Yann Goueffon
email: yann.goueffon@isae.fr

Catherine Mabru

Michel Labarrère

Laurent Arurault

Claire Tonon

Pascale Guigue

Mechanical behavior of black anodic films on 7175 aluminium alloy for space applications

Keywords:

Because of their low outgassing and their thermo-optical properties, black anodized aluminium parts are often used near optical instruments to manage thermal control in space applications. However, critical cases of flaking of the film were observed after simulated thermal ageing. To understand the mechanisms leading to flaking, the influence of the initial porosity of the film on its mechanical behavior during and after the black anodizing process has been investigated. The decrease of limit tensile stress with the porosity, the coloring and the sealing combined to thermal stresses due to the difference of thermal expansion coefficients between film and substrate have been shown to cause crazing in particular conditions. For high initial porosity films, thermal cycling ageing has a detrimental influence on the adhesion measured by scratchtesting. Numerical simulation has been used to simulate the combined effects of thermal stresses and film cracking on the stress field at the interface.

Introduction

Black anodic films on aluminium alloys are used in space industry since many years. Such treatments provide specific characteristics to surfaces (Solar absorptance α s N 0.93 and normal emittance ε n N 0.90) managing passive thermal regulation. The process used to obtain these black films is based on porous anodizing. This is not a deposit of a coating but an electrochemical oxidation of the surface of the aluminium alloy. Therefore, the adhesion of anodic film on its substrate has always been considered very strong. Before the launch, parts of satellites are submitted to thermal cycles under vacuum to simulate the space environment. Cases of flaking of black anodic films have been observed after those tests when pulling-off tapes used to fix thermocouples [1].

In this context, the aim of the present paper is to better understand mechanisms leading to flaking. In order to precisely assess the mechanical behavior of the film, depending on its initial porosity, residual stresses were first measured after each step of the process. The coefficient of thermal expansion was then evaluated by beam bending analysis to determine the thermal stresses occurring during processing and ageing. After, the tensile strength was evaluated thanks to four-point bending tests. Finally, the evolution of adhesion during ageing (thermal cycling) was followed by scratch tests. A finite element model was developed to simulate the combined effects of thermal stresses and film cracking on the stress field at the interface film/substrate.

Experimental procedure

Material

The substrate material was the 7175T7351 aluminium alloy, often used in space industry. Its chemical composition in weight percent is: 1.62%Cu, 0.12%Fe, 2.42%Mg, 0.01%Mn, 0.06%Si, 5.75%Zn, 0.041%Ti, 0.21%Cr and Al the remainder. Its Young modulus is E s = 72 GPa, its Poisson coefficient is ν s = 0.33 and its coefficient of thermal expansion is α s = 2.34 × 10 -5 K -1 .

Black anodizing process

The process of black anodizing followed the ESA Standard [START_REF]ESA ECSS-Q-70-03A, Black anodizing of metals with inorganic dyes[END_REF] for spacecraft design involving four consecutive steps: surface pretreatments, anodizing, coloring and sealing. Samples were rinsed with distilled water after each step.

The alloy sheet (3 × 20 × 40 mm or 10 × 20 × 160 mm) was degreased with ethanol and etched in Na 2 CO 3 (6.2 g/L) and Na 3 PO 4 (12.5 g/L) aqueous mixed solution for 5 min at 93 °C and neutralized in aqueous HNO 3 (50%v/v) for 3 min at room temperature.

The aluminium sheet was used as anode and a lead plate as counter-electrode in the electrochemical cell. The anodizing was run for 60 min in the galvanostatic mode (J a = 1.2 ± 0.1A/dm²) using a sulfuric acid solution (150 g/L) thermally regulated. Changing the anodizing temperature between 5 °C and 25 °C allow controlling the initial porosity (i.e. before coloring and sealing) (Fig. 1) [START_REF] Aerts | [END_REF][4][5]. Porosity was quantified by image analysis after observations with a fieldemission gun scanning electron microscope (FEG-SEM JEOL JSM6700F device). The film thickness is around 20 µm to insert enough black dyes during the coloring.

The anodized part was colored by immersion in two successive baths: firstly for 15 min in a solution of cobalt acetate (200 g/L) regulated at 43 ± 2 °C, and then for 10 min in ammonium hydrosulphide (30 g/L) at room temperature.

The sealing step occurred for 25 min in a bath at 98 ± 2 °C made up of nickel acetate (NiCH 3 COO, 4H 2 O) and boric acid (5 g/L both).

Residual stresses measurements in the film

Residual stresses in the film were measured by beam bending analysis. Thin substrates (200 µm) were prepared by a grinding on SiC paper (240-1200 grits) and polishing with diamond pastes from 6 µm to 1/4 µm; they were anodized on one side (20 mm × 20 mm). The curvatures of the samples induced by stresses were measured ex-situ on 17.5 mm with a Mahr Perthometer Concept with a sensor MFW250.

Coefficient of thermal expansion (CTE) determination

To evaluate the CTE of the film, curvature measurements were realized at different temperatures. Samples were warmed up to 130 °C with a heating stage and the film temperature was controlled with a thermocouple. Curvature measurements were performed during cooling (1 °C/min).

Film tensile strength measurements

The samples (160 × 20 × 10 mm) were anodized on a limited area. The tensile strength was determined with four-point bending tests performed on a tensile testing device (Adamel DY26) (Fig. 2). Acoustic emission measurements were used to determine the cracking load of the film (L c ). Post-experiment SEM observations were performed to detect the presence of cracks in the film.

Adhesion measurements

Adhesion measurements were evaluated using a scratch-test device (CSM Revetest instrument) with a diamond stylus (Rockwell, 200 µm radius tip). Scratch tests were configured with an increasing normal load (loading speed: 30 N/min; advance speed: 5 mm/min). The resulting scratch-print of 5 mm length was observed by SEM to determine the corresponding normal load. Given values are averages of six different scratches on each sample.

Numerical simulation

A finite element simulation was used to evaluate mechanical behavior of a thin film on its substrate. SAMCEF was used to perform isotropic thermo-elastic calculations with 2-dimensional plane strain hypotheses. The geometry used was a substrate (thickness of 3 mm) with an anodic film of 20 µm. A straight crack perpendicular to the surface of 1 micron width and a variable length was added through the film. Mesh size has been validated by comparing K I with theoretical values. Cases of crazing were sometimes observed just after the sealing step. Thus, the impact of the presence of a crack through the coating on the interfacial stresses has been observed when thermally loaded. For a preliminary analysis the interfacial stress is defined as the difference between stresses (direction parallel to the surface) in the coating and in the substrate near the interface.

Results and discussion

Residual stresses

Residual stresses have been measured after each step of the process for films with different porosities. The radius of curvature R of the thin sample is linked with stresses in the film by relation (1) [6]. Stresses are considered to be uniform in the film:

σ Film = E f e f 6Rð1-ν f Þ 1 + 4αβ + 6α 2 β + 4α 3 β + α 4 β 2 ð1 + αÞð1 + αβÞ ð1Þ
with α = es e f and β = Esð1-ν f Þ E f ð1-νsÞ e s and e f are the thicknesses of the substrate and the film respectively. Poisson coefficient of the film, ν f , was assumed to be 0.28 [7]. The Young modulus of the film, E f , was measured by nanoindentation as a function of the porosity and will be the aim of another forthcoming paper [8]. To sum up, the modulus decreased from 80 GPa to 20 GPa for films with porosities increasing from 10% to 50%. The results are reported in Fig. 3, the percentages represent initial porosities.

After the anodizing step, residual stresses are compressive in the film, mainly due to the difference of molar volume between films and substrates [9,10]. Alwitt et al. [START_REF] Alwitt | AAIA Technical Papers[END_REF][START_REF] Alwitt | [END_REF] found tensile stresses (around 15 MPa) after anodizing of 5657 aluminium alloy. They have shown that residual stresses are independent from the film's thickness but are depending on the substrate (5XXX and 6XXX series). The reason for this dependence is not yet well known. Nevertheless, alloying elements, especially copper [13], modify the growth mechanism of the films. The copper content in the AA7175 being about 1.2%, the change of residual stresses could be explained by modification of the film morphology.

After insertion of dyes in the porosity, residual stresses are tensile and increase with initial porosity (i.e. with the quantity of introduced dyes).

The sealing step performed in boiling water (with addition of nickel acetate) is usually considered as a hydration of alumina, inducing a volume expansion of the film and thus large compressive residual stresses. Alwitt et al. [START_REF] Alwitt | AAIA Technical Papers[END_REF] have shown that the compressive stress increases with the sealing time confirming the relation between compressive stresses and the quantity of hydrated alumina.

For initial porosities higher than 30%, stresses are negligible. Actually, residual stresses were released by the presence of cracks through the films with high porosities (Fig. 1).

Causes of crazing

Coefficient of thermal expansion (CTE)

Relation (1) was used to calculate the stresses in the film at different temperatures. The experimental relation between stresses and temperature was established and assumed to be linear in the range of 20-70 °C (stable water content, homogeneous temperature).

The CTE of the film was deduced from relation (2) that gives the stresses in a coating due to a variation of temperature:

Δσ = E f ð1-ν 2 f Þ ½α s ð1 + ν s Þ-α f ð1 + ν f ÞΔθ ð2Þ
with θ the temperature and α f the CTE of the film. Stresses are here considered homogenous in the film.

As an example, Fig. 4 represents stresses in a film versus the temperature. Same results were obtained whatever the porosity. The CTE of uncolored and unsealed films was found to be nearly five times lower (5 × 10 -6 K -1 ) than the aluminium's one confirming previous works of Alwitt et al. [START_REF] Alwitt | AAIA Technical Papers[END_REF] who found the same ratio. In addition, stresses measured once back at ambient temperature are equivalent to residual stresses before heating.

After coloring, the CTE of anodic films is nearly ten times lower than the substrate's one. No modification of residual stresses (at 20 °C) was induced by the test.

During heating and cooling of colored and sealed films, the slope is comparable with the sample colored but not sealed proving that the CTE is not modified by hydration. However, compressive residual stresses are released during the test because dehydration may occur at a temperature over 100 °C [START_REF] Mc Clung | AAIA Technical Papers[END_REF].

The large difference between CTE of the substrate and the film is leading to tensile stresses in the film during sealing at 100 °C. This loading may explain cases of crazing. Nevertheless, the CTE was not significantly varying with the initial porosity.

Tensile strength

All the coated area was submitted to a uniform deformation at the rupture of the film during bending:

ε R = L C ah 4I S E S ð3Þ
a and h are defined on Fig. 1; I S is the quadratic moment of the sample. The film was considered purely elastic and its limit tensile stress (σ R ) is:

σ R = E f ε R + σ residual ð4Þ
Each given value is the average of three different tests.

With increasing porosity, the tensile strength of the anodic film was expected to decrease as described by Ryshkewitch-Duckworth [START_REF] Duckworth | [END_REF]. In addition, a collapse of the tensile strength of ceramics for porosities between 30 and 50% has been shown by Knudsen et al. [16] corresponding to the transition from isolated to interconnected porosity. Nevertheless, Fig. 5 shows that the tensile strength of uncolored and unsealed films increases for porosities from 10% to 30%, while the strength decreases at higher porosities. The porosity was controlled by changing the anodizing temperature that may change the composition of the anodic film and may explain the increase of limit tensile stress. For porosities higher than 30%, the effects of the porosity become preponderant and the tensile strength decreases.

The limit tensile stress is decreased by the coloring step. The quantity of dyes included in the film increases with the porosity, that explains the more important loss of tensile strength for high porosities. The reaction of precipitation occurring inside pores during dying is [17]:

CoðCH 3 COOÞ 2 þ ðNH 4 Þ 2 S→CoS þ 2CH 3 COONH 4 ð5Þ
However, an additional chemical reaction between acetates and the film could explain this weakening. Further tests such as coloring with cobalt nitrates could allow assessing this point.

The hydration of the film enhances this tendency. The transformation occurring during sealing is:

Al 2 O 3 þ ðn þ 1ÞH 2 O→2AlOOH þ nH 2 OðnN1Þ ð6Þ
The resulting hydrated aluminium oxi-hydroxide often looks like a gel having a negligible mechanical resistance. For porosities of 40% and 50%, the film was crazed after the sealing step at 100 °C and therefore measurements are not comparable.

Considering the previous results, at the beginning of the sealing at 99 °C, films with 40% porosity are under tensile thermal stresses of approximately 100 MPa. Added with the tensile residual stresses after coloring (35 MPa), the global stresses in the film are higher than its limit tensile stress (100 MPa).

Thus, the differential thermal dilatations associated with a very low limit tensile stress of highly porous anodic films explains the crazing phenomenon.

Ageing and flaking

The evolution of the adhesion of the two crazed samples was observed after ageing. Thermal cycling between -80 °C and + 80 °C (dwells of 10 min, warming/cooling speed of 10 °C/min) were performed under dried nitrogenous atmosphere. Self-flaking of particles was observed after ageing on samples with 50% porosity.

Scratch-testing was used to compare evolutions of the critical load defined as the lower load provoking a lateral flaking along the scratch [18]. The critical load is constant for black anodic films with an initial porosity of 40% whatever the number of cycles performed (Fig. 6). Thermal cycles between -80 °C and +80 °C have no impact on the measured adhesion. On the contrary, a loss of adhesion occurred after the first thermal cycle for samples with an initial porosity of 50%.

The difference of CTE between the film and the substrate results in tensile thermal stresses in the film at temperatures higher than 20 °C. In addition, the water loss of the sample during ageing would result in tensile stresses in the film. Cracks in samples with 50% of initial porosity may propagate under thermal loading resulting in the loss of adhesion observed.

Numerical simulation

Thermo-mechanical properties of colored and sealed films were used. A thermal load of +80 °C was applied. The influence of the presence of a crack in the film on the variations of interfacial stresses σ int withcrack -σ int withoutcrack σ int withoutcrack is showed on Fig. 7.

Two different behaviors depending on the porosity (i.e. elasticity) of the film are observed. For porosities lower than 30%, the presence of a short crack (b13 µm) has no impact on the interfacial stress considered. When cracks become longer (17 µm), the interfacial stress increases. If cracks are long enough, tensile stresses in front of the crack are transmitted by the rigidity of the film until the interface. Interfacial stresses are enhanced.

For films showing porosity higher than 40%, the presence of cracks causes a high deformation of the film with a low increase of stresses (high elasticity). Tensile stresses are in majority transmitted to the more rigid substrate. Interfacial stresses are released.

The width of cracks has always been found to increase during ageing. This suggests that heating during thermal cycling propagates cracks through the film. However it seems to have a limited impact on the interfacial stresses for high porosity films. The transition between crazing and flaking will be investigated by adding an interfacial crack and observing stress field evolution during thermal cycling.

Conclusion and prospects

Initial porosity of films has been proved to largely influence the mechanical behavior of black anodic films. The coloring step results in low tensile residual stresses and decreases the CTE of the film. The film expansion by hydration during sealing results in large compressive stresses. After heating and water desorption, the behavior is the same as a non-sealed film.

Highly porous films have the lower limit tensile stress, affected by coloring and sealing. Combined to tensile stresses induced by the differential thermal dilatations, it results in crazing of highly porous films during sealing.

Ageing between -80 °C and + 80 °C has an impact on the adhesion only on samples with an initial porosity of 50%. In this case, the low tensile strength permits the crack propagation during these thermal cycles.

Numerical simulation showed that the presence of cracks is less critical for the interface in highly porous films. Further numerical investigations simulating thermal cycles and including an interfacial crack will be performed to understand the transition between crazing and flaking.

Fig. 1 .

 1 Fig.1. FEG-SEM surface views of anodic films with different anodizing temperatures before and after coloring and sealing at nano and meso scales.

Fig. 2 .

 2 Fig. 2. Schema of the four-point bending test.

Fig. 3 .

 3 Fig. 3. Residual stresses measured after each step of the process (anodizing, coloring, sealing) for anodic films with different initial porosities.

Fig. 4 .

 4 Fig. 4. Stress in a film with an initial porosity of 10% as a function of the temperature just after anodizing, after coloring and after sealing.

Fig. 5 .

 5 Fig. 5. Tensile strength of the anodic film at each step of the process and depending on the initial porosity.

Fig. 6 .

 6 Fig.6. Evolution of the scratch-test critical load with the number of thermal cycles for samples of 40% and 50% of initial porosities.

Fig. 7 .

 7 Fig. 7. Variations of the interfacial stress in the direction of the interface due to the presence of a crack through the porous film at 80 °C.


Joseph Morlier
email: joseph.morlier@isae.fr

Guilhem Michon

Virtual vibration measurement using KLT motion tracking algorithm

Keywords: Vibration measurement, video motion tracking, virtual sensor, structural dynamics

This paper presents a practical framework and its applications of motion tracking algorithms applied to structural dynamics. Tracking points ("features") across multiple images is a fundamental operation in many computer vision applications. The aim of this work is to show the capability of computer vision (CV) for estimating the dynamic characteristics of two mechanical systems using a non contact, marker less and simultaneous Single Input Multiple Output (SIMO) analysis. KLT (Kanade-Lucas-Tomasi) trackers are used as virtual sensors on mechanical systems video from high speed camera. First we introduce the paradigm of virtual sensors in the field of modal analysis using video processing. To validate our method, a simple experiment is proposed: an Oberst beam test with harmonic excitation (mode 1). Then with the example of helicopter blade, Frequency Response Functions (FRFs) reconstruction is carried out by introducing several signal processing enhancements (filtering, smoothing). The CV experimental results (frequencies, mode shapes) are compared with classical modal approach and FEM model showing high correlation. The main interest of this method is that displacements are simply measured using only video at FPS (Frame Per Second) respecting the Nyquist frequency.

INTRODUCTION

In mechanical engineering, modal analysis is widely used to assess mechanical systems by means of vibration measurement. Classical modal analysis uses impact hammer or shaker to excite the structure and accelerometer to measure the response. It is time and money consuming.

The goal of the presented work is to develop a method based on instrumentation with an optical camera working with intelligent software in order to continuously assess the dynamic parameters of the structure. Previous works [START_REF] Olaszek | Investigation of the dynamic characteristic of bridge structures using a computer vision method[END_REF][START_REF] Patsias | Damage detection using optical measurements and wavelets[END_REF][START_REF] Poudel | Structural damage detection using digital video imaging technique and wavelet transformation[END_REF][START_REF] Lee | A vision-based system for remote sensing of bridge displacement[END_REF] obtained modal parameters by introducing real targets on the structure or by studying simple structures in ideal conditions. Real time displacement measurement have been done using different approaches of digital image processing techniques (texture recognition algorithm) on a flexible bridge [START_REF] Lee | Real-time displacement measurement of a flexible bridge using digital image processing techniques[END_REF] or using LED targets (colour filtering) [START_REF] Wahbeh | A vision-based approach for the direct measurement of displacements in vibrating systems[END_REF].

The original idea of using optical flow to measure vibration has been demonstrated in [START_REF] Morlier | New image processing tools for structural dynamic monitoring[END_REF]: displacements have been measured on a bridge under harmonic excitation (using openCV framework) and then modal parameters were reconstructed from the vibration test results. Then Ji et Chang have also used this concept of optical flow for vibration measurements [START_REF] Ji | Nontarget Image-Based Technique for Small Cable Vibration Measurement[END_REF][START_REF] Ji | Nontarget Stereo Vision Technique for Spatiotemporal Response Measurement of Line-Like Structures[END_REF]. The applications domain is also the civil engineering: the structures (bridges or cables) are well adapted to the methods/experiments as they have low natural frequencies and high displacement of several centimetres. This paper introduces completely new developments in aerospace domain such as an innovative definition of virtual sensors, or a validation on a complex experiment (helicopter blade) and its correlation with broadband modal analysis and Finite Element Analysis using new technology (high speed camera). The main innovation deals with signal processing enhancements and also in the selection of "stable" virtual sensors. Indeed classical optical flow methods calculate the variation of optical intensity of an arbitrary selected region of interest (ROI) on the image sequence [START_REF] Ji | Nontarget Image-Based Technique for Small Cable Vibration Measurement[END_REF][START_REF] Ji | Nontarget Stereo Vision Technique for Spatiotemporal Response Measurement of Line-Like Structures[END_REF]. Here we propose to select interesting points that satisfy a contrast criteria and use them to measure the displacements at multiple locations simultaneously.

Intuitively, a good feature (a good virtual sensor in our case) needs at least to be a texture or a corner.

A way to detect moving objects is by investigating the optical flow which is an approximation of two dimensional flow field from the image intensities. It is computed by extracting a dense velocity field from an image sequence. The optical flow field in the image is calculated on basis of the two assumptions that the intensity of any object point is constant over time and that nearby points in the image plane move in a similar way [START_REF] Aggarwal | On the computation of motion from sequences of images -A review[END_REF]. Additionally, the easiest method of finding image displacements with optical flow is the feature-based optical flow approach that finds features (for example, image edges, corners, and other structures well localized in two dimensions) and tracks their displacements from frame to frame. The LK [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF] tracker is based upon the principle of optical flow and motion fields [START_REF] Barron | Performance of Optical Flow Techniques[END_REF][START_REF] Lim | Optical flow estimation using temporally oversampled video[END_REF] that allows to recover motion without assuming a model of motion.

For practical purposes, the algorithm developed in [START_REF] Bouguet | Pyramidal Implementation of the Lucas Kanade Feature Tracker[END_REF] is employed on the flexible beam example in order to track the motion of the target pixels and reconstruct displacement signals. It offers various advantages like stable and accurate motion results in a non optimal environment. The paper is organized in three parts, the first one introduces the theoretical background of motion tracking and discusses the optical flow algorithm in the domain of vibration measurement. The second part explains proposed methodology and its validation by a simple harmonic Oberst beam test. Finally the structural characterisation of an helicopter blade is studied using virtual sensors data processing. The dynamic parameters are extracted from FRF reconstruction and experimental results are compared with classical modal analysis and FEM solutions.

THEORETICAL BACKGROUND

KLT

Kanade-Lucas-Tomasi (KLT) features may be used to describe general motions within video images. The KLT algorithm finds several features in each frame of video. It then attempts to find a correspondence between the features in one frame with the features in the next. The origins of the Kanade-Lucas-Tomasi Tracker go back to the work of Lucas and Kanade [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF].

They introduced a way to select features that is explicitly based on the tracking equation. Their intention is to select those features that make the tracker work best. They also proposed using an affine model of image motion to monitor feature dissimilarity between the first and the current frame.

The KLT tracking equation

Most of the time, it is impossible to determine the location of a single pixel in the subsequent frame based only on local information. Due to this, small windows of pixels are used as features.

The goal of tracking is to determine the displacement d of a feature window from one frame to the next (Figure 1).

I(x,y)

J(x,y) The displacement ε is chosen to minimize the dissimilarity between two feature windows, one in image I and one in image J:

∫∫ + = W 2 w(x)dx I(x)] - d) [J(x ε (1)
where w is the given feature window, x = [x, y] T are coordinates in the image and d = [dx, dy] T is the displacement. The weighting function w(x) is usually set to the constant 1. The aim is to find the displacement d that minimizes the dissimilarity. For this, Eq. ( 1) is differentiated with respect to d and equated to zero.

∫∫

= ∂ + ∂ + = ∂ ∂ W 0 w(x)dx d d) J(x I(x)] - d) [J(x 2 d ε (2)
Using the Taylor series expansion of J about x, truncated to the linear term, one obtain:

J(x) y d J(x) x d J(x) d) J(x y x ∂ ∂ + ∂ ∂ + ≈ + (3)
Introducing into Eq. ( 2) yields

∫∫ = + = ∂ ∂ W T 0 g(x)w(x)dx d] g(x) I(x) - [J(x) 2 d ε (4)
where

            ∂ ∂ ∂ ∂ = J y J x g(x)
(5)

Rearranging terms yields a linear 2 × 2 system:

Zd = e (6)
where Z is the 2 × 2 matrix:

∫∫ = W T w(x)dx (x)g(x) g Z and ∫∫ = W (x)dx J(x)]g(x)w - [I(x) e (7)
Equation ( 6) is only approximately satisfied, because of the linearization of Eq. ( 2). However, the correct displacement can be found by minimizing Equation ( 6) using a Newton-Raphson algorithm. An arbitrary feature window does not necessarily contain complete motion information. For a horizontal intensity edge only the vertical motion component is determined. In addition, a feature window inside a homogeneous region contains no motion information at all. This is a fundamental problem regardless of the selected method of tracking. In order to avoid this, it is necessary to choose feature windows carefully. Several interesting operators have been proposed based on intuitive ideas of what good features should be. Shi and Tomasi [START_REF] Lim | Optical flow estimation using temporally oversampled video[END_REF] propose a more principled criterion that is optimal by construction: "A good feature is one that can be tracked well". Thus our virtual sensors should be located on good feature for correct displacement measurements.

Implementation issues

OpenCV means Intel® Open source Computer Vision Library [17]. It is a collection of C functions and a few C++ classes that implement some popular computer vision algorithms. The cvCalcOpticalFlowPyrLK function implements sparse iterative version of Lucas-Kanade optical flow in pyramids [START_REF] Bouguet | Pyramidal Implementation of the Lucas Kanade Feature Tracker[END_REF]. It calculates coordinates of the feature points on the current video frame

given their coordinates on the previous frame with sub-pixel accuracy.

When choosing the feature window size, there is a trade-off between accuracy and robustness.

The accuracy component is related to the local sub-pixel accuracy attached to tracking.

Displacement of individual points in the integration window may vary, especially at occluding boundaries. On one hand a small window size is preferable in order not to "smooth out" too much detail. A smaller size makes it less likely, that displacements vary within the window too much. The robustness component on the other hand, relates to sensitivity of tracking with respect to size of image motion, lighting changes, etc… For the tracking equation to work, the displacement has to be smaller than the integration window size. Therefore to ensure robustness specially when dealing with large image motion, the window should be chosen as large as possible. A solution to this dilemma is to use a pyramidal representation of the image. The lowest level of the pyramid is the original image, while higher levels contain smaller subsampled versions. Features are tracked at the highest level of the pyramid first, to obtain an approximate solution. The displacement is then promoted down to the next level where tracking is continued to improve on the estimate. This is conducted in a recursive way until the lowest level is reached.

This effectively permits to deal with large displacements on top of the pyramid, while maintaining sub-pixel accuracy at the bottom. The resolution of the topmost pyramid level should be selected according to the window size and the maximum expected displacement. The number of pyramid levels may be chosen empirically (2 in practice). The subsampling factor may be computed accordingly. For instance classical implementation uses 7 × 7 feature windows and two pyramid levels subsampled at every fourth pixel. This yields good results for displacements up to 15 pixels.

VIRTUAL SENSORS FOR MECHANICAL SYSTEMS MONITORING

From video motion estimation to dynamic monitoring

An excellent survey [START_REF] Barron | Performance of Optical Flow Techniques[END_REF] explains several classes of optical flow estimation methods and compares their performances. There are several benefits of using high frame rate sequences. First as frame rate increases, the intensity values along the motion trajectories vary less between consecutive frames, when illumination level changes. The frame rate increases the captured sequence and exhibits less motion aliasing. To recover the original continuous spatio-temporal video signal from its temporally sampled version, it is clear that the temporal sampling frequency (or frame rate) fs must be greater than 2Bt (Eq. ( 8)), in order to avoid aliasing in the temporal direction (Nyquist criteria). If global motion is assumed with constant velocities vx and vy (in pixels per standard-speed frame) and spatially band limited image with Bx and By as the horizontal and vertical spatial bandwidths (in cycles per pixel), then the minimum temporal sampling frequency fs (in cycles per speed frame) to avoid motion aliasing is given by

vy 2By vx 2Bx 2Bt fs × + × = = . ( 8
)
The assumptions of optical ideal conditions and ideal blur filter have been employed. Typical high speed camera uses a state-of-the-art CMOS sensor that records images at 1000 FPS (ore more) at 1024x768 pixel resolution (ore more).

Virtual sensors paradigm

This section explains the implementation of a novel vision-based approach for obtaining direct measurements of the absolute displacement time history at selectable locations of mechanical systems. The first step is to record a video of the mechanical system using a high speed digital video camera installed on a fixed point beyond the structure, considered as a fixed point. Then a known length is matched with a pixel correspondence (transform pixels in m). Figure 2 presents the schematics of displacement measurement system using CV techniques. Firstly the measurement points are selected on the first frame of the video using the mouse with a virtual sensor (i.e. targets; good features to track). The relative displacement of each target is calculated using the optical flow algorithm, which requires good points to track. While analysing image frames, the displacement of the target is calculated (targets motion is in pixels, user must have the true scaling factor). The quantity of information depends on the number of pixels per frame and the number of frames per second. The main advantage of this method is the ability to measure the displacements of multiple locations simultaneously. Moreover in our case there is no aperture problem as the estimated motion is well in the same direction that the real motion (Y displacement). An extension of this paradigm from 2D to 3D have been yet demonstrated in measuring three-dimensional motion of a structure using two in-plane measurement systems, which are timely synchronized, and geometrically correlated [START_REF] Chang | Flexible videogrammetric technique for three-dimensional structural vibration measurement[END_REF].

Supervised validation on a simple experiment

The "Oberst beam" is a classical method to characterize damping in materials. As the base beam is made of a rigid and lightly damped material (steel, aluminium), the most critical aspect of this method is to properly excite the beam without adding weight or damping. A cantilever beam has the same dynamical behaviour as a free-free beam of twice the length, excited at its centre by a normal imposed displacement (Fig. 3). In this case, only the first even mode of the free-free beam will be excited, and its modal behaviour will be similar to a clamped beam since the slope and the relative displacement to the imposed motion are zero at this point. Practically the precision of the location of the centre is important to avoid an unbalanced system [START_REF] Wojtowicki | New approach for the measurements of damping properties of materials using Oberst beam[END_REF]. Figure 5 shows the frequency contents of both virtual sensors which highlight the first resonance mode at 3.8 Hz, which verifies the results found previously by an accelerometer.

VIRTUAL SENSORS FOR MONITORING AN HELICOPTER BLADE

Experimental test-rig and pre-processing

For virtual and classical measurement, the experimental set-up is composed of a clamped-free aluminium helicopter blade which is four meters long (Fig 6). For using KLT trackers the blade is excited at its center by an impulsion and the response motion is captured by a high speed camera (Photron Fastcam APX RS3000) at 500 frames/sec.

In this experiment, a pixel represents 2.12 mm using a video at a resolution of 1024x256 pixels ( pixels 47 m 0.1 sf =

), so only the first three bending modes (higher displacements than 2.12 mm) can be measured. After 30 Hz the others modes (displacements inferior to the resolution) induce only noise. The LK optical flow is used to follow 9 targets (green arrows) in bending along Y displacement. These targets become virtual displacement sensors which allow doing a SIMO analysis in only one test. The main difficulty occurs in signal reconstruction (displacement).

Target pixels (which move around x axis) create partial modal data, so displacement signals are irregular data. Thus the small linear displacement hypothesis is used (Fig. 7) to enhance the resolution of the motion and to compensate the missing data. If the absolute value of X (relative displacement on abscissa) is less than 3 pixels the data is used, otherwise it is discarded.

According to the figure 7, the virtual sensors vary only in Y direction excepted a small period for KLT 8. So we can say the selected points belong well to a ROI in the image space, so optical flow must be stable for this virtual sensors. Stability diagram: variation of the displacement in X direction are very small function of the frame number (proportional to time vector). So each target vary only in Y direction excepted for KLT 8 (in blue). The virtual sensors are well "attached" to the virtual structure: the points belong well to a ROI in the image space.

Window functions (Hanning) and low pass filter are applied to the signals. The transfer functions is estimated using tfestimate in Matlab. Figure 8 illustrates the effect of a running moving average (size of the window is 5) on the temporal signals. All these signal processing tools aim at obtaining smoother FRFs for more precise analysis.

Comparison of KLT results with EMA and FEM

The results of the FRFs reconstruction permit to realize a classical modal analysis extracting the frequencies, damping ratios and mode shapes. These parameters which characterize the dynamic behavior of the beam are estimated from FRFs using SDOF frequency method called Rational Fraction Polynomial (RFP, [START_REF] Chang | Flexible videogrammetric technique for three-dimensional structural vibration measurement[END_REF]) around resonances The previous method is compared with classical experimental modal analysis and Finite Element Model. The results are compared Table 2 The mode shape describes the structure's motion when it is vibrating at a particular frequency. In the context of SHM mode shape analysis is widely used [START_REF] Adams | The localisation of defects in structures from measurements of natural frequencies[END_REF][START_REF] Sazonov | Optimal spatial sampling interval for damage detection by curvature or strain energy mode shapes[END_REF][START_REF] Morlier | Diagnosis of a portal frame using advanced signal processing of laser vibrometer data[END_REF] as mode shape is a local parameter which contains spatial information and appears to be more sensitive to the presence of damage zones than natural frequencies. It is also interesting in the boundary conditions analysis or joint characterization [START_REF] Morlier | Diagnosis of a portal frame using advanced signal processing of laser vibrometer data[END_REF][START_REF] Olgac | Modal analysis of flexible beams with delayed resonator vibration absorber: Theory and experiments[END_REF][START_REF] Pai | Identification and applications of boundary effects in beams[END_REF]. The vertical sensitivity is low due to the size of the deflection compared to the length of the beam. It simply allows us to identify only the two first modes with accuracy (Fig. 12); the others displacements are inferior to the pixel resolution. This method can be used according to two other important hypothesis; firstly the number of Frame Per Second (FPS) verifies the Nyquist frequency criteria, secondly the camera axis is perpendicular to the studied 2D structure (to avoid angular errors). One of the limitations is that the influence of the camera viewpoint and calibration has not been taken into account in this study. The other disadvantage is that in some applications the small linear displacement hypothesis is not verified. It induces instabilities of the targets which lead to several partial displacement measurements instead of one fixed (x direction) virtual sensor displacement (in bending; y displacement). Nevertheless it offers a simple measurement tool to understand and visualize structural dynamics problems: no targets need to be placed on the structure and several virtual sensors can be tracked simultaneously.

Hz f i 1 ± . Results are listed in table 1. ) f ( E ) f ( σ ) ( E ξ ) ( ξ σ 2.

CONCLUSIONS

The main goal of this paper is to introduce a practical vibration measurement system developed using available CV tools. KLT trackers are simply used as virtual sensors to measure displacement on video choosing good features to track on the image. The developed method was first validated by using a simple Oberst beam example. Then using several virtual sensors we succeed to estimate the first two main modes of a helicopter blade under impulse excitation. This application can also be used with the help of less expensive tools than high speed camera, e.g. with a classical camera (frequency max is 12.5Hz at image resolution of 1024*768 pixels). One advantage of this non contact measurement method (versus Laser Doppler Vibrometer) is that we can measure in one test several simultaneous outputs for SIMO modal analysis. In order to correct the limitations of KLT tracker, an enhanced version of SIFT (Scale Invariant Feature Transform, [START_REF] Lowe | Distinctive Image Features from Scale-Invariant Keypoints[END_REF]) should be developed for 2D structural dynamics application. Finally by the assumptions of high contrast, high vertical resolution (to obtain sufficient deflection) and camera stability, our method coupled with operational modal analysis [START_REF] Peeters | Stochastic System Identification for Operational Modal Analysis: A Review[END_REF] could also be used to study the flutter behaviour of airplanes under ambient excitation or at a different scale to assess the dynamic parameters of MEMS. Currently work is in progress is to analyse three synchronized video (Multiple camera) for 3D marker less dynamic reconstruction. flow principle. Pixel motion from image I to image J is estimated solving the pixel correspondence problem: given a pixel in I, look for nearby pixels of the same color in J. Two key assumptions are needed: color constancy (a point in I looks the same in J) and small motion (points do not move very far). Virtual sensor paradigm using KLT tracker. The chosen key points are used as virtual sensor to measure relative displacement frame after frame (sampling frequency is inverse of fps). In our experiment simple plane problems are analyzed in measuring bending displacement (Y in pixel). The true displacement (in m) is obtained from the conversion of a known size part of the structure in pixels. Table 1: Estimated mean E and standard deviation σ for frequencies f and damping ratios ξ for two modes (RFP).

Listing of Figures

Listing of Tables

Table 2: Comparison of the estimated frequencies using optical method, classical EMA with burst random excitation (Force sensor not able to detect very low frequency) and Finite Element Analysis using CATIA.

Fig 1 :

 1 Fig 1: Optical flow principle. Pixel motion from image I to image J is estimated solving the pixel correspondence problem: given a pixel in I, look for nearby pixels of the same color in J. Two key assumptions are needed: color constancy (a point in I looks the same in J) and small motion (points do not move very far).

Fig 2 :

 2 Fig 2: Virtual sensor paradigm using KLT tracker. The chosen key points are used as virtual sensor to measure relative displacement frame after frame (sampling frequency is inverse of fps). In our experiment simple plane problems are analyzed in measuring bending displacement (Y in pixel). The true displacement (in m) is obtained from the conversion of a known size part of the structure in pixels.

Fig 3 :

 3 Fig 3: Oberst beam experimental set-up for the free-free beam excited at its centre. Half-period of the first mode is visualized from several successive frames. Virtual sensors are visualized with a small green dot; displacement is measured in Y direction (green arrows).

Figure 4 Fig 4 :

 44 Figure4shows the measurements estimated by virtual sensors on the two masses at the

Fig 5 :

 5 Fig 5: Frequency contents of the virtual sensors which highlight the expected harmonic mode of the beam at 3.8Hz.

Fig 6 :

 6 Fig 6: Helicopter blade example: KLT trackers are used to follow 9 targets in bending (Y displacement). The targets are numbered from 1 to 9, the blade is excited at its center.

  Fig 7. Stability diagram: variation of the displacement in X direction are very small function of the frame number (proportional to time vector). So each target vary only in Y direction excepted for KLT 8 (in blue). The virtual sensors are well "attached" to the virtual structure: the points belong well to a ROI in the image space.

Fig 8 :Fig 9 :

 89 Fig 8: Experimental displacements measured on virtual sensors 1 to 9 sampled at FPS. The zoom on the enhanced displacement signaks shows the effect of the moving average function on the temporal signal. This pre-processing aims at obtaining smooth FRFs for better modal parameter estimation.

Fig 10 :

 10 Fig 10: Half-period is visualized from several successive frames. Virtual sensors are visualized with a small green dot; displacement is measured in Y direction (green dots).

Figure 11 presents

 11 Figure 11 presents filtered FRFs (9 measurement points) and the result of the identification of the

Fig 11 :

 11 Fig 11: Filtered FRFs form 9 measurement points and identification of the second resonance at 15 Hz using SDOF RFP method.

Fig 12 :

 12 Fig 12: Two first mode shapes extracted using KLT on experimental data in black continuous line: The global content of these modes is very close to the two first mode computed using CATIA (red dotted line). Mode 1 is on top of the figure, mode 2 on bottom. The mode shape are normalized in order to be compared to FE modal analysis.

Fig 1 :

 1 Fig 1: Optical flow principle.Pixel motion from image I to image J is estimated solving the pixel correspondence problem: given a pixel in I, look for nearby pixels of the same color in J. Two key assumptions are needed: color constancy (a point in I looks the same in J) and small motion (points do not move very far).

  Fig 1: Optical flow principle.Pixel motion from image I to image J is estimated solving the pixel correspondence problem: given a pixel in I, look for nearby pixels of the same color in J. Two key assumptions are needed: color constancy (a point in I looks the same in J) and small motion (points do not move very far).

Fig 2 :

 2 Fig 2:Virtual sensor paradigm using KLT tracker. The chosen key points are used as virtual sensor to measure relative displacement frame after frame (sampling frequency is inverse of fps). In our experiment simple plane problems are analyzed in measuring bending displacement (Y in pixel). The true displacement (in m) is obtained from the conversion of a known size part of the structure in pixels.

Fig 3 :

 3 Fig 3: Oberst beam experimental set-up for the free-free beam excited at its centre. Half-period of the first mode is visualized from several successive frames. Virtual sensors are visualized with a small green dot; displacement is measured in Y direction (green arrows).

Fig 4 :

 4 Fig 4: Validation of the algorithm using two virtual sensors on the left (blue line) and right mass (red dot line). The displacement (Y) is measured as a function of time. This displacements have an harmonic form according to the excitation frequency in this experiment close to the resonant frequency which appears at 3.8Hz (period of 0.26s).

Fig 5 :

 5 Fig 5: Frequency contents of the virtual sensors which highlight the expected harmonic mode of the beam at 3.8Hz.

Fig 6 :

 6 Fig 6: Helicopter blade example: KLT trackers are used to follow 9 targets in bending (Y displacement). The targets are numbered from 1 to 9, the blade is excited at its center.

Fig 7 .

 7 Fig 7. Stability diagram: variation of the displacement in X direction are very small function of the frame number (proportional to time vector). So each target vary only in Y direction excepted for KLT 8 (in blue). The virtual sensors are well "attached" to the virtual structure: the points belong well to a ROI in the image space.

Fig 8 :

 8 Fig 8: Experimental displacements measured on virtual sensors 1 to 9 sampled at FPS. The zoom on the enhanced displacement signaks shows the effect of the moving average function on the temporal signal. This pre-processing aims at obtaining smooth FRFs for better modal parameter estimation.

Fig 9 :

 9 Fig 9: KLT trackers show classically damped sinusoids containing two main frequency. Tracker 1 (in blue) at the free end of the blade exhibit higher displacement than Tracker 9 just near the clamped end (in green).

Fig 10 :

 10 Fig 10: Half-period is visualized from several successive frames. Virtual sensors are visualized with a small green dot; displacement is measured in Y direction (green dots).

Fig 11 :

 11 Fig 11: Filtered FRFs form 9 measurement points and identification of the second resonance at 15 Hz using SDOF RFP method.

Fig 12 :

 12 Fig 12: Two first mode shapes extracted using KLT on experimental data in black continuous line: The global content of these modes is very close to the two first mode computed using CATIA (red dotted line). Mode 1 is on top of the figure, mode 2 on bottom. The mode shape are normalized in order to be compared to FE modal analysis.

Table 1 :

 1 Estimated mean E and standard deviation σ for frequencies f and damping ratios ξ for two modes (RFP).

	32(Hz)	0.2	1.3 (%)	0.02
	15.02(Hz)	0.04	0.36 (%)	0.04

Table 2 :

 2 . Comparison of the estimated frequencies using optical method, classical EMA with burst random excitation (Force sensor not able to detect very low frequency) and Finite Element Analysis using CATIA.Using the results of table 2 we can see that KLT tracker method are limited in bandwidth compared to classical EMA (Experimental Modal Analysis) but it offers a simple measurement tool to understand and visualize structural dynamics engineering problems.

	KLT Tracker (Hz)	EMA (Hz)	FEA (Hz)	Mode shapes
	2.31	na	2.37	First bending
	15	15.11	14.79	Second bending (1 node)
	na	41.89	41.26	Third bending (2 nodes)
	na	56.04	61.25	Torsion


Yann Goueffon

Laurent Arurault
email: arurault@chimie.ups-tlse.fr

Sandra Fontorbes

Catherine Mabru

Claire Tonon

Pascale Guigue

Chemical characteristics, mechanical and thermo-optical properties of black anodic films prepared on 7175 aluminium alloy for space applications

Keywords:

Black anodic coatings are usually used on space vehicles for their thermo-optical properties. This paper presents a study of the chemical characteristics of these black anodic films, as well as their mechanical and thermo-optical properties as a function of the operational conditions of the process. The influence of the anodizing parameters was seen to affect the morphology of the coating (thickness and porosity). In particular, the electrolyte temperature was found to have a major impact on the porosity as it modifies the chemical kinetics during anodization. The impact of the preparation steps on Young's modulus was also investigated. The colouring and sealing steps thus modified the mechanical properties at the coating surface, resulting in a Young's modulus gradient in the film. Finally, a decrease of film porosity seemed to limit the risks of crazing then flaking, while thermo-optical properties were not affected.

Introduction

Aluminium alloys are usually used because they have a low density and quite good mechanical characteristics. To provide specific superficial properties (against corrosion or wearing for example), many surface treatments are available. In particular, different colouring processes can be used to confer a black colour to the anodic films prepared on aluminium parts: incorporation of organic dyes [START_REF] Wernick | The Surface Treatment and Finishing of Aluminium and its Alloys[END_REF][START_REF] Arurault | [END_REF], electrocolouring with metals [3][4][5][6], anodic polymerization [7], co-deposition of organocomplex anions [8] or impregnation by coloured mineral precipitates, such as cobalt sulphide. This last process is usually limited to applications requiring thermal stability of the dyes, i.e. in two main applications: the preparation of solar selective collectors for energy storage and conversion [9] and in spacecraft [START_REF]ESA ECSS-Q-70-03A, Black-anodizing of metals with inorganic dyes[END_REF][START_REF] Goueffon | [END_REF][12]. In space applications, this colouring process is usually called "black anodizing" and involves sulphuric anodization followed by a colouring step based on the precipitation of solid black CoS, in two successive impregnation baths.

Anodic films coloured black with minerals are used in the space industry to avoid stray light in optical instruments, to limit outgassing in orbit and the pollution of instruments by modifying the thermo-optical superficial properties (solar absorptance, ˛s > 0.93 and emittance, ε > 0.90). The ˛s/ε ratio is a characteristic of passive thermal regulation. Typically, heaters may have a ratio higher than one while for coolers the ratio should be lower. Black anodized parts usually have a ratio close to one, and are considered as warm.

Alternating passages of a satellite in front of the sun and in the shadow of the Earth result in thermal cycling which typically reaches temperatures from -140 to +140 • C. The difference in thermo-mechanical characteristics (thermal expansion and Young's modulus) between a coating and its substrate can lead to crazing and possible flaking of the film. This phenomenon has been observed with black anodic films on aluminium alloys from the 2XXX and 7XXX series [START_REF] Goueffon | [END_REF].

The aim of this work is to study the characteristics and properties of these CoS coloured anodic films with a view to the prevention of flaking and a possible pollution of optical instruments. In particular, the influence of the process parameters (especially of the anodizing step) on the anodic film characteristics (porosity and thickness), as well as the mechanical properties (Young's modulus) and thermo-optical properties (absorptivity and emissivity), were thus highlighted.

Experimental procedure

The substrate material was 7175T7351 aluminium alloy, often used in space industry and on which cases of flaking have previously occurred. Its chemical composition in weight percent is: 1.62%Cu, 0.12%Fe, 2.42%Mg, 0.01%Mn, 0.06%Si, 5.75%Zn, 0.041%Ti, 0.21%Cr and Al the remainder. All reagents used were ana- lytical grade, and aqueous electrolyte solutions were obtained using deionised water.

Preparation process

The preparation of black anodic films including inorganic dyes followed the ESA Standard [START_REF]ESA ECSS-Q-70-03A, Black-anodizing of metals with inorganic dyes[END_REF] for spacecraft. The process involved four main consecutive steps: surface pretreatments, anodizing, colouring and sealing.

The alloy sheet (40 mm × 20 mm × 3 mm) was degreased with ethanol and then etched in a mixed aqueous solution of Na2CO3 (6.2 g L -1 ) and Na3PO4 (12.5 g L -1 ) at pH 11, for 5 min at 93 • C and neutralized in aqueous HNO3 (50%, v/v) for 3 min at room temperature. The sheet was rinsed in distilled water at the end of each step.

The sheet was then used as anode and a lead plate (3 mm × 40 mm × 40 mm) as counter-electrode in the electrochemical cell. The anodizing step was particularly studied: operational conditions were, for a given duration (in the range 20-120 min, usually for 60 min) in the galvanostatic mode (0.5 < Ja < 3.0 A dm -2 , typically 1.2 ± 0.1 A dm -2 ) using a sulphuric acid solution (50 < C < 250 g L -1 , usually 150 g L -1 ) thermally regulated (5 < T < 35 • C, typically at 20 • C). The samples were then immediately rinsed in distilled water.

The anodized part was coloured by immersion in two successive baths: first for 15 min in a solution of cobalt acetate (200 g L -1 ; pH 6.5) regulated at 43 ± 2 • C, and then for 10 min in a bath of ammonium hydrosulphide (30 g L -1 ; pH 9.3) at room temperature. The samples were rinsed in distilled water after each step to remove excess solution. The sealing step occurred for 25 min in a bath regulated at 98 ± 2 • C and made up of nickel acetate (NiCH3COO•4H2O) and boric acid (both at 5 g L -1 ).

Microscopic and chemical analysis

The meso-and nanostructures of the coatings were observed by field-emission gun scanning electron microscopy (FEG-SEM) with a JEOL JSM6700F device. Thicknesses were measured on the cross-sections of the samples and energy dispersive X-ray (EDX) spectroscopy analyses were performed to identify the chemical composition inside the film, and especially the location of the dyes. Moreover, analyses by inductively coupled plasma (ICP) were performed to determine the Al, Zn and Cu ion contents in the electrolyte at the end of anodizing.

Porosity measurement

Porosity was measured before the colouring and sealing steps. The surface of the anodic films was observed using FEG-SEM and the resulting images were analysed with the free software ImageJ. The ratio between the pore areas and the total surface area analysed was taken to be the porosity of the anodic film, i.e. the porosity was considered homogeneous through the whole thickness of the anodic film.

Nanoindentation

The nanoindentation tests were performed on an XP apparatus (Agilent Technology) to evaluate the Young's modulus (E) of our anodic films. A Berkovich tip was used with a CSM (continuous stiffness measurement) modulus to obtain dynamic measurements on the surface of the films. The CSM frequency used was 45 Hz with a strain rate of 0.05 s -1 . The maximum depth on indentation was 2000 nm, i.e. lower than 10% of the total thickness (about 20 m), in order to avoid substrate effects. With homogeneous films, an average value was determined for 20 measurements between the depths of 800 and 1000 nm.

Absorptivity and emissivity

Hemispheric absorptivity measurements were performed on an Elan 510 reflectometer. The light source used was a xenon lamp with wavelengths from 250 to 2500 nm and a spectral distribution comparable to that of sunlight (the main difference is in the near infrared). Normal emissivity measurements were performed on an Elan 520 emissiometer operating at 70 • C.

Results and discussion

The anodizing step

Current density and time of anodizing

The growth of the anodic film occurs according to reaction (1) and is mainly controlled with the current density and the anodizing time:

2Al + 3H 2 O → Al 2 O 3 + 6H + + 6e - (1)
Figs. 1 and2 show the variations of film thickness as a function of the current density and the time of anodizing, respectively. The measured thickness increased with time and current density, in agreement with Faraday's law, while the porosity steadily increased with thickness (Fig. 3).

It is now possible to calculate the faradic rate at the anode [13] taking into account both the experimental thickness and the aluminium content in the anodizing electrolyte. Fig. 4 thus shows the virtual thickness of the compact film (TCF), after correcting the experimental porous film thickness by its porosity, as well as the TCF without acid dissolution (TCFWD), i.e. taking into account the amount of aluminium dissolved in the electrolyte. The corresponding anodic efficiency (i.e. TCFWD/faraday's thickness ratio) in these experimental conditions is decreased from 74 to 58% in the 20-120 min time range, in agreement with previous works [14].

Concentration and temperature of the electrolyte

Fig. 5 shows the influence of the concentration of sulphuric acid on the film thickness. In the 50-250 g L -1 concentration range, the thickness of the anodic film remained constant, while there was an increase of the porosity (Fig. 5) for the same experimental conditions.

Both the thickness and porosity of the anodic films were also studied as a function of the electrolyte temperature (Fig. 6). The film thickness was constant (about 20 m) for electrolyte temperatures between 5 and 27 • C, then drastically decreased between 27 and 30 • C reaching values lower than 6 m at temperatures higher than 27 • C. There is a proportional increase of the porosity from 10 to 50% for electrolyte temperatures in the range 5-25 • C. At higher temperatures, the porosity became so high and irregular that the thickness significantly decreased.

The porosity of the anodic film mainly depends on the chemical dissolution (reaction (2)) occurring at its surface with the acidic electrolyte:

Al 2 O 3 + 6H + → 2Al 3+ + 3H 2 O (2)
So, the porosity changes are influenced by the bath parameters because chemical dissolution kinetics is mainly influenced by the temperature and the concentration of the electrolyte [15]. In contrast, these parameters had a very low influence or no impact at all on the electrochemical reaction (reaction (1)) in this temperature range.

Mechanical characteristics

Nanoindentation measurements were performed on samples with different porosities (10-50%). The dynamic method of indentation firstly shows a homogeneous value of Young's modulus of the anodic film through its whole thickness. Fig. 7 reveals how Young's modulus behaves with increasing porosity.

The relationship between Young's modulus and porosity was previously described by Wang [16] using the following equation:

E = E 0 exp(-(bp + cp 2 ))
where p is the porosity, and b and c are constants.

Experimental points are well fitted for porosities up to 30% with the following parameters (E 0 = 86.2 GPa, b = 0.2, c = 2.48). Nevertheless, at high porosities Young's modulus collapses. Wang [16] described such a phenomenon for high porosities even though they did not fit the theoretical relationship well. For a porosity of 50%, nanoindentation measurements were not exploitable but Young's modulus was extrapolated and evaluated at about 20 GPa.

The colouring steps

Fig. 8 shows the distribution of S, Co and Ni versus the depth inside the anodic film, its surface being located on the left of the curves while the film/substrate interface is on the right. Sulphur is already present in the anodic film (Fig. 8a), before the first colouring step (with cobalt acetate), due to the incorporation of sulphates from the sulphuric acid electrolyte during anodizing [17].

Impregnation by cobalt acetate (first step of colouring) causes the simultaneous alteration of the sulphur content from the film surface and cobalt incorporation through the whole film (Fig. 8b). The alteration of the sulphur content at the film/electrolyte interface could probably be explained by the higher temperature (T = 43 ± 2 • C) of the cobalt acetate electrolyte or by the integration of acetates in the film instead of sulphates. Moreover, because of the impregnation process, cobalt incorporation appears quite irregular due to the diffusion and capillarity difficulties in the tortuous mesopores [START_REF] Goueffon | [END_REF].

After the second colouring step, there was an increase of the sulphur content through the whole film thickness, but especially near the surface (Fig. 8c). The presence of sulphur is due to its introduction during the second colouring step, through impregnation with ammonium sulphide and the precipitation of cobalt sulphide.

The reaction between cobalt acetate (pH 6.5) and ammonium sulphide (pH 9.6) results in CoS precipitation [18] according to:

Co(CH 3 COO) 2 + (NH 4 ) 2 S → CoS + 2CH 3 COONH 4 ( 3
)
In the second colouring bath, i.e. at high pH (about 9.6), the stable ionic species are Co(OH) 2 , CH 3 COO -(pK a = 4.8), NH 3 and NH 4 + (pK a = 9.3) as well as HS -(Fig. 9). However, due to the low solubility product (K s = 1.9 × 10 -27 [19]) of solid CoS, precipitation could be considered as quantitative.

Anodic films with different initial porosities (10-40%) were coloured. EDX measurements were then performed on crosssections to localize the dyes in the different films (Fig. 10a-d).

Sulphur is present throughout the coloured anodic films (before sealing), whatever the initial porosity. Moreover, both the amount of dye that finishes up in the pores and the depth of impregnation, increase with the initial porosity. These results stress that the tor- tuosity of the mesopores restricts the impregnation of the cobalt acetate and ammonium sulphide and that deep dark colouring sometimes requires repeated dips [9].

The sealing step

The sealing step is usually considered as hydration of the anodic film, inducing a volume expansion of the film and hence pore occlu- sion [20,[START_REF] Alwitt | AIAA-922158-CP, AAIA Technical Papers[END_REF]. However, in this case, pore closing also results from the additional precipitation of nickel hydroxide, at the top of the coating, as shown on the EDX profiles (Fig. 11).

Surface observations

Fig. 12a-e shows SEM top views of anodic films prepared at different anodizing temperatures, i.e. with different initial porosities and with subsequent colouring and sealing.

Anodic films, prepared at an anodizing temperature from 15 • C, i.e. with an initial porosity higher than 30%, show microcracks after colouring and sealing (Fig. 12c-e). From the crazing point of view, colouring and sealing are key steps because uncoloured and unsealed anodic films with an initial porosity of 40% exhibited a surface free of microcracks even after heat treatment (150 • C for 25 min) in agreement with previous works [START_REF] Arurault | [END_REF]. This crazing phenomenon could be explained by the coefficient of thermal expansion of coloured anodic films, which is nearly 10 times lower than that of the substrate [12]. Indeed, the sealing temperature is regulated at 98 ± 2 • C and results in tensile stresses occurring in the anodic film. Furthermore, it changes the residual stresses and affects the overall mechanical characteristics [12,[START_REF] Alwitt | AIAA-922158-CP, AAIA Technical Papers[END_REF].

Mechanical behaviour

Fig. 13 presents the evolution of Young's modulus as a function of the indentation depth and at different steps of the preparation process.

Young's modulus appears constant (about 80 GPa), before the colouring and sealing steps. After colouring, the chemical changes occurring in the film, especially dye precipitation, reduced the modulus particularly near the surface, where CoS solids were specifically located. The loss of Young's modulus can be directly due to the decreased sulphur content of the film during the impregnation in cobalt acetate. Sulphates from the films are indeed released into the bath, from near the film surface, which explains the presence of a gradient of modulus causing modifications of the mechanical characteristics. The modulus increased steadily along the film depth, until it reached the initial value (about 80 GPa) when the amount of dyes had become negligible. Film sealing led to an additional decrease of Young's modulus; the decrease was greater when the sealing duration was longer (50 min. instead of 25 min.). The Young's modulus gradient also depends on porosity (Fig. 14a-d) and thus on the amount of dyes included in the films (Fig. 10).

In all cases, the modulus seems to tend to the initial value obtained after anodizing and before any post-treatment. Moreover, the Young's modulus gradient clearly depends not only on the porosity and microcracks but also on the chemical content of the film, i.e. dye location and amount, as well as the hydration level, especially resulting from the sealing step. Below, we study if these chemical or physical changes have or do not have an influence on the desired thermo-optical properties.

Thermo-optical properties

Black anodizing is used in the space industry to change the thermo-optical properties of surfaces, specifications recommending a minimum absorptivity of 0.9 and emissivity of 0.93 [START_REF]ESA ECSS-Q-70-03A, Black-anodizing of metals with inorganic dyes[END_REF]. In the uncoloured anodic films (40% porosity) the absorptivity was evaluated to 0.35 and the emissivity to 0.89. Fig. 15a andb shows the evolution of the normal emissivity and the solar hemispheric absorptivity of black anodic films as a function of their thickness. Two anodizing temperatures (5 and Emissivity and absorptivity were only affected for anodic film thicknesses lower than 6 m, while they were found to be independent of the anodizing temperature (5 and 20 • C), i.e. the porosity (10 and 40%, respectively). These results mean that, apart from the microcracks, the thermo-optical properties are only governed by the presence of CoS dyes.

Conclusion

Black anodic films were prepared according to a four-step process, including pretreatments, anodizing, colouring and sealing. Their characteristics (thickness and porosity) were easily controlled with the anodizing parameters (time, current density, concentration and temperature of the electrolyte). The increase of porosity with the temperature or concentration of the electrolyte has a particularly significant influence on the Young's modulus of the film measured just after anodizing. Wang's model gives a good fit for experimental points at low porosities, while a collapse occurs for porosities higher than 35%.

The colouring step consists in the insertion of CoS precipitates in the tortuous mesopores. The quantity of dyes increased with the porosity but the dyes were still mainly located near the film surface. Young's modulus was directly affected by this colouring step probably because of the degradation of sulphates in the film. The decrease in Young modulus was amplified by the sealing step.

Moreover, the increase in porosity may also have an impact on tensile strength and residual stresses, explaining the formation of crazing patterns. So, reducing the porosity may be a good way to avoid crazing while keeping satisfactory thermo-optical properties.

Fig. 1 .

 1 Fig. 1. Thickness of the anodic film versus current density (t = 60 min, C = 150 g L -1 , T = 20 • C).

Fig. 2 .

 2 Fig. 2. Thickness of the anodic film versus time of anodizing (J = 1.2 A dm -2 , C = 150 g L -1 , T = 20 • C).

Fig. 3 .

 3 Fig. 3. Porosity of the film versus its thickness (controlled by varying the anodizing time; J = 1.25 A dm -2 , C = 150 g L -1 , T = 20 • C).

Fig. 4 .

 4 Fig. 4. Thickness of the anodic film versus anodizing time (from Faraday's law, experimental, theoretical compact film TCF, theoretical compact film without dissolution TCFWD).

Fig. 5 .

 5 Fig. 5. Thickness and porosity of the anodic film versus sulphuric acid concentration (T = 20 • C, J = 1.25 A dm -2 , t = 60 min).

Fig. 6 .

 6 Fig. 6. Thickness and porosity of the anodic film versus the electrolyte temperature (C = 150 g L -1 , J = 1.25 A dm -2 , t = 60 min).

Fig. 7 .

 7 Fig. 7. Young's modulus of non-coloured anodic films versus porosity (T from 5 to 25 • C, C = 150 g L -1 , J = 1.25 A dm -2 , t = 60 min).

Fig. 8 .

 8 Fig. 8. EDX analysis of the cross-section of a film with an initial porosity of 40% (T = 20 • C, C = 150 g L -1 , J = 1.25 A dm -2 , t = 60 min). (a) After anodizing, (b) after immersion in cobalt acetate (first colouring bath) and (c) after precipitation of dyes (second colouring bath).

Fig. 9 .

 9 Fig. 9. Simulation of the sulphide species distribution versus pH.

Fig. 10 .

 10 Fig. 10. EDX analysis of the cross-section of coloured anodic films (T from 5 to 20 • C, C = 150 g L -1 , J = 1.25 A dm -2 , t = 60 min) with an initial porosity of: (a) 10%, (b) 20%, (c) 30% and (d) 40%.

Fig. 11 .

 11 Fig. 11. EDX analysis of the cross-section of a coloured sealed anodic film with an initial porosity of 40% (T = 20 • C, C = 150 g L -1 , J = 1.25 A dm -2 , t = 60 min).

Fig. 12 .

 12 Fig. 12. SEM surface views of anodic films (T from 5 to 20 • C, C = 150 g L -1 , J = 1.25 A dm -2 , t = 60 min) after colouring and sealing with different initial porosities: (a) 10%, (b) 20%, (c) 30%, (d) 40% and (e) 50%.

Fig. 13 .

 13 Fig. 13. Young modulus during indentations for four samples (T = 5 • C, C = 150 g L -1 , J = 1.25 A dm -2 , t = 60 min) with initial porosities of 10% (after anodizing, after colouring, after sealing 25 min, after sealing 50 min).

Fig. 14 .

 14 Fig. 14. Young modulus of coloured and sealed samples (T from 5 to 20 • C, C = 150 g L -1 , J = 1.25 A dm -2 , t = 60 min) with different initial porosities: (a) 10%, (b) 20%, (c) 30% and (d) 40%.

Fig. 15 .

 15 Fig. 15. Normal emissivity (a) and solar hemispheric absorptivity (b) depending on the thickness of the coating and the anodizing temperature (5 • C or 20 • C).

20 •

 20 C) were used to evaluate the influence of the film characteristics (porosity and chemical content) on these two properties.

Acknowledgement

The authors thank Peter Winterton for his helpful comments.


Joseph Morlier
email: joseph.morlier@isae.fr

A pedagogical image processing tool to understand structural dynamics

This paper presents a framework and one pedagogical application of motion tracking algorithms applied to structural dynamics. The aim of this work is to show the ability of high speed camera to study the dynamic characteristics of simple mechanical systems using a marker less and simultaneous Single Input Multiple Output (SIMO) broadband analysis. KLT (Kanade-Lucas-Tomasi) trackers are used as virtual sensors on mechanical systems video. First we introduce the paradigm of virtual sensors in the field of modal analysis using video processing. Then we present a pedagogical example of flexible beam (Fishing rod) video. From KLT tracking we extracted displacements data (virtual sensors) which are then enhanced using filtering and smoothing and then we can identify natural frequency and damping ratio from classical modal analysis. The experimental results (mode shapes) are compared to an analytical flexible beam model showing high correlation but also showing the limitation of linear analysis. The main interest of this paper is that displacements are simply measured using only video at FPS (Frame Per Second) that respects the Nyquist frequency. There is no target needed on the structure only few critical pixels that are good features to track and which become virtual sensors.

Introduction

In order to achieve the right combination of material properties and service performance the dynamic behavior is one of the main points to be considered. To better understand the dynamic behavior of the structure we need to characterize the resonances of the structure. A common way of doing this is to define its modal parameters i.e. natural frequency, damping ratio and mode shape [START_REF] Ewins | Model Testing: Theory and Practice[END_REF]. The goal of our work is to develop a method to replace classical contact accelerometers based instrumentation with an optical camera working with an intelligent software in order to continuously assess the dynamic parameters of the structure. Previous works [START_REF] Olaszek | Investigation of the dynamic characteristic of bridge structures using a computer vision method[END_REF][START_REF] Patsias | Damage detection using optical measurements and wavelets[END_REF][START_REF] Poudel | Structural damage detection using digital video imaging technique and wavelet transformation[END_REF][START_REF] Lee | A vision-based system for remote sensing of bridge displacement[END_REF] obtained modal parameters by introducing real targets on the structure or by studying simple structure in ideal conditions. Real time displacement measurement have been done using different approach of digital image processing techniques (texture recognition algorithm) on a flexible bridge [START_REF] Lee | Real-time displacement measurement of a flexible bridge using digital image processing techniques[END_REF] or using LED targets (colour filtering) [START_REF] Wahbeh | A vision-based approach for the direct measurement of displacements in vibrating systems[END_REF]. In our previous work [START_REF] Morlier | New image processing tools for structural dynamic monitoring[END_REF] displacement of a bridge under harmonic excitation was reconstructed by using video openCV framework. Such bridges have low natural frequencies within 5 Hz and maximum displacement of several centimetres. In a previous paper [START_REF] Morlier | Virtual vibration measurement using KLT motion tracking algorithm[END_REF] we continue in studying the linear dynamic response of a helicopter blade using a broadband excitation. Here the size of the structure is smaller: so the frequencies increase and the experimental setup changes (need of high speed camera to verify the Nyquist criteria) and the high displacement of the fishing rod permit to study more modes. This method can be used in structural dynamics according to three important hypothesis; firstly the number of Frame Per Second (FPS) verifies the Nyquist frequency criteria, secondly the camera axis is perpendicular to the studied 2D structure (to avoid angular errors) and finally the global displacement (in pixels) must be superior to the pixel resolution if not the mode would not appear in the Frequency Response Function (FRF). According to these hypotheses the main drawback of our works is that industrial applications are limited to large structures because they have lower frequencies and larger displacements. But however it remains interesting for educational purposes; it offers a simple measurement tool to understand and visualize structural dynamics engineering problems. From the application point of view, video camera and motion tracking algorithm replaces accelerometers for bending displacement measurements under broadband excitation. One advantage of this non contact measurement method (versus Laser Doppler Vibrometer) is that we can measure in one test several simultaneous outputs for SIMO modal analysis.

A way to detect moving objects is by investigating the optical flow which is an approximation of two dimensional flow field from the image intensities. It is computed by extracting a dense velocity field from an image sequence.

The optical flow field in the image is calculated on basis of the two assumptions that the intensity of any object point is constant over time and that nearby points in the image plane move in a similar way [START_REF] Aggarwal | On the computation of motion from sequences of images -A review[END_REF]. Additionally, the easiest method of finding image displacements with optical flow is the feature-based optical flow approach that finds features (for example, image edges, corners, and other structures well localized in two dimensions) and tracks their displacements from frame to frame. The LK [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF] tracker is based upon the principle of optical flow and motion fields [START_REF] Barron | Performance of Optical Flow Techniques[END_REF][START_REF] Lim | Optical flow estimation using temporally oversampled video[END_REF][START_REF] Bouguet | Pyramidal Implementation of the Lucas Kanade Feature Tracker[END_REF] that allows to recover motion without assuming a model of motion. For practical purposes we use this algorithm on the flexible beam example in order to track the motion of the target pixels and reconstruct displacement signals. It offers various advantages like stable and accurate motion results in non optimal environment. The paper is arranged in three parts, the first introduces the theoretical background of motion tracking and discusses of the optical flow algorithm in the domain of vibration measurement. Then we present the experiment of vibrating fishing rod. Finally the structural characterisation of a flexible beam is studied using virtual sensors data processing. The dynamic parameters are extracted from FRF reconstruction and experimental results are compared with analytical solutions.

KLT motion tracking

Kanade-Lucas-Tomasi (KLT) features may be used to describe general motions within video images. The KLT algorithm finds several thousand features in each frame of video. It then attempts to find a correspondence between the features in one frame with the features in the next. The origins of the Kanade-Lucas-Tomasi Tracker go back to the work of Lucas and Kanade [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF]. They introduced a way to select features that is explicitly based on the tracking equation. Their intention is to select those features that make the tracker work best. They also proposed using an affine model of image motion to monitor feature dissimilarity between the first and the current frame. Most of the time, it is impossible to determine the location of a single pixel in the subsequent frame based only on local information. Due to this, small windows of pixels are used as features. The goal of tracking is to determine the displacement d of a feature window from one frame to the next (Figure 1).

I(x,y) J(x,y)

Figure 1: Optical flow principle. Pixel motion from image I to image J is estimated solving the pixel correspondence problem: given a pixel in I, look for nearby pixels of the same color in J. Two key assumptions are needed: color constancy (a point in I looks the same in J) and small motion (points do not move very far).

The displacement is chosen as to minimize the dissimilarity between two feature windows, one in image I and one in image J:

∫∫ + = W 2 w(x)dx I(x)] - d) [J(x ε (1)
where W is the given feature window, x = [x, y]

T are coordinates in the image and d = [dx, dy] T is the displacement. The weighting function w(x) is usually set to the constant 1. The aim is to find the displacement d that minimizes the dissimilarity. For this, we differentiate Equation (1) with respect to d and equate it to zero.

∫∫

= ∂ + ∂ + = ∂ ∂ W 0 w(x)dx d d) J(x I(x)] - d) [J(x 2 d ε (2)
Using the Taylor series expansion of J about x, truncated to the linear term, we obtain:

J(x) y d J(x) x d J(x) d) J(x y x ∂ ∂ + ∂ ∂ + ≈ + Putting this into Equation (2) yields ∫∫ = + = ∂ ∂ W T 0 g(x)w(x)dx d] g(x) I(x) - [J(x) 2 d ε Where             ∂ ∂ ∂ ∂ = J y J x g(x)
Rearranging terms yields a linear 2 × 2 system:

Zd = e ( 3
)
Where Z is the 2 × 2 matrix: 3) is only approximately satisfied, because of the linearization of Equation ( 2). However, the correct displacement can be found by minimizing Equation (3) using a Newton-Raphson algorithm. [START_REF] Shi | Good features to track[END_REF] propose a more principled criterion that is optimal by construction: "A good feature is one that can be tracked well". Thus our virtual sensors should be located on good feature for correct displacement measurements.

∫∫ = W T w(x)dx (x)g(x) g Z and e is ∫∫ = W (x)dx J(x)]g(x)w - [I(x) e Equation (

Several interest operators have been proposed based on intuitive ideas of what good features should look like. Shi and Tomasi

When dealing with dynamic systems, it is clear that the temporal sampling frequency (or frame rate) fs must be greater than 2Bt (Equation 4), in order to avoid aliasing in the temporal direction (Nyquist criteria). If global motion is assumed with constant velocities vx and vy (in pixels per standard-speed frame) and spatially band limited image with Bx and By as the horizontal and vertical spatial bandwidths (in cycles per pixel), then the minimum temporal sampling frequency fs (in cycles per speed frame) to avoid motion aliasing is given by

vy 2By vx 2Bx 2Bt fs × + × = = . ( 4
)
The assumptions of optical ideal conditions and ideal blur filter have been done here. Typical high speed camera uses a state-of-the-art CMOS sensor that records images at 1000 FPS at 1024x768 pixel resolution (or more).

Experiment : Vibrating fishing rod

In this experiment a pixel represents 2.12 mm using a video at a resolution of 1024x256pixels ( Pixels m 47 / 1 . 0 sf = ), so only the first three bending modes (higher displacements than 2.12 mm) can be measured. After 30 Hz the others modes (displacements inferior to the resolution) induce only noise. Figure 2 presents the cantilever flexible beam experiment. The LK optical flow is used to follow 9 targets (green arrows) in bending along Y displacement. These targets become virtual displacement sensors which allow doing a SIMO analysis in only one test. The main experimental hypothesis is to constrain the flexible beam in Y direction (using blocks). The dynamic behaviour of the fishing rod is more complex (chaotic) as the beam has a tapered geometry, the material is composite and the behaviour remains nonlinear due to large deflections and/or rotations. The main problem occurs in signal reconstruction (displacement). In fact target pixels (which move around x axis) create partial modal data, so displacement signals are irregular data. Thus the small linear displacement hypothesis is used (stability diagram of the Figure 3) to enhance the resolution of the motion and to compensate the missing data,. If the absolute value of x (relative displacement on abscissa) is less than 3 pixels the data is used, otherwise it is not implemented. The type of excitation (pulling down) offers a correct bandwidth at low frequency which allows the identification of the first three modes. The excitation level is the static-force that creates the deflection (1.96N) which induces the free decay.

The results of the FRFs reconstruction permit to do a classical modal analysis extracting the three dynamic parameters (frequencies, damping ratios and mode shapes). These parameters which characterize the dynamic behavior of the beam are estimated from FRFs using classical SDOF frequency method called Rational Fraction Polynomial (RFP, [START_REF] Richardson | Parameter Estimation from Frequency Response Measurements using Rational Fraction Polynomials[END_REF]) around resonances Hz f i 1 ±

. Results are listed in Table 1 and can be compared with very good correlation to previous results from classical accelerometers sensors. The figure 5 shows filtered FRFs (9 measurement points) and the result of the identification of the first resonance at 3.32 Hz for each transfer functions. Thus the FRF correlation between experimental data and identified data is very good for each virtual sensor. Frequency (Hz) Rebuilt FRF m/N (dB) Figure 5: Filtered FRFs form 9 measurement points and identification of the first resonance at 3.32 Hz using SDOF RFP method.

) f ( E ) f ( σ ) ( E ξ ) ( ξ σ 3.

Data Analysis: Fitting theoretical mode shapes

In order to validate our experiment results our first approach was to use the theory of the tapered beam [START_REF] Wang | Tables of natural frequencies and nodes for transverse vibration of tapered beams[END_REF]. The aim was to compare experimental frequency ratio with theoretical to identify the pseudo frequencies i β of the beam. According to the results of table 2, a good correlation of experimental data is found with the tapered beam theory. Taking into account that for uniform and tapered beam the second frequency has almost the same value, it is easy to deduce from these ratios the estimation of i 2).

Frequency ratio

Tapered theory Experimental Error

2 1 2 1 2         = β β ω ω 2.61 2.945 12% 2 2 3 2 3         = β β ω ω 1.81 2.

22%

Table 2: Estimated frequency ratios (tapered theory) compared with experimental results.

We can also notice that higher error occurs for the third parameter identification due to the nonlinear behaviour of this mode. The mode shape describes the structure's motion when it is vibrating at a particular frequency. Classically, the equation used to represent a uniform (regular section) beam under free vibration can be written as follows:

0 ) , ( ) , ( 4 4 2 2 = ∂ ∂ + ∂ ∂ t x v x EI t x v t S ρ
where x is the longitudinal coordinate, v is the transversal displacement of the beam in Y direction (which is perpendicular to X), t is time, E is the Young's modulus, S is the cross section area, I is the planar moment of inertia of the cross section, L is the length and ρ is the density of the beam. Assuming that bending stiffness is independent of time and that the steady state vibration has a harmonic form, we get:

0 ) x ( Y dx ) x ( Y d 4 4 4 = -λ
Using separated variables and solving the differential equation, we can express the mode shape as:

) x sin( A ) x cos( A ) x sinh( A ) x cosh( A ) x ( Y i i 4 i i 3 i i 2 i i 1 i λ λ λ λ + + + = (5)
Where the spatial frequency of the i th mode is defined as

EI m i i ω λ = 2 with n i ... 1 = .
Using Equation 5 experimental data can be fitted with the analytical equation of the dynamic motion of a beam using least square method with very good correlation (

0.98 R 2 f
) for the first two modes (figure 9). All the mode shapes highlight local singularities due to local change in the beam materials properties (beam having 15 variable sections). It can be noticed that the coefficient estimated from the analytical fit are typical of the Fixed-Free bending mode behaviour (opposite coefficients i A 1 / i A 3 and i A 2 / i A 4 [START_REF] Rao | Mechanical vibrations[END_REF]) which correlate well the shape of the mode (Figures 6). Figure 6 : First mode shape extracted using experimental data: The global content of this mode is close to the first mode of an uniform beam. Some important singularities appear at the discontinuities of the fishing rod. For the second mode, the experimental data correlate well with the general model of mode shape. This mode is very close from second mode shape of Fixed-Free beam, expected for several zones that highlight the discontinuities of the fishing rod. The motion of each part of the flexible beam has been assumed to be harmonic as explained before. The spatial part for each section is given by Eq 6 where indice i is the i th mode and j is the j th section [START_REF] Timoshenko | Vibration Problems in Engineering[END_REF]:

) sin( ) cos( ) sinh( ) cosh( ) ( 4 3 2 1 x A x A x A x A x Y ij ij ij ij ij ij ij ij ij λ λ λ λ + + + = (6)
If the number of sections is n then the number of unknown parameters in the final solution is 4n. Matching the boundary conditions at each of the n-1 interfaces, as described in Eq. ( 6), insures continuity between the sections and equilibrium of the total bending moments and internal shear forces (Figure 10). 7: The boundary conditions are matched at each of the (n-1) interfaces between the elements. For each interface there are 4 equations in order to insure continuity and equilibrium for the shear force and bending moment. At the two extremities, the conditions are the usual ones for a beam with a clamped and a free end.

The poor global third mode shape regression can be explained by the fact that Eq 5 is used instead of Eq 6. Then the main influence is the beam geometry (the fishing rod has 15 section) which should be fitted using a piecewise multiple fit tool to adjust the experimental data with enhance model of stepped beam (Eq 6). In addition normalization of this mode shape (Figure 8) increases the nonlinear behaviour (the real amplitude is very small compared to the first mode). Figure 8 : Third mode shape extracted using experimental data highlights a complex behaviour: the 15 zones of the stepped beam are describes by red dotted stem. Each part of the beam should be identified with a different parameter (Eq 6).

In the experimental results the spatial sampling is not regular (good features to track are zones that have good contrast i.e. stepped zones of the fishing rod), so the spatial resolution is very poor but easy to interpolate. One of other limitations is that the influence of the camera viewpoint and calibration has not being taken into account in this study. The other disadvantage is that in some applications the small linear displacement hypothesis is not verified. It induces some instabilities of the targets which lead to several partial displacement measurements instead of one fixed (x direction) virtual sensor displacement (in bending; y displacement). Finally the vertical sensitivity will be low due to the size of the deflection compared to the length of the beam. It allows us to identify with only the two first modes with accuracy; the others have complex behavior or displacement inferior or close to the pixel, so that the noise influences more on the mode shape estimation process. Nevertheless the main interest of this method is that no targets need to be placed on the structure and also no time consuming computation is needed (for real time applications) even if several virtual sensors are to be tracked.

Conclusion

The main goal of this paper is to show a pedagogical image processing tool to understand structural dynamics. OpenCV framework can easily be used for displacemet measurement on a video of a vibrating system according the speed of camera respect the Nyquist criteria. KLT trackers are simply used as virtual sensors to measure displacement on video choosing good features to track on the image. We succeed to estimate the first three main modes of a flexible beam (cantilever composites fishing rod) under broad band excitation. For educational purposes, this simple application can also be used with the help of less expensive tools than high speed camera, e.g. with a classical camera (frequency max is 12.5Hz at image resolution of 1024*768 pixels). Finally it will be interesting to develop a 3D framework with several synchronised camera to continuously monitor an important structure like a bridge.

Figure 2 :

 2 Figure 2: Cantilever flexible beam example: KLT trackers are used to follow 9 targets in bending (Y displacement). The targets are numbered from 1 to 9.

Figure 3 :

 3 Figure 3: Using this stability diagram the linearity hypothesis is checked: Each target has does not vary in X direction.

1 XYFigure 4 :

 14 Figure 4: Effect of the moving average function on the temporal signal. This pre-processing aims at obtaining smooth FRFs.

  General model: f(x) = a.cos(b.x)+c.sin(b.x)+d.cosh(b.x)+e.sinh(b.x

Figure

  Figure7: The boundary conditions are matched at each of the (n-1) interfaces between the elements. For each interface there are 4 equations in order to insure continuity and equilibrium for the shear force and bending moment. At the two extremities, the conditions are the usual ones for a beam with a clamped and a free end.

Table 1 :

 1 Estimated mean E and standard deviation σ for frequencies f and damping ratios ξ for three first modes extracted using RFP.

	32 (Hz)	6E-4	0.93 (%)	3E-4
	9.78 (Hz)	8E-2	0.96 (%)	3E-3
	21.69 (Hz)	3E-2	0.73 (%)	3E-3

Table 3 :

 3 Estimated parameters of the fit of experimental mode shapes with analytical formula (Eq 5). The R² and RMSE coefficients show very good correlation with the general model.

Acknowledgments

The author want to thanks Guilhem Michon for his technical help and the 2 nd year students Paul Sebellin and Emmanuel Godard of ISAE campus-Supaero for being part of this pedagogical research project.