G Michon

A Berlioz

C H Lamarque

A

This paper is concerned with the experimental study of the dynamic behaviour of an inclined cable linked to a continuous beam modelling a simplified version of deck and cable of a bridge. External excitation is exerted on the beam. The cable attached to the end of the beam is submitted to a vertical sinusoidal solicitation due to the response of the finite stiffness beam. A guided device located at the end of the beam ensures the excitation with a variation of the horizontal component of the cable tension that introduces a new parametric excitation. Analysis of preliminary experimental results for main and secondary resonances permits us to consider simple modelling with one degree of freedom systems obtained by projection of the continuous three-dimensional model of the cable on adapted Irvine mode. Analytical treatment of these models involving data from the experimental devices shows a correct qualitative agreement between preliminary experiments and theoretical and numerical results.

INTRODUCTION

Cables oscillations studies have been extensively considered in the past decades. Some of them are devoted to the static behaviour [START_REF] Krenk | Mechanics and Analysis of Beams, Columns and Cables[END_REF], as other ones deal with the dynamic aspect [START_REF] Irvine | The linear theory of free vibrations of a suspended cable[END_REF], [START_REF] Irvine | Cable Structures[END_REF], [START_REF] West | Natural vibrations of suspension cables[END_REF], with a linear approach. Other work were conducted with the necessary description of the nonlinear terms to understand the behavior of cables vibration in real environment.

Meirovitch [START_REF] Meirovitch | Elements of Vibration Analysis[END_REF] shows that the in-plane motion of a not damped taut cable can be described by a Duffing type equation. Takahashi and Konishi [START_REF] Takahashi | Non-linear vibrations of cables in three dimensions, Part I: Non-linear free vibrations[END_REF] are interested in the free or forced three-dimensional nonlinear vibrations. Benedettini et al. [START_REF] Rega | Planar non-linear oscillations of elastic cables under subharmonic resonance conditions[END_REF], [START_REF] Benedettini | Parametric analysis of large amplitude free vibrations of a suspended cable[END_REF], [START_REF] Benedettini | Non-linear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions[END_REF] consider oscillations of an elastic cable suspended between two horizontal supports under various cases of resonances. These studies are also important regarding applications (dampers [START_REF] Xu | Non-linear vibration of cable-damper systems Part I: formulation[END_REF], aerial cable-cars [START_REF] Portier | Dynamic phenomena in ropeways after a haul rope rupture[END_REF], tendons control [START_REF] Achkire | Active tendon control of cable-stayed bridges[END_REF], stayed bridges, etc…).

Inclusion of nonlinear terms is compulsory to highlight other phenomenon (nonlinear modal interaction [START_REF] Pilipchuk | Non-linear modal interactions in shallow suspended cables[END_REF], super-harmonic effects, internal resonances [START_REF] Zheng | Super-Harmonic and Internal resonances of a Suspended Cable with Nearly Commensurable Natural frequencies[END_REF], bifurcations [START_REF] Reilly | Global bifurcations in the forced vibration of a damped string[END_REF], and parametric nonlinearities [START_REF] Lilien | Vibration amplitudes caused by parametric excitation of cable stayed structure[END_REF]). Nayfeh et al. [START_REF] Nayfeh | Linear and Nonlinear Structural Mechanics[END_REF] gives some interesting references.

Gattulli et al. investigates an analytical, numerical and experimental analysis of a cable stayed bridge in [START_REF] Gattulli | Nonlinear interactions in the planar dynamics of cable-stayed beam[END_REF] and [START_REF] Gattulli | One-to-two global-local interaction in a cablestayed beam observed through analytical, finite element and experimental models[END_REF]. Their modeling approach is based on the separate description of both medias (i.e. cable and beam), linked by boundary and relevant mechanical conditions.

The results suggested in these multiple references use all the possible approaches to describe, explain but by condensing the phenomenology of the observations. Usually, after the setting in equations in the form of a three-dimensional system of nonlinear partial derivative equations of evolution governing displacements of the cable, the authors use a discretization by introducing a truncated base of modes of the linear part (Irvine's modes) selected according to the solicitation range. Projecting on a truncated basis, a regular nonlinear spring -masse system is obtained and studied via analytical methods or by purely numerical approaches according to the type of awaited oscillations.

The presented piece of research is in the continuity of previous studies [START_REF] Berlioz | A Nonlinear Model for the Dynamics of an Inclined Cable[END_REF], [START_REF] Berlioz | Nonlinear vibrations of an inclined cable[END_REF], [START_REF] Boujard | Nonlinear parametric resonances in a stay of the Iroise cable-stayed bridge: analytical model and experiments[END_REF] in which this method was also used. In this works, the vibrations of an inclined cable were studied from the point of view of two particular resonances: principal resonance where the sinusoidal external excitation frequency is supposed near to the first in-plane Irvine natural frequency, and then when the external excitation frequency is close to the double of the natural frequency. The results obtained showed a good agreement between the experiment and the analytical description obtained for a one degree of freedom system projected on the suitable Irvine's mode. The adopted analytical method was the familiar multiple scales method.

Nevertheless in this work, the excitation was produced by imposing a displacement directly at the lower end of the cable, with a device that maintained the horizontal component of the cable tension as constant as possible. We considered that this model supposed to represent a small-scale model of a cable-stayed bridge did not truly take into account the deck of the bridge by the introduction of a finite stiffness. This is the reason why we study here a specific experiment of an inclined cable connected to a horizontal beam. The excitation of the beam is transmitted to the cable via a guidance system constituting an added mass. The measurement of the excitation force permits to decompose the dynamic components of the horizontal reaction and the components of the external excitation transmitted to the cable. The measurement of the excitation exerted on the basis of the cable and the measurement of the response of the cable permits to highlight concerned resonances and to consider a simplified model.

The aim of the paper is to understand by measurement the role of this stiffness, and to use a similar model as simple as possible (ideally a nonlinear one degree of freedoms system), the added beam acting simply like "filters" external excitation. The guidance system acts as a filter for the horizontal reaction to introduce in the simplified model.

The paper is organized as follows: after this introduction, the experimental set-up is detailed in the first Section. The second Section presents experimental results concerning reaction and excitation forces, and specific cable response. The simplified model and its reduction are presented in the third Section. The comparison between the experimental and theoretical results, and the capacity of the model to reproduce the observed phenomenon are proposed in the last Section.

EXPERIMENTAL INVESTIGATION

Description of the experimental set-up

The global experimental set-up, see Figure 1, is composed of a composite flexible blade which represents the deck, and an inclined cable (a steel wire surrounded by copper wire, in order to increase the weight per unit length, but not the flexural rigidity modulus). Both elements are linked to a mass, forced to move vertically, and which represents the anchor point and the equivalent mass of a section of the deck. Therefore, the cable has a given initial static tension due to the mass. A 100N electrodynamics shaker applies a force close to blade clumping. The measurement device is composed as follows. The excitation force from the shaker to the structure is measured thanks to a piezo-electric sensor. The transmitted force from the end of the beam to the mass is also measured thanks to a piezo-electric sensor. The vertical displacement of the mass is measured by a laser displacement sensor. The instantaneous cable tension is measured via an S-shape force sensor, which also captures the static tension. Finally a high resolution laser sensor captures without contact the in-plane motion of the cable.

Set-up mechanical characteristics and observations

In this experimental device, the cable length is 1.90m, the inclination angle is 21.7°, and the sag is about 10mm. The static tension is adjustable and set in this application to 62N in the cable.

The presence of the beam at the lower end of the cable has an important influence on the natural frequencies of the system. Therefore, if the cable is clamped-clamped, the out of plane natural frequency is equal to 4.87 Hz and the in-plane natural frequency is equal to 6.06 Hz. But, in this case, where the lower end of the cable is linked to a mass and a beam, the out of plane natural frequency is 6.1 Hz, and the in plane natural frequency is 5.8Hz.

Experimental results

The experimental results present the response amplitude of the cable as a function of the excitation frequency for the first two instability zones.

Primary resonance (Ω ≈ ω 1 )

The primary resonance presents a nonlinear behavior even for low excitation amplitude, see Figure 3.

2.3.2.Sub harmonic resonance (Ω ≈ 2ω 1 )

In the primary parametric instability zone, the response of the cable exhibit large amplitudes, see Figure 4. This instability naturally appears for large excitation amplitudes. A softening behavior is highlighted.

5.9 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 0

MECHANICAL MODEL

Let consider a cable, of length L, clamped at both ends and inclined of an angle θ with respect to the horizontal axis, see Figure 3. The motion at the lower end Z(t) is imposed and given by: ( ) ( )

01 cos Z t Z t = Ω x y X Y L θ Z(t) B A z Figure 3: Inclined cable model.
Classical method consists in considering a nonlinear elastic model of the cable represented by three partial derivative equations system describing the three-dimensional motion of the cable. The different steps are given in [START_REF] Berlioz | A Nonlinear Model for the Dynamics of an Inclined Cable[END_REF] or [START_REF] Berlioz | Nonlinear vibrations of an inclined cable[END_REF].

In our case, and following the experimental observations, the specific boundary condition inverses the classical distribution of in-plane and out of plane natural frequencies. This parameter is essential in the behavior of the response.

Regarding the in-plane vertical observed vibrations, and considering principal nonlinear terms, the retained problem is related to V(s,t), the vertical cable displacement, where t is time and s the curvilinear abscissa identical to the spatial coordinate x. The continuous problem is projected using the following expression for V:

( ) ( ) ( ) ( ) , . . cos V x t f x y t x Z t θ = +
where y is the unknown modal coordinate, θ the angle of the inclined cable with respect to the horizontal axis, ( ) ( )

cos z Z t Z t φ = Ω +
the imposed displacement at the lower end, and f(x) the Irvine function for the first mode defined by: ( )

0 0 0 0 0 0 1 cos tan sin 2 1 cos tan sin 2 2 2 x x L L f x ω ω ω ω ω ω       - -             =       - -            
Static and dynamic deformed shapes of the cable are presented on Figure 4. Among the parameters governing the obtained equation of motion for the one degree of freedom system, appears the horizontal tension H. This tension is composed of a static part H 0 and a dynamic part rH 0 , and is written as follow:

( ) ( ) ( ) 0 1 cos cos r H H r t φ θ = - Ω +
Due to the external excitation Z(t), and the moving point B, the damping can be taken under the general form:

( ) ( )

1 1 c 1 c cos t ξ = ξ + Ω + φ
Where , ,

Z r c

φ φ φ represent the phase delay between excitation and displacement, tension and damping respectively.

Finally, the one degree of freedom system which condensed y oscillation is given by:

( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2
+ ξω + +α - Ω + φ +α - Ω + φ = γ - Ω + φ Ω + φ +γ - Ω + φ Ω + φ + Ω + φ + Ω + φ   β +β - Ω + φ Ω + φ +     β - Ω + φ Ω + φ +β - Ω + φ  
Therefore, this model includes a tension fluctuation H, part of the parametric excitation, and an imposed vertical displacement Z at the lower end of the cable, responsible for the forcing term in the equation of motion, and also for parametric excitation.

The method of multiple scales in time is used to the first order.

1 i i y y y T t,i 0,1 = + ε = ε =
The contribution of nonlinear terms and damping terms are maintained to the same order. So we introduce:

i i 0i i0 i i0 , , i 1,2,3 ,i 1,2 ,i 2,3 β β γ
γ ξ ζ = ε = = ε = α = εα = = ε
This leads to the two differential equations: b T Θ = -σ , we obtain the two equilibrium equations governing amplitude a 0 and phase Θ.

2 2 0 0 1 2 0 y T y 0 ∂ + ∂ ω = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
∂ + ∂ ∂ ω = -∂ ∂ ∂ - ζω + Ω + φ + Ω + φ ∂ -α - Ω + φ -α - Ω + φ +β - Ω + φ Ω + φ +β - Ω + φ Ω + φ +β - Ω + φ +γ - Ω + φ ( ) ( ) ( ) ( ) ( ) ( ) 0
-ζω Θ γ + γ γ - ζω =   -ζω       + - - α       + Θ β + β -β     + Θ
-β β ω Θ γ + γ γ - ζ ω =   - -β + σ-α + β       - - - -α       Θ β + β -β     + Θ
The predicted response amplitude is plotted on Figure 5 for a given set of parameters. The two possible responses merge close to the natural frequency, and move afterwards.

Sub harmonic resonance

In this case, the detuning rule is:

+ Θ α β -β β -β ζω =   -       - - +     + Θ and ( ) ( ) 2
  + β -α -β -β + ω σ       -α - - +     + Θ ζ ω =
The predicted response amplitude is plotted on Figure 6 for a given set of parameters. Three possible responses exist, where the intermediate solution is unstable.

CONCLUSION

This paper was devoted to the experimental and theoretical investigation on the nonlinear behavior of cable stayed bridges. The experimental approach highlighted specific non-linear phenomenon and couplings between the cable and the beam. The softening behavior is due to the presence of the beam stiffness.

The theoretical model is originally based on the classical inclined cables in-plane equation of motion. In the developed model, the cable is both subjected to an imposed boundary displacement and tension fluctuation, which leads to a forced, parametric and nonlinear equation of motion. Both approaches depict the same conclusion: the cable-beam coupling leads to a softening behavior of the system. The perspectives are oriented toward the inclusion of the pile stiffness in the model, and also toward the passive control of the phenomenon.

Figure 1 :

 1 Figure 1: Experimental set-up sketch.

Figure 2 :

 2 Figure 2: Experimental set-up: global view and detail of the excitation system.

Figure 1 :

 1 Figure 1: Experimental response amplitude vs. excitation frequency for the in-plane motion of the cable for two levels of excitation force, for the primary resonance ( 1 ω Ω ≈ ), for both sweep directions (* : up; + : down) Parameters are dimensionless.

Figure 2 :

 2 Figure 2: Experimental response amplitude vs. excitation frequency for the in-plane motion of the cable for two levels of excitation force, for the sub-harmonic resonance ( 1 2ω Ω ≈ ). Parameters are dimensionless.

Figure 4 :

 4 Figure 4: Static (dashed) and dynamic (line) deformed shapes of the cable.

Introducing the classical detuning rule: 1 Ω

 1 ≈ ω + εσ , and eliminating the secular terms in the second differential equation lead to a single complex equation.

Figure 5 :

 5 Figure 5: Predicted dimensionless response amplitude vs. excitation frequency, for the primary resonance (Ω ≈ ω1).

Figure 6 :

 6 Figure 6: Predicted dimensionless response amplitude vs. excitation frequency, for the sub-harmonic resonance ( 1 2ω Ω ≈).


Amir Shahdin
email: amir.shahdin@isae.fr

Joseph Morlier

Yves Gourinat

Correlating low energy impact damage with changes in modal parameters: A preliminary study on composite beams

Keywords: Damage detection, Vibration measurement, Design of experiments, Composites 1

This paper is an experimental study of the effects of multi-site damage on the vibration response of a composite beam damaged by low energy impact. The variation of the modal parameters with different levels of impact energy and density of impact is studied. Specimens are impacted symmetrically in order to induce a global rate of damage. A damage detection tool Damage Index is introduced in order to verify the estimation of damping ratios. Design of Experiments is used to establish the sensitivity of both energy of impact and density of damage. The DOE analysis results (using natural frequency only) indicate that impact energy for 2 nd , 3 rd and 4 th bending modes is the most significant factor contributing to the changes in the modal parameters for this kind of symmetrical dynamic test.

Introduction

The use of composite materials in structural components has increased dramatically in recent years by offering enormous potential and benefits to the aerospace industry and many other sectors. However composite laminates are susceptible to delamination from a wide variety of sources which include fabrication stress, environmental cyclic loading and foreign object impact damage [START_REF] Tsai | Introduction to composite materials[END_REF][START_REF] Li | Influence of Fiber/Matrix interface properties on complementary energy and composite damage tolerance[END_REF][START_REF] Gadelrab | The effect of delamination on the natural frequencies of a laminated composite beam[END_REF]. Delamination may lead to severe degradation of the mechanical behavior of structures due to loss of structural integrity. The detection of delamination and the study of their influences on the mechanical behavior of delaminated composite structures has become an important issue in recent times. Therefore in this article, problems at low impact energy levels are analyzed, which arise before delamination (behavior of the structure being relatively linear). So the aim of this work is to understand the relationship between changes in dynamic parameters (natural frequencies and damping ratios) due to impact damage in composite beam specimens at early stages.

The purpose of structural health monitoring systems is to provide information about the condition of a structure in terms of reliability and safety before the damage threatens the integrity of the structure. So the diagnosis of the damage in structural systems requires an identification of the location and type of damage, as well as the quantification of the degree of damage. Therefore, several techniques have been used to detect damage in structures. In recent years, structural health monitoring using vibration based damage detection has been rapidly expanding and shown to be a feasible approach for detecting and locating damage. Doebling et al. [START_REF] Doebling | Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review[END_REF] gave an overview of the vibration based damage detection methods. The main idea behind this technique is that modal parameters i.e., frequencies, damping ratios and mode shapes change in a detectable manner due to loss of stiffness and mass. Several techniques have been researched for detecting damage in composite materials, many of them showing the effectiveness of dynamic response measurements in monitoring the health of engineering structures [START_REF] Adams | A vibration technique for nondestructively assessing the integrity of structures[END_REF][START_REF] Fugate | Vibration-Based Damage Detection Using Statistical Process Control[END_REF][START_REF] Waldron | Damage Detection Using Finite Elements and Laser Operational Deflection Shapes[END_REF][START_REF] Sundaresan | Methods of distributed sensing for health monitoring of composite material structures[END_REF][START_REF] Leclerc | Impact Detection in an Aircraft Composite Panel -a Neural Network Approach[END_REF][START_REF] Banks | An experimentally validated damage detection theory in smart structures[END_REF][START_REF] Yu | Experimental validation of vibration-based damage detection for static laminated composite shells partially filled with fluid[END_REF][START_REF] Richardson | Detection of damage in structures from changes in their dynamic (modal) properties-a survey[END_REF][START_REF] Tracy | Effect of delamination on the natural frequencies of composite laminates[END_REF][START_REF] Linder | Damage detection for composite materials using dynamic response data, adaptive structures and materials symposium[END_REF][START_REF] Garesci | Experimental results of a damage detection methodology using variations in modal parameters[END_REF][START_REF] Zou | Vibration-based model-dependent damage (delamination) identification and health monitoring for composite structures -a review[END_REF]. These methods are among the earliest and most common, principally because they are simple to implement on any size structure. Structures can be excited by ambient energy, an external shaker or embedded actuators. Accelerometers and laser vibrometers can be used to monitor the structural dynamic responses. Khoo et al. employed different vibration techniques in order to locate damage in structures. They used resonant poles to identify the modes, especially those modes that exhibit relatively large pole shifts are believed to be affected by damage [START_REF] Khoo | Structural damage assessment using vibration modal analysis[END_REF]. S.G. Mattson et al. carried out damage detection based on residuals and discussed the phenomenon of false negatives. According to them, false negatives give no indication of damage when damage is present. [START_REF] Mattson | Damage detection and localization based on outlying residuals[END_REF].

The delamination-induced decrease of natural frequencies is one of the most commonly practiced approaches in damage detection. Several authors have tried to explain structural damage by changes in natural frequencies. Adams and Cawley [START_REF] Adams | The localisation of defects in structures from measurements of natural frequencies[END_REF] localized damage in structure from measurements of natural frequencies. Yam et al. [START_REF] Yam | Damage detection of composite structures using dynamic analysis[END_REF] showed that with the increase in delamination size the natural frequency decreases, by artificially creating a crack and then vibrating the structure with the help of an embedded actuator. Giannoccaro [START_REF] Giannocaro | Fatigue damage evaluation of notched specimens through resonance and anti-resonance data[END_REF] studied the effects of fatigue damage on modal properties of composite specimens and found a considerable shift towards the left in the frequency response functions (FRF) after about 90% of the specimen life. Shang et al. [START_REF] Shang | Effect of fatigue damage on the dynamic response frequency of spot-welded joints[END_REF] showed that the changes in the natural frequencies of spot-welded joint specimens varied non-linearly with damage fraction. Arkaduiz [START_REF] Arkaduiz | Non-linear vibration of a delaminated composite beam[END_REF] found that the presence of a self created delamination reduces the natural frequencies of a composite beam when harmonically excited.

The introduction of damage into a material generally results in an increase of damping, which is related to energy dissipation during dynamic excitation. Therefore, damping has also been proposed as a potentially sensitive and attractive damage indicator, though research works related to damping are fewer in number than those on natural frequency. The main sources of internal damping in a composite material arise from microplastic and viscoelastic phenomena in the matrix together with the interface effects between the matrix and the reinforcement [START_REF] Adams | Damping in composites[END_REF]. Zhang and Hartwig [START_REF] Zhang | Relation of damping and fatigue damage of unidirectional fibre composites[END_REF] recommended damping in the evaluation of damage process which seemed more sensitive than the natural frequencies. Gibson [START_REF] Gibson | Modal vibration response measurements for characterization of composite materials and structures[END_REF] vibration tested composite specimens and found that for higher modes damping loss factor measurements are more significant than frequency measurements. Similarly, Saravanos and Hopkins [START_REF] Saravanos | Effects of delaminations on the damped dynamic characteristics of composites[END_REF] experimented on composite beams, and showed that the delamination has a more profound effect on modal damping than the natural frequencies. Colakoglu [START_REF] Colakoglu | Description of fatigue damage using a damping monitoring technique[END_REF] showed that the damping factor increased with the number of fatigue cycles. Richardson and Mannan [START_REF] Richardson | Correlating minute structural faults with changes in modal parameters[END_REF] found that the loss of stiffness in a structure corresponds to a decrease in modal damping frequency combined with an increase in damping. Another way to correlate damage with change in dynamic parameters is to use mode shapes. Works of Pandey et al can be cited as a first reference in this context [START_REF] Pandey | Damage detection from changes in curvature mode shapes[END_REF]. This domain of Structural Health Monitoring offers a lot of potential coupled with advanced signal processing tools [START_REF] Morlier | Diagnosis of a portal frame using advanced signal processing of laser vibrometer data[END_REF][START_REF] Douka | Crack identification in beams using wavelet analysis[END_REF][START_REF] Hong | Damage detection using the Lipschitz exponent estimated by the wavelet transform: applications to vibration modes of a beam[END_REF].

This paper focuses on the correlation of modal parameters and impact damage in composite beams from an experimental point of view based on low energy impacts. Nowadays, laser vibrometers are employed extensively to measure vibrations instead of accelerometers [START_REF] Amraoui | Laser vibrometry based detection of delaminations in glass/epoxy composites[END_REF][START_REF] Sundaresan | Experimental damage detection on a wing panel using vibration deflection shapes[END_REF][START_REF] B J Halkon | Vibration measurements using continuous scanning laser vibrometry: velocity sensitivity model experimental validation[END_REF]. The article is arranged in such a way that first the material and the experimental procedure are presented then the experimental results are discussed and finally a design of experiments is carried out on the experimental results. A nomenclature for all the symbols used in this article is also provided at the end.

Material and Specimen

Resin-containing carbon-fiber/epoxy prepreg of T300/914 is used to fabricate the test specimens. The material is supplied by Hexcel composites, the physical properties are set out in Table 1). The specimens are processed in a press. The curing cycle of the laminates is 2h at 180°C with a warming-up cycle of 0.5h at 135°C. The laminates are cut into beams using a diamond wheel cutter, following the ASTM D3039/D3470 standards. The vibration tests are carried out with two steel masses attached at the ends [START_REF] Vanhoenacker | The use of multisine excitations to characterise damage in structures[END_REF]. The aim of putting these masses at the ends is to enhance the difference in the modal parameters between the undamaged and the damaged test specimens. The dimensions of these end masses for specimens of Type 1 and 2 are listed in Table 3.

Vibration Tests

The experimental equipment used to obtain the modal parameters discussed in this paper is

shown in Figure 1. 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 8200 which is then assembled on a shaker supplied by Prodera having a maximum force of 100N. A fixation system is used to place the test specimens on the force sensor. The fixation system and the end masses are glued to the test specimens with a HBM X60

Impact Tests

A drop tower is employed to carryout the impact tests as shown in Figure 3 and a detailed cut away of the drop assembly is shown in Figure 4. The impactor tip has a hemispherical head with a diameter of 12.7mm. The specimens for the impact tests are fixed at all ends in the support system. Different sensors are used to measure force, acceleration and the velocity of impact. A force sensor (type 9051A) provided by Kistler is placed between the impactor tip and the free falling mass of 2kg (Figure 4). A Kistler accelerometer (type 8704B5000) is placed at the top of the main block. The impact velocity is measured with the help of an optic sensor having a laser diode. The sensor is positioned in such a way that the main block's grooves pass the laser beam just before the impact. Further details on the impact test methodology of this drop tower can be found in the reference [START_REF] Petit | Impact and compression after impact experimental study of a composite laminate with a cock thermal shield[END_REF].

Figure 4 Detailed cutaway of the drop assembly, the guidance tube and the blocking system

The impact height h between the specimen surface and the impact head is calculated with

impact tip E h 1.1 g.m = ( 1
)
where g is the gravitational constant, m tip is the total weight of the impact head, freefalling mass mass, force sensor and the accelerometer which in this case is 2.03 kg and E imp is the energy of impact. The factor 1.1 is to compensate losses due to friction between the guidance tube and the drop assembly. The impact parameters for the Type 1 (1.04 mm thickness) and Type 2 (3.12 mm thickness) test specimens are listed in Table 4. The composite beam specimens have three states. First one is the undamaged state (UD), the second is the damage state due to 4 impacts (D1) and the third is the damage state due to 8 impacts (D2). Vibration tests are carried out after each of these three states. A simple case is studied where the impact points are chosen as such that the damage is symmetric on both sides of the two axes of symmetry. The impact parameters (Table 4) show that the impact energies can be classified into two major groups. A low impact energy group comprising of impact energies of 3J and 4J, and a high impact energy group with impact energies of 7J to 10J. [START_REF] Doebling | Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review[END_REF] Results and Discussion

Tracking of Poles for Damage Detection

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)*) = ⎧ ⎫ ω = + ⎨ ω - ω - ⎩ ⎭ ∑ ⎬ (2)
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 ( 2) is the complex conjugate term). 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 ) = -σ + ω (3)
The magnitude of each pole is the undamped natural frequency (ω n ). The undamped natural frequency (ω n ) and the modal damping ( σ ) are related to mass, stiffness and damping as follows:

given by

2 2 n d K (k ) M ω = ω + σ = ( 4
) C 2 (k) M σ = (5)
The effect of physical properties on poles in the complex s-plane is illustrated in Figure 5.

Figure 5 Movement of pole due to mass stiffness and damping effect

From Figure 5, 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 ( σ σ ) due to damage in the specimens [START_REF] Richardson | Correlating minute structural faults with changes in modal parameters[END_REF].

Significance of End Masses

As discussed before, the end masses are placed to enhance the difference between the modal parameters for the undamaged and the damaged cases. In order to verify this, vibration tests are carried out on Beam 1, tested in the intact condition (UD), first without end masses and then with the end masses. Then it is impacted at 2.5J first at 4 points (D1) and then at 8 points (D2). After each of these two series of impacts on Beam 1, vibration tests are again carried out both with and without masses. A comparison of the frequency response functions for the measurement point 5 of Beam 1 with and without end masses is shown in Figure 6. From Figure 6, it is clear that the difference in the damped modal frequencies before and after impact is greater for the case with end masses. The disadvantage of Type 1 specimens is that, the maximum energy for which they can be impacted is 5J. Above this energy level, the impactor tip penetrates the specimens creating severe damage (creating a hole). The aim of the impact tests is to damage locally the test specimens instead of damaging them globally. Therefore further vibration tests have been carried out on Type 2 specimens (3.12 mm thickness).

Frequency and Damping Changes

Frequency and damping changes are studied with the help of bending modes as they have the largest amplitudes for the type of test configuration presented in this article. The first three bending mode shapes of the composite beam calculated by the Polymax method integrated in LMS Test Lab are shown in Figure 7. In this section the first bending mode is not studied, because the difference in the damped natural frequencies between the damaged states (D1 and D2) and undamaged state (UD) for the first mode is very small. For the 2 nd and 3 rd bending modes, the variation of damped natural frequency as a function of the undamaged (UD) and the two damage states (D1 and D2) is presented in Figure 8. As discussed before in section 4.1, that damage in the specimens prompts a decrease in natural frequencies so from Figure 8, it is clear that the decrease in the natural frequencies in case of the high energy group (7, 8, 9 and 10J) is more prominent than that for the low energy group (3J and 4J). Beam 4 for the second bending mode (Figure 8a), has a natural frequency lower than that of the other beams for the undamaged case (UD) but this is not the case for the third bending mode (Figure 8b). This anomaly outlines the inherent possibility of false negatives which can arise due to boundary conditions and gives no indication of damage when it is present as discussed in the reference [START_REF] Mattson | Damage detection and localization based on outlying residuals[END_REF].

The shift in the natural frequencies between the undamaged and the damaged cases is more prominent at higher frequencies. This is evident in Figure 9, which shows a comparison of the sum of the frequency response functions for Beam 7 (impacted at 10J) for the undamaged case (UD), damaged at 4 points (D1) and damaged at 8 points (D2). The sum of the FRF can be compared as for each beam 33 symmetric measurement points have been chosen and the eight impact points are also symmetric on both sides of the two major axes of symmetry. The variation of damped natural frequencies with damping ratios (shift in the s-plane) for the Beams 3, 4 and 6 for the 2 nd and 3 rd bending modes is plotted in Figure 10. Figure 10 shows that damping increases whereas the frequency decreases with the increase in damage. The increase in damping is more prominent in specimens impacted at 7J and 9J (Beam 4 and 6) as compared to 4J (Beam 3). So it can be said that the variation of damping ratio validates the theoretical reasoning presented in Section 4.1.

Damage Index Variations

Another way to explain the changes in modal parameters due to damage is with the help of a Damage Index (DI) [START_REF] Cheraghi | A novel approach for detection of damage in adhesively bonded joints in plastic pipes based on vibration method using piezoelectric sensors[END_REF]. The first step is to calculate the integral (I x ) of the amplitude of the frequency response functions (FRF) over a defined frequency range (interval) as follows:

fk int x fk int I H( ) + - = ω ∫ dω (6)
where H(ω) is the frequency response function matrix obtained experimentally, (f k ) is the damped resonant frequency and (int) is the frequency interval chosen.

In the second step, the Damage Index is calculated by taking the percentage difference between the integrals of the resonance peaks of the FRF at the damaged and the undamaged states.

In this study, there are two damaged states (D1 and D2) and one undamaged state (UD). So the two damage indices are calculated as follows:

Damage Index for states D1 and UD = 100 4 For this article, a frequency interval of ± 20Hz (Figure 11) is chosen to calculate the integral of the resonance peaks of the FRF. In order to verify the robustness of this method, similar calculations have been carried out on frequency intervals of ± 10Hz and ± 40Hz as well. It has been observed that if the intervals are changed, the values of the integral (I X ) are different but the damage index always gives the same results being a ratio. The percentage increase in damage index is the percentage difference of DI8 and DI4 based on Equations ( 7) and ( 8) and is calculated as follows:

% increase in damage index = DI8 DI4 x100 DI8 - ⎡ ⎛ ⎞ ⎜ ⎟ ⎢ ⎝ ⎠ ⎣ ⎦ ⎤ ⎥ (9)
The damage index variation is explained in Figure ( 12) by plotting the percentage increase in damage index calculated from Equation ( 9) as a function of the first four bending modes.

Figure 12

Variation of percentage of increase in Damage Index as a function of modes Figure 12 shows that in case of the low energy impact group (3J and 4J), the percentage increase in damage index is between 2% to 13%. But in the cases of high impact energies (7, 8, 9

and 10J), this percentage ranges from 20% to 60%. Therefore, there is a clear increase in damage index with the rise in impact energy levels which corresponds to damage accumulation in the composite beam specimens.

Comparison between Damping Ratio and Integral of the Resonance Peaks (I x )

A comparison between the damping ratio and the integral of the resonance peaks (I x ) is carried out because they both describe the shape of the peak in the frequency response. The damping ratio and the integral (I x ) calculated by Equation ( 6) plotted as a function of damped natural frequency for the 2 nd and 3 rd bending modes, for the Beams 3, 4 and 6 is shown in Figure 13.

The direction of the increase of damage is represented by the increase in marker size for each plot.

The smallest marker size represents the undamaged state (UD), the medium damage at 4 points (D1) and the largest damage at 8 points (D2). It can be seen from Figure 13, that the variation of damping ratio and the integral of the amplitudes of the resonance peaks (Ix) show similar behavior for the same beam specimen which is not surprising because they both describe the form of the resonance peaks. For the other three Type 2 beams (Beam 2, 5 and 7), the damping ratios estimated by the Polymax method are not stable because they do not show an increase with the increase in damage in the specimens. This can be attributed to the fact that damping is a phenomenon that is harder to estimate as compared to the natural frequency [START_REF] Iglesias | Investigating various modal analysis -Extracting techniques to estimate damping ratio[END_REF][START_REF] Morlier | Original Statistical Approach for the Reliability in Modal Parameters[END_REF]. On the other hand, the integral (Ix) values for these three beams (Beam 2, 5 and 7) increase with the increase in damage in the specimens. This variation of the integral (Ix) with the increase in damage for Beam 2, 3 and 5 for the 2 nd and 3 rd bending modes is plotted in confidence curves cross the horizontal red lines, then the effect of that factor is said to be significant on the results. If the confidence curves are asymptotic to the horizontal line then the effect of the factors is marginally significant and finally if the confidence region between the curves contains the horizontal line then the effect is said to be insignificant. The results shown in Figure 14 prove that the effects of impact energy are significant on the natural frequencies for the 2 nd , 3 rd and 4 th bending modes. The effects of the density of impact are partially significant on these modes as compared to the impact energy. However, these two factors have no influence on the natural frequency of the first bending mode because the frequency shift between the damaged and the undamaged cases is insignificant.

Conclusion

Dynamic tests have been carried out on pristine and damaged composite beam specimens using burst random excitation. The variation of modal parameters with damage is studied with the help of changes in natural frequency, damping ratio and damage index. Vibration tests are carried out by placing steel masses at the two ends of the composite beams in order to increase the difference in modal parameters between the undamaged and the damaged cases. Results show that with the accumulation of damage in the specimens, there is a decrease in natural frequency accompanied by an increase in the damping ratio. Damage index tool is also used to study the changes in the modal parameters before and after impact. Damage index is based on the integral of the amplitudes of the frequency response functions (FRF) and therefore estimates the phenomenon of dissipation in the specimens. It is seen that the damage index increases with the increase of damage in the specimens. Therefore the aim of the damage index is to verify the damping ratio results estimated by Polymax and to have an additional tool for understanding the damping behavior of damaged composite beams. Finally a design of experiments is carried out on the results of natural frequencies. It is verified that the energy of impact is the most significant factor and its increase generates a shift in the s-plane.

In the future, similar experiments shall be carried out with sine dwell testing in order to have a better estimation of the damping ratio. The Design of Experiments shall also be expanded by taking into account some other parameters e.g., size and shape of the impactor head, thickness of the specimens and the lay-up of the plies.

  rapid adhesive. The response displacements are measured with the help of a 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.

Figure 1

 1 Figure 1 Diagram of the experimental set-up

Figure 2

 2 Figure 2 Composite test specimen with location of damage, excitation and measurement points

Figure 3

 3 Figure 3 Arrangement of the test equipment for the impact test

Figure 6

 6 Figure 6 Comparison of frequency response functions (FRF) of Beam 1 at measurement point 5 for

Figure 7

 7 Figure 7 Mode shapes of composite beam calculated by the Polymax method (a) first bending, (b)

Figure 8

 8 Figure 8 Variation of damped natural frequencies with damage states for (a) 2nd bending mode and

Figure 9

 9 Figure 9 Comparison of the sum of the frequency response functions for Beam 7 for the

Figure 10

 10 Figure 10Variation of natural frequencies with damping ratio, for an increase in density of

  Undamaged , I x 4Damaged and I x 8Damaged are the integrals of the resonance peaks of the FRF for the undamaged state (UD), damage state D1 (damaged at 4 points) and damage state D2 (damaged at 8 points) respectively.

Figure 11

 11 Figure 11 Methodology for the calculation of integral Ix (area under the curve)

Figure 13

 13 Figure 13 Variation of Damping ratio and Integral (Ix) as a function of frequency for (a) second

Figure 14 .Figure 14

 1414 Figure 14. As both damping ratio and damage index (based on the integral Ix) explain the same

Figure 15

 15 Figure 15 Results of Design of Experiments (Leverage plots with confidence curves)

Nomenclature Type 1 =

 1 Composite beam specimens of 1.04 mm thickness Type 2 = Composite beam specimens of 3.12 mm thickness UD = undamaged state D1 = damaged state at 4 points D2 = damaged state at 8 points h = impact height E imp = energy of impact (J) m tip = Total weight of impactor tip + freefalling mass + force sensor + accelerometer g = gravitational constant 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) σ (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 I x = Integral of the amplitude of the frequency response functions (FRF) int = Frequency interval (user defined) (Hz) f k = Resonance frequency (Hz) DI = Damage Index DI4 = Damage Index for states D1 and UD DI8 = Damage Index for states D2 and UD

Table 1

 1 Physical properties of carbon/epoxy prepreg T300/914

	Properties	Symbol	Value
	Young's modulus in fiber direction Young's modulus in transverse direction E 2 E 1 Shear Modulus G 12	1.44 10 6 MPa 1.0 10 5 MPa 4.2 10 3 MPa
	Poisson ratio Volume density	μ 12 , μ 23, μ 31 0.25, 0.3, 0.017 ρ 1550 kg/m 3
	Two types of composite beam specimens are tested in this article. The first ones have a
	thickness of 1.04mm having 8 plies and are discussed under the name of Type 1 specimens,
	whereas the others have a thickness of 3.12mm having 24 plies and are described as Type 2
	specimens. The geometric configurations and lay-up of these two types of specimens are listed in

Table 2 .

 2 The lay-up is chosen as such, in which the delamination is said to have more profound effects on the dynamic characteristics[START_REF] Saravanos | Effects of delaminations on the damped dynamic characteristics of composites[END_REF].

Table 2

 2 Geometry and lay-up of the two types of composite test specimens

Type 1 Specimens (1.04mm thickness) Type 2 Specimens (3.12mm thickness)

	Length	480 mm	Length	480 mm
	Width	50 mm	Width	50 mm
	Thickness	1.04 mm	Thickness	3.12 mm
	Number of layers (plies)	8	Number of layers (plies) 24
	Thickness of each ply	0.125 mm	Thickness of each ply	0.125 mm
	Lay-up	[0/90/45/-45]s	Lay-up	[(0/90/45/-45) 3 ]s

Table 3

 3 Dimensions of the end masses for the two types of composite test specimens

End Masses for Type 1 Specimens End Masses for Type 2 Specimens

		Length	50 mm	Length	50 mm
		Width	12 mm	Width	30 mm
		Thickness	8 mm	Thickness	10 mm
		Material	Steel	Material	Steel
	3	Experimental Procedure

Two types of experiments are carried out in this work. Vibrations tests for measuring the dynamic characteristics of the composite test specimens and impact tests in order to create damage in the specimens. The experimental procedure for these two types of tests is discussed separately.

Table 4

 4 Impact test parameters

	Beam Thickness Energy of	Height	Velocity of
			impact	(mm)	impact
			(J)		(m/s)
	1	1.04 mm	2.5	138	1.59
	2	3.12 mm	3	166	1.74
	3	3.12 mm	4	221	1.98
	4	3.12 mm	7	387	2.64
	5	3.12 mm	8	442	2.83
	6	3.12 mm	9	498	3.07
	7	3.12 mm	10	553	3.24
	Weight

of mass + impactor tip + force sensor = 2.03 kg

Acknowledgments

The authors gratefully thank M. Labarrère (ISAE -Campus ENSICA) for his technical support during the experimental work.


Amir Shahdin
email: amir.shahdin@isae.fr

Laurent Mezeix

Christophe Bouvet

Joseph Morlier

Yves Gourinat

Monitoring the Effects of Impact Damages on Modal Parameters in Carbon Fiber Entangled Sandwich Beams

Keywords: Entangled Sandwich Materials, Vibration Testing, Impact Testing, Modal Parameters

The aim is to study the impact toughness of two types of entangled sandwich materials (heavy and light) with the help of vibration testing. A simple case of symmetrical impacts is studied in this article as no literature is available regarding impact tests on entangled sandwich materials. The variation of modal parameters with two levels of damage (BVID and Damage not apparent on the surface) is studied. Vibration test results show that the light entangled specimens possessing good damping capabilities seem more sensitive to impact damage than the heavy ones. Furthermore, damping is found to be more sensitive to damage than the stiffness variations, 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.

Introduction

Laminated honeycomb sandwich materials are being used widely in weight sensitive structures where high flexural rigidity is required, such as in the aerospace industry. By inserting a light weight core between the two face sheets, the bending stiffness and strength are substantially increased compared with a single layer homogeneous structure, without adding much weight. When the beam or plate undergoes flexural vibration, the damped core is constrained primarily to shear. This shearing causes energy to be dissipated and the flexural motion to be damped. 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 higher impact energy and retain a high degree of structural integrity. For aeronautical structures, a field where this problem has been quite studied, the components have to undergo (i) low energy impacts caused by dropped tools, mishandling during assembly and maintenance, and in-service impacts by foreign objects such as stones or birds, and (ii) medium to high energy impacts in military aircrafts caused by weaponry projectiles. In a low energy impact (but high enough to produce damage), only a very small indentation will be seen on the impact surface. This level of damage is often referred to as barely visible impact damage (BVID). 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 behavior of composite sandwich panels[END_REF][START_REF] Dear | Impact damage processes in composite sheet and sandwich honeycomb materials[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] Bull | Compressive strength after impact of CFRP-foam core sandwich panels in marine applications[END_REF][START_REF] Abrate | Impact on composite structures[END_REF].

The vibration characteristics of sandwich materials have drawn much attention recently in order to attenuate vibrations and to cushion impact force for structural components and mechanical parts. The dynamic parameters of a structure i.e. natural frequency, damping and mode shapes are determined with the help of vibration testing which provides the basis for rapid inexpensive dynamic characterization of composite structures [START_REF] Gibson | Modal vibration response measurements for characterization of composite materials and structures[END_REF]. D.J Ewins gave a detailed overview of the theory behind vibration testing [START_REF] Ewins | Modal testing theory and practice[END_REF]. A wide amount of literature is present related to vibration testing of composite sandwich structures [START_REF] Hwu | Vibration suppression of composite sandwich beams[END_REF][START_REF] Hu | Free vibration of delaminated composite sandwich beams[END_REF][START_REF] Nilsson | Prediction and measurement of some dynamic properties of sandwich structures with honeycomb and foam cores[END_REF][START_REF] Renji | Experimental modal densities of honeycomb sandwich panels at high frequencies[END_REF][START_REF] Renji | Loss factors of composite honeycomb sandwich panels[END_REF]. The equations that explain the dynamic behavior of sandwich beams are described extensively in the literature and notably in the references [START_REF] Yan | Governing equations for vibrating constrained-layer damping sandwich plates and beams[END_REF][START_REF] Mead | A comparison of some equations for the flexural vibration of damped sandwich beams[END_REF]. One way of damage detection is with the help of vibration testing, as the presence of delamination effect the vibration characteristics of a structure (e.g., natural frequency, damping ratio and mode shape).

In order to carryout effective structural health monitoring (SHM), it is therefore important to understand the performance of delaminated composites and debonded sandwich composites in a dynamic environment [START_REF] Della | Vibration of delaminated composite laminates: A review[END_REF][START_REF] Sanders | Non-destructive evaluation of damage in composite structures using modal parameters[END_REF][START_REF] Shahdin | Correlating low energy impact damage with changes in modal parameters: A preliminary study on composite beams[END_REF][START_REF] Kim | Effect of debonding on natural frequencies and frequency response functions of honeycomb sandwich beams[END_REF][START_REF] Lestari | Damage detection of fiber-reinforced polymer honeycomb sandwich beams[END_REF]. Comprehensive reviews on vibration-based damage detection methods have been presented by Zou et al. [START_REF] Zou | Vibration-based model-dependent damage (delamination) identification and health monitoring for composite structures. A review[END_REF] on the model-dependent delamination identification methods for composite structures, and by Yan et al. [START_REF] Yan | Development in vibration-based structural damage detection technique[END_REF],

Doebling et al. [START_REF] Doebling | A summary review of vibration-based damage identification methods Shock and Vibration[END_REF] and Sohn et al. [START_REF] Sohn | A review of structural health monitoring literature: 1996-2001[END_REF] on general vibration-based damage detection methods. Most of the traditional SHM methods are based on the fact that damage leads to appreciable reduction in the stiffness of a structural element (changes in natural frequencies) [START_REF] Gadelrab | The effect of delamination on the natural frequencies of a laminated composite beam[END_REF][START_REF] Tracy | Effect of delamination on the natural frequencies of composite laminates[END_REF][START_REF] Salawu | Detection of structural damage through changes in frequency: a review[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. When a delamination or debonding failure mode is concerned, friction between the interacting surfaces may occur for small bending deformations. As friction is an energy dissipation mechanism, it is reasonable to assume that damping may be used for SHM, when this type of damage is concerned [START_REF] Montalvao | A method for the localization of damage in a CFRP plate using damping[END_REF][START_REF] Li | Effect of thickness and delamination on the damping in honeycomb-foam sandwich beams[END_REF][START_REF] Adams | Damping in composites[END_REF][START_REF] Saravanos | Effects of delaminations on the damped dynamic characteristics of composites[END_REF].

Enhanced study on the vibration characteristics of composite structures has singled out the importance of material damping in the design process in recent years, as the control of noise and vibration in high precision, high performance structures and machines has become more of a concern. At the same time, composites sandwich researchers have focused more attention on damping as a design variable [START_REF] Wang | Damping of Honeycomb Sandwich Beams[END_REF][START_REF] Berthelot | Damping analysis of composite materials and structures[END_REF]. One way of increasing damping in sandwich materials is by putting a viscoelastic layer as core between the two laminates [START_REF] Hao | Vibration and damping analysis of a sandwich beam containing a viscoelastic constraining layer[END_REF][START_REF] Rao | Dynamic analysis and design of laminated composite beams with multiple damping layers[END_REF]. Jueng and Aref reported that sandwich structures with combined honeycomb-foam cores have higher damping than those with individual honeycomb or solid viscoelastic cores [START_REF] Jung | A combined honeycomb and solid viscoelastic material for structural damping application[END_REF].

These advancements have led to the need for developing materials possessing better damping characteristics. Newer materials like fiber entangled materials can be used as potential dampers and sound absorbers in specific applications like the inner paneling of a helicopter, where structural strength is not the primary requirement. Entangled materials are made from natural materials (wool, cotton etc) as well as artificial ones (carbon, steel, glass etc) and are quickly becoming of widespread use as sound absorbers [START_REF] Garai | A Simple empirical model of polyester fiber materials for acoustic applications[END_REF]. Bonded metal fibers entangled materials offer advantages for use as heat exchanger [START_REF] Golosnoy | Optimisation of metallic fiber network materials for compact heat exchangers[END_REF] or insulation [START_REF] Zhang | Experimental and theoretical studies on hightemperature thermal properties of fibrous insulation[END_REF].

These materials possess low relative density, high porosity and are cost-effective. Sandwich structures normally consist of two thin skins separated by a thick core. Traditional core materials are usually honeycomb, foam or balsa. Recently, a novel type of sandwich has been developed with bonded metallic fibers as core material [START_REF] Markaki | Mechanics of thin ultra-light stainless steel sandwich sheet material, part i. stiffness[END_REF][START_REF] Dean | Energy absorption during projectile perforation of lightweight panels with metallic fibre cores[END_REF][START_REF] Zhou | Mechanical Properties of Fibrous Core Sandwich Panels[END_REF]. This material presents attractive combination of properties like high specific stiffness, good damping capacity and energy absorption. Entangled materials with carbon fibers have also been studied for use as core material [START_REF] Mezeix | A new sandwich structured composite with entangled carbon fibers as core material[END_REF]. Entangled cross-linked carbon fibers present many advantages as core materials i.e., open porosity, multifunctional material or the possibility to reeve electric or control cables on core material. Mechanical testing has also been carried out on entangled specimens made of wood fibers [START_REF] Castéra | Comportement physico-mecanique des matériaux fibreux cellulosiques considère comme des milieux aléatoires[END_REF], glass fibers [START_REF] Baudequin | Identification des mécanismes physiques mis en jeu lors de la reprise d'épaisseur de la laine de verre[END_REF] and various matted fibers [START_REF] Poquillon | Experimental data about mechanical behavior during compression tests for various matted fibres[END_REF].

Characterization of carbon and glass fiber entangled sandwich materials has been carried out both statically (compression and bending tests) and dynamically (vibration test) by L. Mezeix and A. Shahdin [START_REF] Mezeix | Experimental data and modelling of entangled fibers and entangled cross-linked fibers during compression[END_REF][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]. The compression and bending test results show that the entangled sandwich specimens have a relatively low compressive and shear modulus as compared to honeycomb and foam sandwich materials [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.

The main motivation of the work presented in this article, is to carry the research process on entangled sandwich materials one step further by carrying out impact tests on the carbon fiber entangled sandwich materials in order to understand the relation between damage level density and modal parameters. Monitoring the dynamic characteristics of these sandwich specimens allows us in future to study the dectability of impact and to verify that whether damping can be used as a damage indicator tool as it tends to be more sensitive to damage than the stiffness variations.

2.

Materials and fabrication procedure before being sprayed on the carbon fibers. This allows the mixture of resin and hardener to become less viscous.

Resin-containing carbon-fiber/epoxy prepreg of T700/M21 is used to fabricate the skin materials [START_REF] Hexply | Data Sheet[END_REF]. The material is supplied by Hexcel composites, the physical properties are set out in Table 1. The upper and lower skins consist of four plies each with a stacking sequence of [0/90/90/0] [START_REF] Bunyawanichakul | Contribution a l'etude du comportement des inserts dans les structures sandwich composites[END_REF]. The thickness of each ply is 0.125mm.

Fabrication methodology

The fabrication of entangled sandwich specimens is often a tedious and complex process. As these types of materials are still mostly in the research phase, so as such standard fabrication processes do not exist. The fabrication procedure used in this article is the one developed by L. Mezeix et al. at CIRIMAT/LGMT in France [START_REF] Mezeix | Experimental data and modelling of entangled fibers and entangled cross-linked fibers during compression[END_REF]. For the test specimens used in this article, approximately 900 g of fibers of 10mm length are cut manually. The carbon fibers are then treated in a solution of dichloromethane for 24 hours and are then cleaned for 2 hours in methanol. These uncoated carbon fibers are then separated by a blow of compressed air. The mixture of resin and hardener is then sprayed on the separated carbon fibers by a spray paint gun. In case of larger test specimens, the volume of carbon fibers is large, so with the current technology it is not easy to spray the resin equally on the carbon fibers. The fibers vaporized by the resin are then placed in the mold between the two skins of unidirectional composites. In order to polymerize the fiber network sandwich specimens, the mold is then heated in an oven up to 180°C for two hours in a press [START_REF] Hexply | Data Sheet[END_REF]. The dimensions of the test specimens used for vibration and impact testing are 480 x 50 x 11 mm. The thickness of the entangled sandwich specimen core is 10 mm and that of each skin is 0.5 mm.

In order to verify the process of fabrication, a single test specimen is fabricated first by using a small mold (510 x 65 x 11 mm) with a fiber core density of 100 kg/m 3 . This specimen showed that 100 kg/m 3 fiber core density is relatively insufficient for a volume of 480 x 50 x 10 mm 3 , as there are places in the core of the sandwich beam that lack sufficient quantity of fibers. So for the next specimen, a fiber core density of 150 kg/m 3 is chosen. As compared to the previous specimen (fiber core density of 100 kg/m 3 ), the specimen with 150 kg/m 3 fiber core density has a far better fiber distribution. Finally a test specimen with a fiber core density of 200 kg/m 3 is fabricated by using the same small mold. It is seen that a fiber density of 200 kg/m 3 for the core is on the higher side and it is not possible to close the mold properly. So for the fabrication of the next batch of specimens, a fiber core density of 150 kg/m 3 is chosen.

In the next step a larger mold (510 x 250 x 11 mm) is used to produce multiple test specimens having the same characteristics. First the large mold is used to fabricate four relatively identical test specimens referred to as heavy specimens in the article. The average composition of these four heavy test specimens is presented in Table 2. Next, the same large mold is used to produce relatively lighter specimens. They have approximately 25 g less mixture of resin and hardener than the previously produced heavier specimens. These specimens shall be referred to as light specimens in future discussions (Table 2). The ratio of the mixture of resin and hardener in the heavy specimens is approximately 2.5 times more than that of the light specimens. The weight comparison of the vibration test specimens fabricated from the small and large mold is presented in Table 4 (in Section 4.2). The difference in weights observed between the specimens is due to the uneven distribution of the manually sprayed resin.

Vibration tests

The experimental equipment used to obtain the vibration test results discussed in this paper is shown in Fig. 1. 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

Impact tests

The impact test system used to damage the entangled sandwich beams is a drop weight system as shown in Fig. 3, and a detailed cut away of the drop assembly is shown in Fig. 4.

Fig. 3. Arrangement of the test equipment for the impact test

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 impact velocity is measured with the help of an optic sensor. The combined weight of the impact head, freefalling mass, force sensor and the accelerometer is 2.03 kg. In the calculation of impact height, a factor of 1.1 is used to compensate for the losses due to friction between the guidance tube and the drop assembly. 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 clamped. Further details on the impact test methodology of this drop tower can be found in the reference [START_REF] Petit | Impact and compression after impact experimental study of a composite laminate with a cork thermal shield[END_REF][START_REF] Shahdin | Correlating low energy impact damage with changes in modal parameters: A preliminary study on composite beams[END_REF]. The entangled sandwich specimens tested in this article 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 lead to high damage. In the aeronautical domain, BVID corresponds to 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] Petit | Impact and compression after impact experimental study of a composite laminate with a cork thermal shield[END_REF].

As no scientific literature can be found related to impact testing of entangled sandwich materials, so two test specimens of each of the four heavy and four light specimens are used for trial impact tests, in order to determine the BVID levels. The impact energies are chosen in such a way that the heavy and light specimens have the same level of damage. In general, two levels of damage are studied in this studied

• Barely visible impact damage (0.6-0.8 mm of indentation measured just after the impact)

• Damage not apparent on the surface (0.1-0.2 mm of identation measured just after the impact)

These trial impact tests revealed that in case of the heavy specimens, impact energy of 8 J corresponds to the BVID limit. However, in case of light specimens, they have to be impacted at 12 J in order to induce damage corresponding to the BVID limit. As discussed previously, that sometimes damage that is not visually apparent on the surface can prove quite detrimental to the load bearing capacities of sandwich structures. Therefore, during the trial impact tests on the heavy and light entangled sandwich specimens, an indentation depth of 0.1-0.2 mm is found to be undetectable through visual inspection. This indentation depth corresponds to impact energy of 6 J in case of heavy specimens and 8 J for the light ones.

After the trial impact tests, two remaining specimens of each heavy and light specimens are used for the real impact and vibration tests. Each specimen is impacted at four different points (Fig. 2), but the impact energy level is kept the same for each of the specimens. The two heavy specimens are impacted at 6 J (0.1-0.2 mm indentation depth) and 8 J (BVID limit), whereas the two light specimens are impacted at 8 J (0.1-0.2 mm indentation depth) and 12 J (BVID limit). The impact parameters for the two heavy and two light entangled sandwich specimens studied in this article are listed in Table 3. The data obtained during the drop weigh impact tests is shown in Fig. 5.

Fig. 5. Impact test data (a) force-time (b) force-displacement

Four similar impacts have been performed on each specimen. However, in order to clarify these plots, only one impact test result for each specimen is plotted. All the impact curves presented in Fig. 5 are 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] Bull | Compressive strength after impact of CFRP-foam core sandwich panels in marine applications[END_REF][START_REF] Abrate | Impact on composite structures[END_REF]. In Fig. 5 a, the impact forces are drawn as a function of time during impact tests. These curves are globally smooth and almost sinusoidal at low impact energy, with little oscillation due to natural frequencies of the panel. They show an important force signal fall followed by oscillations which is characteristic of delamination onset. This phenomenon is more prominent for higher impact energies. The force-displacement plot (Fig. 5 b) shows the same force signal peak as soon as the delamination begins. These curves also give us an indication about the static strength of the heavy and light entangled sandwich specimens.

They underline the facts that as the light specimens are less dense having less resin as compared to the heavy ones, so in order to attain peak force or maximum energy, more time and displacement are required. The results of the static tests (compression and bending) carried out on the heavy and light specimens have been discussed in detail in the reference [START_REF] Mezeix | Experimental data and modelling of entangled fibers and entangled cross-linked fibers during compression[END_REF].

The entangled sandwich specimens have three states. First one is the undamaged state (UD), the second is the damage state due to two impacts (D1) and the third is the damage state due to four impacts (D2). As impact tests are carried out the first time on entangled sandwich materials, so a simple case with symmetric impacts is chosen. These impact points are shown in Fig. 2. Vibration tests are carried out on the four entangled sandwich specimens after each of these three states.

Results and discussion

Tracking of poles for damage detection

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 )*) = ! " # = + $ % # & # & ' ( ) (1)
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 ( 1) is the complex conjugate term). 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 ) = !" + # (2)
The magnitude of each pole is the undamped natural frequency (ω n ). The undamped natural frequency (ω n ) and the modal damping ( ! ) are related to mass, stiffness and damping as follows: given by

2 2 n d K (k ) M ! = ! + " = (3) C 2 (k ) M ! = ( 4
)
The effect of physical properties on poles in the complex s-plane is illustrated in Fig. 6.

Fig. 6. Movement of pole due to mass stiffness and damping effect

From Fig. 6, 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 entangled specimens [START_REF] Richardson | Correlating minute structural faults with changes in modal parameters[END_REF].

Monitoring through frequency and damping changes

Monitoring of the impact damage in the entangled sandwich specimens is carried out through frequency and damage changes. Frequency and damping results presented in this article are the global parameters of the specimen, and are extracted from the measurements carried out on the 33 measurement points. The frequency and damping changes are studied with the help of bending modes as they have the largest amplitudes for the type of test configuration presented in this article. For the first four bending modes, the variation of damped natural frequency as a function of the undamaged (UD) and the two damage states (D1 and D2) is presented in Fig. 7. As discussed before in section 4.1, that damage in the specimens prompts a decrease in natural frequencies. So from Fig. 7, it is clear that the decrease in the natural frequencies for both the heavy and light specimens is more prominent in case of the higher impact energies i.e., 8 J in case of the heavy (EH 2) and 12 J in case of the light specimens (EL 2). But the interesting fact is that, for the heavy specimen (EH 1) impacted at 6 J which does not produce a visible damage on the surface, the average change in frequency for the first four bending modes between the undamaged and the damaged cases is 6 %. Similarly, for the light specimen (EL 1) impacted at 8 J this change in frequency ratio is 13 %. So it can be seen that the damage not visually apparent can affect the modal parameters resulting in a certain loss of rigidity. Therefore, vibration testing can be an effective tool to carry out non destructive tests for structural health monitoring purposes.

Fig. 7 also shows a dispersion in the natural frequencies between the two heavy (EH 1 and 2) and the two light specimens (EL 1 and EL 2) at the undamaged state. Each of the two heavy and light entangled sandwich specimens is fabricated from the same mold. However, this dispersion is a result of the differences in weight (Table 4) which as outlined previously, is due to the uneven distribution of the manually sprayed resin which highlights the complexity of the fabrication process.

The shift in the natural frequencies between the undamaged and the damaged cases is more prominent at higher frequencies. This is evident in Fig. 8, which shows a comparison of the sum of the frequency response functions (FRF) for the entangled sandwich specimen EH 2 (impacted at 8 J) for the undamaged case (UD), damaged at 2 points (D1) and damaged at 4 points (D2). The sum of the FRF can be compared as for each entangled sandwich beam 33 symmetric measurement points have been chosen and the four impact points are also symmetric on both sides of the two major axes of symmetry. Damping ratios are considerably higher in case of the light specimens, as they are more dissipative in nature due to lesser amount of resin. It can be noticed that with the exception of the 4 th bending mode (Fig. 8d), the change in damping ratio between the undamaged and the damaged states for the two heavy specimens (EH 1 and EH 2) is smaller as compared to the two light specimens (EL 1 and EL 2), which shows that the light specimens are more sensitive to damage than the heavy ones. The affect of damage on the frequencies and damping ratios can be further elaborated by studying the frequency and the damping change ratios between the undamaged (UD) and the damaged cases (D1 and D2) for the two heavy and the two light entangled sandwich specimens, presented in Table 4. The percentage change in frequency and damping ratios between the undamaged and the damaged cases is calculated with the help of Eq. 5 and Eq. 6

Change in frequency between UD and D1, (Δf

) = UD D UD f (k) f (k) f (k) ! 1 (5)
Change in damping between UD and D1, (Δ

! ) = D UD UD (k) (k) (k) ! " ! ! 1 (6)
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. 6 is the same. Furthermore, in order to calculate the frequency and damping change ratios between UD and D2 the same procedure is used.

For all the four specimens studied in this article, it can be seen from Table 4, that the damping change ratios are more prominent than the frequency change ratios. The maximum damping change ratio is 109.6 % and the maximum frequency change ratio is 29.7 % which occur in the case of EL 2 specimen impacted at 12J. It can concluded from the above results that damping seems more sensitive to damage than the natural frequency variations even in the case of entangled sandwich materials. 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. Furthermore, if the frequency and damping change ratios are compared for the two heavy and light specimens (EH 2 and EL 2) impacted at the BVID limits i.e., having the same damage (0.6 -0.8 mm indentation depth), it can be seen from Table 4 that the change in modal parameters is more significant in case of the light entangled specimen. This shows that even if the same level of damage (BVID) is imparted to the two specimens, the lighter specimens seem to be more affected. Similarly, by comparing the heavy (EH 1) and the light entangled specimen (EL 1) having the same lower level of damage that is not visible (0.1 -0.2 mm indentation depth) in Table 4, it is evident that again the light specimen (EL 1) is seen more sensitive to damage than the heavy one. The only exception is the damping change ratio for the 4 th bending mode, which is higher in case of the heavy specimen.

So overall, it can be concluded that the light specimens having lesser amount of resin possess good damping capabilities as seen in Fig. 9, but are more sensitive to impact damage than the heavy ones, even when they have the same amount of damage. Therefore, while selecting the application of these light entangled sandwich materials, their sensitivity to impact damage should be taken into consideration. The vibration results also prove that the damage is more localized in the heavy specimens as they are denser in nature as compared to the light specimens, which results in smaller variations of modal parameters in case of heavy specimens. Furthermore, the damage in light specimens is less restricted to a certain zone and thus the light specimens are more globally affected than the heavy specimens.

Conclusion

Entangled sandwich materials possesses high damping characteristics and can be used for specific applications like the inner paneling of a helicopter cabin as their structural strength is on the lower side. So in this article, impact toughness of these entangled sandwich materials is studied. Due to the lack of available literature on the behavior of entangled sandwich materials due to impact damage, a simple case of symmetrical impacts is studied.

The impact energies are chosen in such a way that the heavy and light specimens have the same level of damage. Vibration tests are carried out after each of the undamaged and damaged states to study the variation of modal parameters with damage. Two types of entangled sandwich specimens (heavy and light) are studied in this article. The light specimens have 2.5 times less resin than the heavy ones. Results show that with the accumulation of damage in the specimens, there is a decrease in natural frequency accompanied by an increase in the damping ratio. Vibration test results prove that the light specimens having better damping characteristics are more sensitive to impact damage than the heavy ones. Therefore, while selecting the application of these light entangled sandwich materials, their sensitivity to impact damage should be taken into consideration. In the heavy specimens, the damage seems to be more localized as compared to the light ones.

Furthermore, it can concluded that damping seems more sensitive to damage than the stiffness variations. 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.

In this article impact toughness has been compared for two types of entangled sandwich specimens only. In the future, the impact toughness of these entangled sandwich materials shall be compared with classical sandwich materials, having honeycomb and foam as cores. The sensitivity of both the energy of impact and density of damage shall be established by making use of the design of experiments (DOE).

f k = Resonance frequency (Hz) for the kth mode ! k = Damping ratio (%) for the kth mode

Fig. 1 .

 1 Fig. 1. Diagram of the experimental set-up

Fig. 2 .

 2 Fig.2. Sandwich test specimen with location of damage, excitation and measurement points

Fig. 4 .

 4 Fig. 4. Detailed cutaway of the drop assembly, the guidance tube and the blocking system

Fig. 7 .

 7 Fig. 7. Variation of damped natural frequencies with damage states for (a) 1st bending mode, (b) 2nd bending mode, (c) 3rd bending mode and (d) 4th bending mode: UD is undamaged state, D1 is damaged at 2 points of impact and D2 is damaged at 4 points of impacts, for the four entangled sandwich specimens

Fig. 8 Fig. 9 .

 89 Fig. 8 Comparison of the sum of the frequency response functions for EH 2 for the undamaged case (UD), damaged at 2 points (D1) and damaged at 4 points (D2)

2.1. Core and skin materials

	The core of the entangled sandwich test specimens used in this article consists of
	carbon fibers (HTS-5631) that are made of a yarn of standard carbon filaments having a
	diameter of 7µm. 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. For the cross-linking of carbon
	fibers, epoxy resin is used. The epoxy resin SR 8100 and injection hardener SD 8824 are used
	provided by SICOMIN. All the test specimens presented in the article are carefully weighed
	using METTLER balance.

A better vaporization is achieved if the resin is heated up to 35°C

Table 1

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

	Young's modulus in fiber direction (E 1 )	125000 MPa
	Young's modulus in transverse direction (E 1 ) 9000 MPa
	Shear Modulus (G 12 )	5000 MPa
	Poisson Ratio (v 12 )	0.4
	Volume density (ρ)	1550 kg/m 3

Table 2 .

 2 Composition of vibration test specimens having fiber core density of 150 kg/m 3 fabricated from the small and large mold (Average of four specimens)

		Density	150 kg/m 3 (Light)	150 kg/m 3 (Heavy)
		Weight of Fibers	39 g	39 g
		Weight of Resin & Hardener	18 g	43 g
		Weight of Skin (Upper + Lower)	38 g	38 g
		Total Weight of Specimen	95 g	120 g
	3.	Experimental procedure
	3.1.

Table 3

 3

	Impact test parameters
	Beam Name	Energy of	Height	Indentation just	Velocity of impact
		Impact (J)	(mm)	after impact (mm)	Measured (m/s)
	Entangled Heavy 1 (EH 1)	6	331.8	0.1 -0.2	2.49
	Entangled Heavy 2 (EH 2)	8 (BVID)	442.3	0.6 -0.8	2.83
	Entangled Light 1 (EL 1)	8	442.3	0.1 -0.2	2.83
	Entangled Light 2 (EL 2)	12 (BVID)	663.5	0.6 -0.8	3.52

Table 4

 4 Frequency and damping change ratios between the undamaged (UD) and the two

	damaged states (D1 and D2) for the two heavy (EH1 and EH2) and the two light (EL1 and
	EL2) entangled sandwich specimens
	Type of	Specimen	Between	Frequency Change Ratios (%)	Damping Change Ratios (%)
	Specimen	Weight (g)	States
				1st	2nd	3rd	4th	1st	2nd	3rd	4th
				Mode	Mode	Mode	Mode	Mode	Mode	Mode	Mode
	EH1 (6J)	127	UD and D1 4.26	2.48	4.89	2.04	7.01	2.21	9.54 31.38
			UD and D2 8.83	5.61 11.33 4.73 23.98 15.95 19.74 47.30
	EH2 (8J)	124	UD and D1 9.11 UD and D2 16.11 7.98 12.45 10.73 40.58 27.69 33.06 30.88 2.96 7.62 6.18 10.13 9.23 14.17 19.63
	EL1 (8J)	95	UD and D1 7.96	9.78 14.65 7.04 24.18 11.43 16.16 20.73
			UD and D2 18.73 12.07 21.76 12.82 86.15 29.29 39.45 40.56
	EL2 (12J)	98	UD and D1 18.20 9.95 17.23 9.36 85.10 15.32 15.96 32.88 UD and D2 29.70 18.85 27.35 17.11 109.6 38.66 86.92 47.49

Acknowledgments

The authors gratefully thank research project students Hanno Niemann from TU Braunschweig for his support during the impact tests and J. B. Lecaillon and A. de Nazelle from Saint-Cyr l'Ecole for their help during the fabrication process


published or not. The documents may come

Linkspotter : outil interactif d'exploration et de visualisation de corrélations

   visualiser la distribution de chaque variable grâce à son histogramme ou à son diagramme en barres.  visualiser un lien entre un couple de variables à l'aide de nuage de points, de boîtes-àmoustaches, etc.En outre, un nouveau coefficient de corrélation que nous appelons Maximal Normalized Mutual Information (MaxNMI) basé sur une discrétisation supervisée Equal-Freq des variables continues est également introduit par Linkspotter. L'intérêt de ce coefficient est qu'il peut être calculé et comparé quel que soit le type de couple de variables (continue vs catégorielle, continue vs continue, catégorielle vs catégorielle).

	Alassane Samba
	Orange Labs
	2 avenue Pierre Marzin, 22300 Lannion, France
	alassane.samba@orange.com
	Mots clefs : Corrélation, Information mutuelle, Graphe dynamique, Visualisation, Interface
	utilisateur.
	L'exploration des relations entre les variables est une étape importante dans le processus
	d'exploration de données. Il permet au Data scientist de mieux comprendre les phénomènes
	décrits dans les jeux de données. Cela aide également à éviter le sur-apprentissage et à se
	prémunir contre le « concept drift » dans la modélisation.
	Tout en offrant de nouveaux algorithmes et méthodes, Linkspotter utilise et combine
	harmonieusement des fonctionnalités provenant d'autres packages R, à savoir infotheo,
	minerva, energy, mclust, shiny, visNetwork et rAmCharts.

Afin de faciliter l'exploration des données, Linkspotter est un package R offrant plusieurs fonctionnalités permettant d'analyser et de visualiser de manière exhaustive en utilisant un graphe toutes les corrélations bi-variées d'un fichier de données.

Ses fonctionnalités principales sont:

 le calcul de plusieurs matrices de corrélation correspondant à différents coefficients  le partitionnement des variables par apprentissage non supervisé.

Il offre également une interface utilisateur complète, conviviale et personnalisable permettant de :  visualiser les corrélations à l'aide d'un graphe (les variables correspondant aux noeuds et les corrélations correspondant aux arcs). C'est une nouvelle approche de visualisation qui est proposée plus conviviale que l'affichage d'une matrice de corrélations coloriée.

Le code [1] et une démonstration [2] de Linkspotter sont disponibles en ligne.

Références

[1] https://github.com/sambaala/linkspotter [2] http://linkspotter.sigmant.net