Amir Shahdin

Joseph Morlier
email: joseph.morlier@isae.fr

Laurent Mezeix

Christophe Bouvet

Yves Gourinat

Evaluation of the impact resistance of various composite sandwich beams by vibration tests

Keywords: Structural Integrity Assessment, Composite Sandwich beams, Vibration Testing, Impact Resistance

Impact resistance of different types of composite sandwich beams is evaluated by studying vibration response changes (natural frequency and damping ratio). This experimental works will help aerospace structural engineer in assess structural integrity using classification of impact resistance of various composite sandwich beams (entangled carbon and glass fibers, honeycomb and foam cores). Low velocity impacts are done below the barely visible impact damage (BVID) limit in order to detect damage by vibration testing that is hardly visible on the surface. Experimental tests are done using both burst random and sine dwell testing in order to have a better confidence level on the extracted modal parameters. Results show that the entangled sandwich beams have a better resistance against impact as compared to classical core materials.

Introduction

The aim of composite sandwich structures is to increase the stiffness and specific strength and to reduce the weight so it is advantageous to employ them in aerospace applications where the challenge is to produce structures lighter and lighter. However damage in these structures may negate many of the benefits of sandwich construction. Impact can induce various types of damage in the structure. The facesheets can be damaged through delamination and fibre breakage; the facesheet and core interface region can be debonded and the core can be damaged through crushing and shear failure mechanisms. Safe and functional effectiveness of stressed sandwich structures can often depend on the retention of integrity of each of the different materials used in its manufacture. Therefore lightweight sandwich materials used in next generation of more advanced aircraft, marine craft, road and rail vehicles must possess the capability to absorb high impacts. For aeronautical structures, a field where this problem has been extensively studied, the components have to undergo low energy impacts caused by dropped tools, mishandling during assembly and maintenance, and in-service impacts by foreign objects such as stones or birds. In these low energy impacts normally, a small indentation is seen on the impact surface. This level of damage is often referred to as barely visible impact damage (BVID).

Although not visually apparent, low energy impact damage is found to be quite detrimental to the load bearing capacities of sandwich structures, underscoring the need for reliable damage detection techniques for composite sandwich structures. In recent years, vibration based damage detection has been rapidly expanding and has shown to be a feasible approach for detecting and locating damage. A detailed and comprehensive overview on the vibration based damage detection methods has been presented in references [START_REF] Doebling | A summary review of vibration-based damage identification methods[END_REF][START_REF] Yan | Development in vibration-based structural damage detection technique[END_REF][START_REF] Sohn | A review of structural health monitoring literature[END_REF][START_REF] Carden | Vibration based condition monitoring: A review[END_REF][START_REF] Van Der Auweraer | International research projects on structural damage detection[END_REF]. The basic principle of vibration based damage detection can be explained as follows. Any structure can be considered as a dynamic system with stiffness, mass and damping. Once some damages emerge in the structures, the structural parameters will change, and the frequency response functions and modal parameters of the structural system will also change. This change of modal parameters can be taken as the signal of initial damage occurrence in the structural system. Shift in natural frequency is the most common parameter used in the identification of damage. A large variety of works can be found in the scientific literature related to the study of structural damage by changes in natural frequencies [START_REF] Salawu | Detection of structural damage through changes in frequency: a review[END_REF][START_REF] Khoo | Structural damage assessment using vibration modal analysis[END_REF][START_REF] Gadelrab | The effect of delamination on the natural frequencies of a laminated composite beam[END_REF][START_REF] Yam | Damage detection of composite structures using dynamic analysis[END_REF][START_REF] Kim | Effect of debonding on natural frequencies and frequency response functions of honeycomb sandwich beams[END_REF]. However, in structures made of composite materials there seems to be a tendency to use damping as a damage indicator tool, as it tends to be more sensitive to damage than the stiffness variations, mainly when delamination is concerned. Therefore damping has been proposed in scientific literature as a more sensitive and attractive damage indicator as compared to natural frequencies [START_REF] Shahdin | Correlating low energy impact damage with changes in modal parameters: A preliminary study on composite beams[END_REF][START_REF] Adams | Damping in composites[END_REF][START_REF] Zhang | Relation of damping and fatigue damage of unidirectional fibre composites[END_REF][START_REF] Gibson | Modal vibration response measurements for characterization of composite materials and structures[END_REF][START_REF] Saravanos | Effects of delaminations on the damped dynamic characteristics of composites[END_REF].

There has been considerable research on the impact performance and damage development in carbon fiber composite materials and sandwich composite materials; see for example references [START_REF] Schubel | Impact and post impact behaviour of composite sandwich panels[END_REF][START_REF] Petit | Impact and compression after impact experimental study of a composite laminate with a cork thermal shield[END_REF][START_REF] Abi Abdallah | Experimental analysis of damage creation and permanent indentation on highly oriented plates[END_REF][START_REF] Abrate | Impact on composite structures[END_REF]. A comprehensive review of low-velocity impact responses of composite materials is presented by Richardson and Wisheart [START_REF] Richardson | Review of low-velocity impact properties of composite materials[END_REF]. Dear et al. [START_REF] Dear | Impact damage processes in composite sheet and sandwich honeycomb materials[END_REF] studied the impact toughness of different lightweight sandwich panels and composite sheet materials.

They emphasized on the degree of damage inflicted on the contact surface, through-thickness and rear surface of the materials when subjected to different impacts. Their aim is to develop lightweight stiff materials with enhanced structural integrity that can absorb higher impact energy. Vaidya et al [START_REF] Vaidya | Impact and postimpact vibration response of protective metal foam composite sandwich plates[END_REF] studied composite sandwich structures composed of aluminum foam core and found it optimal for resisting low-velocity impacts. They also studied the vibration response of composite sandwich plates under a free-free-free-free boundary condition. They concluded that relationships can be established between the vibration response of the sandwich plate (natural frequency and damping ratio) and energy of impact. Similarly several novel sandwiches have been developed in scientific literature with a view to enhance the impact toughness [START_REF] Vaidya | Low velocity impact and vibration response of multi functional sandwich plates[END_REF][START_REF] Hosur | Manufacturing and low-velocity impact characterization of foam filled 3-D integrated core sandwich composites with hybrid face sheets[END_REF][START_REF] Vaidya | Impact response of integrated hollow core sandwich composite panels[END_REF][START_REF] Sutherland | Impact characterisation of low fiber-volume glass reinforced polyester circular laminated plates[END_REF][START_REF] Sutherland | Effect of laminate thickness and of matrix resin on the impact of low fiber-volume, woven roving E-glass composites[END_REF][START_REF] Abdullah | The impact resistance of polyporopylene based fibermetal laminates[END_REF].

In this paper we have made Low Velocity Impact (LVI) and post impact vibration tests to measure the impact resistance. More precisely we evaluated the resistance against impact of entangled sandwich materials in comparison with standard sandwiches with honeycomb and foam cores, is uniquely based on vibration test results i.e., decrease in natural frequency (global parameter of a structure) which signifies loss of rigidity and increase in damping which corresponds to friction in damaged zones. The authors want to clarify here, that impact toughness is only studied through vibration tests and the classical procedure used for determining the impact toughness i.e., Compression After Impact (CAI) has not been implemented as it is outside the scope of this work. Characterization of carbon and glass fiber entangled sandwich materials has been carried out both statically (compression and bending tests) and dynamically (vibration test) by Shahdin et al. [START_REF] Shahdin | Fabrication and mechanical testing of a new sandwich structure with carbon fiber network core[END_REF][START_REF] Shahdin | Fabrication and mechanical testing of glass fiber entangled sandwich beams: A comparison with honeycomb and foam sandwich beams[END_REF]. Vibration tests verify the presence of high damping in the entangled sandwich specimens making them suitable for specific applications like the inner paneling of a helicopter cabin, even if the structural strength of this material is on the lower side. Shahdin et al. [START_REF] Shahdin | Monitoring the effects of impact damages on modal parameters in carbon fiber entangled sandwich beams[END_REF] also carried out the monitoring of impact damage in two types of carbon entangled sandwich beams (heavy and light). The light specimens have 2.5 times less resin than the heavy ones. Results show that the specimens with less resin are more sensitive to impact damage, have poorer static strength but on the other hand possess good damping capabilities. Unfortunately, little scientific literature can be found related to impact testing of entangled sandwich materials. One work that can be cited in this regard is that of Dean et al [START_REF] Dean | Energy absorption during projectile perforation of lightweight panels with metallic fibre cores[END_REF]. Their paper documents an experimental and numerical study of energy absorption in lightweight sandwich panels with entangled stainless steel fiber cores. The results show that the sandwich panel absorbs 40% more energy than the two separated face plates with classical core.

As our research concerning entangled sandwich materials is still in the preliminary phase i.e., we do not master completely the fabrication process of entangled sandwiches, therefore for the instance comparison is only provided with standard sandwiches with honeycomb and foam cores. Comparison with enhanced sandwich structures e.g., honeycomb sandwiches with viscoelastic layer, etc is not in the scope of this work and shall be duly considered in future.

Material and Specimen

Six sandwich beam specimens are studied in this article. The main emphasis of this article is on the two entangled sandwich beams with carbon and glass fibers as core materials.

The honeycomb and foam sandwich beams are only presented for comparison purposes. As the entangled sandwich material being a relatively new material is in the phase of extensive research at this moment. Therefore comparison with standard sandwich beams is essential in order to evaluate the performance of entangled sandwich materials.

Three of the six sandwich beams have entangled carbon fiber, honeycomb and foam as core materials. For the skin, unidirectional carbon-fiber/epoxy prepregs of T700/M21 are used [START_REF] Hexply | Data Sheet[END_REF]. The prepreg sheets are supplied by Hexcel composites. The upper and lower skins consist of four plies each with a stacking sequence of [0/90/90/0]. The thickness of each ply is 0.125 mm. The mechanical properties of the prepreg T700/M21 are given in Table 1.

The other three sandwich beams have entangled glass fibers, honeycomb and foam as core materials. The skins of these three sandwich beams are made of glass woven fabric 20823 supplied by Hexcel composites as well. The glass woven fabric is impregnated with the help of epoxy resin. The epoxy resin SR 8100 and injection hardener SD 8824 are used provided by Sicomin. The upper and lower skins consist of two plies each with a total thickness of 0.5 mm containing 50 % of resin. The thickness of the skins in case of glass woven fabric is kept similar to carbon fiber skins. The sandwich beam specimens are fabricated using an autoclave and an aluminum mold. The skin and the core are cured simultaneously in order to have an excellent bond. The physical properties of the glass woven fabric are set out in Table 2.

The honeycomb and foam cores can be selected from a wide range of metallic and non-metallic honeycomb cores and a variety of non-metallic foams. The honeycomb sandwich beams in this article are made of Nomex-aramid honeycomb core (HRH 10)

supplied by Hexcel composites [START_REF] Hexweb | Honeycomb attributes and properties[END_REF]. The honeycomb core has a nominal cell size of 6.5 mm and a core thickness of 10 mm. In case of the foam sandwich beams, the foam core has also a thickness of 10 mm and is provided by Rohacell (material 51 A). Mechanical properties of the honeycomb and foam cores are listed in Table 3 and Table 4 respectively.

The core in case of the carbon entangled sandwich beam consists of carbon fibers (HTS-5631) that are made of a yarn of standard carbon filaments having a diameter of 7 µm.

The length of the carbon fibers is 10 mm and their elastic modulus is 240 GPa. The fibers are provided by the company Toho-Tenax. In case of the glass entangled sandwich beam, the core consists of glass fibers made of glass filaments having a diameter of 14 µm. The length of the glass fibers is 10 mm with an elastic modulus of 73 GPa. The fibers are provided by the company PPG Fiber Glass Europe. For the cross-linking of carbon and glass fibers, epoxy resin SR 8100 and injection hardener SD 8824 are used provided by Sicomin as used in case of glass woven fabric for the skin material. A better vaporization is achieved if the resin is heated up to 35°C before being sprayed on the carbon and glass fibers. This allows the mixture of resin and hardener to become less viscous. The properties of the carbon and glass fibers used in the core of entangled sandwich beams are presented in Table 5.

All the test specimens presented in the article are carefully weighed using Mettler balance. The weights of the six sandwich beams are presented in Table 6.

The fabrication of honeycomb and foam sandwich beams shall not be explained as the fabrication process is simple and very well known. However, the fabrication of entangled sandwich beams is a relatively complex process. The carbon and glass fibers are cut with the help of a fiber cutting machine supplied by Matrasur Composites. The fibers are then separated by a blow of compressed air. The mixture of resin and hardener is then sprayed on the separated glass fibers by a spray paint gun. The fibers vaporized by the resin are then placed in the mold between the two skins. To produce good quality sandwich beams reliably, cure cycle is adopted as follows: 1 hour from the ambient temperature to 125°C, 1.5 hours at 125°C and 1 hour from 125°C to the ambient temperature. In case of the glass sandwich beam, the core has 26 g of glass fiber and 17 g of epoxy resin approximately. For the carbon sandwich beam, the core has 30 g of carbon fiber and 22 g of epoxy resin approximately. The same mold, cure cycle and skins are used for the fabrication of honeycomb and foam sandwich specimens. An aluminum mold (300 x 65 x 11 mm) is used for the fabrication. A fiber core density of approximately 200 kg/m 3 is chosen for the entangled sandwich core in case of both carbon and glass fibers.

The six sandwich test beams are shown in Fig. 1.

Experimental Methods

High quality vibration tests

The experimental equipment used for vibration testing is shown in Fig. 2. The experimental set-up is that of a free-free beam excited at its center, based on Oberst method [START_REF] Wojtowicki | New approach for the measurements of damping properties of materials using oberst beam[END_REF]. The Oberst method states that a free-free beam excited at its center has the same dynamical behavior as that of a half length cantilever beam. The test specimen is placed at its center on a B&K force sensor (type 8200) which is then assembled on a shaker supplied by Prodera (type EX) having a maximum force of 100 N. However the force sensor is not capable of measuring reliable response below 5 Hz. A fixation system is used to place the test specimens on the force sensor. The fixation is glued to the test specimens with a HBM X60 rapid adhesive. The response displacements are measured with the help of a non-contact and high precision Laser Vibrometer OFV-505 provided by Polytec. The shaker, force sensor and the laser vibrometer are manipulated with the help of a data acquisition system supplied by LMS Test Lab for burst random testing and Ideas Test (B&K) for sine dwell testing.

The center of the test specimens is excited at Point 14 as shown in Fig. 3. Each sandwich specimen is tested with two types of excitations i.e., burst random and sine dwell.

For both the testing systems (LMS and B&K), the resolution is kept 0.25 Hz to allow a good shape of the resonance peaks at low frequency range and to have a reliable comparison of modal parameters between the two systems. Response is measured at 27 points that are symmetrically spaced in three rows along the length of the beam to have reliable identifiable mode shapes. The level of the excitation signal for both the excitations is chosen as 1N which is kept fixed during all the vibration tests conducted in this paper. With the help of LMS by using burst random excitation, we have the advantage of having in quick time the overall dynamic (modal) response of our structure if we are mostly concerned with frequency and mode shapes. In addition, this broadband type of testing helps us identify the modes that we can use later on for sine-dwell testing. However if we need precise damping measurements then sine-dwell testing becomes inevitable but the problem with it is the lengthy acquisition times.

Burst random excitation is a broadband type excitation signal. 50% burst percentage is used for burst random excitation. Normally burst random excitations are leakage free but the first author after trying different window functions found out that by putting Hanning windows on both the excitation and response signals, better quality signals FRFs are obtained.

The signal is averaged 10 times for each measurement point and the frequency band chosen is 0-2650Hz. Disadvantages of burst random excitation are that they removes distortion, has a medium signal to noise ratio, and are not able to characterize nonlinearity [START_REF]The Fundamentals of Modal Testing[END_REF]. Jumping phenomenon (Sine dwell test with sweeping frequency up and down) is another indication of nonlinear vibration due to damage [START_REF] Shahdin | Damage monitoring in sandwich beams by modal parameter shifts: A comparative study of burst random and sine dwell vibration testing[END_REF].

Sine-dwell excitation is the discrete version of sine sweep. The frequency is not varied continuously, but is incremented by discrete amounts at discrete time points. The advantage of sine-dwell testing is its capability of detecting non linear structural dynamic behavior unlike the broadband excitations i.e., with sine dwell excitation we can differentiate between linear and non-linear effects because the excitation is at a single frequency [START_REF] Gloth | Influence and characterisation of weak non-linearities in sweptsine modal testing[END_REF]. As sine dwell testing requires larger acquisition times, so instead of studying the whole frequency band (0-2650 Hz), acquisition is carried out only around the first four bending modes previously identified by burst random testing by keeping the same resolution.

The modal parameters are extracted with the help of Polymax and Polyreference, integrated in the data acquisition systems, for burst random and sine-dwell testing respectively. The Polymax estimation method used by LMS acquisition system is a new noniterative frequency domain parameter estimation method based on weighted least squares approach. This Polymax least-squares complex frequency domain method is implemented in a very similar way as the industry standard Polyreference used by B&K system, which is a time domain least squares frequency domain method. Both of these methods work in similar fashion as follows:

• Firstly we compute the sum of 27 FRFs then compare resonance peaks with MIF indicator.

• We select a reduced band around each resonance separately. For consistent assessment of damping, a frequency interval of ± 20 Hz is chosen for each resonance peak for both Polymax and Polyreference, because by changing the frequency interval damping values can be affected.

• We also use the polyreference stability diagram to ensure the accuracy of modal parameters. Indeed for reliable damping measurement, that value of pole should be chosen which displays a stable value for several model orders and if possible for each mode the value of poles should be chosen at the same model order to ensure that there is minimum uncertainty while comparing the damping values between different damage states.

One of the specific advantages of these two techniques lies in the very stable identification of the system poles and participation factors as a function of the specified system order, leading to easy-to-interpret stabilization diagrams. This implies a potential for automating the method and to apply it to "difficult" estimation cases such as high-order and/or highly damped systems with large modal overlap. As discussed previously, both Polymax and Polyreference are based on least-squares complex optimization methods, so both of them calculate the optimal pole value (frequency and damping) based on the 27 measurement points. We do not have access to the average values, variances or standard deviations for the 27 FRFs as the estimated modal parameters are the results of an optimized process.

The reference [START_REF] Peeters | The PolyMAX frequencydomain method: a new standard for modal parameter estimation[END_REF] explains these two estimators in detail.

So from the above discussion it can be said that both Polymax and Polyreference methods work in similar fashion, so the difference in the resulting modal parameters if it is the case, is due to the difference in excitations than due to the different estimation methods.

Low Velocity Impact tests

The six sandwich beams tested in this article are damaged by drop weight impacts below the barely visible impact damage limit (BVID), in order to simulate damage by foreign impact objects such as stones or birds. The impact tests are carried out by a drop weight system as shown in Fig. 4, and a detailed cut away of the drop assembly is shown in Fig. 5.

The impactor tip has a hemispherical head with a diameter of 12.7 mm. A force sensor (type 9051A) provided by Kistler is placed between the impactor tip and the free falling mass of 2 kg. The velocity before the impact is measured with the help of an optical velocity sensor from which the energy of impact can be verified. The combined weight of the impact head, freefalling mass, force sensor and the accelerometer is 2.03 kg. The size of the impact window is 80 x 40 mm 2 which allows all the impact points to have the same boundary conditions and all the four ends are fully clamped. Further details on the impact test methodology of this drop tower can be found in the references [START_REF] Sohn | A review of structural health monitoring literature[END_REF][START_REF] Carden | Vibration based condition monitoring: A review[END_REF].

A simple case of symmetrical impacts is studied in this article as unfortunately very little literature is available regarding impact tests on entangled sandwich materials. If satisfactory results are obtained, then asymmetric damage shall be studied in the future. The sandwich beams are impacted by taking into account the barely visible impact damage limit (BVID). BVID corresponds to the formation of an indentation on the surface of the structure that can be detected by detailed visual inspection and can indicate high damage. In the aeronautical domain, BVID means an indentation of 0.3 mm after relaxation, aging etc (according to Airbus certifications). In this study, it is decided to take 0.6-0.8 mm of penetration depth as detectability criterion just after the impact [START_REF] Sohn | A review of structural health monitoring literature[END_REF][START_REF] Carden | Vibration based condition monitoring: A review[END_REF] which corresponds to an indentation depth of approximately 0.3 mm by taking into account the above mentioned factors such as relaxation, aging, humidity etc. The idea behind the impact tests is to damage the specimens below the BVID limit, in order to detect by vibration testing the damage that is not visible through naked eye.

However in case of honeycomb sandwich beams, it is difficult to induce the same amount of damage at different points in the same specimen, even if it is impacted with the same energy i.e., impacting at the honeycomb cell center and at the corner leads to different damages. Therefore, it is not possible to have the same density of damage in the honeycomb sandwich beams at the two impact points. This phenomenon is explained by taking the example of the carbon honeycomb sandwich beam impacted at 4 J shown in Fig. 6. The indentation depth at the impact point 1 is 0.1 mm due to impact at the honeycomb cell corner.

Whereas the indentation depth at the impact point 2 is 0.5 mm because the impactor head has induced severe damage due to impact at honeycomb cell center. This phenomenon introduces asymmetry in the beams and highlights the difficulty in inducing a global symmetric damage.

The impact energy is chosen in such a way that each sandwich beam has approximately the same level of damage i.e., below the BVID limit which is nearly invisible on the surface. The impact parameters and the indentation depths measured for the six sandwich beams are listed in Table 7. It shall be noticed that the dispersion in damage between the two impact points is smaller in case of the entangled and foam sandwich beams as compared to the honeycomb sandwich beams due to the phenomenon explained above.

The data obtained during the drop weight impact tests carried out on the six sandwich beams is presented in Fig. 7.

Two similar impacts have been performed on each sandwich beam. However, in order to clarify these plots, the three glass and three carbon sandwich beams are plotted separately and in addition, only one impact test result for each specimen is plotted. All the impact curves presented in Fig. 7 are low-pass filtered at 15 kHz to avoid a free frequency of the impactor at about 20 kHz. These curves, representative of all performed impact tests, are very classic in the literature [START_REF] Sohn | A review of structural health monitoring literature[END_REF][START_REF] Carden | Vibration based condition monitoring: A review[END_REF][START_REF] Van Der Auweraer | International research projects on structural damage detection[END_REF]. In Fig. 7 a and 7 b, the impact forces are drawn as a function of time for the six sandwich beams. These curves are globally smooth and almost sinusoidal at low impact energy. It can also be seen, from the force-displacement plots of both the carbon and glass sandwich beams (Fig. 7 c and7d), that after the first damage in the classical sandwiches with honeycomb and foam cores there is a decrease in the force signal followed by oscillations which signifies damage and loss of rigidity in the material. But in case of both glass and carbon fiber entangled sandwich beams (Fig. 7 c and 7d), after the appearance of first damage the material continues to rigidify which is shown by a progressive increase of force signal. If we speak in terms of energy dissipation, it can be observed from the force displacement curves of the honeycomb and foam core sandwiches (Fig . 7c and7 d) that the energy dissipation seems mostly due to the rupture mechanism. However in case of entangled sandwiches the behavior is different, it is possible that the energy dissipation might be predominantly due to damping as no oscillations or force signal loss is observed i.e., no apparent damage signs.

This behavior can also be seen on the stress-strain curves of compression tests which are very classical in scientific literature [START_REF] Shahdin | Fabrication and mechanical testing of a new sandwich structure with carbon fiber network core[END_REF][START_REF] Shahdin | Fabrication and mechanical testing of glass fiber entangled sandwich beams: A comparison with honeycomb and foam sandwich beams[END_REF]. Compression test results in case of standard sandwiches with honeycomb and foam cores exhibit that there is a progressive decrease in the stress level once the maximum elastic limit is attained. However for the entangled sandwiches, normally there is a densification phase after the maximum elastic limit which explains the rigid nature of the entangled sandwiches as compared to the honeycomb and foam sandwich beams tested in this paper. This rigid nature of entangled sandwich beams is also verified further ahead in this article while evaluating the resistance against impact by shifts in natural frequency.

The sandwich beams tested in this paper have two states. First one is the undamaged state (UD) and the second is the damage state due to two impacts (D1). Vibration tests are carried out on the six sandwich beams after each of these two states. The effect of impact damage on the modal parameters of the three types of sandwich beams is studied in the following sections of this paper with the help of frequency and damping changes between the undamaged (UD) and the damaged case (D1) with the help of Eq. 1 and Eq. 2.

Change in frequency between UD and D1, ) (

) ( ) ( 1 k f k f k f f UD UD D ! = " (1)
Change in damping between UD and D1, ) (

) ( ) ( 1 k k k UD UD D ! ! ! ! " = # ( 2
)
where f UD (k) is the damped natural frequency for the undamaged specimen for the k th mode and f D1 (k) is the damped natural frequency for the specimen damaged at two impact points (D1) for the k th mode. Nomenclature in case of Eq. 2 is the same.

Shift in modal parameters due to damage

Modal parameter estimation is a special case of system identification where the a priori model of the system is known to be in the form of modal parameters. The identification process consists of estimating the modal parameters from frequency response function (FRF) measurements. Modal identification uses numerical techniques to separate the contributions of individual modes of vibration in measurements such as frequency response functions. Each term of the FRF matrix can be represented in terms of pole location and a mode shape. The FRF matrix model is represented mathematically by:

[ ] [ ] [ ] mod es k 1 R(k ) R(k ) * H( ) (j (k ) p(k )) (j (k ) p(k )*) = ! " # = + $ % # & # & ' ( ) (3)
The numerator R(k) is the residue of the FRF and is a function of the product between mode shape components at all points. The denominator gives the modal frequency and modal damping (second term in Equation ( 3) is the complex conjugate term also known as poles).

The poles p(k), are the roots that satisfy this equation and are related to modal frequency and damping as follows:

p(k ) (k ) j (k ) = !" + # (4)
The magnitude of each pole is the undamped natural frequency (ω n ). The undamped natural frequency (ω n ) is related to the modal damped frequency (ω d ) and the modal damping ( ! ) and also to the mass, stiffness and damping as follows:

2 2 n d K (k ) M ! = ! + " = (5) C 2 (k ) M ! = (6)
The effect of physical properties on poles in the complex s-plane is illustrated in Fig. 8.

From Fig. 8, it can be observed that a change in stiffness affects only the frequency, while changes in mass and structural damping affect both modal damped frequency (ω d ) and modal damping ( ! ). For this study, the primary interest is to study the decrease in the modal damped frequency (ω d ) and the increase in modal damping ( ! ) due to damage in the sandwich specimens [START_REF] Schwarz | Experimental modal analysis[END_REF].

Results and discussions

Effect of impact damage on modal parameters

The effect of impact damage on the three types of sandwich beams is studied with the help of modal parameter shifts for the first three bending modes, as they have the largest amplitudes for the type of test configuration presented in this article. Frequency and damping ratios are the global parameters of the specimen, and are extracted from high quality measurements carried out on the 27 measurement points. The modal parameters (natural frequency and damping) help in monitoring globally the health of a specimen. For the first three bending modes, the variation of damped natural frequency as a function of the undamaged (UD) and the damage state (D1) for the six sandwich beams for both burst random (BR) and sine-dwell (SD) testing is presented in Fig. 9.

Fig. 9 shows that as a result of impact damage, there is a decrease in the natural frequencies for the six sandwich beams as discussed before in section 3.3. It can be noticed that this decrease is less prominent in case of both the carbon and glass entangled sandwich beams as compared to the honeycomb and foam sandwich beams. It is also evident from Fig. 9 that in case of natural frequencies, both burst random and sine-dwell testing give similar results. But the interesting fact is that for all the sandwich beams, even as the impact damage does not produce a visible damage on the surface, the change in frequency between the undamaged and the damaged cases is quite noticeable. This proves that there is a notable loss of rigidity without any signs of damage on the surface (with the exception of some impact points in case of honeycomb sandwich beams as discussed previously). It is particularly in these cases that vibration testing becomes a very useful tool for damage detection.

The fact that the change in natural frequency between the damaged and the undamaged case is small in the entangled beams as compared to the foam and honeycomb beams can be seen in Fig. 10, which presents a comparison between the frequency response functions of the undamaged and the damaged cases for the six sandwich beams for the 1st bending mode. The frequency response functions presented in Fig. 10 are obtained with the help of sine-dwell testing. For the six sandwich beams, it can be concluded that the shift in natural frequencies is slight in case of entangled sandwich beams proving that they have a loss of rigidity that is less pronounced as compared to the honeycomb and foam sandwich beams. Furthermore in case of the three carbon sandwich beams (Fig. 10 a), the frequency response functions of the carbon entangled sandwich beam are more acute (smaller in width)

as compared to the honeycomb and foam sandwich beams. This phenomenon is less evident in case of the glass entangled sandwich beams (Fig. 10 b).

It is noticed that in case of honeycomb sandwich beams, the level of damage is not the same on both sides of the beams as it depends on whether the honeycomb cell center or corner is impacted as discussed previously and shown in Fig. 6. So this asymmetric damage leads to distortion of the resonance peaks or the appearance of twin peaks instead of one. This is evident in Fig. 11, which shows a comparison of the sum of the frequency response functions (FRF), estimated by burst random testing, for the six sandwich beams for the undamaged (UD) and the damaged cases (D1) for the 3rd bending mode. The sum of the FRF can be compared as for each sandwich beam 27 symmetric measurement points have been chosen

that are symmetric on both sides of the two major axes of symmetry. in the presence of damage. We can say that frequency estimation is then reliable and so on we are able to detect damages from frequency shifts in the FRF. Estimating damping with reliability is more difficult [START_REF] Morlier | Original statistical approach for the reliability in modal parameters estimation[END_REF]. That's why we use two Modal Parameter Estimators POLYMAX in the frequency domain and POLYREFERENCE in the time domain. Both algorithms are based on least-squares optimization on 27 experimental FRFs to fit poles from analytical data (complex conjugate poles of Eq 3) with experimental FRFs. Finally damping ratios can be wiewed as equivalent linearized damping ratios. However, for the estimation of damping ratios for the damage state D1, there is a notable difference between the results of burst random and sine dwell testing as shown in Fig. 12, because sine dwell excitation can differentiate between the linear and non-linear effects as the excitation is at a unique frequency.

It can be seen in Fig. 12 that in general the damping increases with the increase in damage in the sandwich beams. Nonetheless in case of burst random testing, for the carbon entangled and the carbon honeycomb sandwich beams (Fig. 12 b) and for the carbon foam sandwich beam (Fig. 12 c), the damping decreases with damage. However sine dwell testing shows a logical increase of damping for these beams. Furthermore, the estimation of damping by sine-dwell testing for the damage state (D1) is always notably higher as compared to burst random testing. It can be said that sine dwell testing is more capable of detecting non linear structural dynamic behavior (due to accumulation of damage as in state D1) unlike the broadband excitations. Furthermore, the change in damping ratios between the undamaged and the damaged case is smaller again in case of entangled sandwich beams. Therefore it can be concluded that the entangled sandwich beams show a better resistance to impact as compared to the honeycomb and foam sandwich beams, whereas all the beams have more or less the same level damage. This trend verifies somewhat the results of the impact tests carried out on these sandwich beams in Section 3.2. By taking into consideration these results, in terms of application the authors think that the entangled sandwich materials can replace the aluminum parts at the wing tips used to resist impact, as these aluminum parts are not load carriers. Hence the entangled materials can be used in these types of specific applications where static strength is not required but the main objective is the resistance to impact.

The effect of impact damage on the natural frequencies and damping ratios can be further elaborated by studying the frequency and the damping change ratios presented in Tables 8 and9 between the undamaged (UD) and the damaged case (D1) for the six sandwich beams based on Eq. 1 and Eq. 2.

Tables 8 and9 show that the shift in modal parameters is less in case of the entangled sandwich beams which signifies that they possess better impact toughness as compared to the standard sandwich beams with honeycomb and foam as core materials. It can also be noticed that the change in damping ratios is greater in case of sine dwell testing, because the damping ratio estimated by sine-dwell testing is always higher in case of the damaged state (D1) as compared to the burst random testing. Furthermore, the results in Tables 8 and9 underline the fact that the damping change ratios are more prominent than the frequency change ratios. The maximum damping change ratio is 310 % whereas the maximum frequency change ratio is 25 %. It can be concluded from the above results that damping seems more sensitive to damage than the natural frequency variations in case of honeycomb sandwich beams. So it is reasonable to assume that damping may be used instead of natural frequency as a damage indicator tool for structural health monitoring purposes. However, the fact that damping is a parameter that is relatively difficult to estimate as compared to natural frequency has to be taken into account.

Conclusion

The aim of the article is to evaluate the resistance against impact of entangled sandwich beams by the pole shift which signifies loss of rigidity (decrease in natural frequency) and increase in friction in damaged zone (increase in damping ratio). A simple case of symmetrical impacts is studied and Low Velocity Impacts are done below the BVID limit in order to detect damage by vibration testing that is hardly visible on the surface. From the results it can be concluded that both carbon and glass fiber entangled sandwich beams show a better resistance to impact (based only on decrease in natural frequency) as compared to the honeycomb and foam sandwich beams, whereas all the beams have more or less the same level damage.

Nomenclature

Fig. 11 b

 11 Fig. 11 b and Fig. 11 e show that in case of honeycomb sandwich beams due to asymmetric distribution of damage the shape of peaks become distorted. A slight distortion of peak is also observed in case of the entangled glass beam in Fig. 11 d. But on the whole for the entangled and foam sandwich beams the damage is pretty much symmetric and peaks remain intact. Both burst random and sine-dwell testing give similar natural frequency results

  BR = Burst random testing SD = Sine-dwell testing UD = undamaged state D1 = damaged state at 2 points FRF = Frequency Response Function H(ω) = Frequency Response Function matrix j = Imaginary axis in the complex plane * = Complex conjugate ω(k) = Modal damped frequency for kth mode (rad/s) p(k) = Pole location for the kth mode R(k) = Residue magnitude (FRF/s) ! (k) = Modal damping for kth mode ω n = Undamped natural frequency (rad/s) ω d = Damped natural frequency (rad/s) C = Structural damping matrix (force/velocity) K = Stiffness matrix (force/displacement) M = Mass matrix f k = Resonance frequency (Hz) for the kth mode ! k = Damping ratio (%) for the kth mode

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .

 1234 Fig. 1. The six sandwich beams tested in this article with (a) carbon prepregs and (b) glass woven fabric as skin materials

Fig. 5 .

 5 Fig. 5. Detailed cutaway of the drop assembly, the guidance tube and the blocking system

Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .

 6789 Fig. 6. Dispersion of damage between the two impact points in case of honeycomb carbon beam, impacted at 4 J

Fig. 10 Fig. 11 Fig. 12 .

 101112 Fig. 10 Comparison of the frequency response functions estimated by sine-dwell testing for the undamaged case (UD) and damaged at 2 points (D1) for the 1 st bending mode for (a) Point 11 for the three carbon sandwich beams and for (b) Point 21 for the three glass sandwich beams

Table 1

 1 Physical properties of carbon/epoxy prepreg T700/M21 used as skin material in carbon

	sandwich beams
	Homogenized elastic modulus in the longitudinal direction (E x ) 67000 MPa
	Homogenized elastic modulus in the transverse direction (E y )	67000 MPa
	Shear Modulus (G 12 )	5000 MPa
	Poisson Ratio (υ 12 )	0.4
	Volume density (ρ)	1550 kg/m 3

Table 2

 2 Properties of glass woven fabric used as skin in glass sandwich beams

	Elastic modulus in the longitudinal direction (E x )	23000 MPa
	Elastic modulus in the transverse direction (E y )		23000 MPa
	Shear modulus (G)		2900 MPa
	Poisson ratio (υ)		0.098
	Table 3
	Properties of Honeycomb core (Hexcel-aramid)
	Cell size	6.5 mm
	Density	31 kg/m 3
	Compressive strength	0.89 MPa
	Compressive modulus	75.8 MPa
	Shear strength in longitudinal direction ( xz ! ) 0.65 MPa
	Shear modulus in longitudinal direction (G xz ) 29 MPa
	Shear strength in width direction ( yz ! )	0.31 MPa
	Shear modulus in width direction (G yz )	13.8 MPa

Table 6

 6 Types and weight of the six sandwich beams tested in this article Type of Sandwich Beam Weight of Sandwich Beam (g)

	Entangled Carbon	63
	Honeycomb Carbon	29
	Foam Carbon	27
	Entangled Glass	52
	Honeycomb Glass	24
	Foam Glass	29

Table 7

 7 Impact test parameters of the sandwich beams

	Type of Specimen Energy of Impact	Indentation just after	Velocity of impact
		Measured (J)	impact (mm)	Measured (m/s)
			Point 1	Point 2
	Entangled Carbon	4.9	0.1	0.15	2.21
	Honeycomb Carbon	3.9	0.1	0.5	1.98
	Foam Carbon	3.8	0.1	0.2	1.98
	Entangled Glass	6.2	0.2	0.15	2.49
	Honeycomb Glass	3.8	0.2	0.4	1.98
	Foam Glass	3.8	0.15	0.25	1.98

Table 8

 8 Frequency change ratios (%) between the undamaged (UD) and the two damaged states (D1 and D2) for the carbon and glass sandwich beams for both burst random (BR) and sine dwell

	(SD) testing
	Type of Specimens	Between	Mode 1	Mode 2	Mode 3
		States
			BR	SD	BR	SD	BR	SD
	Foam Glass (4J)	UD and D1 14.2	14.2	21.4	21.6	18.0	18.2
	Honeycomb Glass (4J)	UD and D1 10.4	9.7	8.0	7.9	9.7	9.4
	Entangled Glass (6J)	UD and D1	5.8	5.5	7.2	7.1	8.2	7.6
	Foam Carbon (4J)	UD and D1 25.2	25.6	15.3	15.5	22.6	22.7
	Honeycomb Carbon (4J) UD and D1 14.4	14.9	7.2	8.6	12.8	12.4
	Entangled Carbon (5J)	UD and D1	2.2	2.2	2.1	2.2	2.8	2.9

Acknowledgments

The authors gratefully thank research project student Hanno Niemann from TU Braunschweig for his technical support during the impact tests.

List of tables

Damping change ratios (%) between the undamaged (UD) and the two damaged states (D1 and D2) for the carbon and glass sandwich beams for both burst random (BR) and sine dwell


Joseph Morlier
email: joseph.morlier@isae.fr

Dimitri Bettebghor

Compressed sensing applied to modeshapes reconstruction

Modal analysis classicaly used signals that respect the Shannon/Nyquist theory. Compressive sampling (or Compressed Sampling, CS) is a recent development in digital signal processing that offers the potential of high resolution capture of physical signals from relatively few measurements, typically well below the number expected from the requirements of the Shannon/Nyquist sampling theorem. This technique combines two key ideas: sparse representation through an informed choice of linear basis for the class of signals under study; and incoherent (eg. pseudorandom) measurements of the signal to extract the maximum amount of information from the signal using a minimum amount of measurements. We propose one classical demonstration of CS in modal identification of a multi-harmonic impulse response function. Then one original application in modeshape reconstruction of a plate under vibration. Comparing classical ! 2 inversion and ! 1 optimization to recover sparse spatial data randomly localized sensors on the plate demonstrates the superiority of ! 1 reconstruction (RMSE).

Introduction

Compared to nondestructive testing (ultrasound techniques, analysis of magnetic fields, radiology, thermal methods) vibration-based analysis allows a mixed global/local analysis of the structure with the potential of being applied in situ (laser vibrometer, optical sensor, etc) [START_REF] Doebling | Damage Identification and Health Monitoring of Structural and Mechanical Systems From Changes in Their Vibration Characteristics: A Literature Review[END_REF][START_REF] Sohn | A Review of Structural Health Monitoring Literature form 1996-2001[END_REF]. Deciding on an optimal sensor placement and optimal frequency sampling is a common problem encountered in many engineering applications and is a critical issue in the construction and implementation of an effective Structural Health Monitoring system (SHM). As a first example we study the modeshape reconstruction from grid placement. It highlights the fact that mode shapes visualization is often biased due to spatial aliasing. On figure 1, we can see that 9 grid point measurement are not enough precise to reconstruct the (3,1) mode shape. In a first paper [START_REF] Morlier | Original statistical approach for the reliability in modal parameters estimation[END_REF] we use Monte Carlo approach to demonstrate the ability of Kriging method at high spatial density to reconstruct mode shape with accuracy using regular or random grid. In this paper we are not analyzing a problem of Sensor Placement Optimization (SPO), which aims at identifying the sensor layout that will optimize one or more of the probabilistic performance measures. A detailed literature about SPO can be found in [START_REF] Sohn | A Review of Structural Health Monitoring Literature form 1996-2001[END_REF][START_REF] Guratzsch | Sensor placement optimization under uncertainty for structural health monitoring of hot aerospace structures[END_REF]. We prefer to have a "signal processing" approach. According to well known theorem sampling theorem, Stubbs and Park [START_REF] Stubbs | and Sikorski Optimal sensor placement for mode shapes via Shannon's sampling theorem[END_REF] introduced this theorem for spatial data for avoiding well know problem called "aliasing". Schulz et al. [START_REF] Schulz | Structural Health Monitoring Using Frequency Response Functions and Sparse Measurements[END_REF] address the issue of damage resolution as a function of spatial distribution of sensors. They show that damage can be located within a spatial resolution equal to the distance between sensors on a structure. Sazonov and Klinkhachorn have developed an optimal sampling theory [START_REF] Sazonov | Optimal spatial sampling interval for damage detection by curvature or strain energy mode shapes[END_REF]. This approach allows us to estimate high resolution modeshapes taking into account experimental noise and enables to evaluate even small damages. This method use the curvature mode shape properties to find relationship between optimal sampling data and Signal to Noise Ratio and have been adapted for wavelets approach by Morlier et al [START_REF] Morlier | Diagnosis of a portal frame using advanced signal processing of laser vibrometer data[END_REF].

The first works combining acoustic measurement and parcimonious approach were presented last year at the French Congress of Acoustics [START_REF] Chardon | Compressed sensing" pour l'holographie acoustique de champ proche -I : Aspects algorithmiques et simulations[END_REF][START_REF] Peillot | Le[END_REF]. The advanced mathematical techniques so-called Compressive Sensing (CS) benefit fields as diverse as sensors, signal processing, image compression etc ... CS is used to find some kind of underlying structure behind most of the analog signals on the condition that these signals are sparse [START_REF] Candès | Stable signal recovery from incomplete and inaccurate measurements[END_REF][START_REF] Donoho | Compressed Sensing[END_REF][START_REF] Candès | An Introduction To Compressive Sampling[END_REF]. It is then possible to acquire signal at lower sampling frequency, and therefore no longer verify the Shannon/Nyquist frequency.

Compressive sensing: learning by numerical examples

The least-squares solution to such problems is to minimize the ! 0 norm-that is, minimize the amount of energy in the system. This is usually simple mathematically (involving only a matrix multiplication by the pseudo-inverse of the basis sampled in). However, this leads to poor results for many practical applications, for which the unknown coefficients have nonzero energy. To enforce the sparsity constraint when solving for the underdetermined system of linear equations, one can minimize the number of nonzero components of the solution. The function counting the number of non-zero components of a vector was called the ! 0 norm by David Donoho. Candès. et. al. [START_REF] Candès | Stable signal recovery from incomplete and inaccurate measurements[END_REF][START_REF] Donoho | Compressed Sensing[END_REF], proved that for many problems it is probable that the ! 1 norm is equivalent to the ! 0 norm, in a technical sense: This equivalence result allows one to solve the ! 1 problem, which is easier than the ! 0 problem. Finding the candidate with the smallest ! 1 norm can be expressed relatively easily as a linear program, for which efficient solution methods already exist [START_REF]L1-MAGIC is a collection of MATLAB routines[END_REF].

The traditional approach to data acquisition is based on the Shannon-Nyquist theorem: to acquire a signal with a bandwidth of size W must be sampled at a higher frequency 2W. The compressive sensing exploits the fact that many real signals can be expressed in a sparse way and the inconsistency between certain types of bases to reduce this number of samples. A vector Ssparse is a vector that has at most S nonzero components. Many natural signals, when expressed in a particular base, have a representation with many significant coefficients. Data compression exploits this fact by removing these low coefficients, which slightly reduces the signal quality. These real-world signals (e.g. sound, images, video) can be viewed as an ndimensional vector. To acquire this signal, we consider a linear measurement model, in which we measure an m-dimensional vector b = Ax ∈ R m for some m × n measurement matrix A (thus we measure the inner products of x with the rows of A). For instance, if we are measuring a time series in the frequency domain, A would be some sort of Fourier matrix.

This leads to the following classical question in linear algebra : How many measurements m do we need to make in order to recover the original signal x exactly from b?

The classical theory of linear algebra is as follows:

• If there are at least as many measurements as unknowns (m ≥ n), and A has full rank, then the problem is determined or overdetermined, and one can easily solve Ax = b uniquely (e.g. by gaussian elimination).

• If there are fewer measurements than unknowns (m < n), then the problem is underdetermined even when A has full rank. Knowledge of Ax = b restricts x to an (affine) subspace of R n , but does not determine x completely. However, if one has reason to believe that x is "small", one can use the least squares solution : x = argmin x:Ax=b x ! 2 = A * (AA * ) !1 b as the "best guess" for x.

Compressed sensing is advantageous whenever signals are sparse in a known basis. So the advantage is obvious when measurements (or simulations) are expensive and mathematical inversion is cheap. Such situations can arise in imaging (e.g. the "single-pixel camera"), Sensor networks, MRI Astronomy etc…and modal analysis.

In fact, the above proof also shows how to reconstruct an S-sparse signal x ! R n from the measurements b = Ax .

x is the unique sparsest solution to Ax = b . In other words,

x = argmin x:Ax=b x ! 0 (1)
where

x ! 0 := x i 0 i=1 n ! =#{1 " i " n:xi # 0} is the sparsity of x.
Unfortunately, in contrast to the ! 2 minimisation problem (least-squares), ! 0 minimisation is computationally intractable (in fact, it is an NP-hard problem). In part, this is because ! 0 minimisation is not a convex optimisation problem. A simple, yet surprisingly effective, way to do so is ! 1 minimisation (or basis pursuit); thus, our guess x* for the problem Ax = b is given by the formula :

x ! = argmin x:Ax=b x ! 1 (2)
This is a convex optimisation problem, However, the ! 1 norm is not differentiable and this prevents from using classical optimization algorithm from differentiable optimization (such as SQP,...). Nonetheless, this non-differentiable convex optimization problem can be transformed in a linear programming optimization problem

x ! = argmin

x,y"# n ,y >0:Ax=b

y i i=1 n $
and then can be solved fairly quickly by linear programming methods. Note that compressed sensing has equivalent formulations in the statistical field of regression. Best subset regression methods for instence seek amongst all the predictors the best set that explain the best the outcome to be predicted. This equivalent to ! 0 minimization. Several heuristics exist to solve it through for instance best forward and best backward subset selection. The same way ! 1 minimization has equivalent in regression, these are for instance lasso or LAR (Least Angle Regression) techniques. A valuable reference on these specific regression techniques is [START_REF] Friedman | The Elements of Statistical Learning[END_REF]. The main assumption behind these statistical techniques, which is precisely the same underlying assumption for compressed sensing, is known in statistics as bet on sparsity.

In a recent paper [START_REF] Bao | Compressive sampling for accelerometer signals in structural health monitoring[END_REF], the authors use CS on accelerometer signals of vibration of a bridge. The interest is obvious when engineers try to analyse vibrations during a year, using hundreds of sensors (MEMS and wireless network). We can note that the choice of the basis (wavelets, Fourier) is crucial with this type of approaches. In aeronautic we have the same interest in continuous monitoring of structures over a long period (flutter detection).

We try to illustrate a basic application of CS in the next example (using Matlab code ! 1 -Magic of the University of Caltech [START_REF]L1-MAGIC is a collection of MATLAB routines[END_REF]). Let's take the example of an analog signal (ie Fs = 400Hz >> Nyquist frequency) to 4 frequency components 30, 60, 100 and 130 Hz. We will compare in figure 2 the reconstruction of the signal regularly sampled at Fs=150 Hz using linear reconstruction formula (Shannon) and compressive sensing with fewer points of observations (but random sample). We see that the spectrum (DCT) has four resonances in the continuous signal, and 4 also in the digital signal but aliasing appears because Fs is too low: the time signal reconstruction will not be correct. When we solve this problem using Moore-Penrose pseudo inverse, we can note the appearance of noise in Figure 3b (whereas CS imposes zero coefficients).

It is then easy to compare the result of the spectrum reconstructed by ! 2 norm and the solution "magic" by using the ! 1 norm given in figure 3. We shall then compare the reconstruction on zoomed time signal (Figure 4) : N Random varies from N/5 to N/10, where N is number of samples. We see that the ! 2 norm solutions and sampling the 'classic' is not correct with respect to the accuracy of ! 1 method. Of course reconstructions ! 2 and ! 1 inversion become better with increasing the number of observations. The current trend is to develop sampling acquisition cards for CS, but the requirements go beyond the technological limits of today's sensors.

Modeshapes reconstruction

When dealing with modeshapes reconstruction the principle is classicaly to make a regular grid of sensors. CS principles will permit to make sensor placement random. In numerical computation, surface interpolation functions create a continuous (or prediction) surface from sampled point values. The continuous surface representation of a raster dataset represents height, concentration, or magnitude. In Matlab, zi=griddata(x,y,z,xi,yi) fits a surface of the form z=f(x,y) to the data in the (usually) non uniformly-spaced vectors (x,y,z). Here we study empirically the feasibility of using. Shannon's sampling theorem is well known in 1D (time domain) but it also exists in 2D as demonstrate in the figure 5. Generally, the estimation accuracy will increase together with the sampling density. We propose to compare least square ! 2 inversion with CS method using randomly chosen sensors on the vibrating structures.

A FEM modal analysis of a (Simply Supported) SSSS plate was done using following geometrical and material properties (length=width= 0.8m; height= 0.01m;E = 210e9 Pa; nu = 0.33; rho = 7700). We choose to illustrate the CS principle using few sensors and a well-chosen (physical) dictionary basis (Fourier Basis). In these numerical experiments, we used 8 sensors and chose a natural basis of the first 25 eigenmodes of simply supported rectangular plate. The ! 2 norm minimization is computed through Moore-Penrose pseudo-inverse of the observation matrix A, while the ! 1 norm minimization is performed through simplex method. It should be noted whenever the dimension of x remains low, there is no substantial difference in terms computational burden between the two methods. However, for larger dimension (ranging from 100 for instance), the dimension of linear program, which is the double of the dimension of x, can lead to a prohibitive execution time. For such large problem decompositionaggregation methods can help to save computational time. Indeed, the linear problem has a specific structure (often denote block-angular) that can be exploited through classical decomposition algorithm of large-scale linear programming (such as Benders decomposition or Dantwig-Wolfe decomposition).

Experimentaly modeshapes are commonly estimated from the residues obtained by curve fitting algorithm from set of FRFs [START_REF] Ewins | Modal Testing: Theory, Practice and Application[END_REF]. This numerical study can be compared to experimental test where Laser Doppler Vibrometer can be moved automatically and so control the succession of acquisition for each point of the grid (regular or random). One can notice there are a variety of ways to derive a prediction for each location; each method is referred to as a model. With each model, there are different assumptions made of the data, and certain models are more applicable for specific data (for example, one model may account for local variation better than another). The existing methods of interpolation method can be found in [START_REF] Funkhouser | [END_REF][START_REF] Forrester | Engineering Design via Surrogate Modelling : a Practical Guide[END_REF]: Inverse Distance, Polynomial Regression, Kriging, Nearest Neighbour, Minimum Curvature, Radial Basis Function etc. Error due to sensor placement uncertainties can be defined by thrust regions (different radius depending of each sensor accuracy). This is not taken into account in this study but clearly affect the modeshape estimation.

Conclusion

We present an interesting approach for modal analysis using signals (1D, 2D) that do not respect the Shannon's theorem. A simple multi harmonic vibration signal is proposed to illustrate the CS principles ( ! 1 minimization). A more realistic example based on modeshape reconstruction has been done and the reconstruction appears to be enhanced by the use of CS method comparing to classical ! 2 inversion. We also exhibit on the plate example (modal analysis) the crucial choice (physical) of dictionary basis (Fourier Basis). Future works will test the CS ability on more complex structures (thin-walled structure) using 'optimized' dictionary basis adapted for experimental modal analysis using random sensors placement.

Fig. 1 .

 1 Fig. 1. : (3,1) mode of vibrating plate plus regular grid distribution of sensors in white circles (a) and The cubic interpolation which shows a spatial aliasing in mode shape reconstruction (b). A regular grid of 9 sensors permits only to reconstruct the (1,1).

Fig. 2 .

 2 Fig. 2. Analog signal (in blue), discretized signal (magenta) respecting Nyquist frequency (N points) and randomized signals at low resolution (N/10) (a), and DCT spectrum comparison (b)

Fig. 3 .

 3 Fig. 3. Comparison of DCT spectrum of reconstructed signals by ! 1 inversion (a) and ! 2 inversion (b) of randomized signals. The ! 2 inversion is not capable of good reconstruction (noise)

Fig. 4 .

 4 Fig. 4. Comparison of reconstructed signals by ! 1 inversion (green) for different sampling N/10 (a) and N/5: (b) of randomized signals. From the time domain (zoom) The ! 2 inversion (red) is not capable of good reconstruction of the continuous signal (blue) whereas ! 1 optimization (green) is reliable even for low sampling.

Fig. 5 .

 5 Fig. 5.: In general artefacts are due to under-sampling or poor reconstruction: Temporal aliasing (Shannon's theorem) (a), Spatial aliasing (b) due to limited spatial resolution and induce loss of details.

Fig. 6 .

 6 Fig. 6. First modeshape reconstruction of a SSSS plate. The sensors are in red (randomly chosen) on the 'continuous' modeshape (a) one can see the ! 2 inversion results on bottom. (b) We demonstrate the CS ability to reconstruct the modeshape, on bottom one can see the error versus continuous modeshape (maximum error of 5E-3).


Joseph Morlier
email: joseph.morlier@isae.fr

Mauricio Bergh

Laurent Mevel

Modeshapes recognition using Fourier descriptors: a simple SHM example

The main objective of this study is to develop an alternative criterion for modeshape classification, as the currently available one, MAC (Modal Assurance Criteria), is only a vector correlation representing modeshape similarities. This new method is developed to provide a set of features (Fourier Descriptors) for comparing modeshapes with "local" similarities of higher interest than "global" similarities using nodal lines. These lines are able to characterize modeshapes very easily. So when damage occurs, we are able to track the few descriptors changes to localise the damage. We validated our method on a CFCF plate demonstrating the quality of the damage localisation and possible use in a "mode tracking" application (space structure).

Introduction

For aeronautical structures, a field where this problem has been quite studied, the components have to undergo low energy impacts caused by dropped tools, mishandling during assembly and maintenance, and in-service impacts by foreign objects such as stones or birds. In these low energy impacts normally, a small indentation is seen on the impact surface. This level of damage is often referred to as Barely Visible Impact Damage (BVID). Although not visually apparent, low energy impact damage is found to be quite detrimental to the load bearing capacities of sandwich structures, underscoring the need for reliable damage detection techniques for composite sandwich structures. In recent years, vibration based damage detection has been rapidly expanding and has shown to be a feasible approach for detecting and locating damage [START_REF] Doebling | A summary review of vibration-based damage identification methods[END_REF][START_REF] Yan | Development in vibration-based structural damage detection technique[END_REF][START_REF] Carden | Vibration based condition monitoring: A review[END_REF]. Any structure can be considered as a dynamic system with stiffness, mass and damping. Once some damages emerge in the structures, the structural parameters will change, and the frequency response functions and modal parameters of the structural system will also change. This change of modal parameters can be taken as the signal of early damage occurrence in the structural system. Shift in natural frequency or in damping ratio is the most common parameter used in the identification of damage [START_REF] Khoo | Structural damage assessment using vibration modal analysis[END_REF][START_REF] Gadelrab | The effect of delamination on the natural frequencies of a laminated composite beam[END_REF][START_REF] Yam | Damage detection of composite structures using dynamic analysis[END_REF][START_REF] Shahdin | Correlating low energy impact damage with changes in modal parameters: A preliminary study on composite beams[END_REF]. Vibration mode-shape comparison between numerical models and experimental data is an essential step in the study of structural finite element model (FEM) updating. The Modal Assurance Criterion (MAC) is the most popular method for such comparison at the moment, which works perfectly well for small and medium sized structures. MAC provides a measure of closeness between the predicted and measured eigenvectors but contains no explicit information on shape features.The main objective of this study is to develop an alternative criterion for modeshape classification, as the currently available one, MAC (Modal Assurance Criteria), is only a vector correlation representing modeshape similarities. This new method is developed to provide a set of features (Fourier Descriptors) for comparing modeshapes with "local" similarities of higher interest than "global" similarities. This new method uses Image Processing and Pattern Recognition tools and is validated on a simple plate under vibration. A procedure for damage detection comparing two successive states (undamaged-damaged) is also proposed by classifying most interesting modes in term of dissimilarities.

The main hypothesis in this work is that the modeshape are acquired using very fine spatial resolution. According to well known theorem sampling theorem, Stubbs and Park [START_REF] Stubbs | and Sikorski Optimal sensor placement for mode shapes via Shannon's sampling theorem[END_REF] introduced this theorem for spatial data for avoiding well know problem called "aliasing". Schulz et al. [START_REF] Schulz | Structural Health Monitoring Using Frequency Response Functions and Sparse Measurements[END_REF] address the issue of damage resolution as a function of spatial distribution of sensors.

They show that damage can be located within a spatial resolution equal to the distance between sensors on a structure. Sazonov and Klinkhachorn have developed an optimal sampling theory [START_REF] Sazonov | Optimal spatial sampling interval for damage detection by curvature or strain energy mode shapes[END_REF]. This approach allows us to estimate high resolution mode shapes taking into account experimental noise and enables to evaluate even small damages. This method use the curvature mode shape properties to find relationship between optimal sampling data and Signal to Noise Ratio and have been adapted for wavelets approach by Morlier et al [START_REF] Morlier | Diagnosis of a portal frame using advanced signal processing of laser vibrometer data[END_REF].

Theoretical background

Modal analysis is a process whereby one can describe a structure in terms of its natural characteristics, which are the frequency, damping and mode shapes -its dynamic properties [START_REF] Ewins | Modal Testing: Theory, Practice and Application[END_REF]. MAC provides a measure of consistency (degree of linearity) between estimates of a modal vector. This provides an additional confidence factor in the evaluation of a modal vector from different excitation (reference) locations or different modal parameter estimation algorithms. The modal assurance criterion's values go from zero, representing no consistent correspondence, to one, representing a consistent correspondence. In this manner, if the modal vectors under consideration truly exhibit a consistent, linear relationship, the modal assurance criterion should approach unity and the value of the modal scale factor can be considered reasonable. Note that, unlike the orthogonality calculations, the MAC is normalized by the magnitude of the vectors and, thus, is bounded between zero and one. But the MAC can only indicate consistency, not validity or orthogonality. If the same errors, random or bias, exist in all modal vector estimates, this is not detected by the modal assurance criterion [START_REF] Allemang | The Modal Assurance Criterion -Twenty Years of Use and Abuse[END_REF]. MAC can be interpreted as the cosine of the angle between the numerical and measured eigenvectors. However, MAC index carries no explicit information on shape features. A new technique, based on the well-developed paradigms of Image Processing (IP) and Pattern Recognition (PR) are considered in this paper. IP is a set of computational techniques for analyzing, enhancing and reconstructing images. A typical PR approach involves the estimation of a series of shape attributes or features with good discriminative capability. The mapping from the space of shapes to the space of shape descriptors should determine the distance between descriptors of two models as a meaningful measure of the underlying similarity of their shapes [START_REF] Nixon | Feature Extraction and Image Processing Second edition[END_REF]. Fourier Descriptors (FD) describe a family of related image features. Generally, it refers to the use of a Fourier Transform to analyze a closed planar curve. FDs were originally proposed in 1960 by Cosgriff [START_REF] Cosgriff | Identification of Shape[END_REF], and thereafter became popular among the pattern recognition community through publications like Zahn [START_REF] Zahn | Fourier descriptors for plane closed curves[END_REF], Persoon and Fu [START_REF] Persoon | Shape discrimination Using Fourier Descriptors[END_REF] and are among the most popular shape representation methods for vision and pattern recognition applications. The basic idea underlying this approach consists in representing the shape of interest in terms of a 1D, 2D or even 3D signal. Many studies have been lead on the use of the Fourier Descriptors as a mechanism for shape identification [START_REF] Lin | New Forms of Shape Invariants from Elliptic Fourier Descriptors[END_REF]. Some work has also been done using Fourier descriptors to assist in Optical Character recognition (OCR) [START_REF] Granlund | Fourier preprocessing for hand print character recognition[END_REF][START_REF] Kuhl | Elliptic Fourier Features of a Closed Contour[END_REF][START_REF] Trier | Feature extraction methods for character recognition -A survey[END_REF]. In the context of OCR, the planar curve is generally derived from a character boundary. Since each character's boundary is a closed curve, the sequence of (x, y) coordinates that specifies the curve is periodic. This makes it ideal for analysis with a Discrete Fourier Transform (DFT). There are several variations of Fourier Descriptor features and of their use in shape recognition. For example, the formulation used by [START_REF] Wang | Shape descriptors for mode-shape recognition and model[END_REF] applies the Fourier Transform to the sequence of angular differences between line segments in the curve, while the method used by [START_REF] Lin | New Forms of Shape Invariants from Elliptic Fourier Descriptors[END_REF] is to apply the transform to the sequence of complex numbers formed by x + i y, where the point on the curve is (x, y). The method used in this project is most similar to the Elliptic Fourier Descriptors [START_REF] Kuhl | Elliptic Fourier Features of a Closed Contour[END_REF]. This method involves applying separate Fourier transforms to the sequence of x components and the sequence of y components of each curve. This allows the curve to be reconstructed exactly from the feature data, provided that all components of the frequency spectrum are saved. It is not typically necessary to do this, however, as most of the information about the curve is contained in the low frequency components of its transform.

Elliptic Fourier descriptors

The SD of an image (modeshape) may be considered as a point in the shape-feature vector space. 2D mode shapes are considered in this paper. Thus, the general form of SD can be expressed as :

! D = f I(x, y) [ ] (1
) where I(x,y) denotes the displacement modeshape function and f [*] is the tranformation extracting the shape features.

More specifically, we can project the image onto the kernel function R(x,y) as :

! D = R(x, y)I(x, y)dxdy " # (2)
where ! " is the domain of definition.

Shape descriptors are so dependant on the kernel function. Fourier, Wavelets and Zernike moments have been successfully used [START_REF] Wang | Mode-shape recognition and finite element model updating using the Zernike moment descriptor[END_REF][START_REF] Wang | Vibration mode shape recognition using image processing[END_REF][START_REF] Wang | Shape descriptors for mode-shape recognition and model[END_REF] for modeshape recognition and finite element model updating. Here We choose to analyse modeshapes using the Elliptic Fourier Descriptor.Some properties of the FDs directly follow from the underlying theory of the Fourier transforms and series, for instance, the invariance to geometric transformations. The FD is based on the frequency components from Fourier Transform (FT) of the images. According to the well-known theory of the FT, the kernel function of the SD is the complex valued sinusoid,

! D f (u,v) =
e "i2# (ux +vy ) I(x, y)dxdy

"$ +$ % "$ +$ % (3) D f (u,v
) is a continuous function having the same cardinality as I(x,y), and for real applications, this needs to be reduced whilst retaining as much information as possible. Using lower frequency and higher energy components we can describe the shape.

The cumulative angular function transforms the two-dimensional (2D) description of a curve into a one-dimensional periodic function suitable for Fourier analysis. In contrast, EFD maintain the description of the curve in a 2D space [START_REF] Granlund | Fourier preprocessing for hand print character recognition[END_REF]. This is achieved by considering that the image space defines the complex plane. Each pixel is represented by a complex number. The first coordinate represents the real part, while the second represents the imaginary part. Thus, a curve is defined as

! c(t) = x(t) + jy(t) (4)
Here, we consider that the parameter t is given by the arc-length parameterization. Figure 1 shows an example of the complex representation of a curve. This example illustrates two periods of each component of the curve. In general, ! T = 2" , thus the fundamental frequency is ! " = 1. It is important to notice that this representation can be used to describe open curves.

To obtain the elliptic Fourier descriptors of a curve, we need to obtain the Fourier expansion of the curve. The Fourier expansion can be performed by using the complex or trigonometric form.

In general, the equation linking curve and Fourier expansion terms is expressed in matrix form as

Each term in this equation has an interesting geometric interpretation as an elliptic phasor (a rotating vector). That is, for a fixed value of k, the trigonometric summation defines the locus of an ellipse in the complex plane. We can imagine that as we change the parameter t the point traces ellipses moving at a speed proportional to the harmonic number k [START_REF] Nixon | Feature Extraction and Image Processing Second edition[END_REF].

We compute the dynamic behavior of a Cantilever Free Cantilever Free (CFCF) plate on ABAQUS. The modeshapes (eigen vectors) are obtained using a standard eigenvalue solver. The figure 1 plots the 6 first mode shapes. The nodal lines are obtained by thresholding (displacement close to zero). For each modeshapes we can save nodal lines in a bitmap file (image) for further analysis with image processing tools. Using a classical EFD algorithm (Figure 3), contours are extracted from the nodal lines data using IP thresholding. From the complex curve, we compute Normalized Fourier Transform, then Fourier Descriptors and finally we are able to reconstruct each mode shape. One drawback of the method is that the image resolution should be enhanced (from 40 by 50 pixels to 200 by 250 pixels, so the resolution is increased by a factor 5 by splines interpolation). Retrieval of the original modeshape can accurately be obtained by the inverse Fourier transform (Figure 3b). Good approximation may reasonably be achieved by retaining only a small number of high energy terms (50 terms). We can see clearly the approximation is enhanced with number of Fourier Descriptors (Figure 4).

Application in damage detection

The MAC is used to analyse the correlation between the undamaged plate mode shapes with each other, the following results are obtained (Figure 4). From the MAC results on a simple plate we can observe high similarities between mode 10 and 15 for example (so a coefficient close to 0.7 in the MAC matrix in figure 4). But just comparing FD of these two modes (Figure 5), we can see that important dissimilarities exist. We can also conclude that reconstruction from FD can help to have "local" information of dissimilarities.

Fig. 5. Dissimiliraties can be seen between mode 10 and 15 comparing relative amplitude of Fourier Descriptors

An advanced criteria "correlation of Fourier Descriptor" adapted from correlation of Zernike moment [START_REF] Wang | Mode-shape recognition and finite element model updating using the Zernike moment descriptor[END_REF][START_REF] Wang | Vibration mode shape recognition using image processing[END_REF][START_REF] Wang | Shape descriptors for mode-shape recognition and model[END_REF] should be developed to distinguish close modes using the advantage of comparing at different levels of approximation (From Fourier to Wavelets Descriptors).

Changes in modal parameters (frequency, damping, mode shapes) are commonly used in SHM to detect, localize and identify damages in structures. FRF updating process can help to localize damages [START_REF] Shahdin | Correlating low energy impact damage with changes in modal parameters: diagnosis tools and FE validation[END_REF] but modeshapes are often difficult to use as a tool for localization of damages. In fact, direct comparison of modeshapes of damaged and undamaged plate gives no reliable information about the damage location because many false damages can appear in the map of absolute value of difference of 2 states of modeshapes (damaged, undammaged).

This method will help us to select "interesting modes" in the entire modal basis. The question is: How do we classify them? J Just in selecting modes with the lowest correlation using a mixed indicator. This indicator is just the combination of coefficient of correlation (R) and slope of the polynomial regression (order 1) between two sets of FD (state 1, undamaged, state 2, damaged) with equal weights).

For Finite Element Analysis with ABAQUS we use following dimension and properties: Plate dimension are 236 x 291 mm, and material properties are close to T700/M21 (laminate composites): density =1550kg/m3, E1=110.3GPa, E2=E3=7.69GPa, G12=G13=4.75GPa, G23=2.746GPa.

The delamination is localised on the right down part of the plate. We then compute eigenvalues and eigenmodes (modal frequencies and modeshapes) in the [0-2030Hz] frequency bandwidth (15 modes).

Experimentaly modeshapes are commonly estimated from the residues obtained by curve fitting from set of FRFs [START_REF] Ewins | Modal Testing: Theory, Practice and Application[END_REF]. In a first approach, comparing all the modeshapes between the two states gives lots of false damage. Then we compute the MAC between state 1 and 2 (Figure 6). We can not conclude about the damage existence and location only using this result (very close from the MAC using only undammaged case on Figure 4).

Fig. 6. Modal Assurance Criteria (MAC) of modeshapes of undamaged plate Vs damaged plate gives no information about the damage location

In a second approach we compare FD of the 15 modeshapes between the 2 states (undammaged, damaged). It also seems to have high correlation between the 2 cases (Figure 7). It is interesting to note that the results obtained from classification from R and slope are slightly different (Mode 9 is always the most dissimilar), so we decide to average the influence of these two indicators by creating an indicator which give the same weight to R and slope (Figure 8). It aims at quantifying the "local dissimilarities" of the modeshapes. The mapping of a potential damage zone (MD) is then created by a weighted indicator based on the absolute difference between selected modeshapes ! U state mod e (x, y)(step 2 is damaged case, step 1 is undamaged case) taking into account only dissimilar modes in the modal basis (in fact just a thresholding using lower values of the mixed indicator In fact even if each difference localizes the damage by a peak, it exists in every difference false damages (with lower values). So MD has been developed to intersect the location where changes are high and then, the more modes have this change, the more MD is high (Figure 9).

Conclusion

Shape Descriptors show the desirable properties of computational efficiency and ease of image reconstruction using a small number of SD terms. The Elliptic Fourier Descriptors are more general and very effective at extracting mode-shape features by virtue of their sinusoidal patterns. Nodal lines are able to characterize modeshapes very easily. We can wonder if experimentally, is it more convenient (or reliable) to measure displacement or to measure nodal line (null displacement)? Finally, we also validated our method on a CFCF plate demonstrating the quality of advanced for both modeshapes tracking and damage localisation. Futur works will focus on wavelets decomposition as enhanced feature descriptors.

Fig. 1 .

 1 Fig. 1. 6 first modeshapes of CFCF plate

Fig. 2 .

 2 Fig. 2. Elliptic Fourier descriptor are used to reconstruct mode 2 (a) based on 50 Fourier descriptors (d).

Fig. 3 .

 3 Fig. 3. A schematic contour gradual reconstruction of mode 2, using, respectively, 2, 6, 12 and 50 descriptors

Fig. 4 .

 4 Fig. 4. MAC exhibits high similarities (>0.7) betweens mode 10 and 15 (red circle), but comparing mode 10 and 15 we can easily see the difference in the center of the plate

Fig. 7

 7 Fig. 7 Comparison of Fourier Descriptors of mode 9 for undamaged and damaged cases : on top regression which allows to establish indicators to classify interesting mode shapes

Fig. 8

 8 Fig. 8 Mixed indicators versus modes, high changes of this indicator permits to see the influence of damage on the nodal line changes. In red, extracted mode shapes for damage localization using lower values of the mixed indicator.

!Fig. 9 .

 9 Fig. 9. Difference between selected modeshapes (5,9,13). Top mode 9 only, middle combination of mode 5,9, down combination of mode 5,9,13. Classifying most interesting modes permits to reduce influence of false damages.

Acknowledgments

This research was fund by EPICEA project (SAPES composites).