any function u ∈ H 2 (0, L) verifying the boundary conditions, the relation ( 5) must be verified by the new normal modes φn :

L 0 EI(x) φ ′′ n (x)u ′′ (x) + T φ ′ n (x)u ′ (x) -λn µ(x) φn (x)u(x) dx = 0 (9)
As we want to perform an estimation to the first order, the following approximations are made:

λn μn φn -λ n µ n φ n ≈ λ n µ n ∆φ n + (∆λ n µ n + λ n ∆µ n )φ n
The same type of approximation is made for T : T φn -T φ n = T ∆φ n + ∆T φ n ≈ T ∆φ n + ∆T φ n , and the same is done for EI.

The difference of ( 9) minus ( 5) is zero for any u ∈ H 2 (0, L) verifying the boundary conditions. The variation ∆φ n can be expressed in the basis of normal modes {φ k } ∞ k=1 , as:

∆φ n (x) = ∞ ∑ k=1 α nk φ k (x) (10)
This yields the following first order estimate of the variation of the eigenvalue:

∆λ n L 0 µ(x) φ 2 n (x)dx ≈ L 0 ∆EI(x)φ ′′2 n (x)dx + ∆T L 0 φ ′2 n (x)dx -λ n L 0 ∆µ(x) φ 2 n (x)dx (11)
This equation can be modified so as to express the eigenvalue relative variation:

∆λ n λ n ≈ γ n 1 + γ n L 0 ∆EI(x) (φ ′′ n (x)) 2 dx L 0 EI(x) (φ ′′ n (x)) 2 dx + ∆T T 1 (1 + γ n ) - L 0 ∆µ(x)φ 2 n (x)dx L 0 µ(x) φ 2 n (x)dx (12)
For the beam without tension, it reads:

T = 0, ∆λ n λ n ≈ L 0 ∆EI(x) (φ ′′ n (x)) 2 dx L 0 EI(x) (φ ′′ n (x)) 2 dx - L 0 ∆µ(x)φ 2 n (x)dx L 0 µ(x) φ 2 n (x)dx (13)
EWSHM 2014 -Nantes, France

The external modifications (mass and/or bending stiffness) are assumed to be localized around x 0 on an interval Ω of length ∆L with ∆L very small with respect to L, the total length of the beam. Moreover, since λ n varies as the square of the frequency f n then 2 ∆ f n f n = ∆λ n λ n . In the simple case of rectangular functions and if the bending stiffness and the density are assumed to be initially constant (EI(x) = EI and µ(x) = µ), it leads to:

2 ∆ f n f n ≈ γ n 1 + γ n ∆EI ∆L EI (φ ′′ n (x 0 )) 2 φ ′′ n 2 2 + ∆T T 1 (1 + γ n ) - ∆µ ∆L µ φ 2 n (x 0 ) φ n 2 2 ( 14
)
This relation is established for each mode. It permits to link the ratio of the variation of the frequency f n to the ratio of the variation of the bending stiffness, of the mass density and of the tension as soon as the corresponding initial mode shape and mode curvature are known. It can also be rewritten as:

∆ f n f n ≈ 1 2(1 + γ n ) ∆T T + G n (x 0 ) ∆EI∆L EIL -H n (x 0 ) ∆µ∆L µL (15)
with

G n (x) = γ n 2(1 + γ n ) Lφ ′′2 n (x) φ ′′ n 2 2 , H n (x) = 1 2 Lφ 2 n (x) φ n 2 2 (16)

LOCALIZATION PROCEDURE

The objective of this part is to use the relation [START_REF] Esfandiari | Theoretical and experimental structural damage diagnosis method using natural frequencies through an improved sensitivity equation[END_REF] concerning the variation of each frequency to localize the modifications. We assume that we know the p first frequencies of two states (an initial state and a modified one) and that the corresponding mode shapes of the initial state are known. The modifications of the mass and of the bending stiffness are assumed to be localized around the abscissa x 0 on a small interval of length ∆L (with ∆L ≪ L) and the tension is assumed to be measurable in the two states. For simplicity the modification is assumed constant on the interval [x 0 -∆L 2 , x 0 + ∆L 2 ], in case the relation [START_REF] Esfandiari | Theoretical and experimental structural damage diagnosis method using natural frequencies through an improved sensitivity equation[END_REF] leads to:

∆ n (x 0 ) ≈ G n (x 0 ) ∆EI∆L EIL -H n (x 0 ) ∆µ∆L µL (17)
with

∆ n (x 0 ) = ∆ f n f n - 1 2(1 + γ n ) ∆T T ( 18
)
If the frequencies and the tension are measured and as γ n depends on the characteristics of the beam and can be evaluated, then ∆ n is a known quantity. The functions G n (x), H n (x) depending of the mode shapes of the initially state are assumed to be known too. The terms ∆EI∆L EIL and ∆µ∆L µL are unknown and represent the relative variation of the bending stiffness and of the mass.

To localize the modification, the idea is to use [START_REF] Morassi | Identification of a crack in a rod based on changes in a pair of natural frequencies[END_REF] for three different frequencies and to take into account the fact that these three equations with two unknowns are compatible, which means that the determinant of order 3 formed by the coefficients ∆, G and H is equal to zero for x = x 0 . So if we define the function F m,n,k (x) by:

F m,n,k (x) = G m (x)H n (x)∆ k (x) + G n (x)H k (x)∆ m (x) + G k (x)H m (x)∆ n (x)+ -G n (x)H m (x)∆ k (x) -G k (x)H n (x)∆ m (x) -G m (x)H k (x)∆ n (x) (19)
this function should be null for x = x 0 . The value x 0 is a minimizer of |F m,n,k (x)| for any triplet (m, n, k). So, for the states before and after the modification, if the three first eigenvalues are measured and the corresponding mode shapes are known, then it is possible to evaluate x 0 as x m = ArgminF 1,2,3 (x). Besides, if eigenvalues of higher order are known, it is possible to improve the identification of the possible values of x 0 by taking into account these complementary informations. So if the p first eigenvalues (with p ≥ 3) are available, the possible values of x 0 can be obtained as the values of x that minimize the quantity Q p (x) defined below:

Q p (x) = ∑ 1≤n≤m≤k≤p |F m,n,k (x)| (20)
If the frequencies are not distinct, the function F m,n,k (x) is equal to zero, so only the case of distinct frequencies is relevant.

It should be noted that the quantity Q p (x) increases with the number p of used frequencies for each fixed value of x. It also can be observed that the function Q p increases globally with the importance of the variation of the mechanical parameters. The maximum of the function Q p is linked to the amplitude of the modification. It will be interesting to establish a relation between Q p ∞ and a measure of the modification.

APPLICATION TO BEAMS WITHOUT TENSION

The experimental tests consist of a free-free beam that have been described and used in [START_REF] Morassi | Damage detection and generalized Fourier coefficients[END_REF]. The specimen was suspended from above by means of two soft springs, so as to simulate free-free boundary conditions. It should be recalled that the free-free beam has a double multiplicity zero eigenvalue, corresponding to two independent rigid body motions. These vibrating modes are insensitive to damage and will be omitted in the sequel. The damage consisted of two symmetric notchs placed at the crosssection at 0.255 m from the left end (see Figure 1). Their depth was progressively increased by 1mm at a time from the undamaged configuration to a final level of damage corresponding to a depth of 6mm on both sides of the cross-section. For each level, the lowest seven natural frequencies were measured according to an impulse technique (Table 1).

In this case the only unknown quantity is the relative variation of the bending stiffness so: The use of two different frequencies is sufficient to eliminate the unknown. This case with a single unknown has been addressed in [START_REF] Morassi | Identification of a crack in a rod based on changes in a pair of natural frequencies[END_REF][START_REF] Dilena | The use of antiresonances for crack detection in beams[END_REF] and a similar relation has been used in slightly different cases (rod with a crack or a beam with a crack represented by the insertion of a massless rotational spring at the damaged cross section). In our case, the measure obtained for other low frequencies are also taken into account in the function Q p (x) which is defined by:

∆ f n f n ≈ G n (x 0 ) ∆EI∆L EIL = Lφ ′′2 n (x) φ ′′ n 2 2 ∆EI∆L EIL (21)
Q p (x) = ∑ 1≤n≤m≤p |F m,n (x)| (22)
with in the case of a single modification

F m,n (x) = ∆ f n f n G m (x) - ∆ f m f m G n (x) (23)
As in the general case, an estimate of the localization should be obtained by x m = Argmin Q p (x) and an indication of the magnitude should be given by the maximum of the function Q p (x).

In [START_REF] Morassi | Damage detection and generalized Fourier coefficients[END_REF], the aim was to localize the modification by estimating the value of the bending stiffness along the beam using Fourier type series. The partial development uses from 2 to 7 frequencies. The notch is supposed to correspond to the minimal value of the bending stiffness. The results obtain for notches of 3 mm depth are shown in Figure 2 and the localization is obtained approximatively for p = 3, 4 and 5 but completely fails for p = 2, 6, 7. As the number of used frequencies increases, a lot of perturbations appears in the estimation of the bending stiffness. Moreover this method does not succeed to localize small defects of 1 or 2 mm.

We now apply our method to this beam with a 3 mm notch. In Figure 3, the graphes of Q p and of the normed function Q p , are represented for p = 3, 4, 5, 6, 7. Due to the symmetry of the initial problem, the square of the initial mode shape and mode curvature are even functions with respect to the mid-point and so two symmetric minima can be observed.

The precision of the localization of x m is quite good for less than five eigenfrequencies and it increases with p as it can be seen on the graph of the normed function.

CONCLUSION

This paper discusses the development of a procedure for the localization of modifications from the knowledge of the transverse vibrations of an Euler-Bernoulli beam with or without tension. The first natural frequencies, the corresponding mode shapes and curvature mode shapes of the initial state, and only the first natural frequencies of the modified state are assumed to be known. Expanding the variation of the eigen vectors of the modified state on the basis of the eigen vectors of the unmodified state, a first order relation is first established in the general case, for each mode, between the relative variation of the natural frequency, of the tension, of the bending stiffness and of the mass density.

When internal modifications (of bending stiffness and/or mass density) are localized around a point x 0 , these analytical estimations of the variations of the natural frequencies as function of the external modification, allowed us to build a function of the position x along the beam, which is theoretically null when x = x 0 . This function can be normed so that its value varies between 0 and 1.

A procedure of localization based on the searching of the global minimum of this function was performed and it has proved to provide a good estimate of x 0 .

Figure 1 :

 1 Figure 1 : Experimental model of bending vibrating beam and damage configuration (Lengths in mm)

Figure 2 :

 2 Figure 2 : Estimated bending stiffness EI for 3mm with p frequencies used and damage location at x 0 = 0.225m (a) p=2, (b) p = 3, (c) p = 4, (d) p = 5, (e) p = 6 and (f) p = 7.

Figure 3 :

 3 Figure 3 : Free-free beam with a notch of 3mm depth at x 0 = 0.225m

Table 1 :

 1 Experimental frequencies of a beam with notches of varying depth

	Frequency Undamaged	1 mm	2 mm	3 mm	4 mm	5 mm	6 mm
	f 1	72.19	72.19	72.16	72.16	72.06	71.94	71.53
	f 2	198.40	198.31	198.06	197.68	196.44	194.69	189.25
	f 3	387.73	387.50	386.84	385.33	381.56	374.97	360.96
	f 4	639.72	639.38	638.41	636.28	630.81	623.78	607.03
	f 5	951.47	951.31	950.75	950.03	947.03	943.47	935.16
	f 6	1320.56	1320.56 1320.34 1320.25 1319.97 1319.97 1319.16
	f 7	1747.03	1746.81 1746.63 1746.13 1742.88 1739.41 1728.28

EWSHM 2014 -Nantes, France


Julián Sierra-Pérez

M.-A Torres-Arredondo

Guénaël Cabanes

Alfredo Güemes

Luis E Mujica

Claus-Peter Fritzen

DAMAGE DETECTION IN METALLIC BEAMS FROM DYNAMIC STRAIN MEASUREMENTS UNDER DIFFERENT LOAD CASES BY USING AUTOMATIC CLUSTERING AND PATTERN RECOGNITION TECHNIQUES

Keywords: Fiber Optics Sensors, Self-Organizing Maps, Neural Networks, Principal Component Analysis, Strain Field Pattern Recognition

In general, the change in the local strain field or global stiffness caused by damage in a structure is very small and the strain field tends to homogenize very quickly in the field close to the defect. Moreover, other environmental effects can fade the slight changes in the strain field. Only by comparing the response of the structure at several points some information about damage may be unveiled. By means of pattern recognition techniques based on the strain field, this task can be achieved. This is the basis of the strain measurements data-driven models. The main limitation of the strain field pattern recognition techniques lies in the susceptibility of the strain field to change depending on the load conditions. In the case of dynamic loads, this may reflect even a greater limitation. Robust automated techniques are required to manage these limitations. In first instance, automatic clustering techniques are needed so that data can be classified according to the load conditions and secondly, a dimensional reduction technique is needed in order to obtain patterns that often underlie from data. Within the context of this paper, a combination of Local Density-based Simultaneous Two-Level (DS2L-SOM) Clustering based on Self-Organizing Maps (SOM) and Principal Components Analysis (PCA) is proposed in order to firstly, classify load conditions and secondly, perform strain field pattern recognition. The clustering technique is the basis for an Optimal Baseline Selection. An experimental validation of the technique is discussed in this paper, comparing damages of different sizes and positions in an aluminum beam, under a set of combined loads under dynamic conditions. Strains were measured at several points by using Fiber Bragg Gratings.

INTRODUCTION

When a solid body is subjected to external stimulus (like forces, pressure or changes in temperature for example), it deforms. Deformation implies a change in the dimensions of the solid body and therefore, a strain. Commonly, the engineering structures are made of elastic materials and therefore, the deformation lie in the elastic range. Associated with this elastic deformation are changes in the stress/strain distribution across the structure and changes in the storage of elastic strain energy. As the load increases, some permanent deformations and damage (cracking) may occur, which is accompanied by a release of the stored energy and a change in the strain field. As its name suggests, the strain field refers to the distribution of strains through a region (field) of a body. When a variation occurs suddenly, in in very short distance, the intensity of stress/strain increases abruptly. This condition is known in the literature as "stress concentration". It is possible to estimate the stress/strain concentration in the region close to a disturbance, that is, the stress/strain concentration in "the near field". However, when one moves away from the near field to the far field, the scenario changes.

The strain-based damage detection rely on the redistribution of loads and strains in a structure as consequence of the damage apparition. The far field strain in a structure is usually only changed by large scale damage. Damage detection requires measurements very near to the area of damage. For this reason, methods based on fiber optic strain sensors have been usually used only for monitoring structural "hotspots" where high stresses are expected and therefore, a higher probability of damage occurrence. However, also the small changes produced in the far field can be detected by means of the use of appropriated techniques and damage assessment can be performed through the information extracted from such slight changes in the strain field.

The change in the local strain field can be studied experimentally in a very easy way. By mean of studying all the possible correlations between all the possible couples of sensors may be possible to reconstruct the strain field in structure for a given state (i.e healthy structure or a damage case). Using this information in conjunction with pattern recognition techniques would be possible to infer some change in the global behavior of the structure associated to any defect apparition.

An additional issue is the strain field is dependent of the load conditions (i.e. the inertia of the section). For this reason, it is necessary to classify the operational conditions in order to isolate different "strain fields" of a structure due to different operational conditions. This task can be achieved by mean of manual methods (i.e. if the operational conditions are known) or by mean of automatic classification methods.

STRAIN FIELD PATTERN RECOGNITION TECHNIQUES AND AUTOMATIC CLUSTERING

A time response or spectrum obtained through several experimental measurements and features extraction, which is the result of signal processing, allows to create data sets that can be seen as patterns. The study of these groups leads to damage detection based on pattern recognition techniques. The features extraction can be defined as the process of identifying damage-sensitive parameters from the gathered data. This process usually results in some form of data reduction. When pattern recognition techniques are used like a damage detection approach, it must be assumed that each pattern represents a particular damage condition or structural state. The main idea is then, to determine whether a structure is damaged or not and try to assess the severity of damage.

There are two classical categories of approaches to damage detection by means of pattern recognition: "statistical methods" and "syntactic methods". Perhaps, the statistical methods are the most used in SHM. The main reason is that over the last decade recent advancements like the application of Artificial Neural Networks (ANN) or the so called Machine Learning Techniques (MLT) have proven their effectiveness in real Structural Health Monitoring (SHM) applications. In fact, the pattern recognition techniques are themselves, a subset of MLT.

Many techniques for statistical analysis have been developed for building models under uncertain conditions. However, in SHM applications, all the measurements must be studied together in order to increase the probability of damage detection. Then, it is necessary to use multivariate statistical tools in order to getting some valuable information about the system behavior. [START_REF] Staszewski | Signal processing for damage detection[END_REF] In MLT, computational rules are inferred or "learned" on the basis of experimental data (evidences), on the contrary to what happened in classical computation where, rules was imposed only by the programmer without an automatically feedback that could modify the rules. The MLT can be divided into two major groups. The supervised MLT and the unsupervised MLT. [START_REF] Staszewski | Signal processing for damage detection[END_REF] In supervised MLT, examples of excitations and responses are required for specific postulated relationship, in this way, real associations between excitations and their associated responses might be learned and the possible errors can be corrected. However, supervised MLT present some issues when are implemented in SHM systems. The main problem is the difficulty in acquiring enough information to ensure that all the possible damage cases are covered. On the other hand, the unsupervised learning is concerned with the characterization of a data set on the basis of measurements and therefore, determining the underlying structure. Common examples of unsupervised techniques include the Learning Vector Quantization (LQV) and the Self Organizing Maps (SOM).

SOM is a class of unsupervised learning ANN, which purpose is to discover significant patterns in the input data without a target set. In its basic form, SOM allows to convert the nonlinear relationships between high dimensional data into simple geometric relationships of their image points on a low dimensional display, usually, a regular two dimensional grid of nodes. [START_REF] Kohonen | Self Organizing Maps[END_REF] SOM compresses the information while preserving the most important topological and/or metric relationships of the primary data elements on the display, it may also be thought to produce some kind of abstractions. One of the most widely used SOM methodologies is the one developed by Kohonen. The goal of the Kohonen SOM is to transform an input pattern of arbitrary dimension in a bidimensional discrete map. [START_REF] Reed | Dimensionality reduction using linear and nonlinear transformation[END_REF] The main advantage of the SOM is its ability of permitting the grouping of input data into clusters. In order to achieve this goal, the SOM internally organizes the data based on features and their abstractions from input data. SOM uses the training process to organize the two dimensional maps consisting in the topological links between neurons connected by means of weights connections. A clustering process can be formally defined as the task of partitioning a set of objects into a collection of mutual disjoint subsets.

Cabanes and Bennani proposed an efficient method of clustering based on the learning of a SOM. In the first phase the process, a standard SOM is used to compute a set of reference vector representing the local means of the data (weight vectors). Later, in a second phase, the obtained weight vectors are grouped in order to form the final partitioning. A traditional clustering method like K-means or hierarchical methods. This approach is called a two-level clustering method. Perhaps, one of the most important task in clustering is to determine the number of clusters K. This task is also known as the model selection problem. If no previous knowledge about the data structure, there is no a simple way to estimate the number of clusters [START_REF] Cabanes | Learning the number of clusters in self organizing maps[END_REF].

The methodology is based on learning at the same time the structure of the data and its segmentation by using both, distance and density information. The algorithm assumes that a cluster is a dense region of objects surrounded by a region of low density. The main advantage of this methodology lies in the ability of the algorithm to determine automatically the number of clusters during the learning process.

Then, no a priori hypothesis for the number of clusters is required. They called this particular methodology as "Local Density-based Simultaneous Two-Level Clustering" or DS2L-SOM. The main idea of the two-level clustering technique based on SOM, consists in combine the dimensionality reduction and learning capabilities of SOM with another clustering method applied to the reduced space, in order to produce a final set of clusters. The mapping between the input space and the network space is then constructed in such way that two close observations in the input space would activate two close cells of the SOM. To achieve a topological mapping, the neighbors of a winner neuron can adjust their weight vectors towards the input data vector as well, but at a lesser degree, depending on how far away they are from the winner neuron.

In the algorithm each neighborhood connection is associated with a real value v which indicates the relevance of the connected neurons. This value is called "neighborhood value" and is adapted during the learning process. For each data, both best close reference vectors are linked by a topological connection. The value of this connection will be increased, whereas the value of all other connections from the BMU will be reduced. At the end of the training, the set of interconnected reference vectors will constitute an artificial image of well separated clusters.

In order to perform online monitoring of multivariate data, diagnostic and fault detection, several methods have been reported in the literature. These methods are known as multivariate statistical projection methods. To avoid the course of dimensionality (understood like the need of low dimensionality in the feature vectors), data are often projected onto a lower dimensional feature space using specially designed mapping functions. This process is called "data reduction" or "data condensation". Among the most used projection methods is the Principal Component Analysis (PCA). PCA provides arguments on how to reduce complex data set to a smaller dimension and also reveals simpler patterns or "structures" that may be hidden under the data. The ultimate goal of the technique is to discern which data represent the most important dynamics of a particular system and which data, on the other hand, are redundant or just noise. This is achieved by determining a new coordinate space. This space is based on the covariance of the original data set. For a detailed description of PCA technique the reader is directed to [START_REF] Sierra | Damage Detection by Using FBGs and Strain Field Pattern Recognition Techniques[END_REF] PCA is a mathematical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components. This transformation is defined in such a way that the first principal component has the largest possible variance (that is, accounts for as much of the variability in the data as possible). Usually, the number of principal components can be much smaller than the number of original variables. Each succeeding component in turn has the highest variance possible under the constraint that it be orthogonal to (i.e., uncorrelated with) the preceding components.

There are statistical tools that used along with PCA, allow detection of anomalous behavior in systems. The two most common tools are Q the index (or Squared Prediction Error) and 𝑇 2 index. The index Q indicates how well each sample fits the PCA model. It is a measure of the difference between a sample and its projection in the main components retained by the PCA model. [START_REF] Sierra | Damage Detection by Using FBGs and Strain Field Pattern Recognition Techniques[END_REF], [START_REF] Jackson | Control procedures for residuals associated with principal component analysis[END_REF], [START_REF] Villez | Use of Unfold PCA for On-line Plant Stress Monitoring and Sensor Failure Detection[END_REF], [START_REF] Burgos | Active piezoelectric system using PCA[END_REF]

EXPERIMENTAL SETUP

To conduct this experiment, a square cross section aluminum beam was used. The square cross section had dimensions of 20 mm by 40 mm and 1 mm thickness. The beam was attached to a testing bench in cantilever mode. C-clamps were used for fixing the beam. The cantilever length was 120 cm. The beam can be appreciated in Figure 1a. To prevent deformation of the beam in the fixed zone, a wooden block with same dimensions as the inside of the aluminum beam was introduced along the clamping (a) (b) length. Another piece of wood was used to distribute the pressure applied with the C-clamps. A guideline system was used in order to guarantee the clamping conditions were always the same. Different hook elements were added to the beam at different positions to introduce the loads. The first set of hook elements was fixed at 24 mm of the tip. The second hook element was fixed at 250 mm from the first ones and the third one, 250 mm from the second one. A transversal member (at the end of cantilevered end) was also included in order to introduce torque loads. This member had a length of 300 mm. Four optical fibers were bonded to the beam, each one having eight FBGs. 32 sensors were used in total, one of them intended to measure the temperature. The first sensors were located at 50 mm from the clamping end. From this point, all sensors were spaced uniformly each 150 mm. For interrogating the FBGs, a four channels Micron Optics SM130 was used. The test consisted in measuring the strains under different load configurations (load cases) using different load magnitudes for each load case. First, experiments were performed for healthy structure (initial condition), and subsequently, after inducing different artificial damages in the beam (damage cases).

The procedure consisted in taking the reference level (zero strain) without any load, then, the structure was loaded. The load was released by means of a trigger device. This device also allows adjusting the displacement at the cantilever tip of the beam. In this way, it was possible to reach the reference level (zero strain) regardless of the load magnitude used. Once the reference level was reached, the load was released instantaneously guaranteeing that load conditions was always the same for all the experimental trials. The acquisition equipment was programmed to save data 200 milliseconds before any sensor reaches more than 5 µε. The strains were measured for five seconds at a sampling rate of 100 Hz. In this scenario, each experiment was performed five times and, for the undamaged case, 20 times (5 experimental trials for baseline and 15 for test it). This procedure was repeated for all the performed experiments. Six load cases were tested and three load magnitudes were used (3.25, 3.75 and 4.25 kg).

The damages consisted in holes at several positions. In total, three damages were induced in the structure. Damage 2 (D2) with 5.1 mm, damage 4 (D4) with 7 mm and damage 6 (D6) with 8 mm. It is interesting to note that the most severe damage (damage number 6) only reduces the cross area of the beam in the section where is located, by less than 7%; this gives an idea of the severity of damages sought to be detected.

RESULTS

In order to classify the information according to the operation variables (load case), the two-level clustering method based on SOM, proposed by Cabanes and Bennani was used in this work [START_REF] Cabanes | Learning the number of clusters in self organizing maps[END_REF]. The methodology can be summarized as follows:

 For each damage case data for all load magnitudes and all load conditions were put together in a matrix where the rows was experimental trials (measurements) and the columns variables.  The matrices were parametrized by subtracting from each element, the mean of its corresponding row and later, rows were randomly reordered.  Standard SOM algorithm was implemented on the baseline matrix. The SOM developed by Vesanto et al. was used for this purpose.

[9]  The Two-level clustering proposed by Cabanes and Bennani was implemented on the SOM for the baseline matrix. The DS2L-SOM algorithm developed by Cabanes was used for this purpose.

A new "improved SOM" with separated clusters (corresponding to different load cases) was obtained for the baseline. Each cluster is a new specific baseline for one load case. In this sense, this methodology can be considered as an "Optimal Baseline Selection" or OBS.  All the matrices for the damage cases were projected into the new improved SOM for the baseline. In this way, the data in those matrices was compared with the data into the baseline. As a result, the data for all damage cases could be classified according to the load cases initially separated for the baseline.  After having the different baselines corresponding to each load condition and the damaged cases classified according to the same load conditions than the baseline, a PCA study was performed.

The Ds2L-SOM algorithm automatically calculates the optimal number of cells, resulting in a 5 × 39 map. A 90º rotated view of the U-Matrix for this map can be appreciated in Figure 2a. The results for the clusters found are depicted in Figure 2b. In this case, four clusters were recognized. After the data for the baseline were clustered, the data corresponding to the different damage cases were classified according to the four clusters previously mentioned. Only the results for the cluster number 4 are presented in this article. It is important to notice that the best results of this experiments were obtained for the cluster 1 since all the measurements included in "cluster 1", really belonged to load case included in this cluster. For instance, the cluster 4 has a combination of similar samples which really do not belong to the load cases included in the cluster 4. This can be explained because the damage occurrence modifies the strain patterns in the structure and it is possible these patterns become more similar to other patterns for the healthy structure (baseline) as product of the disturbances induced by damage. The Q index is depicted in Figure 3a. As it can be seen, there is a good separation of all the damaged cases from the baseline in both models. Some undamaged indices lie outside the confidence intervals or in within 95% and 99% of confidence. However, is possible to classify these indices as abnormal indices. The 𝑇 2 index for both models can be seen in Figure 3b. Some deviations can be appreciated with respect to the baseline.

Figure 3: a) Q index with damage thresholds for 95% and 99% of confidence (dashed line and solid line respectively). b) 𝑇 2 index with damage thresholds for 95% and 99% of confidence (dashed line and solid line respectively).

Finally, Q index vs. 𝑇 2 is presented in Figure 4a and the the two first scores are presented in Figure 4b.

Figure 4: a) Q index vs. 𝑇 2 index with damage thresholds for 95% and 99% of confidence (dashed line and solid line respectively). b) Projection into the two first principal components with damage thresholds for 95% and 99% of confidence (dashed line and solid line respectively).

CONCLUSIONS

In order to achieve the first level of SHM (damage detection) by using strain measurements in a structure under different load cases and different load magnitudes, readings gathered from FOS and by means of application of PCA and different damage indices, several scenarios were experimentally analyzed.

A PCA baseline model was built using the responses for the healthy structure. In subsequent steps, several experiments were performed by inducing different artificial damages to the structures. All these experimental data were projected into the PCA model, for which, a selected number of principal components were retained. Finally, different damage indices and thresholds were calculated.

The FOS offer unique advantages including small size, easy of embedment in composite structures, immunity to electromagnetic interference, excellent multiplexing capabilities, excellent accuracy and sensitivity. All this advantages make the FOS the ideal choice for strain-based SHM techniques. Precisely these advantages convert the strain-based techniques, based in turn in FOS, in a promising field of research in SHM. FBGs showed to be very sensitive to small strain changes in the structure, which makes it suitable for the proposed technique.

By including several load conditions at the same time in one model, caused a decrease in the damage detection sensitivity since a "more general" model was built instead of "specific models" for each load condition. This propitiated the necessity of developing the proposed classification and clustering techniques and the "Optimal Baseline Selection" methodology. The automatic classification technique based on SOM and DS2L-SOM algorithms probed to be very accurate and promising when classifying the data according to operational variables is not possible.

In all the experiments performed it was possible to detect deviations among different indices associated to the baseline (and the undamaged case) and the different damage cases. The Q index showed more sensitivity in this study to detect anomalies. On the other hand, the 𝑇 2 showed a good potential to discern if a model is well defined since it is able to give an idea of the variability inside the model.

Figure 1 :

 1 Figure 1: a) Aluminum beam. b) Location of FOS at cross section. Dimensions in mm.

Figure 2 :

 2 Figure 2: U-matrix and cluster groups. a) U-matrix. b) Cluster groups.

EWSHM 2014 -Nantes, France

EWSHM 2014 -Nantes,

ACKNOWLEDGEMENTS

The research included in this document was partially supported by the Spanish Ministry of Research through the project DPI2011-28033-C03-03


Francis Berghmans
email: fberghma@vub.ac.be

Thomas Geernaert

Camille Sonnenfeld

Sanne Sulejmani

Geert Luyckx

Nicolas Lammens

Joris Degrieck

Karima Chah

Hugo Thienpont

Microstructured

Microstructured Optical Fiber Bragg Grating Sensors for Structural Health Monitoring Applications

Keywords: microstructured optical fiber, optical fiber sensor, fiber Bragg grating, structural health monitoring

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Microstructured optical fiber Bragg grating (MOFBG) sensors are becoming increasingly popular owing to the peculiar characteristics of microstructured optical fibers (MOFs) that cannot be achieved using conventional optical fiber technology. More specifically, the design flexibility of such microstructured fibers allows developing sensors that exhibit selective sensitivities to e.g. axial strain, transverse strain or even shear stress, whilst being negligibly cross-sensitive to temperature changes. This is a great asset for structural health monitoring (SHM) applications. Section 1 of this paper will shortly summarize the state-of-the-art on the use of MOF based sensors for SHM purposes.

Section 2 focuses on a specific microstructured fiber design to which we refer as 'Butterfly' MOF. This fiber is highly birefringent and encodes the transverse strain into the spectral distance  between the two Bragg peaks reflected by a fiber Bragg grating (FBG) fabricated in this fiber (Figure 1). Since the birefringence of that MOF is not sensitive to temperature changes, the transverse strain measurement is independent of temperature variations. In Section 3 we show that by using this MOF, we can simultaneously measure the normal strain in 3 dimensions inside a composite -the axial strain, the transverse in-plane strain and the transverse out-of-plane strainwith µstrain resolutions using commercially available FBG interrogator equipment. Finally, Section 4 deals with the potential of our MOFBG sensor for other structural health monitoring related applications, in particular cure monitoring during and residual strain quantification following composite material manufacturing, shear stress measurements in lap joints and temperature insensitive vibration monitoring.

STATE-OF-THE-ART OF MOF FOR SHM-RELATED APPLICATIONS

As stated above, MOF technology allows achieving optical fibers with sensing characteristics that cannot be obtained with conventional fiber technology. MOF sensing approaches and techniques have been reviewed for example by Frazao et al. [START_REF] Frazao | Optical sensing with photonic crystal fibers[END_REF], Canning [START_REF] Canning | Properties of Specialist Fibres and Bragg Gratings for Optical Fiber Sensors[END_REF] and Pinto et al. [START_REF] Pinto | Photonic Crystal Fibers for Sensing Applications[END_REF]. With respect to SHM related applications, MOF based sensors have been mostly combined with gratings inscribed in these fibers to enable measurements of mechanical quantities. A far from exhaustive list of examples includes pressure, transverse force, strain and bend sensing, with emphasis on avoiding strain and temperature cross-sensitivity or discriminating between strain and temperature effects [START_REF] Frazão | Discrimination of strain and temperature using Bragg gratings in microstructured and standard optical fibres[END_REF]- [START_REF] Tenderenda | Longitudinal strain sensing with photonic crystal fibers and fiber Bragg gratings[END_REF]. Grating inscription techniques in MOF have been reviewed recently in [START_REF] Berghmans | Challenges in the fabrication of fibre Bragg gratings in silica and polymer microstructured optical fibres[END_REF].

THE BUTTERFLY MOF AND GRATINGS WRITTEN THEREIN

We have obtained the cross-section of our highly birefringent Butterfly MOF following extensive modeling that considered both the optical and the mechanical characteristics of the fiber [START_REF] Martynkien | Highly birefringent microstructured fibers with enhanced sensitivity to hydrostatic pressure[END_REF]. We have modeled the sensitivity of the phase modal birefringence B to external thermo-mechanical perturbations with the commercially available COMSOL Multiphysics® software [17].

The phase modal birefringence B is defined as:

( ) (1)
where  x and  y are the propagation constants of the orthogonally polarized fundamental modes propagating through the MOF. When fabricating a FBG with a period  in a birefringent MOF, the grating will return two reflection peaks with a wavelength separation  (see also Figure 1) given by: (

The polarimetric pressure and temperature sensitivities K p and K T are defined as:

( ) (3)
If we want to obtain a MOF that is very sensitive to pressure whilst being -in as much as possible -insensitive to temperature changes, the figure-of-merit K p /K T should be maximized. To do so one has to be able to calculate K p and K T . A full description of how this can be done is out of the scope of this paper. We refer to [START_REF] Martynkien | Highly birefringent microstructured fibers with enhanced sensitivity to hydrostatic pressure[END_REF], [START_REF] Berghmans | Applying optical design methods to the development of application specific photonic crystal fibres[END_REF] and [START_REF] Sulejmani | Control Over the Pressure Sensitivity of Bragg Grating-Based Sensors in Highly Birefringent Microstructured Optical Fibers[END_REF] for a more detailed discussion.

The "Butterfly" fiber that we have obtained, with an ideal and as-built cross-section as shown in Figure 2, fulfills our requirements in terms of sensitivity, fiber manufacturability and optical guiding properties.

THREE-DIMENSIONAL STRAIN MEASUREMENTS IN COMPOSITE MATERIALS

Due to its particular structure, the sensitivity of the butterfly MOFBG to transverse strain varies in a sinusoidal manner with the angular orientation α of the microstructure with respect to the direction of the applied load. It is most sensitive when the load is applied along α = 90°, which corresponds to the direction of the slow-axis indicated in Figure 2(a) [START_REF] Sonnenfeld | Microstructured Optical Fiber Sensors Embedded in a Laminate Composite for Smart Material Applications[END_REF]. This dependence of the sensitivity on the angular orientation can be exploited to quantify the strain along the 3 directions of space inside a carbon fiber reinforced polymer (CFRP) composite material. To do so, one needs to relate the normal strains in the 3 directions of space in the composite material to the measured changes in the 4 Bragg wavelengths returned by two closely spaced MOFBG sensors embedded in a composite coupon with different angular orientations, as schematically illustrated in Figure 3.

The relation between the strains and the temperature in the material at the location of the sensors and the Bragg wavelength shifts is given by Equation (4).

( ) [ ] [ ] ( ⁄ ⁄ ⁄ ⁄ ) (4)
This equation uses the Transfer Coefficient matrix [TC] that relates the temperature and strain distributions in the host composite material to those in the optical fiber sensor in the coordinate system (1 h ,2 h ,3 h ) of the host composite material [START_REF] Luyckx | Multi-axial strain transfer from laminated CFRP composites to embedded Bragg sensor: I. Parametric study[END_REF], [START_REF] Voet | Multi-axial strain transfer from laminated CFRP composites to embedded Bragg sensor: II. Experimental validation[END_REF]. More specifically, is the axial strain, is the transverse in-plane strain and is the transverse out-of-plane strain (see also Figure 3). We obtained the [TC] matrix using extensive finite-element modelling. The sensitivity matrix [K] relates the temperature and strain changes in the optical fiber core to the output of the optical signal [START_REF] Lawrence | A fiber optic sensor for transverse strain measurement[END_REF], [START_REF] Nelson | Determination of the Kmatrix for the multi-parameter fiber grating sensor in AD072 fibercore fiber[END_REF]. This sensitivity matrix is obtained in an analytical form.  B1,1' and  B1,2' denote the initial unstrained Bragg wavelengths of the first MOFBG, and  B2,1' and  B2,2' are the initial unstrained Bragg wavelengths of the second MOFBG. ∆ B1,1' , ∆ B1,2' , ∆ B2,1' and ∆ B2,2' are the shifts of these Bragg wavelength peaks that stem from thermo-mechanical load applied to the composite material. The coordinates 1' and 2' associated with the two Bragg wavelengths returned by one MOFBG correspond to the directions of the slow and fast axes of the MOF, respectively (see Figure 2). , and are the results of our finite element simulations. is the strain in the direction through the thickness, calculated from the load cell reading and from the material parameters of the laminate.

Figure 4 (b) shows a typical result obtained for a transverse out-of-plane load applied to a CFRP coupon in isothermal conditions, as illustrated in Figure 4(a). We obtain a good agreement between our finite element modelling results and the strains measured and derived using Equation (4). The calculated axial strain in the composite material shows a relatively larger relative error of up to 50% when compared to the experimental data of the strain gauge. However, and since the axial strains are very low, this corresponds to a difference of only 24 µε in the MOFBG configuration 0°/90° as sketched in Figure 3. The values of the transverse in-plane strain also exhibit some differences with respect to the measured strains, which can be explained by their low values. This stems from the fact that the expected strains along the 1 h -and 2 h -directions have the same magnitude as the strain measurement resolutions of about 5 µε (these resolutions are not discussed in this paper). The calculated transverse out-of-the plane strains exhibit an excellent agreement with the measured strains, with a relative error around 10%.

External ply External ply

Center plies

1 h 2 h 3 h

OTHER SHM RELATED APPLICATIONS OF THE BUTTERFLY MOF

Composite cure monitoring and residual strain quantification

In earlier work we have shown that it is possible to efficiently integrate Butterfly MOFBG based sensors in CFRP material to monitor its manufacturing process. The sensors allow assessing the onset of polymerization and quantifying the internal residual strain in the composite material that was created as a result of the vacuum bagging autoclave process [START_REF] Berghmans | Opportunities for Structural Health Monitoring of Composite Material Structures with novel Microstructured Optical Fiber Sensors[END_REF]- [START_REF] Sonnenfeld | Cure cycle monitoring of laminated carbon fiber-reinforced plastic with fiber Bragg gratings in microstructured optical fiber[END_REF].

Figure 5(a) shows a scheme of the CFRP specimen in which the Butterfly MOFBG was integrated and the evolution of the Bragg wavelength separation  during the cure cycle. The temperature variation recorded by thermocouple TC6 placed in close vicinity to the MOFBG is shown as well. Figure 5(b) shows a first drop of  (part B) which corresponds to the polymerization of the sample. The cooling phase (part D) features a large decrease of associated with the build-up of residual strains during the consolidation phase. Since the phase modal birefringence of the Butterfly MOF is inherently insensitive to temperature, the changes in are due to thermally induced transverse strain resulting from the changing strain state in the CFRP material as it cures.

In part B  decreases with 22 pm which corresponds to a compressive transverse strain of about -100 µstrain. In region C, the sensor signal remains constant meaning that no transverse strain is measured by the MOF sensor. One can therefore reasonably assume that the cure reaction has been completed and that the composite material is formed. The cooling down to room temperature (part D) is associated with a large decrease of and thus with the development of substantial transverse residual strain of about -1100 µstrain in the composite material. The conversion from change in  to actual transverse strain values is obtained using data from previous sensor calibrations, as explained in [START_REF] Sonnenfeld | Cure cycle monitoring of laminated carbon fiber-reinforced plastic with fiber Bragg gratings in microstructured optical fiber[END_REF].

Shear strain monitoring in glued lap joints

We have also shown that it is possible to measure shear strain in glued joints with a butterfly MOF that is embedded such that the transverse strain sensing axes of the MOF are aligned with the directions of principal stress in a shear loaded single lap adhesive joint (SLJ) [START_REF] Sulejmani | Shear stress sensing with Bragg gratingbased sensors in microstructured optical fibers[END_REF]. The scheme of the embedded sensor and the measured change of the Bragg wavelength separation with applied load are shown in Figure 6. We obtained a shear stress sensitivity of about 60 pm/MPa, which corresponds to a shear strain sensitivity of 0.01 pm/µε. Compared to conventional birefringent fibers and as discussed in [START_REF] Sulejmani | Shear stress sensing with Bragg gratingbased sensors in microstructured optical fibers[END_REF], our dedicated MOF design has a fourfold larger sensitivity. These results also show that by changing the angular orientation of the butterfly MOF sensor it can be used for either shear strain sensing or for transverse strain sensing, which provides opportunities for multi-axial strain sensing with the same type of sensor [START_REF] Sulejmani | Fiber Bragg grating-based shear strain sensors for adhesive bond monitoring[END_REF].

Temperature insensitive vibration monitoring

We have also built a polarimetric vibration sensor based on the Butterfly MOF [START_REF] Chah | Temperature-insensitive polarimetric vibration sensor based on HiBi microstructured optical fiber[END_REF]. A section of the MOF was subjected to a transversal force by means of a transducer that locally affects phase modal birefringence. This transducer is nothing else but a mechanical beam that transversally crushes the fiber over a 3 mm long section. The force is applied along one of the fiber's birefringence axes. When the transducer vibrates with a certain frequency, it induces a transversal force and hence a stress birefringence that also varies with that same frequency. To analyze the sensor response we used the so-called Stokes formalism (see also [START_REF] Chah | Temperature-insensitive polarimetric vibration sensor based on HiBi microstructured optical fiber[END_REF] for more details). The varying birefringence leads to a phase shift induced between the two polarization eigenmodes and results in a time varying state of polarization (SOP) at the output of the fiber. A polarization analyzer converts the varying SOP fluctuation into a varying optical output power. Depending on the vibration spectrum (frequency and acceleration amplitude), the stress induced birefringence varies as well as the optical output power. This allowed showing that the high sensitivity of the Butterfly MOF to transversal load leads to a more sensitive vibration sensor than 'Panda' type polarization maintaining fiber (PMF). The effect of temperature variations was also much less detrimental to the sensor response than when using Panda PMF. For an input sine-wave vibration, the peak-to-peak optical output variation of the sensor was only 10% over a temperature range from 25°C to 120°C.

CONCLUSION

We have introduced the principles of operation of a fiber Bragg grating based sensor inscribed in a highly birefringent microstructured optical fiber that has been designed so as to feature an enhanced sensitivity to transverse strain whilst being almost insensitive to temperature changes. We have illustrated how this sensor can be used in various applications that may support structural health monitoring, in particular three-dimensional strain measurements inside composite materials, cure process monitoring of and residual strain quantification in composite materials, shear stress monitoring in adhesive joints and temperature insensitive vibration monitoring.

Figure 1 :

 1 Figure 1: Sensing principle of the butterfly MOFBG. Load applied transversally to the fiber is encoded in the spacing  between the two Bragg peaks. (a) Response of the sensor to temperature changes or axial strain -both peaks move in the same way and remains unchanged. (b) Response of the sensor to transverse load -the peaks move in opposite direction and  changes.

Figure 2 :

 2 Figure 2: (a) Ideal cross-section of the Butterfly MOF (b) Ideal parameters of the Butterfly MOF (c) Scanning Electron Microscope photograph of the cross-section of the fabricated Butterfly MOF. (d) Close-up of the core region of the MOF.

FastFigure 3 :Figure 4 :

 34 Figure 3: Illustration of 2 closely space MOFBGs embedded in the middle of a laminate composite (not drawn to scale) with the longitudinal direction of the optical fibers aligned with the reinforcement fibers of the embedding plies. The left MOFBG is oriented with α = 0°, whilst the right sensor is oriented with α = 90°.The 2 MOFBGs are stripped over a few centimeters at the grating location. The axis system (1 h ,2 h ,3 h ) represents the coordinate system within the host composite material. The directions of the carbon reinforcement fibers in the plies are indicated by the black double arrows.

Figure 5 :

 5 Figure 5: (a) Schematic of the optical fiber network embedded in the composite specimen. The coordinate system of the laminate is indicated. TC stands for thermocouple. (b) Changes in the temperature and in the peak separation  during the entire cure cycle. The temperature profile is recorded with thermocouple TC6(adapted from[START_REF] Sonnenfeld | Cure cycle monitoring of laminated carbon fiber-reinforced plastic with fiber Bragg gratings in microstructured optical fiber[END_REF]).

Figure 6 :

 6 (a) Configuration of the tested and modeled SLJ with an optical fiber embedded in the center of the adhesive layer. (b) The Bragg peak separation increases due to tensile loading with a sensor response of 67.4 pm/kN. Results from 2D FEM modeling of the SLJ are in very good agreement with the experimental results [28].

EWSHM 2014 -Nantes, France

ACKNOWLEDGEMENTS

The authors would like to acknowledge financial support from the Agency for Innovation by Science and Technology (IWT) for funding this research with the SBO Project grant 120024 'Self-Sensing Composites'. The authors also acknowledge the Research Foundation -Flanders (FWO), the Methusalem and Hercules Foundations Flanders, and the COST action TD1001.