Christophe Droz
email: christophe.droz@gmail.com

Olivier Bareille

Mohamed N Ichchou

A new procedure for the determination of structural characteristics of sandwich plates in medium frequencies

Keywords: Honeycomb, sandwich panel, inverse identification, group velocity, wave

This paper presents a new wave-based material identification strategy for sandwich panels, able to estimate their tensile and equivalent shear moduli from single shot measurements. The proposed algorithm is based on the determination of the maximal wave speed in the bending-to-shear transition bandwidth. A definition is proposed for the transition phenomenon resulting from the conversion of flexural waves in the composite waveguide, and the existence of a local energy velocity maxima is demonstrated using the 4 th -order sandwich theory. Analytical expressions are derived for the equivalent bending and shear parameters in terms of the transition frequency and the maximal group velocity. An practical iterative procedure is also proposed for the experimental identification of the transition bandwidth, while a Hilbert Transform magnitude is employed to measure time of arrivals of the flexural wave pulses. The robustness of the method is highlighted using a numerical case study and an experimental validation is finally conducted on a sandwich panel with composite faces. Results are compared with static measurements and a wave-based technique (Inhomogeneous Wave Correlation) and show significant accuracy improvements while ensuring reduced post-processing times.

Introduction

Composite assemblies have become a crucial component in transportation and aerospace industries due to their exceptional strength-to-weight ratios. Sandwich panels have the most widespread utilization due to their wide range of possible configurations. They are usually composed of a soft and lightweight core made of foam or honeycomb construction, surrounded by thin fibrereinforced skins. However, the design and optimization of these structures in vibroacoustic environments remain challenging, especially due to the difficulty to perform an accurate modelling of the composite assemblies in the medium frequency range [START_REF] Mace | Mid-frequency methods in sound and vibration-part a[END_REF], too high for deterministic finite element models to apply while the main assumptions of energy distribution underlying the Statistical Energy Analysis (SEA) are not met [START_REF] Lyon | Theory and application of statistical energy analysis[END_REF]. Yet, although the design of structurally advanced composites does not usually requires models of the local scale's components (such as the aramid fibers of a honeycomb construction, or the resin of its skins), accurate predictions of the overall dynamical behaviour at the global scale are fundamental.

Although a large number of mechanical characteristics of sandwich's components can be found in the literature, the increasing research effort on the development of composite skins and structurally advanced core's constructions yields a challenge in terms of prediction of the equivalent core's and skin's material parameters [START_REF] Hohe | Effective stress-strain relations for two-dimensional cellular sandwich cores: Homogenization, material models, and properties[END_REF]. Besides, the same nominal sandwich panel (same core size, materials and stacking sequence) can exhibit different mechanical characteristics, depending on the manufacturing process or epoxy resin density, which can reach more than 30% of the total mass. Additionally, the use of material tables defined using static experiments for the modelling of sandwich panels in the vibroacoustic range can become an issue. Indeed, although frequencyindependence of the mechanical parameters is a fair assumption in a given bandwidth, static and frequency identifications often show great discrepancies.

Various methods were devoted to the determination of effective core properties of sandwich panels [START_REF] Cunningham | A new measurement technique for the estimation of core shear strain in closed sandwich structures[END_REF][START_REF] Schwingshackl | Determination of honeycomb material properties: existing theories and an alternative dynamic approach[END_REF], Karakoc ¸and Freund [START_REF] Karakoc | Experimental studies on mechanical properties of cellular structures using nomex R honeycomb cores[END_REF] proposed an experimental procedure to determine the compliance matrix of Nomex structures. For dynamical identification, modal methods are often employed, Matter et al. [START_REF] Matter | Identification of the elastic and damping properties in sandwich structures with a low core-to-skin stiffness ratio[END_REF] proposed an identification technique for evaluating the elastic and damping properties of sandwich laminates with soft cores. Other methods based on free modes and comparisons with analytical and FE methods were also developed [START_REF] Ip | Material characterization for orthotropic shells using modal analysis and rayleigh-ritz models[END_REF][START_REF] Schwaar | Modal numerical-experimental identification method for characterising the elastic and damping properties in sandwich structures with a relatively stiff core[END_REF][START_REF] Jiang | An approach on identification of equivalent properties of honeycomb core using experimental modal data[END_REF] to determine elastic properties of sandwich structures. Recently, wave-based methods were developed for the determination of these equivalent properties. They advantageously avoid the uncertainties produced by the boundaries on the modal responses and enable effective characterization at higher frequencies, where modal overlap becomes too important to use modal data (see [START_REF] El-Sabbagh | Characterisation of flax polypropylene composites using ultrasonic longitudinal sound wave technique[END_REF] for an example of high-frequency characterization of polypropylene composites).

The propagation of guided waves in sandwich structures has indeed been the subject of intense research in the recent years. Many studies are concerned with exploiting wave dispersion characteristics (also called dispersion curves) to understand wave's physics in heterogeneous structures and design composite waveguides with advanced capabilities [START_REF] Liu | Locally resonant sonic materials[END_REF][START_REF] Castaings | Guided waves propagating in sandwich structures made of anisotropic, viscoelastic, composite materials[END_REF]. One can cite, among other: the design of auxetic pyramidal cores [START_REF] Scarpa | Kirigami auxetic pyramidal core: mechanical properties and wave propagation analysis in damped lattice[END_REF], the vibroacoustic optimization of sandwich's geometries [START_REF] Droz | Vibro-acoustic optimisation of sandwich panels using the wave/finite element method[END_REF][START_REF] Ranjbar | Vibroacoustic optimization of anti-tetrachiral and auxetic hexagonal sandwich panels with gradient geometry[END_REF], or the inclusion of internal resonators to improve damping properties [START_REF] Antoniadis | Hyper-damping properties of a stiff and stable linear oscillator with a negative stiffness element[END_REF] and a very large number of applications [START_REF] Rose | Ultrasonic guided waves in solid media[END_REF][START_REF] Potel | Characterization of composite materials by ultrasonic methods: modelization and application to impact damage[END_REF] can also be found for defect detection in sandwich structures.

For these new types of composites, studies were conducted on experimental wave-based methods for the identification of effective elastic properties of sandwich constructions: an inverse wave approach for estimating structural wave parameters was proposed by McDaniels and Shepard [START_REF] Mcdaniel | Estimation of structural wave numbers from spatially sparse response measurements[END_REF]. The Inhomogeneous Wave Correlation (IWC) method was proposed in [START_REF] Ichchou | Identification of effective sandwich structural properties via an inverse wave approach[END_REF] for the identification of the dispersion curves in two dimensions, also called k-space. Least mean square algorithm is then applied to sandwich plate theory to obtain the homogenized panel properties. Despite a rather good accuracy, these methods require to measure and post-process the entire displacement field to extract the wavenumber space, leading to expensive and time-consuming characterizations. One of the consequences is a possible inaccuracy when using standard sandwich material tables provided in the literature.

This paper addresses the genuine need of wave-based mid-frequency identification methods for sandwich panels, involving simple experimental set-ups. A strategy is investigated for the elastic parameter identification of sandwich panels using Time-of-Flight (ToF) measurements [START_REF] Papadakis | The measurement of ultrasonic velocity[END_REF][START_REF] Song | Guided wave propagation in honeycomb sandwich structures using a piezoelectric actuator/sensor system[END_REF], since measuring the energy velocity in plates is a mature technique extensively used in the literature (interested reader may refer to [START_REF] Su | Guided lamb waves for identification of damage in composite structures: A review[END_REF][START_REF] Loosvelt | A wavelet-based processing method for simultaneously determining ultrasonic velocity and material thickness[END_REF][START_REF] Chimenti | Review of air-coupled ultrasonic materials characterization[END_REF] or Sikdar et al. [START_REF] Sikdar | Guided wave propagation in a honeycomb composite sandwich structure in presence of a high density core[END_REF] for a recent application to honeycomb composites). The proposed method focuses on a phenomenon called transition, which is a specific type of wave conversion in waveguides [START_REF] Chillara | Guided wave mode conversions across waveguide transitions: A study using frequency domain finite element method[END_REF][START_REF] Droz | A reduced formulation for the free-wave propagation analysis in composite structures[END_REF], occurring in sandwich structures subjected to flexural vibrations.

In section 2, the practical consequences of the transition on flexural waves' velocities are studied and an definition based on 4 th -order sandwich theory is proposed for the transition frequency. The existence of a local velocity maximum around this frequency is exploited to dissociate the shear and bending contributions to the wave motion in the composite. An identification procedure based on transient wave pulses velocities is proposed in section 3 and applied to a sandwich waveguide. The numerical results are compared with analytical results and the accuracy of the identification method is discussed. In section 4, an experimental case study is proposed. The mechanical characteristics of a sandwich panel involving a Nomex honeycomb core surrounded with carbon-reinforced skins are estimated using the proposed method and compared with the IWC results and manufacturer's data.

Transition frequency in symmetric sandwich

Theoretical background

The governing equations for the anti-symmetric motion of sandwich waveguides are described hereby [START_REF] Nilsson | Prediction and measurement of some dynamic properties of sandwich structures with honeycomb and foam cores[END_REF][START_REF] Mead | The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions[END_REF]. Consider a sandwich plate whose normal deflection is assumed connected to the core and shear rotations β and γ. The total deflection w in the direction x produced by β and γ is:

∂w ∂x = β + γ (1)
Considering a linear Lagrangian H = T -U, where U and T stand for the potential and kinetic energies, Hamilton's principle yields the following variational expression for a conservative system :

δ H dt = 0 (2)
Combining Eqs.( 1) and (2) gives the differential equations for β and w:

2D 2 ∂ 4 w ∂x 4 - ∂ 3 β ∂x 3 -G c h c ∂ 2 w ∂x 2 - ∂β ∂x + µ ∂ 2 w ∂t 2 = 0 (3) 2D 2 ∂ 3 w ∂x 3 - ∂ 2 β ∂x 2 -D 1 ∂ 2 β ∂x 2 + I ρ ∂ 2 β ∂t 2 -G c h c ∂w ∂x -β = 0 (4)
where G c is the core's shear modulus, h c and h s are the core's and skin's thickness. The overall flexural stiffness D 1 , the layer's flexural stiffness D 2 and the mass moment of inertia per unit width I p and mass per unit area µ are written:

D 1 = E c h 3 c 12 + E s h s h 2 c 2 + h c h s + 2h 2 s 3 , D 2 = E s h 3 s 12 , (5)
I p = ρ c h 3 c 12 + ρ s h s h 2 c 2 + h c h s + 2h 2 s 3 , µ = 2h s ρ s + h c ρ c . (6)
where E s and ρ s are the skin's Young modulus and density, and ρ c is the equivalent core's density.

Eliminating the angular displacement β in Eqs.( 5) and 6) and assuming plane waves propagate along the x-direction : w(x, t) = w 0 e -j(ωt-kx) , with k the wavenumber and ω the circular frequency yields:

2D 1 D 2 k 6 -2D 2 I ρ k 4 ω 2 -µ + 2D 2 µ + I ρ G c h c k 2 ω 2 + G c h c D 1 k 2 -µω 2 + I ρ µω 4 = 0 (7)
Two main assumptions are introduced: the rotation inertia is negligible compared with the shear effects I ρ ω 2 Gh c , and the bending stiffness per unit width of the skins are small in comparison with the one of the plate: D 2 D 1 . Therefore the 4 th -order dispersion relation in the sandwich becomes:

S k 4 -µω 2 k 2 -µω 2 S /D = 0 (8)
where

S = h c G c 1 + h s h c 2 and D E s h s h 2 c 2 + h c h s + 2h 2 s 3
are the transverse shear rigidity and bending stiffness of the plate. Note that Eqs.(1-8) are derived for propagation in x-direction. In the case of a honeycomb sandwich with composite face sheets, the equivalent bending and shear coefficients depend on the considered direction k θ = f (µ, S θ , D θ ).

Asymptotic solutions for shear and bending modulus identification

The solution of Eq.( 8) gives the flexural wavenumber:

k 2 = µ 2S          ω 2 + ω ω 2 + 4S 2 µD          (9)
When D, S and m are fixed, two asymptotic behaviours can be distinguished: at low frequency, the sandwich plate is ruled by the skins' Young modulus and has classical plate behaviour. The 'bending' wavenumber and group velocity are written:

k bending = k(ω → 0) = 4 µ D √ ω, ; c g (ω → 0) = 2 4 D µ √ ω (10)
while in higher frequencies the sandwich's motion is mainly governed by the shear coefficient S . The asymptotic wavenumber and group velocity become:

k shear = k(ω → ∞) = µ S ω, ; c g (ω → ∞) = S µ . (11)
Between these two frequency domains, both the shear and bending modulus contribute to the flexural motion. In sandwich structures, the transition frequency refers to the passage from a flexural wave governed by the skins' stiffness to one governed by the core's transverse shear. The asymptotic expressions given in Eqs. [START_REF] Jiang | An approach on identification of equivalent properties of honeycomb core using experimental modal data[END_REF] and [START_REF] El-Sabbagh | Characterisation of flax polypropylene composites using ultrasonic longitudinal sound wave technique[END_REF] were used by Ghinet and Atalla [START_REF] Ghinet | Vibro-acoustic behaviors of flat sandwich composite panels[END_REF] and more recently in [START_REF] Droz | Vibro-acoustic optimisation of sandwich panels using the wave/finite element method[END_REF][START_REF] Guillaumie | Vibroacoustic flexural properties of symmetric honeycomb sandwich panels with composite faces[END_REF][START_REF] Baho | On global bendingshear core transition effects for the vibroacoustic of sandwich structures: Analytical and numerical investigations[END_REF] to give interpretations of the transition in sandwich panels. In [START_REF] Guillaumie | Vibroacoustic flexural properties of symmetric honeycomb sandwich panels with composite faces[END_REF], the transition frequency ω T was defined by the geometric intersection between the asymptotic wavenumbers, k bending (ω T ) = k shear (ω T ), leading to a transition frequency ω T = S √ µD . It should be noted that a similar definition based on the asymptotic modal densities [START_REF] Ghinet | Vibro-acoustic behaviors of flat sandwich composite panels[END_REF] (or group velocities)

yields: ω T = 1 2

S

√ µD .

Transition and maximum wave velocity

The definitions given above for the transition can be used for example to define the bandwidth on which regression models based on Eqs.( 10) and ( 11) can be applied. It avoids the use of nonlinear regression models based on Eq.( 8) but requires a priori knowledge of the transition to ensure the accuracy. Although the effective bending modulus can be approximated from the dispersion curves below the transition, this is not the case for the shear modulus. Indeed, the shear velocity cannot be measured in sandwich plates due to the presence of the second transition, shown in figure 1a. This transition is associated with a localization of the flexural wave in the skins and occurs once the hypothesis D 2 D 1 is no longer applicable. Consequently the asymptotic velocities cannot be measured and used to determine a core's shear coefficient.

However, it can be seen that the group velocity reaches a local maximum at 1 kHz. Since the group velocity describes the average local velocity of energy transport in the wave diffusion process, the physical meaning of this value can be seen as the maximum energy flux for a normalized energy density [START_REF] Schriemer | Energy velocity of diffusing waves in strongly scattering media[END_REF].

Since the shear velocity is not reachable, this maximum of energy velocity is the only point of the dispersion curves which is independent of the bending stiffness within the transition bandwidth (see figure 1b.), hence can be considered a pure core's shear effect. In the following, the transition is defined by the frequency at which the wave velocity reaches this local maximum. With this definition, the group velocity can be written from Eq.( 9):

c g = ∂ω ∂k = 2Ω µ 2S (ωΩ + ω 2 ) µ 2S 2ωΩ + Ω 2 + ω 2 , ( 12
)
where Ω = ω 2 + ω 2 0 and ω 0 = 2S √ µD . The transition frequency defined above requires the frequency derivative of the velocity, written:

∂c g ∂ω = ω 2 Ω -Ω 3 + 2ω 2 0 ω ω 2 Ω + Ω 3 + 2ω 0 ω 2 + 2ω 3 µ 2S (ωΩ + ω 2 )

Solving

∂c g ∂ω = 0 gives the solution for the transition frequency:

ω T = 2S 3µD . (13)
Therefore the maximum group velocity is written:

c g (ω T ) = 4 3 2S 3µ . (14)
Noteworthy, the velocity at the transition is independent of the bending stiffness D, as observed in figure 1b. The shear coefficient S can therefore be determined without the use of regression models requiring the entire dispersion curves in the considered bandwidth. Besides, the asymptotic and transition velocities are related by the coefficient

c g (ω T ) c s =        4 3 2 3      
 which is independent of the sandwich material parameters. This standard exceedance of approximately 8.87% of the shear velocity can lead to a significant underestimation of the time-of-flight in the transition bandwidth. As the maximal energy velocity of a wave pulse can be easily determined in a given frequency range, the shear coefficient can be derived using the following expression:

S = 27µ 32 max(c g ) 2 . ( 15
)
The bending modulus can then be determined using Eqs.( 13) and ( 15):

D = 4S 2 3µω 2 T . ( 16
)
Both S and D can therefore be determined in the direction of propagation using the maximal velocity and its frequency in a sandwich panel. In the following, a numerical validation is proposed for a sandwich waveguide whose material characteristics are known.

Numerical validation

In this section, the group velocity and transition frequency are determined in a sandwich structure to identify its equivalent mechanical characteristics. Consider the waveguide defined in table 1. Vertical motion is applied at x = X 0 to produce a flexural motion while the displacements are acquired at x = X 1 where X 1 -X 0 = 3 m. The excitation signal is a tone burst of amplitude U 0 = 10 -6 m involving n 0 cycles at the frequency f 0 with Hanning window defined by: Table 1: Dimensions and material characteristics of the sandwich beam. Note that the core is described as an homogeneous medium. The beam's width is l y = 10 mm and the damping is not considered in this model.

u(t) =                U 0 sin π f 0 t n 0 sin (2π f 0 t) , t ≤ n 0 f 0 0, t > n 0 f 0 (17
Note that the value of n 0 has a significant influence on the bandwidth of the excitation spectrum, hence on the possible spatial dispersion of the wave pulses. An illustration of the acquisition procedure is given in figure 2. The propagation of wave pulses with different frequencies and durations is computed using FEA on a distance of 3 m. The Hilbert Transform H[x(t)] of a time signal x(t) is commonly used to calculate the main attributes of x(t). It is defined by:

H[x(t)] = 1 π +∞ -∞ x(t) t -τ dτ. ( 18
)
Acquisistion X1 3m z x Acquisistion X0

Pulse excitation

Figure 2: Illustration of the numerical procedure for measuring the group velocities in the beam. Simulations are conducted using the FEA package Abaqus CAE V6.13 with Abaqus/Explicit solver. Linear C3D8 elements are used and a minimum mesh density of 15 elements per wavelength is applied, leading to a 120000 elements model at 13.2 kHz. Note that the choice of a beam is motivated by the elevated computational times associated with 2D transient simulations. The signal envelope is determined at emission, X 0 and arrival, X 1 using Hilbert transform while the time delay is defined by the time difference between the maximal amplitudes of the envelopes.

Results are shown in figure 3 for six wave pulses with different frequencies. At 1.4 kHz the pulse exhibits higher dispersion due to the reduced number of cycles, while negligible dispersion occurs above 4 kHz. A converted wave is observed at 13.2 kHz, due to the application of a vertical motion at the free edge, which is not associated with the shear wave motion. The accuracy obtained on the wave's speed clearly increases with frequency, although the absence of reflected and converted waves at low frequencies allows a reliable determination of the envelope's maximum. Exact values for the equivalent bending and shear coefficients can be obtained from Eqs.( 5) and ( 6): D = 7, 778 Nm, S = 1.2805 × 10 6 N/m and the mass per unit area is µ = 2.196 kg.m -2 . The group velocities obtained through 4 th -order theory are compared with the transient simulations in figure 4. Noteworthy, the velocities obtained thought transient response are in very good agreement with the analytical results. The accuracy is good, considering the frequency spectrum bandwidth of the excitation signals (only 6 cycles at 1.4 kHz). A 1% error is observed at 13.2 kHz, which is consistent with the appearance of a second transition frequency (around 14 kHz) resulting in a dissociation from the asymptotic shear velocity. The analytical shear velocity is c s = 763.6 m.s -1 . Using this approximated velocity and Eq.( 11) for the determination of S yields a 2.1% overestimation of the shear coefficient.

The transition occurs at 1.8 kHz and the maximum group velocity is c max = 831.6 m.s -1 . Using the relations derived in the previous section yields D = 7, 792.4 Nm and S = 1.2811 × 10 6 N/m which reduces the error to 0.05%. The mechanical characteristics of the panel obtained through transition analysis are therefore : [START_REF] Ranjbar | Vibroacoustic optimization of anti-tetrachiral and auxetic hexagonal sandwich panels with gradient geometry[END_REF] GPa and G trans = 50.04 MPa [START_REF] Potel | Characterization of composite materials by ultrasonic methods: modelization and application to impact damage[END_REF] which is a very close from the the exact values for E s and G c .

E trans = 81.
Based on these simulations, the maximal velocity can be used more effectively than the shear velocity to determine equivalent material properties of the considered sandwich panel. It can be noted that the second transition occurs in high frequencies in the considered example: a reduced core's thickness and thicker skins would result in an even more advantageous use of the transition compared with the shear velocity. It should also be mentioned that the results of the simulations presented in figure 4 could be obtained from a fewer number of pulse acquisitions involving broadband spectrum excitations, combined with Fourier Transform to determine the frequency components. Although this approach can produce faster results in simulations, a better accuracy is expected (see section 4) from tunable quasi-monochromatic wave pulses for the identification of the transition frequency.

Experimental case study

Experimental set-up and mechanical characteristics

The experiment is carried out on a rectangular sandwich panel measuring 60 cm × 288 cm, placed in a vertical position. The constitutive materials are a 10 mm-thick Nomex honeycomb core involving a 3.2 mm cell size, while propagation is considered in the W-direction. The core is surrounded by 0.6 mm-thick Hexforce skins with multi-axial carbon-reinforced fibres mixed with SR1700 epoxy resin. The density of the skins is ρ s = 1451 kg.m -3 and the core's density is measured using the overall panel's weight ρ c = 99 kg.m -3 (manufacturer: 96 kg.m -3 ). A picture of the components is shown in figure 5 and a description of the experimental set-up is shown in figure 6. The shaker is placed at the edge of the plate while a laser vibrometer is used to measure normal displacements at the surface. The IWC method [START_REF] Ichchou | Identification of effective sandwich structural properties via an inverse wave approach[END_REF] is employed to estimate the equivalent shear and bending modulus in the plate. The shaker is used to produce the harmonic excitation while the displacement field is measured using 66 points (referred as IWC measurements in figure 6), distributed along 50 cm, providing a spatial resolution of 7.5 mm and the wavenumbers between 0 and 4000 Hz.

Measured k IWC results

Figure 7: Real wavenumber in the main direction measured in the sandwich plate. A significant error is expected in the low frequencies, due to the limited number of periods for wavelengths determination. The convergence increases in higher frequencies, providing equivalent material characteristics and a good correlation with analytical results.

The wavenumbers measured by the IWC method are shown in figure 7. The resulting material characteristics are:

E iwc = 62 GPa and G iwc = 37.8 MPa (20)
while the manufacturer provides the following mechanical characteristics:

E stat. = 70 GPa and G stat. ∈ [30 -38] MPa (21)
which correspond to an 11% error on E s while G c is located in the upper range of the prediction interval.

ToF estimation of the shear and bending coefficients

The method proposed in section 2 is hereby applied to estimate the Young modulus and shear coefficient from the group velocity at the transition. First, the shaker is used to generate wave pulses in the plate at x = 0 (as shown in figure 6). The duration of the pulse is chosen above 10 cycles to reduce dispersion.

The procedure is conducted as follows: Pulse signals of frequency ω are generated in the structure and controlled by trigger. The velocity is measured for each pulse at 1 m from the source using scanning laser vibrometer. The pulse frequency is increased by steps of 20 Hz between 500 Hz and 1000 Hz and 50 Hz between 1000 Hz and 1500 Hz in order to approximate the transition frequency. It is characterized by a decreased of group velocity with frequency. Each pulse is averaged between 30 and 50 times to reduce experimental noise. The proposed procedure is summarized in figure 8. Note that an increased propagation distance d is required for sandwich structures whose transition frequency is low compared with their group velocity. Although a first estimation of the transition frequency can be performed without signal postprocessing, it is required for determining the velocity. This step allows an approximation of the transition frequency between 800 Hz and 950 Hz. A refined sweep is then conducted between 830 Hz and 900 Hz to obtain a more accurate value. The time signal is then extracted and postprocessed using Hilbert Transform and compared with the excitation signal to compute the time delay. A comparison between the Hilbert Transform of the time signals for various pulses between 500 Hz and 1200 Hz is presented in figure 9.a. It can be seen that the signal amplitude is influenced by the pulse frequency. Above 5 ms, a gap can also be observed between two pulses in the time signal. These effects are produced by the reflections at the edges (y = 0 and y = 600 mm) and at the free end of the plate (x = 2880 mm). It is therefore clear from figure 9.a that an extensive signal processing work would be required to measure the mean energy velocity while taking into account these reflections.

The main advantage of the proposed method lies in the absence of dispersion of the wave pulse in the forward direction, resulting from the fact that the transition is associated with the maximum group velocity. Indeed, the spectrum contributions of a finite pulse centred on the transition frequency (see for example figure 4) will exhibit velocities below or equal to the maximum group velocity. A practical consequence is a negligible dispersion of the first arriving components of the pulse. It means that the time of flight can be estimated using the difference between the FPA of the input and measured signals.

The Hilbert transform of the FPA is presented in figure 9.b. Noteworthy, a refined frequency sampling for the pulses frequencies and a zoom on the first arrival allows to locate the transition at 880 Hz with a 10 Hz accuracy. Further accuracy would be achievable with a better frequency resolution, but it will not be needed in the considered case study. Indeed, the second consequence of the dispersion characteristics at the transition is that the maximal velocity will not be affected by inaccuracies on the transition frequency, provided the excitation spectrum bandwidth covers the exact transition. The time of arrival therefore reaches its minimum at 880 Hz for t = 2.373 ms after a 1 m propagation so the maximal velocity is: c max = 421.4 m.s -1 . Considering the surface density µ = 2.7402 kg.m -2 and using Eqs.( 15) and ( 16), the bending and shear coefficients become: D = 2, 683 Nm and S = 4.1058 × 10 5 N/m. The equivalent Young modulus of the skins and shear coefficient of the honeycomb core in the main direction are therefore:

E ToF = 69.8 GPa and G ToF = 36.5 MPa [START_REF] Papadakis | The measurement of ultrasonic velocity[END_REF] which is a significant improvement compared with the IWC results since E ToF and G ToF are both in very good agreement with the manufacturer's values. The authors emphasize that in the case of an orthotropic panel, this procedure only provides the directional values for E and G in the propagating direction.

Concluding remarks

This paper has explored a new characterization technique based on the wave dispersion characteristics of guided waves in sandwich plates. The method provides the tensile and shear modulus of sandwich structures in the medium-frequency range by taking advantage of the singular wave dispersion characteristics of flexural waves within the so-called transition bandwidth. The bending-to-shear transition phenomenon was studied using the WFEM and compared with theory. It was shown that this transition is associated with a local maximum of the group velocity and analytical expressions were derived for the maximal velocity and its frequency. It has also been observed that this maximum wave speed is independent of the shear coefficient.

It is emphasized that the proposed wave-based method has significant advantages: the accuracy of the structural parameters is not altered by the presence of dissipative or uncertain boundaries since the data is obtained locally, through single-shot measurements. It is a considerable advantage compared with a number of stationary or other existing methods, since it can then be applied in situ, without requiring additional sample of the structure. The use of the transition bandwidth results in the absence of wave pulse dispersion in the forward direction of propagation, hence enables the use of an FPA measurement. The consequence is a significant increase of the ToF robustness in low frequency while the post-processing effort becomes almost negligible.

To summarize, the paper resulted in the following findings:

• A definition for the transition phenomenon using the existence of a local energy velocity maxima,

• A FPA strategy for determining this velocity from single shot measurements at low frequencies,

• An in situ procedure for measuring effective mechanical parameters of sandwich plates in the vibroacoustic range.

It can be noted that this strategy however requires the existence of a transition which may not always occur when the skins' thickness is large compared with the core's. The flexural waves in the sandwich are therefore converted into flexural waves in the skins while the core's shear effect is negligible. In general the width does not have any influence on the ToF results. Yet, is some highly directive structures the first incoming phase can be produced by reflections on the edges with higher velocities than in the main direction. A reduction of the distance between the acquisition points is then recommended. Furthermore, note that the proposed method may also apply to other wave types, since it is reminded that the bending-to-shear transition in sandwich panels is a specific type of wave conversion.

Figure 1 :

 1 Figure 1: (a) Comparison of the group velocities obtained through 4 th -(--) and 6 th -order theory (---). Sandwich parameters: D/µ = 1100m 4 .s -4 , S /µ = 172420m 2 .s -2 , h c = 10mm, h s = 0.6mm. (b) Influence of the bending modulus (D 0 = 2653Nm) on the frequency and value of the maximal group velocity. Shear velocity: c shear = 415.2m.s -1 , maximum velocity c max = 452m.s -1 .

  ) Material Thickness Density (kg.m -1 ) Young modulus (GPa) Shear modulus (MPa) Skin 0.3 mm 1451 E s = 81 G s = 2780 Core 25 mm 53 E c = 5.23 G c = 50

Figure 3 :

 3 Figure 3: Time acquisition at x = 0 (---) and x = 3m (---) in the sandwich waveguide. The number of cycles is: 6 at 1400 Hz, 8 between 1.8 kHz and 3 kHz, 10 at 4 kHz and 12 at 13.2 kHz. The time of flight is measured at the maximal amplitude of Hilbert transform (solid lines) signal.

Figure 4 :

 4 Figure 4: Comparison between theoretical dispersion curves and the velocities obtained by ToF with transient pulse simulations at 1.4 kHz, 1.8 kHz, 2.5 kHz, 3 kHz, 4 kHz and 13200 kHz. The half-amplitude bandwidth of the excitation spectrum varies from 396 Hz for the first pulse (1.4 kHz) to 1870 Hz for the last pulse (13.2 kHz).

Figure 5 :

 5 Figure 5: Picture of a section of the sandwich panel, the width is the W-direction.

Figure 6 :

 6 Figure6: Experimental set-up for the IWC and ToF measurements. The pulse input is generated using the shaker while the time delay is measured at the ToF measurement point. An illustration of the time signal and its Hilbert Transform is shown. Note that due to wave reflection at the edges, it is preferable to use the first phase arrival (FPA) to compute the group velocity, rather than the time delay of the mean wave pulse.

1 Figure 8 :

 18 Figure 8: Experimental procedure for the determination of c max and ω T summarized as a flow diagram. Note that the comparison between T n and T n-1 can be achieved on the first arrival of the pulse (FPA), without requiring signal processing and the computation of the group velocity.

Figure 9 :

 9 Figure 9: (a): Displacement amplitudes (Hilbert transform) at a distance x 0 = 1 m from the excitation source. The amplitude is normalized at 4 ms to highlight the FPA of the pulse. The first incoming wave pulse is observed at 880 Hz (---) while the time of arrival is t = 2.373 ms. Note that additional filtering procedures can significantly increase the accuracy of the measured velocity (see for example [36, 37]). (b): Comparison of the different measured pulses arrivals between 800 Hz and 900 Hz. A 10 Hz resolution is sufficient to obtained a good evaluation of the transition frequency.


MD Oula Itani
email: oula.itani@aphp.fr

MD G Monsel

MD M Mrad

PhD T Ruf

MD L Paris

MD, PhD E Caumes

Gnathostomiasis in a traveller returning from Madagascar

Gnathostomiasis is a foodborne parasitic zoonosis known to be endemic in Southeast Asia, India, Central and South America, with recent cases reported in Zambia, Botswana and South Africa. We report a case of gnathostomiasis acquired in Madagascar and alert clinicians of the emerging risk in southern Africa.

Case report

A 38-year-old Latvian female with no previous medical history presented to our clinic in Paris, France, in December 2020 with a 3-year history of intermittent migratory swellings of the left upper limb. Two years prior the onset of symptoms, she had travelled to Madagascar for a 3-months humanitarian mission. She was mostly based in Mahajanga, but roughly visited the whole island. She ate local food including seafood and fish which were always well-cooked, but also reported eating an insufficiently cooked frog that she had caught in a freshwater stream. She had no other history of travel in Sub-Saharan Africa, Asia or South America. The symptoms being intermittent, there were no abnormal findings on physical examination. Photos taken by the patient showed localized erythematous swellings of the left upper limb (arm, elbow, forearm, wrist), consistent with panniculitis (Photo 1). Episodes recurred every 3 to 4 months and lasted 2 to 4 weeks. She reported no pruritis or pain, no fever or other symptom. Blood count was within normal range, including eosinophils.

Filariasis serology was positive by enzyme-linked immunosorbent assay (Acanthocheilonema vitae, Bordier Affinity Products, Crissier, Switzerland) but was not confirmed by immunoelectrophoresis (in house method, antigene Ascaris suum), and testing for peripheral blood microfilariae was negative. Serologic testing for gnathostomiasis by immunoblot, performed at the Swiss Tropical and Public Health Institute, was positive. The patient was treated with ivermectin, 0.2 mg/kg, for two consecutive days. Follow-up is ongoing without any relapse up to now.

Gnathostomiasis is a parasitic zoonosis acquired after consumption of raw or undercooked freshwater fish, shellfish, frogs, reptiles or birds containing larvae of the nematode Gnathostoma (G.spinigerum, G.hispidum...). It is known to be endemic in Southeast Asia, India, Central and South America, with recent cases reported in Zambia, Botswana and South Africa. The classic triad of intermittent migratory swellings usually affecting the trunk or limbs, eosinophilia, and a history of travel to an endemic area is highly suggestive of the diagnosis 1 . Other less common manifestations include creeping eruption, nodules or visceral involvement 2 . Diagnosis is rarely confirmed by the isolation of the larvae from the lesions.

Serology with immunoblot is a reliable diagnostic tool. Effective treatments are albendazole (400 mg twice a day for 21 days) or ivermectine (0.2 mg/kg for one or two consecutive days).

Relapses are frequent, mostly after treatment with albendazole 3 .

In our case, the infection was confirmed 5 years after the patient's travel to a non-endemic area, and this long timeframe is not unusual. This is the second reported case of gnathostomiasis acquired in Madagascar 4 and should alert clinicians of the emerging risk in southern Africa.

Authors declaration:

The authors have declared no conflicts of interest. There are no funding to report for this submission.

  ., G.M., M.M., L.P. and E.C. managed the patient's case. T.R. performed the serological test. O.I. and E.C. drafted the manuscript.

Raharisoa A, Izri A, Andrianjafy RL, et al. Autochthonous gnathostomiasis in Madagascar. Emerg Infect Dis, 2020; 26: 1875-1877.


Christophe Droz
email: christophe.droz@ec-lyon.fr

Olivier Robin
email: olivier.robin@usherbrooke.ca

Mohamed Ichchou
email: mohamed.ichchou@ec-lyon.fr

Noureddine Atalla
email: noureddine.atalla@usherbrooke.ca

Improving sound

Keywords: Vibration control, Curved panel, Sound Transmission Loss, Small-scale resonator, Ring frequency, Diffuse Acoustic Field

Introduction

The design of lightweight and efficient soundproofing packages is one of the top priorities in the general transportation industry for noise, vibration and harshness objectives. A very common solution to this weight-efficiency compromise is the use of light porous material attachments on structures, the latter being usually simplified or modelled as flat whereas they can have a variable curvature radius in reality (a ship hull, an aircraft cabin). In usual sound transmission loss (STL) results for cylindrical shells under a diffuse acoustic field (DAF) excitation 1 , a first noticeable STL reduction occurs at the cylinder ring frequency which is a function of the shell radius and longitudinal waves velocity. A second dip in STL results happens at the coincidence frequency where structural and acoustic wavenumbers coincide (common to both curved and plane structures). Classical porous treatments that mainly provides broadband sound attenuation are not optimal for solving these reductions of sound insulation performance in narrow frequency bandwidths. Moreover, for typical aircraft sidewalls, the ring frequency-related dip occurs at low frequency (typically 400 to 800 Hz).

The design of ribbed or stiffened structures can be optimized to modify the frequency bandwidth affected by the two previously cited specific frequencies [START_REF] Fu | Optimization of sound transmission loss characteristics of orthogonally stiffened plate[END_REF] . A large variety of passive or active solutions have also been proposed. A typical example of an active approach for controlling acoustic transmissibility of aircraft panels is decentralized active control with distributed units [START_REF] Boulandet | Aircraft panel with sensorless active sound power reduction through virtual mechanical impedances[END_REF][START_REF] Yuan | Decentralized active control of turbulent boundary induced noise and vibration: a numerical investigation[END_REF] . Concerning passive solutions and apart from the classical solution of increasing the panel's structural damping, innovative approaches to tonal or narrow-band problems have been largely inspired by recent research on locally resonant metamaterials and often involve the use of periodic add-on units. The locally resonant bandgaps obtained using either Helmholtz resonators [START_REF] Doutres | Transfer matrix modeling and experimental validation of cellular porous material with resonant inclusions[END_REF]6 , periodically distributed tuned mass dampers 7 or surfacemounted resonators [START_REF] Xiao | Sound transmission loss of metamaterial-based thin plates with multiple subwavelength arrays of attached resonators[END_REF][START_REF] Liu | Broadband locally resonant metamaterial sandwich plate for improved noise insulation in the coincidence region[END_REF] can be used to improve the STL. For composites plates in which the coincidence frequency zone is usually larger than for a single layer homogeneous plate, a possible solution is then to use multi-modal resonators to broaden stop-bands 10 . Excepting Ref. [START_REF] Doutres | Transfer matrix modeling and experimental validation of cellular porous material with resonant inclusions[END_REF] , a commonality in previously cited references 6-10 is that only plane structures and then coincidence frequency are considered. To the best of the authors' knowledge, the reduction of the negative impact of the ring mode on the STL of a curved panel by means of a passive, tunable and multi-resonant approach has not yet been published. In this work, a tunable resonator concept involving 3D-printed cantilever beams with variable tip end magnets is experimentally studied on an aircraft sidewall panel including stringer and ring frame attachments under a DAF excitation. Several spatial distributions of the resonators including a multi-resonant configuration as well as possible addition of this narrow-band solution to a broadband sound absorbing treatment are investigated.

Proposed design of tunable 3D-printed cantilever small-scale resonators

The proposed small-scale resonators are based on a 3D-printed core structure using polycarbonate polymer consisting in a beam of section 10×4 mm 2 and length 23 mm supported by a stiffener of section 4×3.5 mm 2 (see various illustrations in Figs. 1(a-e)). The base part is a cube of 10×10×8 mm 3 dimensions on which the beam and the stiffener are both connected.

The connection to the structure is made at the base using a sealant (see Fig. 1(d)). The overall unit volume occupied by a resonator without a tuning mass is finally 33×10×8 mm 3 , with a unit weight of 2.72 g. A 1 mm-thick base magnet is glued at the tip end and the resonators are then tuned using different neodymium magnets of known masses. The tuning masses and corresponding resonance frequencies for the resonators shown in Figs. 1(a-c) are 2.24 g and 670 Hz (F1), 1.11 g and 820 Hz (F2), and 0.74 g and 980 Hz (F3), respectively. These average resonance frequencies were estimated on a few numbers of resonators when installed on the tested panel using a laser Doppler vibrometer and a shaker excitation.

Experimental methods and tested configurations

According to standards [START_REF]Acoustics -Measurement of sound insulation in buildings and of building elements using sound intensity -Part 1: Laboratory measurements[END_REF][START_REF] Astm E | Standard Test Method for Laboratory Measurement of Airborne Sound Transmission Loss of Building Partitions and Elements Using Sound Intensity[END_REF] , STL is determined in coupled reverberant-anechoic rooms using measurements of the spatially averaged sound pressure level in the source room L p and of the spatially averaged average sound intensity level L i over a scanning surface S m on the receiving side (both in dB), STL = L p -L i -6 -10 log 10 (S m /S) (with S the effective panel area, considered equal to the scanning area S m considered equal so that the last term was neglected). All measurements were conducted in the coupled reverberant-anechoic rooms at groupe d'acoustique de l'Université de Sherbrooke. A DAF excitation was generated in the reverberant room (7.2 × 6.5 × 3 m 3 dimensions) using a loudspeaker with a white noise input in the 50 -5000 Hz frequency range. The average sound pressure level in the reverberant room L p was obtained by rotating a half-inch PCB microphone with more than a complete rotation of its supporting arm during the signal acquisition time (of 120 seconds for each test). The average radiated sound intensity level L i was measured in the anechoic room (6.8 × 6.5 × 3 m 3 dimensions) using a Bruel & Kjaer sound intensity probe composed of two half-inch microphones and a 12 mm spacer. Manual scanning was performed at a distance of 10 cm from the panel surface following recommended scan patterns [START_REF]Acoustics -Measurement of sound insulation in buildings and of building elements using sound intensity -Part 1: Laboratory measurements[END_REF][START_REF] Astm E | Standard Test Method for Laboratory Measurement of Airborne Sound Transmission Loss of Building Partitions and Elements Using Sound Intensity[END_REF] . Standards 11,12 also recommend not exceeding a pressure-intensity indicator value of 10 dB for a sound-reflecting test specimen which was verified in each measurement. Finally, all the results presented hereinafter are provided in 12 th octave bands between 150 and 5000 Hz.

The structure under test is a curved rectangular fuselage panel equipped with axial and circumferential stiffeners (see Figure 2 The tested configurations are visually displayed and summarized in Figs. 2(a-f). The positioning of a sound absorbing treatment over the panel area is finally shown in Figure 2(e) (including possible combination with the D3 distribution of resonators). A melamine foam of 50.8 mm thickness was chosen to study its effect on measured STL when used alone or combined with resonators. This material has a mass density of 6.1 kg/m 3 , a static air flow resistivity of 7920 Ns/m 4 , a tortuosity of 1, a porosity of 0.98 and viscous and thermal lengths of 132 µm and 149 µm, respectively. Four foam layers of dimensions 1.34 m × 0.37 m are inserted between the circumferential stiffeners, supported by stringers and kept in place with tape (see Figure 2(e)). Since the resonators have a smaller height than the stringers, the resonators are not in contact with the foam layer. The total added mass for the foam is 0.6 kg. Figure 2(f) presents a close-up view of the panel equipped with resonators prior complete installation of the melamine foam.

All the tested configurations are summarized in Table 1 with corresponding percent added mass and figure result number. Using only F1 resonators, the D1, D2 and D3 distributions are first tested (see Figs. 2(b,c,d), respectively). Based on the D2 distribution (Fig. 2(c)), combinations of F1, F2 and F3 resonators are then used to produce a multimodal locally resonant add-on covering the anti-resonance region. Note that although the number of resonators remains unchanged for the three configurations, the added mass is slightly decreased in the multi-modal configuration, as F2 and F3 resonators are lighter than F1 resonators (see Table 1). The effect of the sound absorbing treatment is finally tested alone and in combination with the D3 distribution of resonators in single mode (F1).

Experimental results and discussions

Results obtained are reported in Figs. 3(a-d). The reference STL curve for the bare panel is systematically included in Figs. 3(a-c) in order to ease comparison between test cases described in Table 1. In Figure 3(d), the effect of four key configurations is also plotted in terms of insertion loss so as to highlight STL improvement brought by each setup.

A theoretical ring frequency f r of approximately 530 Hz is calculated using the simplified relation for a non stiffened thin shell 1 f r = E/ρ(1 -ν 2 )/(2πr), with E the Young's modulus (= 70 GPa), ρ the mass density (= 2700 kg/m 3 ), ν the Poisson's ratio (= 0.3) and r the curvature radius. This approximated f r is indicated by a vertical arrow in Figure 3(a), and the largest decrease of measured STL is obtained above this frequency at the 630 and 670 Hz twelfth octave bands. This bias might be explained by the fact that the mass density of the panel skin alone is considered in the previous calculation and that stiffeners and their effect are not accounted for. Nevertheless, the F1 frequency of the resonators is tuned for enlarging the STL in the frequency band where it is found to be the lowest. distribution. Nevertheless, the D2 distribution provides a good compromise in terms of overall STL improvement in the 500-800 Hz frequency range and is therefore chosen for studying the effect of different resonators frequencies. Combinations of F1, F2 and F3 resonators are used with D2 distribution and the obtained results are provided in Figure 3(b). Results show that the 2-modes and 3-modes configurations help limit the negative effect of the anti-resonance and enlarge the STL in the 800-950 Hz bandwidth. Compared with a single frequency configuration, the three modes configuration leads to an average STL improvement of 3.3 dB per 12 th -octave with a maximum of 5.7 dB at 800 Hz. This multi-resonant solution even reduces the add-on's weight from 5.7% to 4.9% (see Table 1). A multi-modal configuration of the resonators clearly increases the upper limit of the covered bandwidth from 750 Hz to 900 Hz. Interestingly, changing the frequency distribution within the fixed number of resonators does not modify the STL results in the rest of the frequency spectrum (see the result given in terms of insertion loss in Figure 3(d)).

In figure 3(c), the results for the D3-F1 configuration is recalled (see Figure 3(a)), and compared to the obtained results with the melamine foam alone and combined with the D3-F1 configuration. The addition of the foam treatment has for consequence an enlargement of the measured sound insulation starting at 250 Hz but nevertheless fails to fully compensate the STL loss resulting from the panel's ring mode as a result of its broadband behaviour (limited to a masking effect). When combined with the D3-F1 resonators configuration, the broadband effect of the melamine is almost additive to the narrow-band effect of the resonators. Compared with the foam alone case, the obtained STL result is enlarged at all frequencies (at the exception of two STL reduction that can be found at 400 and 800 Hz frequencies). It is noteworthy that the additive effect is only limited at the peak enhancement brought by the resonators (see the result given in terms of insertion loss in Figure 3(d)). The maximum STL increase is found at the 630 Hz frequency band, with values of 11.8 and 13.5

dB for the foam alone and combined with the resonators, respectively.

Concluding remarks

The performance of a locally resonant solution for enlarging sound transmission loss of curved panels at ring frequency is demonstrated on an aircraft sidewall panel under a diffuse acoustic field excitation. The STL reduction occurring at and near the ring mode frequency is entirely suppressed with up to 10 dB gain with resonators tuned at a single frequency. The use of a multi-resonant configuration proves to be a solution for both enlarging the efficiency bandwidth and limiting the anti-resonance negative effect. It is finally demonstrated that the proposed concept can be combined with a broadband sound insulation treatment, the resonators' narrow-band and foam's broadband effects being nearly always additive in terms of STL benefits. Based on this first proof-of-concept, the on-going steps of this work are (1) a numerical model guiding an optimization of both the frequency tuning and number of resonators so as to be able to draw a 'cost function-like' map to propose reachable dB enhancements vs a target added mass, and (2) the design of a solution for plane panels so as to solve ring and coincidence frequency STL reductions. Finally, this compact and low-cost solution of 3D-printed small-scale cantilever beams could be also tested on arbitrarily shaped complex industrial-scale structures for solving other vibro-acoustic issues.

Fig. 1 .

 1 Fig. 1. (Color online) Pictures and schematic of the small-scale resonators. (a) Resonator F1. (b)

  (a)). The panel is made of 1.27 mm-thick aeronautic grade aluminum with a 21.4 kg total weight. Its length is 1.7 m with outer and inner circumferences of 1.45 m and 1.3 m, respectively. Its curvature radius is approximately 1.6 m. The equivalent mass per unit area of the fuselage equipped with attachments is approximately 8.68 kg.m -2 while the bare aluminum shell has a theoretical 3.43 kg.m -2 mass per unit area. The panel was mounted in the test window using a frame made of plywood with acoustic sealing made of neoprene adhesive and silicone. The frame and surrounding surfaces were covered with a sound insulating barrier including an open-cell sound absorbing material and a PVC heavy layer.

Fig. 2 .

 2 Fig. 2. (Color online) Pictures of the tested configurations. (a) Bare panel. (b) D1 distribution

Fig. 2 (

 2 Fig.2(a) corresponds to the reference configuration, i.e. the bare panel, with measure-

Fig. 3 .

 3 Fig. 3. (Color online) Summary of absolute and relative STL results obtained. (a) Effect of resonators distribution with single mode tuning (the ring-mode frequency is illustrated by a vertical arrow). (b) Effect of single to three-modes tuning with fixed distribution. (c) Separated and cumulative effect of melamine foam and resonators D3-F1 distribution. (d) Insertion loss (difference between considered solution and bare panel) for four key configurations.

Table 1 .

 1 Summary of tested configurations.

	Tested configuration	Number of resonators Total and tip end Foam Total added
	[results figure number]

Acknowledgments

The help of Patrick Levesque at Université de Sherbrooke for the transmission loss experimental setup was greatly appreciated. The authors would also like to acknowledge Benoit Minard and Navdeep Sharma at Ecole Centrale de Lyon for their support in designing and manufacturing the resonators.

JASA-EL/Improving sound transmission loss of curved panels

The STL of the fuselage is measured before and after placing the resonators tuned on a single mode (F1, 670 Hz) with distributions D1, D2 and D3 (see Table 1). Corresponding results are provided in Figure 3(a). For the three distributions, the largest STL enhancements are not obtained in the 670 Hz twelfth octave band but in the preceding one, centered on a frequency of 630 Hz. Compared with the bare panel, the obtained STL gains for D1, D2 and D3 distributions are 6.6, 10.3 and 11.8 dB, respectively. The more concentrated D1 distribution produces a lower and smoother STL pic than when the resonators are more uniformly spread (i.e. D2 and D3 distributions). This result might indicate that local sound radiation is predominant around the ring mode's frequency bandwidth and that a spread spatial distribution of the resonators improves the sound insulation properties in the tuned frequency's bandwidth. This could also be a consequence of the spatial averaging done when measuring radiated sound intensity, that includes untreated bays in the case of the D1 distribution. The STL reduction observed around between 800 and 900 Hz in Fig. 3(a) is a known consequence of the tuned mass dampers anti-resonance. The dip nevertheless appears at different center frequencies depending on the considered D1, D2 or D3 distribution, with reduction of 5.1, 2.9 and 2.1 dB at twelfth octave bands frequencies of 900 Hz, 850 Hz and 800 Hz, respectively (with the bare panel result as a reference). Note that at the exception of this STL reduction, no other alteration of the sound insulation occurs in the 100-5000 Hz frequency range (see the result given in terms of insertion loss in Figure 3(d)).

In summary, results presented in Figure 3(a) indicate that the largest STL enlargements near the ring frequency (and the lowest alteration above) are reached with the D3


Christophe Droz

Olivier Bareille

Mohamed N Ichchou

Generation of long-range, near-cut-on guided resonances in composite panels

In this paper near-cut-on Guided Resonances (GR) are generated in a composite structure using an array of piezoelectric transducers. A GR is a type of high-order uni-directional wave mode, guided between the finite boundaries of a periodic waveguide's cross-section. Near their cut-on frequency, these waves often exhibit high chromatic dispersion and spatial decay, which compromise their potential use for wave-based inspection and Structural Health Monitoring (SHM) of advanced composite or metamaterials. A custom GR actuator is designed to achieve an accurate shape appropriation of the predicted 3D Floquet-Bloch solutions in a 3m-long sandwich structure. An original wave conversion phenomenon is then exploited to generate almost non-dispersive selected GR pulses near their cut-on frequency. Full wavefield measurements are conducted in time-domain using Laser Doppler Velocimeter to provide new insights on the GR near-cut-on scattering effects in the low vibroacoustic bandwidth.

I. INTRODUCTION

The propagation of elastic waves in complex structures have been the subject of intense research in the last decades, particularly driven by the development of non-destructive testing (NDT) (e.g. in pipes [START_REF] Lowe | Defect 418 detection in pipes using guided waves[END_REF] and rails [START_REF] Hayashi | Guided wave dis-421 persion curves for a bar with an arbitrary cross-section, 422 a rod and rail example[END_REF]3]), or the development of embedded structural health monitoring (SHM) systems for the critical aircraft and aerospace components [4][5][6]. In this field, wave-based techniques are still promised to a variety of applications [7], such as predictive maintenance solutions for critical infrastructures and automated defect evaluation [8,[START_REF] Raghavan | Guided-wave signal processing using chirplet matching pursuits and mode correlation for structural health monitoring[END_REF].

The prediction of propagative phenomena has also become more challenging, due to the increasing complexity of new lightweight components employed in aerospace and transportation industry. These structures are often based on advanced composites, which involve either heterogeneous assemblies or periodic patterns. An abundant literature can also be found on bio-inspired [START_REF] Zhang | Bioinspired engineering of honeycomb structure-using nature to inspire human innovation[END_REF], meta-materials [START_REF] Xiao | Flexural wave propagation in beams with periodically attached vibration absorbers: band-gap behavior and band formation mechanisms[END_REF][START_REF] Zhu | A chiral elastic metamaterial beam for broadband vibration suppression[END_REF] or locally resonant waveguides [START_REF] Liu | Locally resonant sonic materials[END_REF], to cite a few. In these structures, the prediction of guided waves raises challenging issues as well, such as the appearance of local modes or highly scattering behaviors above the first Brillouin zone. In automotive and aerospace industries, similar phenomena can typically be encountered in the low-or medium frequency ranges [START_REF] Mace | Mid-frequency methods in sound and vibration-part a[END_REF], since a variety of macro-scale periodic designs [START_REF] Denli | Structural-acoustic optimization of sandwich structures with cellular cores for minimum sound radiation[END_REF][START_REF] Ouisse | A piezo-shunted kirigami auxetic lattice for adaptive elastic wave filtering[END_REF] can be used to enhance a structure's vibroacoustic properties.

In this context, the development of numerical methodologies able to study wave propagation in continuous or periodic waveguides from a unit-cell analysis, has given new insights on wave's physics inside these structures. One popular implementation is known as the Wave Finite Element Method (WFEM) [START_REF] Zhong | On the direct solution of wave propagation for repetitive structures[END_REF][START_REF] Mencik | Multi-mode propagation and diffusion in structures through finite elements[END_REF][START_REF] Mace | Finite element prediction of wave motion in structural waveguides[END_REF]. One can cite among recent examples the wave scattering effects * christophe.droz@gmail.com in undulated [START_REF] Trainiti | Wave propagation in periodically undulated beams and plates[END_REF], textile composites [START_REF] Thierry | Multiscale wave propagation modelling for two-dimensional periodic textile composites[END_REF], honeycomb 52 [START_REF] Droz | A new proce-495 dure for the determination of structural characteristics 496 of sandwich plates in medium frequencies[END_REF][START_REF] Baid | Disper-499 sion of lamb waves in a honeycomb composite sandwich 500 panel[END_REF] or poroelastic [START_REF] Serra | Wave properties in 502 poroelastic media using a wave finite element method[END_REF] waveguides. Implementations 53 of the WFEM also extends to uncertain or slowly varying 54 [START_REF] Fabro | Wave propagation in one-dimensional 506 waveguides with slowly varying random spatially cor-507 related variability[END_REF]26], piezoelectric [27] or locally resonant metamate-55 rials [START_REF] Nobrega | Vibration band gaps for elastic metamaterial rods using 519 wave finite element method[END_REF][START_REF] Claeys | Design and validation of metama-523 terials for multiple structural stop bands in waveguides[END_REF].

56

More importantly, the development of model order re-57 duction strategies (e.g. [START_REF] Droz | A re-526 duced formulation for the free-wave propagation analysis 527 in composite structures[END_REF][START_REF] Zhou | Wave 530 finite element method based on reduced model for one-531 dimensional periodic structures[END_REF][START_REF] Droz | A hybrid 534 wave-mode formulation for the vibro-acoustic analysis of 535 2d periodic structures[END_REF]) has allowed the handling 58 of large unit-cell FE models. In particular it has given tom actuation system and a prior computation of its dispersion characteristics and deformed shape. Otherwise additional propagating or evanescent waves may be produced as well. Actuation systems based on ring arrays of embedded ultrasonic transducers have been extensively used for guided wave testing applications in pipes. However the proposed actuation strategy is more similar to mode force appropriation techniques [START_REF] Cooper | Experimental 563 evaluation of normal mode force appropriation methods 564 using a rectangular plate[END_REF][START_REF] Wright | Normal-mode 567 force appropriation theory and application[END_REF] in the understanding that generating a guided resonance can be seen as the appropriation of a cross-sectional wave shape within a bandwidth predicted by its dispersion curves.

2) In the proposed realization, the same waveform must be actuated within a broad bandwidth, below and above the cut-on frequencies. Therefore the actuation system has to produce higher loads than for standard modal appropriation techniques. Gxy = 1×10 9 νxy = 0.05 TABLE I. Material properties of the sandwich panel. The honeycomb core is represented as an homogeneous medium and an equivalent loss factor η = 0.2% is assumed in the composite plate. Added mass due to the reflexive tapes (see Sec.III) is included in the upper density.

have higher velocities and are not visible in figure 1. Before reaching their shear asymptotic behavior, guided resonances are subjected to a noticeable veering effect above their cut-on frequencies, as shown in figure 4. This phenomenon yields a smallest frequency-dependence of their group velocities: for GR 8 wave, the predicted group velocity deviates by less than 2% between 3600 Hz and 4800 Hz. Noteworthy, this means that a finite wave pulse generated within this bandwidth is expected to have a lower spatial dispersion than in higher frequencies.

III. DESIGN OF AN EMBEDDED

PIEZOELECTRIC TRANSDUCER FOR GR 8

WAVES

In this section, a piezoelectric actuation system is designed to create the GR 8 wave in the composite panel.

An array of piezoelectric transducers is the preferred solution for generating high-amplitude elastic waves from an embedded system. Although punctual loads could be distributed at the anti-node locations to create the required waveform, it is preferable to apply bending loads using thin d 13 piezoelectric patches. Indeed, applied loads have to be distributed more uniformly along the cross-section than for modal appropriation, where a single mode usually exist at a given frequency. The existence of multiple propagating wavetypes with close deformed shapes requires an accurate load distribution. In thick sandwich laminates, high-amplitude bending motions are obtained by applying in-plane loads on opposite skins, as depicted in figure 6. This strategy presents the advantage of facilitating a future embedding of the actuation device while avoiding the excitation of high-or first-order waves.

The design proposed in figure 6 involves 7 patches on each skin and is able to produce 7 anti-node curvatures, corresponding to the GR 8 wave presented in figure 3.

The array is installed at a free edge of the structure and while energy propagates under several first-order waves. was implemented to compute and analyze wave behav-372 ior, while TFC method was used to achieve in-situ struc-373 tural parameter identification from ToF measurements.

374

The dispersion analysis highlighted the presence of a found in [START_REF]Piezoelectric solutions, pi ceramic gmbh[END_REF].

59

  researchers the ability to handle the entire finite cross-60 section of large-scaled 1D waveguides, such as helicopter 61 rotor blades, acoustic pipes, complex ducts or stiffened 62 panels. Consequently, it is now possible observe a variety 63 of complex wave phenomena, broadening the concepts of 64 Bloch waves and from which Lamb and ring modes can 65 be regarded as a peculiar subclass of waves. One usu-66 ally defines the concept of guided resonance as the mode 67 of a waveguide's finite cross-section, propagating as a 68 plane wave along its main direction. These waves can 69 be seen as a generalization of high-order circumferen-70 tial modes, well-known in axi-symmetric structures, to 71 any kind of uni-directional waveguide exhibiting a finite 72 cross-section. Numerical investigations recently shown 73 that these waves could be of interest in the fields of SHM, 74 energy harvesting or for wave beaming applications. In 75 composite sandwiches, the authors recently [33] shown 76 that well chosen low-frequency GR exhibit higher sen-77 sitivities to small-scaled damages than classical waves, 78 even generated at higher frequencies. 79 However, apart from the aforementioned Lamb and 80 ring modes [34-37], and the recent investigations of Serey 81 et al. [38] on ultrasonic wave mode selection, such guided 82 resonances have been seldom, if ever, generated experi-83 mentally in composites. To the authors' knownledge, GR 84 have never been studied, nor observed near their cut-on 85 frequencies. Despite the recentness of applications for 86 selective GR techniques, the lack of literature on the 87 topic can be explained by the many practical challenges 88 raised by the generation of guided resonances within the 89 low acoustic bandwidth: 1) The selective generation of 90 a complex guided resonance requires the design of a cus-91

3 )

 3 Lamb waves are usually exploited above their cut-on frequency, hence often exhibit very low dispersion (A0, S0 and all first-order waves are considered as having a cut-on at 0 Hz). The present work challenges the generation of long-range guided resonances close to their cut-on frequency, i.e. near their zero group velocity. Dispersion curves therefore have to be thoroughly analyzed to identify bandwidths where transient pulses can be generated with low dispersion. This paper describes the design strategy and implementation of an array of piezoelectric actuators on a carbon-reinforced sandwich panel, in order to investigate the generation of guided resonances in the low vibroacoustic bandwidth. A periodic model of the panel is developed in Sec.II A, where the material properties are retrieved using state-of-the-art wave-based methodologies [22, 41]. In Sec.II B the periodic unit-cell model is then processed through a WFE Framework to analyze wave dispersion characteristics and investigate veering and deformed shapes of the guided resonances. A concept of piezoelectric system is then introduced in Sec.III to achieve waveform appropriation and generate a 8 th -order bending guided resonance (GR 8). The system is implemented on the composite panel in III B. The propagation, conversion and dispersion of generated guided resonances is eventually measured using full-field velocimetry in Sec.IV and the various scattering effect occurring within the cut-on bandwidth are interpreted. Some conclusions are drawn in Sec.V with guidelines for the efficient generation of guided resonances in arbitrarily complex waveguides. II. ANALYSIS OF GUIDED RESONANCES IN THE COMPOSITE WAVEGUIDE A. Description and modeling of the composite panel The composite structure is a 600×2890 mm sandwich panel made of a 10 mm-thick Nomex honeycomb core (ECA, 3.2 mm cell size) surrounded by two 0.6 mm-thick, 3-plies CFR skins. The panel has free edges and is ori-ented in the vertical position, where x, y and z axes 146 refer to the panel's length, width and normal directions, 147 respectively. Wave-based methods are preferred to mea-148 sure equivalent material properties of large-scaled struc-149 tures in the mid-frequencies. These in-situ characteri-150 zation techniques provide important mechanical param-151 eters (i.e. tensile modulus of the skins and equivalent 152 transverse shear modulus of the core layer) within the 153 typical dynamic range where guided resonances are ob-154 served: between 1 kHz and 5 kHz. Mechanical parame-155 ters of the sandwich panel are retrieved using Transition 156 Frequency Characterization (TFC) [22] and Inhomoge-157 nenous Wave Correlation (IWC) [41] methods. A finite 158 element model of the waveguide's representative unit-cell 159 is built based on the results presented in table I. 160 The periodic unit-cell is modeled using FEM with a 161 regular mesh involving 80 linear 3D brick elements along 162 the width and 12 elements in the core's thickness. This 163 mesh (see Figure 3) corresponds to a minimum of 10 164 elements per wavelength for the highest wavetype's order 165 predicted at 5 kHz and 20 elements per wavelength for 166 the solution analyzed in Section III. The total number 167 of degrees of freedom of the representative periodic 3D 168 unit-cell is 6318. 169 B. Prediction of guided resonances using unit-cell 170 modeling 171 The WFEM exploits periodic structure theory to de-172 rive the propagation constants Λ and eigenvectors Ψ 173 from the direct Bloch formulation: 174 [D RL (ω)Λ -1 + (D RR (ω) + D LL (ω)) + D LR (ω)Λ]Ψ = 0 (1) 175 where the dynamic stiffness sub-matrices are denoted 176 D ab (ω) = K + jωC -ω 2 M with a and b the left and 177 right sides of the unit-cell while we denote K, C and 178 M the finite element stiffness, damping and mass ma-179 trices of the periodic cell. Interested reader can refer

FIG. 1 . 1 )FIG. 2 .FIG. 3 .

 1123 FIG. 1. Group velocities of guided waves in the sandwich structure. Colorbar represents imaginary part of the complex wavenumbers, which indicates wave attenuation.

A.

  Concept of wave-mode appropriationModal appropriation techniques (also called phase resonance or normal mode testing) are used to perform stationary modes actuation from a set of distributed external loads. This concept can be applied to perform appropriation on the unit-cell (or cross-section) of a periodic waveguide, enabling the generation of guided resonances at different frequencies, as illustrated in Fig.5. The figure shows how a same GR can be generated at different frequencies, at the condition that the deformed shape of the cross-section (i.e. waveform) remains the same over the considered bandwidth.

260FIG. 4 .FIG. 6 .

 46 figure 3 between 2500 and 4000 Hz. A PVC mold is 273

315GR 4 Fig. 9 343 FIG. 9 .

 493439 FIG. 8. Transient wave pulse propagation time frames (1.12, 2.12, 4.15, 6.49) ms.

375

  group velocity plateau bandwidth (around 4 kHz), where 376 near cut-on guided resonances exhibit an almost non-377 dispersive behavior, despite being close to the wave's cut-378 on frequency. It is emphasized that the analysis of low-379 frequency, near cut-on propagation is generally a chal-380 lenging task, as waves exhibit high group velocity disper-381 sion (GVD), leading a transient pulse to exhibit short-382 range and highly scattering propagation. This group 383 velocity plateau phenomenon was exploited to achieve 384 the generation and propagation of a GR 8 wave along 385 the nearly 3 m-long sandwich structure, while the pulse 386 was slightly above the cut-on frequency. Measurements 387 were achieved using full-field velocimetry, which provide 388 an interesting insight on the guided resonance transient 389 scattering process in a sandwich composite, as it is em-390 phasized that observed bending-to-shear transition ef-391 fects only exist in sandwich waveguides. It is hoped that 392 future work will help further understand the near-cut-on 393 scattering process, providing design actuation systems 394 for more complex GR appearing in arbitrarily shaped 395 periodic structures with finite cross-sections.

  figure 11, which provide the detailed locations of each 404

407 ACKNOWLEDGMENTS 408

 408 The authors gratefully acknowledge the support from 409 the European Research Council through the H2020-EJD-410 VIPER (GA 675441) project. The Labex CeLyA is also 411 acknowledged for their support. The authors would like 412 to thank Stephane Lemahieu and Lionel Charles (Ecole 413 Centrale de Lyon) for their work and contribution during 414 the handling of the experimental set-up and Guillaume 415 Inquiété (Airbus Helicopters) for the fruitful discussions.

  416

wing with embedded piezoelectric sensor/actuator net-