M Jean

Marc Lancelin
email: lancelin@hikari.cpe.fr

Insa : R Gourdon

M Alain

Nicolas Insa

M Laboune

M C Havgoudoukian

M Jean-Pierre

H Charles

M Jean-

Pierre Flandrois

Safia Boudjema

M Didier

M Lagarde

M Alain Mille

C Dayeyan

Materiaux De Lyon

Marc Pelletier
email: marc.pelletier@insa-lyon.fr

C Bernavon

ENERGETIQUE Mega Mecanique

Genie Civil

Acoustique M Jean

Louis Guyader

Scso Scso

M Obadia

Lionel Insa

J Y Toussaint

Matériaux

La détection et la quantification de chocs sur une structure mécanique reste une attente majeure dans de nombreuses thématiques, notamment celle du contrôle de santé structurel (Health monitoring). Les domaines les plus demandeurs sont sans nul doute l'aéronautique et l'aérospatiale où l'utilisation de plus en plus fréquente de structures composites rend leur suivi et leur contrôle de santé indispensables pour d'évidentes raisons de performances et de sécurités de fonctionnement. Savoir détecter et localiser des chocs, c'est avant tout savoir dresser un historique des impacts générés sur la structures et permettre ainsi de gérer au mieux le risque d'apparition d'endommagements structurels (casse, délaminage, déformation…). L'objectif central de ces travaux de thèse a porté sur la détection d'impact par l'estimation du flux d'énergie élastique induit (vecteur de Poynting) lors du choc. Ces estimateurs sont déduits des tensions électriques fournies par des éléments piézoélectriques collés sur la structure. Le chapitre introductif dresse un état de l'art des techniques de détection d'impact. La pertinence des présents travaux est alors justifiée en premier lieu par le besoin de disposer de techniques simples à mettre en oeuvre et basées sur des estimateurs mathématiques ne nécessitant pas de systèmes de calculs complexes. Ceci dans le but ultime de pouvoir proposer des techniques à très faibles besoins énergétiques, pouvant donc être autoalimentables. Dans le 1 er chapitre, la méthode à été évaluée théoriquement et expérimentalement sur une structure 1D (poutre) accueillant un réseau de capteurs piézoélectriques uniformément répartis. Le flux énergétique peut alors être estimé à partir de la différentielle de l'énergie électrique extraite par 2 capteurs consécutifs. Cette différentielle s'annule si l'impact a eu lieu sur la zone délimitée par les 2 capteurs, sinon son signe indique la localisation (i.e. gauche ou droite) de l'impact. Cette technique simple à mettre en oeuvre et totalement passive, possède néanmoins une résolution faible car limitée à la distance inter-capteurs du réseau. La suite des travaux à consister à étendre ce concept de détection à des structures 2D (type plaques ou coques) plus représentatives des structures Resumé --ii -communément utilisées dans les domaines d'application visés. Afin d'avoir les outils théoriques requis, le 3 eme chapitre est consacré au calcul de la réponse impulsionnelle liant la force d'impact à la tension sur les éléments piézoélectriques en considérant une plaque mince infinie. La réponse à un choc quelconque s'obtient en convoluant la réponse impulsionnelle par une force représentative du choc. Dans le 4 eme chapitre, un estimateur de la direction du vecteur de Poynting est d'abord théoriquement établi à partir de la tension d'éléments piézoélectriques. Ceux-ci adoptent ici un arrangement géométrique en maille, délimitant ainsi une surface fermée unitaire. Si le choc a lieu à l' extérieur de la maille, l' estimateur est nul car l' intégrale temporelle du flux au travers de la surface est nul ( flux entrant = flux sortant). Dans le cas contraire, l'estimateur est non nul.

Cette approche est validée par des résultats expérimentaux obtenus sur une plaque métallique de grande dimension (120 x 80 cm 2 ).

Le chapitre 5 introduit un nouvel estimateur du vecteur de Poynting plus performant que celui introduit précédemment.

Le but ici est de disposer non seulement d'une information concernant la direction du flux, mais également d'estimer l'amplitude de ce flux, et de pouvoir au final quantifier pleinement l'impact. Une nouvelle approche théorique est donc développée pour approximer au mieux le vecteur de Poynting, toujours à l'aide des tensions générées par les capteurs piézoélectriques. Les simulations et les résultats expérimentaux obtenus sont concordants et ouvrent d'intéressantes perspectives. En conclusion, ces travaux ont montré qu'il est envisageable de développer des estimateurs du flux d'énergie élastique induit par un choc sur une structure élastique et de pouvoir en tirer des informations concernant sa localisation ,voire son amplitude. L'approche théorique est complexe et nécessite une connaissance globale des divers couplages mis en jeu. Quoi qu'il en soit, ces techniques, de part leur simplicité de mise ne oeuvre et leur Contents -4 ----

sensors. This differential is nil if the impact is inside of the area delimited by the 2 sensors, otherwise its sign indicates the impact location (i.e. left or right of the area).

This simple technique, easy to implement and totally passive, has a low spatial resolution limited by the inter-sensors distance.

The 3 rd chapter is devoted to the impulse response determination of an infinite plate in order to have the theoretical tools useful to estimate the impact force from the

Resumé

--iii -faible demande de moyens de calcul, constituent une proposition intéressante pour de futures techniques de détection d'impact passives ayant de plus la faculté d' être autoalimentables ; autrement dit, ayant la faculté de puiser directement dans les vibrations de la structure l'énergie suffisante pour assurer le calcul de l'estimateur et sa transmission par liaison sans fils.

I would like to express my sincere gratitude and appreciation to two of my advisors Pro. Daniel I would also like to express my gratefulness to my colleagues (Thibaut, Adil, Christophe, Abdelowahed, Saber, Rabah, Akram, Chatchai, Pierre-Jean, Jiawei, Kaixiang, Hongying, Masae, Dan, Qin…) I will not be able to list the names of all whom I feel grateful to, but I would like to express my gratitude to all of the members of LGEF, scientific and administrative for their help and support during my work.

Finally, I would like to take this opportunity to express my deepest thanks to my family for their endless encouragement and affection.

Introduction

Detection and quantification of impacts on elastic structure stays a major preoccupation in many fields, especially for Structural Health Monitoring (SHM).

Aerospace and aeronautic are probably the more applicant industry sectors because the increasingly use of composite material enforces an obligatory monitoring for security survey.

Detecting impacts permits to prevent and supervise the risk of structural damage such as deformation, delamination, breakage… This work concerns with the impact detection based on induced elastic energy flow estimators. These quantities are here deduced from voltage of a piezoelements network embedded on the structure.

The introduction is a state of the art in impact detection techniques. The basic common idea in these works is to indirectly estimate the impact force (amplitude and location) by convolution techniques from measurements of mechanical responses such as displacement, acceleration, and strain. These techniques generally require high computation capabilities, noise treatments and an estimation of the transfer function.

The pertinence of the present work is then justified by the need of a simplest approach using light calculation mathematical estimators in order to develop a low-power technique.

In chapter 2, a new method is theoretically introduced and experimentally evaluate for a 1D structure (beam), exploiting electric response of a piezoelectric sensors network. The energy flow is estimated from the energy differential of 2 consecutive Introduction -7 ----piezoelectric sensor voltage. The response from a particular impact is derived from the convolution of this impulse response by the representative impact force.

.

In chapter 4, an estimator of the Poynting vector direction is theoretically established from the voltage of piezoelectric elements for a 3D structure. These elements have a specific geometrical meshed arrangement, thus delimiting an unitary closed surface of energy flow integration. If the impact is outside of this closed area, the estimator is nil because the integration in time of the flow is nil (ingoing flow=outgoing flow).

In the contrary case, the estimator is not nil.

This approach is validated by experimental results obtained with a large metallic plate (120 x 80 cm 2 ).

In chapter 5, a new estimator of the Poynting vector is introduced. In addition of the energy flow direction, this estimator gives additional information on the energy flow amplitude in order to fully quantify the impact.

A new theoretical approach is developed here in order to refine the approximation of the Poynting vector by the piezoelectric sensor voltage. Simulations issues from this new model and experiments are in good agreement and then open new design perspectives.

In conclusion, these works show that new estimators of the energy flow induced by an impact can give informations on its location and even on its amplitude. The theoretical approach stays complex and needs a global knowledge of the relating mutiphysic couplings.

However, these techniques are simples, easy to implement and do not have a high calculation capacity requirement. They are a new interesting offer for future passive impact detection techniques with moreover the capacity to be self-powered.

Chapter 1 General -8 ----

Chapter 1 Generalities

Introduction

This chapter introduces the generalities concerning the impact detection techniques.

The existing ways of research on this topic are presented and finally the principle of our approach, based on the energy flow estimation, is introduced.

At the end, some basics generalities on piezoelectric systems are given.

Chapter 1 General -9 ----

Motivation for impact identification

Damage due to an external impact is a major preoccupation for structure design in many domains (aerospace, aircraft, automobile, building engineering…). Detecting damages is then essential to guarantee performances and security in such structures.

In that way, many Structural Health Monitoring (SHM) approaches [START_REF] Chang | A new look in design of intelligent structures with SHM[END_REF] have been developed. For example, ultrasonic inspections such as A-Scan, B-Scan ,C-Scan or Lamb Wave techniques 2-3-4-5 detect damages in metallic or composite structures with high efficiency. But most of these techniques are built around time-consuming and/or high need of memory algorithms and finally, they cannot be performed in real-time. To lighten the process, the structure health monitoring is often operated by periodical inspections rather than a continuous inspection. Another drawback of the SHM techniques is that an only eventual damage caused by impact is detected and not the impact itself. Yet it is important to have an impact history, even if they don't induced detectable damage, because it permit to predict damage occurrence. This point largely justify the need of impact detection techniques, not for supplant SHM ones but rather to complete them.

Numerous works have been carried out on impact location estimation. The basic concept relies on a combination of innovative sensors, advanced analytical techniques and artificial intelligence to provide a continual real time health assessment of a structure. When an impact occurs, these smart structures using information from the sensors can determine its location and eventually its amplitude. Exploiting the structural response to infer other information about the structure is referred to as an inverse problem. Before introduce our new technique, an overview of the various types of inverse problem is introduced in following part.

Overview of impact detection techniques

When considering the mechanics of structures deformations, the variation of the applied force is one of the most critical factors. The force applied to a structure cannot be measured directly but indirectly by the induced deformations variations. In order to overcome this difficulty, lots of researchers have studied this subject during several decades. This section will present a review of methods of inverse analysis for the indirect measurement of impact force.

Chapter 1 General -10 ----

The basic idea is to indirectly estimate the impact force from measurements of mechanical responses such as displacement, acceleration, and strain...

To completely define the impact force, it requires knowing its temporal response and its location.

Identification of temporal response of an impact force

The simplest approaches generally employed are based on deconvolution techniques.

Deconvolution technique

If the elastic behaviour of the structure can be considered as linear during the impact process, the response e(t, r) at a particular point is given by the linear convolution of the impact force f(t) by the impulse response function of the linear system h(t) :

   t d f r t h r t e 0 ) ( ) , ( ) , (    (1.1)
It is assumed here that f(t),h(t),e(t) equal to zero when t < 0. This supposed that the impulse response h(t) is known and that the response e(t) can be measured. In that case, the impact force f(t) is obtained by the deconvolution of (1.1).

Deconvolution in the time domain

The equation (1.1) is expressed in the time domain as

f h e   (1.2)
where e and f are vectors composed of discrete values of e(t) and f(t). h is a matrix composed of discrete values of h(t). The impact force history can be obtained by solving equation (1.2). From this approach, Goodier et al [START_REF] Goodier | An experimental surface-wave method for recording force-time curves in elastic impacts[END_REF] obtained the impact force from the strain measured at a point on the surface of structure. Doyle [START_REF] Doyle | An experimental method for determining the dynamic contact law[END_REF] (1984) also estimated the impact force acting on a beam from the bending strain measured by strain gauges. In these studies, the impact force is obtained by solving equation (1.2) using Gaussian elimination. But these conventional Gaussian elimination methods can lead to severely noisy or unstable measurements. The h-matrix coefficients is often ill-conditioned, small errors in the response vector e induces large errors in vector f. For this reason the above approach is not so effective to obtain an accurate and stable Chapter 1 General -11 ----estimation of the impact force.

To overcome this problem of noise amplification, the least-squares method had been used in several works. Chang and Sun 8 made the number of equations greater than the number of unknowns in equation (1.2) by imposing a response duration measurement longer than the impact force one. They solved this least-squares problem by the conjugate gradient method, which is an iterative optimization technique. But, too much iteration may cause an unstable estimation for the same reason as in conventional methods. Chang and Sun did not specify the iteration number.

The least-squares method was also used in Wu et al [START_REF] Wu | Two methods for determining impact-force history on elastic plates[END_REF] works. They considered several responses records at multiple locations simultaneously. To get a stable estimation, they consider the non-negative impact force (compressive or zero force). This problem with the non-negativity constraint was solved by an iterative optimization technique named the gradient projection method. Yen and Wu 10 demonstrated the efficiency of the non-negativity constraint. In their next works, Wu et al carried a numerical simulation for a laminated plate and showed that this technique can improve the estimation pertinence, even when the response records are very noisy.

They also showed that too many iterations cause an unstable estimation. They proposed a criterion for the optimal iteration number, which makes an accurate and stable estimation. But, the criterion is only qualitative.

Tanka and Ohkami [START_REF] Tanaka | Estimation of impact force on a space vehicle based on an inverse analysis technique[END_REF] proposed a technique using Truncated Singular Value Decomposition (TSVD) to detect the impact force acting on a pipe from its response measured by an accelerometer.

Fan Jiangling et al proposed a technique which estimates multiple periodical impact force from single point response in time domain [START_REF] Jiangling | Multiple periodical impact force identification from single point response[END_REF] .

Deconvolution in the frequency domain

Another basic technique is to transform the convolution in the time domain into frequency domain using Fourier transforms as:

) ( ) ( ) ( w F w H w E  (1.3)
When the transfer function H(w) is known, the impact force can be estimated by evaluating the Fourier Transform of the measured response. The advantage of this technique is to reduce the computational task compared to that required in time Chapter 1 General -12 ----domain.

Holzer [START_REF] Holzer | A technique for obtaining compressive strength at high strain rates using short load cells[END_REF] is the one of precursor to use the frequency domain deconvolution. His method involves the use of a short load cell and a fiber optics displacement transducer.

The analysis of the force/time data allows signal correction for the effect of the dynamic characteristic of the force measurement system. The method involves frequency transformation of the force data by a Fast Fourier Transform algorithm, and can correct the dynamic ringing of the load cell and the phase lag of the recording system Doyle [START_REF] Doyle | Further developments in determining the dynamic contact law[END_REF] claimed that frequency domain is better than time domain deconvolution because of simplicity. Some researchers also applied this technique on various structural components (Doyle 15-16-17 Martim and Doyle [START_REF] Marti | Impact force identification from wave propagation responses[END_REF] Rizzi and Doyle [START_REF] Rizzi | Force identification for impact problems on a half plane, Computational Techniques for Contact, Impact, Penetration, and Perforation of Solids[END_REF] ).

In practical computation, the data vector has a finite length, but Fourier Transforms are defined by infinite integrals. Thus, it causes a discontinuity at the end of the data, which is leakage. An exponential window exp(t  ) applied to the data can reduce the leakage.

Inoue et al [START_REF] Inoue | Measurement of impact force by the deconvolution method[END_REF][START_REF] Inoue | Measurement of impact force applied to a plate by the deconvolution method[END_REF] have used this technique to estimate impact force on a beam and plate.

A Fourier transform with an exponential window is the same as Laplace transform [START_REF] Wilcox | Numerical Laplace transformation and inversion[END_REF][START_REF] Inoue | Numerical Laplace transformation and inversion using fast Fourier trans form[END_REF] .

Inoue & al 24 used Tikhonov regularization to numerical Laplace inversion to improve the accuracy of the estimation.

It should be note that a small noise involved in the response data may cause a large change in deconvolution results. Wiener filter was used to overcome this problem by Inoue et al 25 . Wiener filter is an optimal inverse system to reduce the amount of noise by comparison with an estimation of the desired noiseless signal. They demonstrated the effectiveness of the Wiener filter by numerical simulations on a rod.

Kishimoto & al 26 also showed the effectiveness of the Wiener filter on cantilever beam impact detection.

In addition, lots of researchers have used deconvolution in the frequency domain such as Whiston [START_REF] Whiston | Remote impact analysis by use of propagated acceleration signals, I: Theoretical methods[END_REF] , Jordan and Whiston

Identification of the transfer function

The perfect knowledge of the transfer function h(t) is essential to obtain a precise estimation of the impact force.

Chapter 1 General -13 ----

There are basically three approaches to have the transfer function: by theoretical analysis, by experimental analysis, or by numerical analysis. Each of them has some advantages and disadvantages.

The theoretical analysis approach needs a precise modelling of the structure including significant parameters (shape, size, boundaries conditions, etc...). This generally stays a difficult approach because numerous parameters are hard to identify, and generally many approximations are indispensable.

In experimental identification, the transfer function is obtained by a calibration procedure, as described fro example in 33-34-35 . However, this also has many difficulties because small errors in the calibration data cause significant errors in the transfer function. Thus, it makes an unstable estimation of impact force.

The numerical analysis is another way for the transfer function estimation. This approach also needs experimental investigations and is difficult to carry [START_REF] Harrigan | Accurate measurement of impact force pulses in deforming structural components[END_REF] .

Other techniques

Other techniques exist for the impact force detection. The most significant are given here:

The Sum of Weighted Acceleration Technique (SWAT) was proposed by Bateman et ai [START_REF] Bateman | Force reconstruction for impact tests of an energy-absorbing nose[END_REF] . However, the main disadvantage of SWAT is the difficulty of determining the weighting factors.

A Neural network technique was also proposed [START_REF] Chandrashekhara | Estimation of contact force on composite plates using impact-induced strain and neural networks[END_REF] . It can be easily applied in the case of nonlinear response. The estimation process is very simple. The parameters of the network must be determined appropriately to get an accurate estimation. In addition, a larger number of data are required to train the network adequately.

Estimation of impact force location

To estimate the impact location is basically a nonlinear problem even in elastic structures. In most of the researches, the main approach is that the impact location can be obtained from the velocities of stress waves and from their arrival times at several points of the structure. However, it is not easy to determinate the wave velocities and arrival time in complex shapes because of waves reflections and dispersions. The majority of the proposed methods are developed for basic structures as beams or

Chapter 1 General -14 ----plates.

Impact location in beams

In the field of estimation of force location, some effective methods have been proposed. For example, the identification methods based on infinite beam using Timoshenko theory 39 . The response to impulse force of 1  -duration can be calculated as:

1 0    s w c x c x (1. 4)
where x is the distance from the impact location to the measuring point.

Solving Equation (1. 4), x can be obtained as:

    0 0 1 w c c w c c x s s    (1. 5) Chapter 1 General -15 -- --
Relation(1. 5) provides the distance from the impact location. However, the main restriction of this approach is that the impact force must be pulse-type and that human intervention is required for determinate the slowest pulsation component and the duration time 1  , which makes the approach not so simple. Based on Bernoulli-Euler beam theory, Doyle 41 proposed other technique to determine the impact location. The phase information obtained from a spectral analysis of the dispersive waves is used here to locate the source of the waves. Choi & Chang [START_REF] Choi | Identification of foreign object impact in structures using distributed sensors[END_REF] have proposed an impact load identification method using distributed built-in sensors. The identification system consists of a system model and a response comparator. The system model characterizes the dynamic response of the structure subject to a known impact force.

The comparator compares the measured sensor outputs with the estimated measurements from the model and predicts the location and force history of the impact. If the given experimental results are satisfactories, there are a large number of variables to optimize, which might cause difficulties in the optimization process.

To overcome the shortcoming of the above techniques, Doyle [START_REF] Doyle | A genetic algorithm for determining the location of structural impacts[END_REF] improved his initial technique based on the spectral-element method by combined it with a stochastic genetic algorithm to give a scheme that can locate the source of structural impacts.

In this approach, two independent estimations are carried from two responses by deconvolution where the transfer function is predicted by Timoshenko theory. These two estimations were well coinciding if the impact location estimation is correct.

A genetic algorithm was considered to search for a global maximum of the correlation.

A genetic algorithm (GA) is a research technique used in computing to find exact or approximate solutions to optimization and search problems.

Martin & Doyle [START_REF] Martin | A genetic algorithm for determining the location of structural impacts[END_REF] used this technique on a frame structure. First, they developed a method to predict the transfer function as a function of estimated impact location.

Second, they proposed another objective function based on the differences between two estimations of the impact force. However, an appropriate tuning of parameters is essential for obtaining a sufficient fast convergence of the searching process.

Inoue & al [START_REF] Inoue | Experimental waveiet analysis of flexural waves in beams[END_REF] proposed a technique that did not rely on beam theory. In the prior research [START_REF] Kishimoto | Time-frequency analysis of dispersive waves by means of wavelet transform[END_REF] , it was shown that if a propagating pulse in a dispersive medium is recorded at a point, the arrival time of each frequency component of the pulse at that point can be estimated by a time-frequency analysis with the help of wavelet transform. The main advantage of this approach is that the impact location estimation [START_REF] Inoue | Experimental wavelet analysis of flexural waves in beams[END_REF] . There is also an energy extraction-based algorithm has been proposed . [START_REF] Guyomar | Reference -131 ----estimation using piezoelectric sensors[END_REF][49] .

Impact location in plates

The oldest works on impact location in plate must probably be attributed to Yen and Wu [START_REF] Yen | On the inverse problem of rectangular plates subjected to elastic impact, Part I: method development and numerical verification[END_REF][START_REF] Yen | On the inverse problem of rectangular plates subjected to elastic impact, Part Ⅱ: experimental verification and further application[END_REF] . They proposed a technique from the strain responses at particular points on isotropic plates based on Greens's functions. The impact location can be obtained to transverse impact from strain responses measured at multiple points on the plate.

Ohkami and Tanaka [START_REF] Ohkami | Estimation of impact force and its location exerted on spacecraft[END_REF] developed a two-step technique based on classical plate theory.

First, they determine an elliptic region that may contain the true impact location.

Secondly, it is linked with the estimation on the impulse force history. Unfortunately, human interventions are required both in determining the elliptic region and in selecting a final estimation of the impact location.

Gaul & Hurlebaus [START_REF] Gaul | Identification of the impact location on a plate using wavelets[END_REF] proposed an experimental method for detecting flexural waves in plates by the use of piezoelectric films. They solved a simple system of nonlinear equations based on arrival times of flexural waves at several points of a plate. The advantages of the approach are its accuracy. However, it should be possible to extend this approach to composite material.

A triangulation procedure incorporating a Genetic Algorithm (GA) to localize the impact location was proposed by Coverly & Staszewski [START_REF] Coverley | Impact damage location in composite structures using optimized sensor triangulation procedure[END_REF] and Haywood & al [START_REF] Haywood | An automatic impact monitor for a composite panel employing smart sensor technology[END_REF] . They demonstrated the promising potential of GA in impact load identification.

Unfortunately, they only considered the impact location in these researches. There was no mention about how to identify the temporal impact force.

Gang Yan and Li Zhou 56 provide a GA-based approach to identify the impact location and reconstruct the impact force history simultaneously in composite panel.

Base on minimizing the error between measured strains responses in PZT sensors and numerically evaluated ones, a technique for force location was proposed 57 . Not only these theoretical approaches have been developed, many experimental works also exist, mainly acoustic or stress-strain methods using optical fibers or PZT as sensor Chapter 1 General -17 ----materials 58-59 .

Shin [START_REF] Shin | Real-time recovery of impact force based on finite element analysis[END_REF] proposed a technique using modal displacement and accelerometers information.

Another method introduced by Akhavan & al [START_REF] Akhavan | Prediction of impact contact forces of composite plates using fiber optic sensors and neural networks[END_REF] .use fiber optic strain sensor data as inputs to a neural network to obtain contact force history. They had designed and built an instrumented drop-weight impact tower to facilitate the measurement of contact force during an impact event. The impact head assembly incorporates a load cell to measure the contact forces experimentally. An in-house finite-element program is used to establish the validity of the fiber optic sensor contact force response. The finite-element model is based on a higher-order shear deformation theory and accounts for geometric nonlinearity.

Motivation of the present works

The previously introduced existing approaches for impact detection generally require high capabilities for computation, noise treatments and sometimes human interventions. Moreover, they have high electric power consumption, which is not the actual tendency for low-power systems especially for the embedded systems domain.

The motivation of present work gives raise in the need for a simple, low -power method to identify impact. The main goal here is not to increase the resolution of detection, but rather to have a method making the compromise between an acceptable detection resolution, an easy to implement computation algorithm and a low power requirement. The principle of the approach is to estimate the elastic energy flow induced by an impact in a structure with the help of piezoelectric sensors network.

Previous works 62-63-64 discuss on the energy flow in the structure but have not extended the work to impact location estimation.

The main difficulty in our approach is to reasonably estimate the power flow characterised by the Pointyng vector with the help of the electrical response of the embedded piezoelements.

This work has been done through various progressive steps:

In a first step, the work focuses on simple 1D problem. It has been shown that a specific shunted piezoelectric network bonded on the structure (Figure 1.2) could permit to determine the energy flow direction.

Chapter 1 General

-18 -- -- PZT1 PZT2 PZTi x E in E p1 E p2 E pi … E b1 - E b1 + E b2 + E bn + E b2 - impact A Figure 1.
2 sensor network for the detection of the energy flow direction in a 1D structure.

In the second step, the problem is extended to a 2D problem (infinite plate).

Considering a closed surface (S) (Figure 1.3a), when the impact is located inside frame surface S (I-point), the mechanical energy flows out the structural surface. The time domain integral of this outgoing energy is then positive with energy.

In the case of an outside frame impact (O-point), the induced energy quantity flows in and out through the closed surface (S). The out going energy quantity and energy losses in the structure consist ingoing energy. The time domain integral is then nil, considering a lossless plate. is null for an outside impact (O-point) and not null for an inside impact (I-point). For a finite structure (b), reflection on the boundaries induce secondary sources (s1..si) with a null associated energy flow.

In the case of a finite plate, the reflection phenomenon on plate edges has to be taken into account (Figure 1.3b) by considering them as secondary external sources. The flow associated to the reflected energy leads to a null value and hence does not degrade the initial information carried by the energy flow associated to the initial impact. Our works had thus focused on the development of a technique based on this approach. The basic idea is firstly to use a closed piezoelectric sensor and secondly, to estimate the energy flow trough this closed sensor from the piezoelectric sensor voltage induced by impact.

Chapter 1 General -20 ----

Generalities on piezoelectric systems

Many books and papers deal with piezoelectricity and we don't give here more details. We just give here the useful relation for the present work. After introducing the general constitutive equations for a piezoelectric element, main equations for a sensor deforming in bending mode is given, because it corresponds to the case funded in this work: piezoelectric sensors embedded on a bending beam or plate.

We finally show the interest in using piezoelectric technology for impact detection in terms of integration and performances.

Constitutive equations

The general constitutive equations for an anisotropic piezomaterial are given in (1.6) and -1.7).

m mi j T ij i T g D E    (1.6) m mi j E ij i E d T s S   (1.7)

Where

• S is the vector of deformations • T is the vector of stress consists of 6 components T i = 1 .. 6 • E is the electric field vector with 3 components

E i = 1 .. 3 • D d is the electric displacement field vector of 3 components D i = 1 .. 3 • s E is compliance of 6X6 matrix at constant electric field.
• d is 3X6 matrix of piezoelectric charge coefficients.

• g is 3X6 matrix of piezoelectric voltage coefficients. is exploited in actuation applications.

Our task mainly concerns with bending sensors (Figure 1.4). The piezoelement is poled in the 3-direction. The 2 electroded faces are parallel arranged along the 1-direction.

We supposed here that the strain in the 3-direction are negligible compared to the 1-direction one. This permits to say that only lateral deformation (1-direction) induces the electric field E. In that case, the relation (1.6) becomes:

3 33 3 31 1 32 2 33 3 T E D g T g T g T      (1. 8)
It should be note that T 3 equals to zero for flexural wave.

If the sensor is in open-circuit condition, the open voltage is:

3 31 1  V g T e (1. 9)
Chapter 1 General -22 ----

A wide range of forms

Piezoelectric materials exist under four primary forms: massive ceramics, single crystals, fibbers or films. These 4 primary forms can be directly integrated into the structure or they can be used in advanced systems such as summarized This large choice of piezoelectric materials or systems permits to have an adapted response in terms of integration for the wanted applications. This point is crucial for impact detection techniques because the piezoelectric sensors are the core of the detection system. If massive ceramics are often an adapted solution, piezofibers, films or composites floppy patches are very promising alternatives, especially for integration in structures deforming in bending mode.

We focus here on the 2 common used piezoelement for sensing applications:

 Massive ceramics

These materials exist under a multitude of geometries. Columns, tubes, plates are hence commonly available in industry standards (figure 1.9).

Chapter 1 General

-23 ---- The advantages of fibers are low acoustic impedance and high piezoelectric coefficients. In addition, many works focus on a newly developed active fiber composites (AFC), which have a big potential benefits over conventional piezoelectric sensing and actuating devices.

Chapter 1 General

-24 ---- Chapter 1 General -25 ----

Conclusion

This chapter presents the objectives of this thesis and the some basic generalities on piezoelectric systems.

In order to develop an easy to implement low power technique, a new method is introduced.

The starting point of our work is that the impact detection can be given by an estimation of the pointing vector, which represents the energy flow in an elastic structure. This new estimator is deduced from voltage response of piezoelectric sensor embedded or bonded on the structure.

Chapter 2 Preliminary results on beam -26 ----

Preliminary results on beam 2.1 Introduction

In this chapter, the possibility to detect impact by estimating the energy flow is validated in a simple 1D problem. The proposed technique consists in estimating the energy flow evolution in a monitored beam through a network of resistively shunted piezoelectric inserts.. These sensors do not need any external power supply.

The impact location is simply deduced from a dissipated energy comparison given by the shunted resistor voltage.

The first part of this chapter presents the principle of this method. Second part introduces the theoretical approach. Final part, theoretical and experimental measurements are carried out to demonstrate the method.

Chapter 2 Preliminary results on beam -27 ----

Principle

This study considers a one-dimensional beam with resistively shunted piezoelectric elements bonded on the structure as shown Figure 2.1. Besides, the beam is considered as lossless and infinite. The piezoelements are uniformly bonded on the beam with a x  step. Considering an impact at the point A on the beam, the longitudinal waves, as well as the induced energy (E in ) flow generated by this force would propagate in opposite directions (E in+ and E in -) along the beam. Going through a piezoelectric insert (PZT i ), a part of the mechanical energy (E i ) is extracted and converted into electrical energy (E pi ) and finally dissipated in the resistor. Thus, the energy of the wave decreases each time it passes a piezoelectric insert i and can be estimated as, assuming that each shunted piezoelectric insert has the same conversion ratio:

E bn -=E n-1 -E pi (2.1)
Considering that the 2 induced energy flows are identical (E in1 + = E in1 -), the force location can be estimated by simply comparing the dissipated energy by each resistor. Effectively, the gradient energy (E p1 -E p2 ) of 2 consecutive piezoelements would be nil only in the case where they surrounded the impacted area.

The impact spatial resolution of the proposed technique stays lower than x its performance relies on its fast processing and on the fact that the sensors do not need any external power.

PZT1 PZT2 PZTi x E in E p1 E p2 E pi … E b1 - E b1 + E b2 + E bn + E b2 - impact A Figure 2.1:
The studied structure. The PZT elements are uniformly bound on the beam with a xstep. E in is the stored energy in the beam due to the impact at the point A.

Epi is the energy extracted by the i-element and dissipated in the resistor. Eb is the mechanical energy in the beam.

Chapter 2 Preliminary results on beam -28 ----

Modelling

Lumped model

The nondispersive wave propagation in an infinite lossless beam can be modeled using spring-mass systems as shown in K respectively. Applying the fundamental dynamic equation and using Taylor expansion thus yields Equation (2.2) for non-piezoelectric parts and Equation (2.3) for piezoelectric parts, with  the force factor of the piezoelement defined as Equation (2.4), where e, A and l are defined as the voltage coefficient, the surface and the thickness of the piezoelement. Because there are the second derivative of x with respect to displacement u, Figure 2. 2 has u x , u (x-dx)) , u (x+ds) three part in composite section.

        2 2 2 ) , ( ) ( , 2 , , ) , ( x t x u dx K t x u t dx x u t dx x u K t x u M x x x x x           (2.2) dx x t x V x t x u dx K t x u M x x       ) , ( ) , ( ) ( ) , ( 2 2 2    (2.3) l eA   (2.4) x out u V C I      . 0 (2.5)
Chapter 2 Preliminary results on beam -30 ----

The current flowing out of the piezoelements can be derived from the macroscopic piezoelectric equations yielding Equation (2.5), with 0 C the clamped capacitance defined as Equation (2.6), with  S the permittivity of the piezoelement.

The resistive shunt thus leads to Equation (2.7), with R the value of the resistor.

l A C s   0 (2.6) R V u V C x      . 0 (2.7)

Energetic balance

The purpose of this section is to expose the dissipated energy over the two different sections of the system. The energy stored by the structure is given by Equation (2.8), where ) , ( t x F ext are the external applied force.

     dt t x u t x F E x ext in ) , ( ) , (  (2.8)         dt R t x V E n n p 2 , ) , ( (2.9)           dt t x V t x V R E n n n pn ) ) , ( ) , ( ( 1 2 2 1 1 , (2.10)
Because a resistor is connected to the piezoelement (giving Equation (2.7)), the energy E p can be written as Equation (2.9). Consequently this energy extraction reduces the wave energy available for the next parts of the beam. is negative (

0 12   p E
), sensor 2 to 3 is null (

0 23   p E
), 3 to 4 and 4 to 5 is positive It should be note that the signum of the criterion depends both on the wave direction and the direction the estimation is performed. If the wave and the criterion are the same direction (resp. opposite), the criterion is positive (resp. negative). shows the evolution of the extracted energy and the criterion (2.10) for a wave propagating from the point x 0 =7.5.

( 45 34 23 p p p E E E      )
The patches are assumed to be placed at integer values of x. As expected, the criterion is null near impact location, then reaches a minimal and maximal values, and starts decreasing in absolute value, with a signum depending on the wave direction and criterion direction.

Performance Discussion

This technique is not so precise compared to standard method because the impact location estimation accuracy is given by the inter-piezo size x. But, it is much faster and computationally efficient. It is necessary to note that all received signals are not processed by any filters. The noise influence is quite limited. In addition, there is no need of external power supply to the sensors. Finally, taking into account drifts in the coupling coefficient caused by the bonding is simple by adjusting the associated shunted resistance the energy extraction ration for each piezoelectric insert.

Numerical analysis-finite element simulation

Simulations using numerical Finite Element Analysis have been carried out in order to confirm the model. The modeled beam (Figure 2.5) received by five identical bonded piezoelectric patches. The dimensions and the physical properties are given in Table 2.2. The poling direction for the PZT patches is on the y-direction.

The beam with surface mounted piezoelectric patches is meshed (zoomed view in freedom equal to zero). Moreover, the electrical potential is forced to zero on interface between PZT and the beam. A force with a frequency band of 500 kHz in is applied to the beam. As illustration, the Figure 2.7 shows the PZT2 voltage when an impact is applied in part 0. Chapter 2 Preliminary results on beam

-35 -- -- (a)
Force is applied at part 0, all the

) , 1 ( n pn E   is negative, minimal value is 21 p E 
and the absolute value decreases with distance. Chapter 2 Preliminary results on beam

-37 -- -- (c) Force is applied at part 2, 0 21   p E , 0 32   p E ; The others n pn E , 1   are negative, minimal value is 32 p E 
and maximum value is

21 p E  . 1-2 2-3 3-4 4-5 -2 -1 0 1 2 x 10 -14  E (pn+1,n)
Normalized value force at part2 Chapter 2 Preliminary results on beam

-38 -- -- (d)Force is applied at part 3, 0 21 32     p p E E , 0 43   p E . The others n pn E , 1   are negative, minimal value is 54 p E 
and maximum value is

32 p E  . 1-2 2-3 3-4 4-5 -2 -1 0 1 2 x 10 -14  E (pn+1,n)
Normalized value force at part3 Chapter 2 Preliminary results on beam -41 ----

      p p p E E E , 0 54   p E 1-2 2-3 3

Experimental Results

Experimental Set-up

Experimental measurements were done considering a 350×50×3mm

Results & Discussion

The impact is generated by a source moveable piezoelectric patch excited in a sinus-burst-mode (10 periods at a frequency of 530 KHz), corresponding to the beam first flexural mode. Experiment results are shown in Figure 2.10. Results demonstrate that the location of impact is quite well estimated by the zero crossing of the criterion, and follows theoretical predictions in a similar fashion, with a response depending on criterion and wave directions. In addition, although the internal damping coefficient can be quite small, it gives rather precise information of the impact location.

Comparison between experimentally with simulation results, these results shows a good agreement to accurately estimate the impact location.

Conclusion

We suggest here that impact location can be performed by estimating the energy flow in the structure with the help of a network of piezoelectric sensors.

The principles of the method rely on the comparison of the extracted electrical energy by each sensor. The piezoelectric elements can be perceived as local electromechanical dampers that give insight of the energy flow. This research considers infinite lossless and 1-D beam. Following chapter would consider 2-D structure, which is more useful in reality situation. In addition, there are also some other field of investigative such as multiple impacts, resistive shunts, inductive or nonlinear shunts. This study considers infinite lossless beam, as well as resistive shunt.

Future work considering a lossy finite beam would lead to a more realistic model, and therefore to a finer impact location estimation. Besides, it is believed that taking into account internal losses in the beam can predict more accurately the impact location by extrapolation.

Chapter 3 Transient bending wave modeling in an infinite plate with bonded piezoelectric inserts -44 ----

3 Transient bending wave modeling in an infinite plate with bonded piezoelectric inserts

Introduction

In previous chapter proposed a simple efficient technique to evaluate the impact location in a 1D structure (beam). Estimating the location and energy of impacts in plate can be obtained with the same approach, that is to say in estimating the energy flow in the structure. But, the knowledge of the piezoelement electric response is a non trivial problem for a 2D structure. However, when dealing with mechanical wave propagation, a common approach consists in using numerical methods that are often time-consuming.

This chapter proposes a new approach using a Fourier transform for modeling the impulse response of an infinite plate with surface-bonded piezoelectric elements. This step is requisite to deduce the relationship between energy flow and piezoelectric response.

The proposed analytical formulation therefore allows bypassing numerical analysis drawbacks, in this specific case, while giving the response for the whole time

Impulse response modelling: theoretical approach

In this study, a two-dimensional lossy plate with piezoelement bonded on its surface as shown in Figure 3.1 is considered. This Section leads to establish the solution of the flexural traveling wave generated by a pulse force.

Determination of the displacement under pulse force

The general equation for linear flexural wave in thin undamped plates is well-known and can be found in Timoshenko [START_REF] Timoshenko | Theory of Plates and Shells[END_REF] for example and follows the equation:

) , , ( 2 t y x q u D u m z z      (3.1)
where z u is the flexural displacement ) , , ( t y x q
is the applied load.

 is the Laplacian operator m is defined as the surface mass density D represents the bending stiffness that can be expressed as a function of the Young modulus Y, the Poisson ratio v and the plate thickness h by:

) 1 ( 12 2 3    Yh D (3.2)
We may also consider that losses occur in the plate. Assuming loses that depend only Chapter 3 Transient bending wave modeling in an infinite plate with bonded piezoelectric inserts -46 ----on the displacement time derivative leads to the modified motion equation:

) , , ( 2 t y x q u D u u m z z z         (3.3)
where  is the damping factor.

In order to get the impulse response of the system, the space-domain Fourier transform is applied to Equation (3.3), along the x and y axis, which leads to (Guyomar and Powers, 1984 67 , Guyomar and Power, 1985 68 ):

) , , ( ) ( ~2 2 2 t k k q u k k D u u m y x z y x z z         (3.4)
where :

z u ~ and ) , , ( ~t K K q y x
are respectively the expression of the flexural displacement and the load in the space-domain Fourier transform x k , y k are the wave vectors along the x and y axis, Therefore, if the applied load is a pulse excitation, q(x, y, t) can be expressed as:

) ( ) ( ) ( ) , , ( t y x t y x q     (3.5)
where  is the delta distribution. This leads to the spatial Fourier decomposition of the load:

) ( ) , , ( ~t t k k q y x   (3.6)
Therefore the spatial frequency domain equation of motion (3.4) turns to:

) ( ) ( ~2 2 2 t u k k D u u m z y x z z          (3.7)
Such equation is a well-known second order system, whose elementary solution is given by:

) 1 sin( 1 1 1 ) , , ( ~2 2 2 2 2           t k e k m t k k u t k y x z (3.8) with the coefficients  , 2
k and  defined as:

m D   ; 2 2 2 y x k k k   ; 2 2 k m    (3.9)
It is interesting to note that Equation (3.8) shows that the propagation acts as a spatial low-pass filter, with a strong reduction of high spatial frequency components, and whose the cut-off frequency is decreasing with the time. This last observation partly Chapter 3 Transient bending wave modeling in an infinite plate with bonded piezoelectric inserts -47 ----expresses the dispersive nature of the wave.

We knowing the expression of ( , , ) xy u K K t . A 2D inversed Fourier transform leads to the expression of the flexural wave in the physical space (x, y, t).

            ) 1 sin( 1 1 2 1 ) , , ( 2 2 2 2 2        t k e k m t
            ) 1 sin( 1 1 2 1 ) , , ( 2 2 2 2 2         t k e k m t r u t k z dk kd e k i     ) sin(  (3.11)
where  and  are obtain from:

) cos( r x  ; ) sin( r y  ) sin( k k x  ; ) cos( k k y  (3.12)
Using Bessel functions of the first kind, Equation (3.11) becomes:

      0 0 2 2 2 2 ) ( ) 1 sin( 1 1 2 ) , , ( 2 dk rk kJ t k e k m t r u t k z         (3.13)
Expanding the sine term of this expression using Euler's formula and deriving according to the time leads to:

        0 2 2 ) 1 ( [ 2 1 1 2 ) , , (        j i m t r u z  dk rk kJ e j t k ) ( 0 ) 1 ( 2 2       ] ) ( ) 1 ( 0 0 ) 1 ( 2 2 2         dk rk kJ e j j t k       (3.14)
Therefore, referring to [Abramowitz and Stegun, 1964] [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] (Equation (11.4.29)), the Chapter 3 Transient bending wave modeling in an infinite plate with bonded piezoelectric inserts -48 ----expression of the flexural phase velocity yields:

) ) 1 ( 4 sin( ) 1 ( 2 2 1 ) , , ( 2 2 4 2 2          D m t r e D m tm t r u D m t r z  (3.15)
The flexural displacement resulting from a pulse force can be obtained by integrating Equation (

) ( t r q t r u t r u imp z arb z        (3.16)
One can note that the impulse response is independent from the angular value , which is a logical consequence of the wave propagation through an isotropic structure. This can also be observed in Equation (3.15) where the wave instantaneous frequency depends on the product ) (

2 2 t r
, demonstrating the dispersive nature of the wave.

Particularly, as 0  t the instantaneous frequency tends to  . This is a strong limitation when using classical numerical analysis due to the fine spatial and time meshing required.

The wave expression may also be rewritten as:

) ) 1 ( 4 sin( ) 1 ( 2 2 1 ) , , ( 2 2 4 2 2 2 2          D m t r e D m tm r r t r u D m t r z  = 2 2 1 () r f rt (3.17) where 2 2 2 2 2 4 2 12 ( ) sin( ( 1
)) 2 (1 ) 4 rm tD r r m r m fe t t m D t D        .
When t tends to zero, the term As well, this expression shows that the wave propagation is dependent to the scaling factor t r 2 , and consequently it is possible to obtain the whole response in the spatial and time domain as soon as the response at a given location or for a given time instant is known.

As shown in the Figure 3.1, when an impact is applied at the origin (0, 0), two points of the phase velocity are obtained in impact is applied at origin (0, 0) Equation (3.15) can also be expressed in the Cartesian coordinates as a function of x and y as:

   ) 1 ( 2 2 1 ) , , ( 2   D m tm t y x u z  ) ) 1 ( 4 ) ( sin( 2 2 2 4 ) ( 2 2       D m t y x e D m t y x (3.18)

Derivation of the piezoelectric sensor voltage

This part exposes the derivation of the voltage of a piezoelectric element bonded on the structure. The principles of the analysis consist in using the previously calculated flexural displacement velocity in order to estimate the piezovoltage in Cartesian coordinates. It will also be assumed that the piezoelement does not modify the local behavior of the structure (negligible effect of the added mass and stiffness). e defined as the permittivity of the piezoelement and piezoelectric coefficient respectively. For bending wave, we assume that the strain is zero inside the plate ( 0

zz S  )
Let us now consider the specific arrangement of the piezoelectric sensor, as given therefore turns this expression to:

) ( 1 2 1 2 2 2 2 2 31 33 y u x u h e l V D z z S E d             (3.20)
with V the voltage across the piezoelectric element and l its thickness.

The current I flowing out the piezoelectric element is obtained using the relationship between the electric induction field and the current:

  A d dA D I  (3.21)
where A is the surface of the piezoelectric element.

In open-circuit condition the current I is null, and considering null initial conditions of Chapter 3 Transient bending wave modeling in an infinite plate with bonded piezoelectric inserts -52 ----

the voltage and the second-order derivative of x u and y u leads to the expression of the voltage as

0 ) ( 1 2 1 2 2 2 2 2 31 33           A z z S dA y u x u h e l V       0 ) ( 1 2 1 2 2 2 2 2 31 33           A z z S dA y u x u h e l V       0 2 2 2 2 31 33 ) ( 1 2 1 2 Q dA y u x u h e l V A z z S              (3.22) (Assuming electric quantity is null with time is zero 0 0 Q  ):        A z z dA y u x u k V ) ( 2 2 2 2 (3.23) with ) 1 ( 2 ) 2 1 ( ) 1 2 1 ( 2 31 33 31 v c v h e A h le k p S          p c is blocked pizo capacitance.
Finally, the impulse response is calculated in 2 steps: knowing the impact characteristics, the displacement is firstly derived from Eq. (3.18). The resulting piezoelectric voltage is then calculated with the help of Eq.(3.21) and (3.22).

Simulations-Numerical analysis

The propagation of impulse bending waves is simulated with the help of Matlab® software in a infinite plate (Figure 3.4) with the physical properties and dimensions given in Table 3.1and Table 3. 2 The sensor is a square piezoelectric element poled in the Oz direction.

Comparison with finite element simulation

In order to validate the model, this Section proposes a comparison between predicted voltage response from the model with the one obtained by numerical analysis through the use of Finite Element Method (FEM) with the commercial software ANSYS. The plate and the piezoelectric elements are modeled using the SOLID5 and SOLID45 elements, respectively. As well, performance discussion about computation requirements of the proposed method will be exposed.

In the case of the FEM analysis, the infinite dimension condition is fulfilled by the use of a plate with very large dimensions, preventing reflection phenomena in the transient stage in the considered spatial domain.

Secondly, the ratio a/h (ratio between length and thickness of the plate) is chosen high [Ventsel and Krauthammer, 2001] The simulated model is shown in Figure 3.7. Other parameters are those used in the previous simulations.

Transient analysis was achieved considering a burst impact force located at point I (inside frame) or point O (outside frame). As example, the evolution of the plate deformation is shown in Figure 3. 8 for an outside impact at different time. Finally, the piezovoltage evolution can be deduced from these simulations. Figure 3.9and In terms of computational requirements, the FEM resolution takes more than 10 minutes for 60 points, while the derivation of the voltage using the analytical model lasts only 30 seconds for 3000 points, demonstrating the performances of the exposed analysis.

Chapter 3 Transient bending wave modeling in an infinite plate with bonded piezoelectric inserts

Comparison with experimental measurement

This Section exposes the comparison of resolution using the previously exposed analytical model, FEM predictions, and experimental measurements made on a thin plate.

Experimental set-up

The experimental set-up consists of a steel thin plate with surface-bonded piezoelectric devices. The plate is suspended vertically at its four corners by elastic fasteners. Experimental parameters are the same to the ones of previous simulations except for the following factors. The plate is finite with dimensions of 1.2m 0.8m.

The piezoelectric sensor thickness is 0.5mm. The frame formed by the piezoelectric sensors is made of 16 pieces, 4 on each side. The 16 elements are connected to each other in parallel. These elements are used as sensors. Therefore, they should not modify the behavior of the structure. Actually, the voltage sensors are composed by several parts so that there is flexibility on bending mode. Each part is small enough compared to plate dimensions and the mass is negligible. Thus, the modification of structure factors might be ignored. The driving force is generated by the use of an impulse hammer (Dytran~5850B). The range of force is 50-8000 lbs. The output voltage of the connected piezoelectric sensors is observed and recorded using an Chapter 3 Transient bending wave modeling in an infinite plate with bonded piezoelectric inserts -61 ----

Results & Discussion

The Figure 3.12 depicts the comparison between FEM and analytical predictions with the experimentally measured voltage on the square sensor. This As well known in the three-dimensional theory of elasticity, the Poynting vector (P-vector) represents the three-dimensional energy flow in a elastic media.

Here, the basic idea is to deduce in a first step an estimator of the power flow (i.e. P-vector) direction from the sensor voltage response.

We voluntary limit the analysis to the P-vector direction estimation.

This approach is conducted in three different steps:

 First, the energy flow theoretical analysis is proposed with the Poynting vector.

Then, knowing this theoretical expression, an estimator is derived from the electric response. It should be note that this new estimator does not estimate the energy flow amplitude (i.e. Poynting vector amplitude) but only its direction.

 Secondly, numerical simulations are performed to verify the above theory with piezoelectric sensor outputs.

Finally, the experimental confirmation of the theory is demonstrated using piezoelectric sensors integrated on a thin steel plate. All the received signals here are not processed by any filters and no environmental noises need to be handled with special care. The most important advantages of the present method are the low energy requirement and the simplicity. Besides, there is another important advantage. Any boundary limit must be concerned. It means that the structural change does not effect for identification.

Chapter 4 Force detection using energy flow direction estimator with piezoelectric sensors -64 ----

Principle

For lossy system, the value of the energy density is a decreasing function of the distance from the impact location. Considering a 1D-propagation, the energy flow direction can be obtained from the sign value of the energy density difference estimated by piezoelectric sensors, as shown in Figure 4.1a. On can easily appreciate that an arrangement in 2 sensor clusters (1&2; 3&4) permits to localize an impact, in considering again a 1D propagation (Figure 4.1b). In other word, if an impact occurs inside of the area delimiting by the 2 clusters (I-point), the energy density differences E 12 and E 34 of the 2 clusters are opposite sign terms. Effectively, cluster (1&2) sees a Right impact where cluster (3&4) sees a Left impact.

Conversely, an outside impact (O-point) leads to energy density differences E 12 and E 34 of the same sign.

The sum of this 2 energy density differences then gives information (Inside or Outside) on the impact localization. Effectively, an Inside impact induces a sum result leading to zero where an Outside impact leading to no-zero result.

The extension to a 2D propagation (Figure 4.1c) can be intuitively done by placing 2 others clusters (5&6; 7&8) perpendicular to the 2 primary clusters.

It will be shown thereafter that this specific arrangement permits to detect the direction of the flow energy.

Chapter 4 Force detection using energy flow direction estimator with piezoelectric sensors (c) extension for a 2D propagation .

E ij is the energy density difference from sensor i to sensor j.

Chapter 4 Force detection using energy flow direction estimator with piezoelectric sensors -66 ----

Theoretical approach

Two main theoretical steps are needed to obtain the link between the impact location (inside or outside) and the piezoelectric sensor electric outputs:

 Express the energy density difference by the Poynting vector approach.

 link this energy density difference with the piezoelectric sensor outputs.

Expression of the energy density difference

In order to estimate the spatial derivative of the Poynting vector, the energy flow vector for a lossy infinite isotropic plate is firstly established. The strain components in cylindrical coordinates are given by ::

r u S r r           u r r u S r 1 r u r u u r S r r            1 (4.1)
Where s i is the strain component in the i direction u i is the displacement component in the i direction r, the radial distance  is the incident angle of the Poynting vector ( i.e energy lfux)

The stress components are:

) ( 1 2 r u r u E T r r rr        ) ( 1 2 r u r u E T r r         ) 1 ( ) 1 ( 2 r u r u u r E T r r              (4.2)
where E is the Young modulus and  is the Poisson ratio.

The general equation of energy flow vector for an isotropic elastic medium can be expressed by the Poynting vector r P as (Romano et al, 1990 [START_REF] Anthony | A Poynting vector formulation for thin shells and plates, and its application to structural intensity analysis and source localization[END_REF] ) :

Chapter 4 Force detection using energy flow direction estimator with piezoelectric sensors

-67 -- -- ) ( z rz r r rr r u T u T u T P          (4.3)
we consider here an axisymmetric problem to simplify the theoretical approach. This involves that the components  r T and rz T are zero.

Assuming the strain zz T is zero:

r u z u z r     (4. 4)
where z is the half thickness of the plate. Thus, from Equations (4.2) (4.3) (4. 4) the Poynting vector can be expressed as a function of plate deflection:

r u r u r r u z E P z z z r            ) ( 1 2 2 2 2   2 2 1 ( ( )) 1 zz uu E zr υ r r r r        (4.5)
The value of the derivative of the displacement in the space domain Besides, the energy flow can be considered as parallel lines as shown in Figure 4.2.

Because the incident angle  is minuteness changed. Since incident angle  is constant when radial distance r is great, the first term of Equation (4.5) becomes much larger than the second one. By neglecting this second term, Equation (4.5) becomes:

r u r u E z P z z r         2 2 2 2 1  (4.6)
The Where :

                    ) (
 ab l is the vertical length of the considered surface.

 the coefficient A is : The case when incident angle  is different to 0 is explained in following section. The voltages of these sensors are then given by (the value of the derivative of the displacement in the space domain ): L pie , l pie , S pie , 31 e , h pla , s

Poynting vector and piezoelectric sensor outputs

           1 1 1 1 ) 1 ( 2 2 1 r r r r pie z z dr L r u r r u V          1 1 1 1 ) ( 2 2 r r r r pie z dr L r u            2 2 2 2 ) 1 ( 2 2 2 r r r r pie z z dr L r u r r u V         

33

 are respectively defined as the piezoelectric sensor length, the piezoelectric sensor thickness, the piezoelectric sensor surface, the voltage coefficient, the plate thickness and the piezoelement permittivity. The voltage difference is :

) ( 1 2 2 2 2 2 1 2 r r z r r z pie r u r u S V V           3 3 r u l S z ad pie     (4.16)
The spatial integral of the time derivative of the displacement as a function of the voltage can be expressed from Equation (4.13) as:

2 1 ) ( 1 2 1 2 2 2 1 V V L dr r u pie r r z          2 ) ( 1 2 1 ad l V V S      (4.17)
where S is the surface between these two sensors corresponding to the covered area shown in Figure 4 From Equations (4.17) (4.18) and, the Equation (4.19) can be obtained as:

2 ) ( 1 2 1 1 2 ad r r z r r z l V V S r u r u               (4.19)
Assuming that the distance l ad between the two center lines of the piezoelements is small enough compared to the bending wavelength, the output voltage of the piezoelectric can be approximated by a linear function of the radial distance, so that

                     ) ( 2 1 1 2 1 1 2 V V l S l l S V V ad ad ad pie          ) )( ( 2 1 2 1 1 2 2 V V V V S S l pie ad       (4.20)
As well, the derivative of the second term of the right member of Equation (4.12) is given as:

) ( 1 ) ( 1 2
                      ) ( 1 1 1 1 2 2 2 2 V V V V S l pie ad      (4.21)
The spatial derivative of the energy difference

out in P r   
expressed by Poynting vector can be rewritten as:

) V V )( V V ( S S 2 1 ( A ) p ( r 2 1 1 2 2 piezo out in          + ) V V V V ( ) S ( 1 1 1 2 2 2 2 piezo     (4.22)

If l

ad is small enough as the assumption before, S piezo becomes equal to S, and the Equation (4.22) can be rewritten as :

) ) )( ( 2 1 ( 1 ) ( 1 1 2 2 2 1 1 2 2 V V V V V V V V S A P r pie out in              (4.23)
Integrating Equation (4.23) in the time domain, the energy density difference is expressed by: Chapter 4 Force detection using energy flow direction estimator with piezoelectric sensors

-73 -- --          T pie T out in V V V V S A dt P r 0 2 1 1 2 2 0 ) )( ( 2 1 ( 1 ) (    dt V V V V ) 1 1 2 2    (4.24)
We finally obtain here an expression between the energy difference initially expressed from the pointing vector (cf (4.10) and the piezoelements voltage V 1 and V 2 . The left term of Equation (4.24) decreases with the distance from the impact location.

When an impact is applied to the left area of piezoelectric sensor 1 the energy density decrease from piezoelectric sensors 1 to 2. Thus, the right term of Equation ( 4.24) will be negative. On the other hand, when an impact is applied to the right area of sensor 2 the energy density increase from sensor 1 to 2. Thus, the right term of Equation ( 4.24) will be positive.

Impact area location detection with piezoelectric sensors output

Let us consider now the specific arrangement in 2 clusters of 2 sensors as introduced in figure 1b. This arrangement, reminded in Figure 4.5 permits to evaluate the impact location by the areas related to the location of piezoelectric sensors1 to 4. Area 1 is located as the left side of sensor 1. Area 2 is located between sensor 2 and 3. Area 3 is located as the right area of sensor 3. Because of the system symmetry, it is possible to connect sensor 2 with 3 and 1 with 4 (Figure 4.6) to reduce computational requirements. When an impact is applied between these pairs, area 2, the right term of Equation (4.24) becomes positive. On the other hand, the cases of impact applied in outer of these pairs, area 1 or 3, the right

1 3 4 area 1 area 2 area 3 V 2 V 3 V 4 V 1 2
Chapter 4 Force detection using energy flow direction estimator with piezoelectric sensors -74 ----term of Equation (4.24) becomes negative. Two pairs of sensors are used in each vertical and horizontal direction. Area 1 is defined the inner surface of these sensors. This arrangement defines 9 areas (1 to 9)

and permits to know if the impact is located in area 1 or not.

The disadvantage of this structure (

Q Q Q  
within the surface in the time domain from Equation (4.9) and (4.24) as: Q , however, do not depend on  and. Here, 1 Q and 2 Q values are negative, which lead to a negative value of Q. Analogous results can be obtained when the impact is applied in area 2, 4 or 6. Moreover, if the impact is applied in left area to that surface, area 5, or right area, area 9, the absolute 1 Q value becomes higher than 2

      T dt V V V V V V V V Q 0 1 1 2 2 2 1 1 2 1 ) ) )( ( 2
1 (      dt V V V V V V V V Q T       0 4 4 3 3 4 3 4 3 2 ) ) )( ( 2
1 (      (4.
Q value because of the great radial distance. Thus, the value of Q becomes negative. The same operation can be applied for the area 3 and 7. Based on the above results, we can simplify the energy density difference as (Figure 4.8), where only two sensor outputs are necessary:

      T dt V V V V V V V V q 0 1 1 2 2 2 1 1 2 ) ) )( ( 2
1 (     (4.26)
In other words, 1 Q and 2 Q depend on the 1 V , 2 V and 3 V , 4 V voltages respectively. It is therefore possible to know the impact location whether inside or outside with the sign of q simply from two piezoelectric sensors output. When an impact is applied inside of the structure (Figure 4.8), q value is positive.

When an impact is applied outside of the structure, q value becomes negative. The absolute value of q increases with the force magnitude (and same impact location if the impact is outside), because more energy flows through the sensors. When an impact is applied outside, the q amplitude also decreases with the distance of impact location for the same magnitude impact, since less energy flows to the sensors.

However, it seems that this technique has limited sensitivity for low damping.

Impact location and q value determination

In order to validate the theoretical model, simulations using Matlab ® software were carried out.The modelled infinite plate is shown in Figure 4.9. Two piezoelectric framed sensors are glued on the plate. Coordinate origin is on the centre of the sensors contour. The physical properties and dimensions are given in Table 4.1. The impact as shown in Figure 4.10 is applied near the centre of the contour (coordinate (0.01, 0.01)), inside the sensors contour. To avoid symmetry problem, the impact was not applied in the coordinate origin. When the energy flows from inside to outside of the contour, q value becomes positive. When all the energy pass the sensors, there are no energy flow anymore so that q becomes a constant value. The simulated result which On the other hand, when the same impact force is applied outside of the contour, the q becomes negative as shown in Figure 4.12 and Figure 4.13.

In Figure 4.12, the force is applied at location (0.38, 0.35), outside area of the contours. In Figure 4.13 the force is applied at location (0.40, 0.40), father distance from contours than the force of Figure 4.12 q tends to a constant value from a certain time t out as there is no more energy flow through the sensors from that time. t out is greater than t in because of the distance. It is also seen that the absolute value of q magnitude decreases with the distance of impact location. It is because less energy flows through the sensors when the distance increases.

Chapter 4 Force detection using energy flow direction estimator with piezoelectric sensors -78 ---- It is clearly seen that the max/min peak q value increases with the force magnitude.

The reason is that more energy flows through the sensors. The value of q is four times greater since the energy depends on the squared force magnitude. Finally the q-estimator is sensitive to the impact amplitude. Thus, this approach can estimate quantity of the input energy.

q-estimator value with different location inside

A same impact applied at different locations in the inside area (Figure 4.16) leads to the results shown in Figure 4.17 and Figure 4.18. The peak value of q stays constant with different the impact inside location. Hence, this shows that the criterion also gives an idea of the relative impact force magnitude for the same impact location when outside, and whatever the location inside. An impulse hammer (Dytran 5850B) was used for applying the impact. The hammer had a flat head of 6mm in diameter and the impact force exerted by this hammer was approximated as a localized load. It is important to note that all received signals were not processed by any filters, nor were the environmental noises handled with special care.

Chapter 4 Force detection using energy flow direction estimator with piezoelectric sensors -84 ----

Result & Discussion

Four impact tests were performed. One was performed using a steel hammer tip and three using plastics hammer tip, which gives lower pseudo-frequency impacts. First, the steel tip was used with impact waveforms shown in Figure 4.20. This impact is applied both inside and outside of area delimited by the sensors. The coordinate origin is placed at the centre of the surface. The results are given in Figure 4.21. These results demonstrate that the positive (resp. negative) value of q effectively corresponds to an impact applied inside surface of sensor (resp. outside), allowing an effective impact monitoring as expected. These figures also show good agreement between simulation and experimental results.

From these results, it seems that high pseudo-frequencies have more reflections due to fewer losses. Thus, the reflection on the plate boundaries makes the value of q oscillating.

As well, these series of tests show that the force magnitude can also be estimated by the value of q. The peak value gives the impact energy received by the sensor (if the impact is applied inside). All these results show that the value of q decreases with the time because of the reflection in the plate and dielectric losses of the piezoelectric sensors. The reflections can besides be thought as the outside impact.

Conclusion

This chapter introduced a new model for detecting impact location by estimating the energy flow direction with piezoelectric sensors.

Based on a Poynting vector approach, we show that the energy flow difference can be approximated from the electric response of piezoelectric sensors with a specific geometric arrangement.

This approach is a low cost technique with low computational requirements and it is easy to embed in structures.

Simulations and experimental measurements show that the proposed model gives a interesting tool for estimating impact location and eventually impact force magnitude.

Chapter 5 Force detection using energy flow estimator with piezoelectric sensors -88 ----5 Force detection using energy flow estimator with piezoelectric sensors

Introduction

As the before chapters introduce, we can identify the force using energy flow direction with piezoelectric sensors. It is seems that the approach proposed in before chapter is not sensitive in few energy density derivative case. Therefore, this chapter proposes an approach using energy flow with piezoelectric sensors. This approach is more responsive for low loss system.

The present chapter is organized as follow. First, energy flow analysis is proposed in bending wave. Secondly, numerical simulation is shown to verify the theoretical approach. Thirdly, an approach using piezoelectric outputs to express the energy flow is proposed. Finally, simulation and experimental confirmation is demonstrated using some piezoelectric devices put on a thin steel plate.

Chapter 5 Force detection using energy flow estimator with piezoelectric sensors -89 ----

Principles

Application of Poynting vector for power flow expression has been investigated in the previous sections. Extension for impact energy transmission will be developed in this chapter. To obtain the impact location and history, the estimation of the power flux through the closed surface (S) is considered as shown in is null for an outside impact (O-point) and not null for an inside impact (I-point). For a finite structure (b), reflexion phenomena induce secondary sources (s 1 ..s i ) which are the associated null energy flux total

Poynting vector applied with bending wave

The Poynting vector associated with the bending equation is calculated by a standard procedure that consists to start from the propagation general equation to get the energy equation. We start from the bending equation 72 , ) , , (

2 t y x f u D u m z z      (5.1)
where  is the flexural displacement, f is the applied load,  is the Laplacian Chapter 5 Force detection using energy flow estimator with piezoelectric sensors -91 ----operator and m is defined as the surface mass density. D represents the bending stiffness that can be expressed as a function of the Young modulus Y , the Poisson ratio  and the plate thickness h .

) 1 ( 12

2 3    Yh D (5.2)
In order to get the energy flow in the plates, Equation (5.1) is multiplied by the displacement yields: can be also written as: where the second right member term is given by:

z z z z z z z z u f u y x u u y u u x u D u u m                    ) 2 (
x u x u u x u x u x u z z z z z z                       
2 2 2 2 2 2 3 3 x u x u x u x u x x u x u z z z z z z                            (5.5) since 2 2 2 2 2 2 2 2 1                  x u t x u x u z z z  (5.6) Equation (5.4) turns to; 2 2 2 2 2 3 3 4 4 2 1                                  x u t x u x u u x u x u x u z z z z z z z    (5.7)
We can get the same result for the y coordinates since the plate is isotropic.

                                 y u t y u y u u y u y u y u z z z z z z z    (5.8) One term in Equation (5.3) z z u y x u  2 2 4 2  
 is still rest to solve. It can be written as following.

Chapter 5 Force detection using energy flow estimator with piezoelectric sensors The last right term of Equation (5.9) can be given as:

-92 -- --                                                    y u y x u x u y x u u y x u y u y x u x u y x u z z z z z z z z z z     
y x u y x u x u y x u y x u y x u z z z z z z                                2 2 2 2 3 (5.10) with y x u y x u y u y x u x y u y x u z z z z z z                                2 2 2 2 3 (5.11)
It yields the expression:

2 2 2 2                     y x u t y x u y x u z z z  (5.12)
Therefore, the Equation (5.9) can be developed as:

                        y u y x u u y x u x u y x u z z z z z z    2 2 3 2 2 4 2 2 2 2 2 3                                  y x u t x u y x u u y x u y z z z z z   (5.13)
Equation ( 5.3) can be, thus, rewritten as: where w is the energy supplied to the plate.

dt dw u f x u y x u u y x u y u y u u y u y D y u y x u u y x u x u x u x u x D y x u D y u x u D u t m z z z z z z z z z z z z z z z z z z z z                                                                                                                        
The volume integral of Equation (5.14) leads to:

dV dt dw dV E E t m divPdV D V V V c p         ] [ or dV dt dw dV E E t m PdC D C V V c p         ] [ (5.15)
Chapter 5 Force detection using energy flow estimator with piezoelectric sensors

-93 -- --
where C is the surface closing the volume V.

The P corresponds to power flow. p E , c E correspond to potential and kinetic energy of the bending wave. If there are no losses, the second left term of Equation (5.15) equals to zero. The right term of Equation (5.15) corresponds supplied energy.

Thus, the Poynting vector expressed by following two equations:

y u y x u u y x u x u x u u x u P z z z z z z z z x                      
P z z z z z z z z y                       2 2 3 2 2 3 3 (5.17)
It is should note that this Poynting vector expression is not same with the general one [START_REF] Anthony | A Poynting vector formulation for thin shells and plates, and its application to structural intensity analysis and source localization. Part I: Theory[END_REF] because we calculated from flexural wave equation in thin plate. Other methods derive express directly from the stress and strain expression. It seems that the expression of the Poynting vector is not unique.

Four terms in Equation (5.16) and (5.17) correspond to three wave types.

z z z z u y x u u x u   2 3 3 3       , x u x u z z      2 2 , x u y x u z z       2
correspond to shearing wave, bending waves and twisting waves respectively.

It can be clearly seen that the Poynting vector is related only to the flexural displacement u of the plate.

Let's see again the Equation (5.15). We assume that the system is static state at the beginning time. If the impact is applied to the integral volume; the first left term equal to the supplied energy. If the impact is not applied to the integral volume, the first left term equal to zero. The vertical coordinate (P z ) is zero since we consider a flectional structure coupling. The integral volume can be considered as integral surface (A)

shown in Figure 5.2

Now, the integral Poynting vector from inside surface to outside in time domain can be obtained as:

                      T z z z z z z z z y u y x u u y x u x u x u u x u Q 0 2 2 3 2 2 3 3 ) (     dt A d x u y x u u y x u y u y u u y u z z z z z z z z      ) ( 2 2 3 2 2 3 3                  (5.18)
where A is the integral surface shown

Numerical simulation based on Poynting vector

Simulations using Matlab R2006a are carried out in order to confirm the present theoretical model. The plate presented in Figure 5.2 is modeled using Matlab. The physical properties and dimensions of the target plate are given in Table 5. 1. The impact as shown in Figure 5.3 is supposed to be applied in the center of the contour (the coordinate is (0, 0), see Figure 5.2). The energy flows from inside to outside the contour, Q value becomes positive. When all the energy pass the contour, there are not energy flow anymore, Q becomes a constant value. The simulated result which is shown in Figure 5.4 gives a good correspondence. other is in amplitude. It is clearly seen that Q achieves quickly to its first peak when the distance is smaller. It is also clearly demonstrated that the amplitude of Q is bigger when the distance is smaller. As it was mentioned above, the reason for the second case is supposed that there is damping effect. It seems that longer the distance, less energy propagate through the surface. To verify it, the relation between Poynting vector Q and damping coefficient variation are investigated. Figure 5. 7 shows its result.

It is seems that saturated Q value decreases with increase of damping factor. This simulation result verifies well that the overall magnitudes of Q curves would be decreased if the magnitude of the damping ratio increases.

Approximation of Poynting vector and piezoelectric sensors outputs

How to link Poynting vector with piezoelectric sensor outputs is an open issue. If the Equation (5.18) can be linked from piezoelectric sensor outputs, the energy flow can be obtained with easy expression from the sensor voltages. This section proposes the connection between the Poynting vector and piezoelectric output voltage.

From the simulation result, the third and fourth terms of Equation (5.16) are negligible compare to the first and second terms.

The velocity of flexural deflection can be expressed by: Chapter 5 Force detection using energy flow estimator with piezoelectric sensors -99 ----

) ) 1 ( 4 sin( ) 1 ( 2 2
From the Equation (5.19), we can be obtained: Because it is transient problem, when time t turns to zero and r is big enough compare to the time t, the velocity of deformation can be simple as: As the same assumption, If the half width of sensor w and the distance between two sensors l ad are small enough compared to wavelength of bending wave, the two piezoelectric sensor voltages turn to: The spatial integral of the time derivative of the displacement as a function of the voltage can be expressed as:

            ) 4 2 ( ) 1 ( 2 2
           D m t y x z e D m tm x u    4 2 2 2 ) 1 ( 2 2 1         ) 1 ( 4 2 ) ) 1 ( 4 cos(
     ) 1 ( 4 2 ) ) 1 ( 4 cos( 2 2 2 2      D m t x D m t y x 2 2 2 2 2 4 ) ) 1 ( 4 2 )( ) 1 ( 4 sin( ( 2 2           D m t x D m t y x e D m t y x          ) 1 ( 4 2 ) ) 1 ( 4 cos(
           D m t y x z e D m tm x u    4 2 2 2 2 2 ) 1 ( 2 2 1          )) 1 ( 4 2 )( ) 1 ( 4 sin( ( 2 2
           D m t y x z e D m tm x u    4 2 3 3 2 2 ) 1 ( 2 2 1           )) 1 ( 4 2 ( ) 1 ( 4 2 ) ) 1 ( 4 cos( (
2 1 ) ( 1 2 1 2 2 2 1 V V L dr r u pie r r z          2 ) ( 1 2 1 ad l V V S      (5.31)
From the Equation (5.27), (5.30) and (5.31), the Poynting vector P

x can be express by: Chapter 5 Force detection using energy flow estimator with piezoelectric sensors

-102 -- -- 2 2 2 2 1 2 1 2 2 1 1 2 2 )) 1 ( 4 2 ( ) 1 ))( )( ( ) )( ((             pie x S S D m t x V V V V V V V V P     2 2 2 2 2 1 1 2 2 1 2 1 2 ) 1 ( 4 2 ( )) )( ( ) )( ((            pie S S D m t x V V V V V V V V     (5.32)
Integrating the Equation (5.32) in the time domain, the energy flow is given:

      t t o x V V V V dt P 0 2 1 2 1 ) )( ((   dt S S D m t x V V V V pie 2 2 2 2 2 1 1 2 2 )) 1 ( 4 2 ( )) )( (          (5.33)
Because the term

       t dt V V V V V V V V Q 0 2 1 1 2 2 1 2 1 )) )( ( ) )( ((     (5.34)
When an impact is applied the area 1 (in Figure 5.9), the Q value is positive (resp.

negative).

When 4 piezoelectric sensors is used as shown in Figure 5.10,using Equation (5.34), the impact location can be obtained in area 1, 2 or 3. Because of the symmetry of the system, it is possible to connect sensor 2 with 3 and 1 with 4 shown in to reduce computational requirements. When an impact is applied in the area 2, the Q value is positive. Otherwise, for the case of an impact applied in area 1 or 3, the Q value is negative.

When eight sensors are connected shown in Figure 5.12, the impact location can be known in area 1 or not with the energy flow value

x Q and y Q :

       t x dt V V V V V V V V Q 0 2 1 1 2 2 1 2 1 )) )( ( ) )( ((     (5.35)        t y dt V V V V V V V V Q 0 4 3 4 3 4 3 4 3 )) )( ( ) )( ((     (5.36) Sensor 1 Sensor 2 V 2 V 1 Area 1 V 4
Area 2 Area 3

Chapter 5 Force detection using energy flow estimator with piezoelectric sensors -104 ----Figure 5.12: Integrated surface with 8 piezoelectric sensors-1

However, this necessitates four sensors output. In order to reduce computational requirements, x P and y P are considered together and independent from each other because of the far field assumption. In the other word, 1

Q and 2 Q depend on the 1 V , 2 V and 3 V , 4 V respectively.
Thus, we can simplify the energy flow as shown in Figure 5.13. Chapter 5 Force detection using energy flow estimator with piezoelectric sensors -105 ----inside (resp. outside) of the structure (Figure 5.13), Q is positive (resp. negative). The absolute value of Q increases with the force magnitude (and same impact location if the impact is outside), because more energy flows through the sensors. When applying a force outside, the absolute value of Q also decreases with the distance of impact location for the same magnitude impact, since less energy flows to the sensors.

Numerical simulation and experimental results based on piezoelectric sensors outputs

Numerical simulation

The simulation parameters are those used in the previous simulations (Table 5. 1) except for the piezoelectric sensor bonded in the integral surface seen Figure 5.14.

The other parameters are shown in Table 5.2.

Experimental set-up

The experimental set-up consists of a steel thin plate with surface-bonded piezoelectric device, it is suspended at its four corners by elastic fasteners.

Experimental parameters are the same in the Table 5. 1 and Table 5.2. The plate dimension is 2 . 1 8 . 0  m. The squared piezoelectric sensor of size is made of 16 elements. They are connected separately, the one of inside and the one of outside.

The impact force was exerted at the target by an instrumented hammer (Dytran company, 5850B, minimum force 50 lbs, maximum force 8000 lbs, three impact tips) which contained a force transducer to measure the impact force directly. The Integrated Circuit Piezoelectric (ICP) Conditioning Module M32 provided the necessary power supply for the electronic circuit of the sensor. The hammer had a flat head of 6mm in diameter and the impact force exerted by this hammer was approximated as a point load. It is important to note that all received signals were not processed by any filters, nor were the environmental noises handled with special care.

Based these reasons, the approach can easily use as self-power system.

Result & Discussion

A simple impact force as shown in Figure 5.15 is applied to the plates. Because the impact is given by hammer, the impact amplitude and the instantaneous frequency are Chapter 5 Force detection using energy flow estimator with piezoelectric sensors -107 ----not the exactly same every times, thus, it is not care about the amplitude.

The approximation results based on the simulation and experimental data are given in The max peak of Poynting vector increases with the force amplitude, because the more energy flows through the sensors and the energy depends on the squared force magnitude. The simulation results as shown in Figure 5.17 correspond to the theory very well. It is interesting because it mean that we can quantity the energy of the impact.

The pointing vector decreases with increase of damping factor, because the less energy flows through the sensors. The simulation results verify well that the overall magnitudes of Poynting vector would be decreased if the magnitude of the damping ratio increases. When the pseudo-frequency of impact is decrease, if there are bending wave in the plate, the approach can detect the force location area inside the sensor or not. Based on above ideal, the plastic tip is used to give low frequency impact because is not as stiff as the steel tip.

Conclusion & Further Work

A simple, fast and efficient technique for identifying the impact location and force amplitude using Poynting vector is proposed and an approximation way to calculate the Poynting vector is given in this research. The present work demonstrates that it is quite easy to estimate whereas the impact location is inside or outside the chosen contour. When the value of the energy flow Q is positive, the impact location is inside of the contour, and the opposite. In the future, the model of piezoelectric sensors will be empirical in theory, or maybe find another better approach. Future work may be considering anisotropic materials.

Anisotropic plate

Using the approach (5.35) and (5.36), we also do some investigation in anisotropic plate. We have good experiment result, and the theory approach is developing.

A composite plate (fabric 0 / 90, rate of fibre 60%, Density: 1600 kg/m3) is considered in current research as shown in Figure 5.26. The plastic tip is used in this experiment. The result is shown in Figure 5.27. This result demonstrates that the positive (resp. negative) value of approximation vector corresponds to an impact applied inside frame. This result also show that the impact location area can be efficient detected from the value of the proposed criterion. This result show proposed model gives a interesting tool for estimating location and magnitude impact.

Conclusions

The objective of this work was to develop a technique to detect the location of an impact on a structure (plate) and quantity the shock energy. The identification should be simple enough to lead to a self power wireless system. This construct restrict the data processing to basic mathematic operation.

We selected piezoelectric elements to detect the shock history. To get these requirements, we followed several routes that are presented in chapter 2, 4, 5. The basic idea of the research was to extract energy, by means of a resistance in parallel with the piezo elements, bonded on a beam (1D) and aligned with a constant spacing.

A straight comparison between succinct piezoelemets response leads to the impact location on the beam. This processes can be self power quite easily (Energy can be harvested on the beam or the plate shunt)

.

In chapter 3, we develop a modelling of the mechanical impulse response between a force applied at some location in an infinite plate ant the displacement (or the displacement speed) at a different location. We consider the bending mode and ignore all the other bands modes. This modelling, based on a 2D Fourier transform of the bending equation lead to a closed form expression of the displacement speed. The modelling allows getting the displacement response to any arbitrary signals and overcome the problem encountered with the finite element approach. Once the impulse response knowing, the voltage on element bonded to the plate can be easily calculated. It must be stressed that we consider the piezoelement as a pure that we neglect the coupling between the piezoelectment and the plate structure.

The method presented in chapter 4 is based on the variation of the global wave energy measured successive points. Clearly the energy variation is directly related to the wave losses deriving into the propagation. Using a set of piezoelement measuring the voltage different, the wave propagating direction can be estimated. Four sets of these set of piezoelement can be used to form a squared frame thus the generating the wave Chapter 6 Conclusions -120 ----direction detect in 2D. Based on this simple concept, we can derive an approximate relating the power flow (Poynting vecter) to the voltage on the piezoelements. These approximate takes a positive value when the impact occurs inside the piezo frame and negative when the impact is located outside the piezo frame, thus providing an easy way to locate the impact location. Clearly the impact location revolution is of the order of the frame dimension but the processing is very simple and demands small energy requirement since only basic operation involved.

In chapter 5, we introduced a technique based on the power flow for the impact location and energy quantification. We first derived an expression of the Poynting vector from the equation supporting bending wave propagation. Then approach leads to a theoretical energy equation and clearly shows that the power flow is different from zero if the impact occurs inside the frame and zero if it occurs outside. A parallel with the Gauss theorem applied to location of an electrical charge can be made.

From the above mentioned approach, we can derive a new estimation that relates the power to the piezo voltage of the elements constituting the frame. It leads to experimental results that confirm the detection scheme. In this case also, the power requirements to drive the system are low. The approach developed for isotropic plate was used on anisotropic plate (composite). The experimental results show that detection is also quite straightforward in this case.

To conclude, we would say that we developed a set of method to estimate the impact location and quantification on a plate structure. The three proposed approaches show the feasibility of the impact location with detection schemes that requires a small amount of energy to be self-powered and wireless. Extension of this work will concern a more precise quantification estimation. So far we worked with plate structure. It would be very interesting to develop an approach for 3D structure or plate with variable thickness.

List of figure

-121 ----

Lists of figure

List of table

-124 ----

Lists of table

Figure 1 . 1 .

 11 Figure 1.1. Beam acceleration response under impulse force. The beam response to an impulse force is shown in Figure 1.1 (from 40 ). The high frequency compounds (fast wave) propagate velocity close to the shear velocity c s in Timoshenko theory. Information on impact location are carried by the slowest significant component velocity c(w 0 ). This component is observed at the pseudo-pulsation w 0 ( =

Figure 1 . 3 :

 13 Figure 1.3: principle of the energy flux expressed by Poynting vector in an infinite structure (a) without consideration of losses. Sum of energy flux trough the surface (S)



  is the 3X3 matrix of dielectric constants measured at constant stress The Equation (1.6) denotes the direct piezoelectric effect, where an electric field E arises under a mechanical solicitation T. This direct effect is exploited in sensing applications.The reverse piezoelectric effect is represented by (1.7): when an external electric field E is applied on the piezoelement, it deformed itself. This indirect effect

Figure 1 . 4 .

 14 Figure 1.4. Piezoelectric element used as bending sensor.

Figure 1 Figure 1 . 5 .

 115 Figure 1.5. primary forms of piezoelectric material

Figure 1 . 6 :

 16 Figure 1.6: Massive ceramic: industrial standard materials.

Figure 1 . 7 .

 17 Figure 1.7. piezoelectric fibers (a) and an example of integration (b)

Figure 2 . 2 .Figure 2 . 2 :

 2222 Figure 2. 2: Spring-Mass representation of an infinite beam. K and M are respectively the stiffness and the lineic mass of an simple (mechanical) section (only beam) Kp and Mp are respectively the stiffness and the lineic mass of a composite section (beam + piezoelement).

  Differentiating the energy over two consecutive piezoelements thus gives Equation(2.10). Therefore, the Equation (2.10) that performs a differentiation of the energy is almost null considering the two shunted patches that are closed to the impact location. The location estimation of the wave source can be done by determining the zero crossing of that the first criterion. The reason is the first two shunted patches Chapter 2 Preliminary results on beam -31 ----absorb more energy than the other patches. Therefore, from the origin of the wave, the energy dissipated through the shunt resistor is a decreasing value of the distance from the patch to the energy source. Therefore, the criterion (2.10) that performs a differentiation of the dissipated energy is null considering the two shunted patches that are closest to the impact source. Because, the energy decrease with the distance and the each patch absorbs a part of the mechanical energy of the wave, the criterion reaches a minimal and a maximal value for the next set of patches, then the absolute value decreases as shown in Figure2.3.

Figure 2 . 3

 23 Figure 2.3 Energy flow in the beam, differentiating the energy over sensor 1 to 2

Figure 2 . 4

 24 Figure 2.4 Normalized evolutions of the extracted energy (dotted) and the proposed criterion (plain)-force location at x 0 =7.5.

Figure 2 . 4 Chapter 2

 242 Preliminary results on beam -32 ----

Figure 2 . 6 )

 26 Figure 2.6) using the commercial FE code ANSYS 9.0. The structure is meshed with six nodes plane quadratic elements. The patches are meshed with PLANE13, and the beam with PLANE42 (Figure 2.6). PLANE13 is defined by four nodes with up to four degrees of freedom per node (x and y directions and the electric potential V for piezoelectric elements). Moreover, CIRCU94 elements are used to model the resistors, which are connected to each piezoelement electrode. The mechanical boundary conditions of the structure are clamped at both end of the beam (all degrees of

Figure 2 . 5 :Figure 2 . 6 :

 2526 Figure 2.5: The simulated model. Dimensions are in mm1

Figure 2 . 7 :

 27 Figure 2.7: Example of a sensor (PZT2) response for an impact in part 0.

Figure 2

 2 Figure 2.8: simulation results:2-1 means

3

 3 BulkMolding Compound (BMC) beam to confirm the proposed method. Seven resistively shunted piezoelectric inserts (PZT P-18 Saint -Gobain9) are bonded onto the upper surface of the beam (Figure2.9). The dimension of inserts is 30×3×0.5 mm 3 and each separated by a distance x=40mm. Because of the variability of the bonding condition, the coupling of each piezoelectric insert is quite different. The value of each resistor has been tuned to have the same ratio of extracted energy for each piezoelectric patch. The experimental parameters are shown in Table 2.3, except for the value of the resistance, which is tuned to equalize the energy extraction ration due to the bonding.

Figure 2

 2 Figure 2.9 Experimental set-up

Figure 2 . 10 :

 210 Figure 2.10: Experimental (plain) and results for the proposed criterion using a-530KHz piezoelectric moveable source.

  and space domains. The proposed model relies on flexural wave decomposition over the spatial frequency domain and corresponds to a time generalization of the angular spectrum theory, thus introducing flexural wave propagation as a time-varying spatial filter. The inverse Fourier transform is then applied and leads to the impulse response in the physical domain. From this model is also derived an analytical expression of the impulse voltage response of piezoelectric transducers. The predicted impulse response is then compared to FEM simulation and experimental measurements in order to asses the model. Chapter 3 Transient bending wave modeling in an infinite plate with bonded piezoelectric inserts -45 ----

Figure 3 . 1 :

 31 Figure 3.1: Schematics of the considered infinite plate

  is interdependence between space and time domains (which is not the case when dealing with classical propagation). The coefficient 2 1 r expresses the energy conservation theorem. Chapter 3 Transient bending wave modeling in an infinite plate with bonded piezoelectric inserts -49 ----

  Figure 3.2 (0.2, 0.2) and Figure 3.3 (0.4, 0.4). The time to reach maximum value in Figure 3.2 is close to 0.01 second and that in Figure 3.3 is close to 0.04 second. The reason is that the distance between impact location and measure point is two times different. Because of the scaling factor t r 2 , the Figure 3.2 scale down Figure 3.3.

Figure 3 . 2 :Figure 3 . 3 :

 3233 Figure 3.2: Impulse response at the point (0.2, 0.2) --impact is applied at origin (0, 0)

  The piezoelectric voltage can be derived considering the constitutive piezoelectric Chapter 3 Transient bending wave modeling in an infinite plate with bonded piezoelectric inserts -51 ----equations and the previous assumptions, giving the electric induction field D d along the z-axis as a function of the electric field E E along the z-axis and the strain as:

Figure 3 Figure 3 . 4 :

 334 Figure 3.4

  Figure 3.5 depicts the evolution of the voltage as a function of the time when a unitary pulse force is applied at the origin. Other excitation cases in different locations can of course be considered. The instantaneous frequency decreases as a function of time due to the presence of the dispersive 1 sin t    term in Eq (3.18). However, the spatial filtering performed by the insert (and represented by the surface integral in Equation (3.22) significantly changes the envelope of the response. From the time delay in Figure 3.5, it can also be shown that the propagation nature is also well described by the model.

( a )Figure 3 . 6 :

 a36 Figure3.6: Simulated sensor voltage for a half period-burst impact (a), 1 period-burst impact (b) and 2 periods-burst impact (c). The impact location coordinates are (0, 0).

Figure 3 .Figure 3 . 7 :Figure 3 . 10 ,

 337310 Figure 3.10 respectively show the comparison between the two resolutions methodswhen the excitation is a 5-kHz, one cycle sine burst applied at the location (13mm, -13mm) (i.e. inside the piezoelectric element location) and when the excitation is a 5-kHz, one cycle sine burst applied at the location (-110mm, 38mm) (i.e. outside the

Figure 3 . 8 :Figure 3 . 9 :Figure 3 . 10 :

 3839310 Figure 3. 8: Deformation of the plate obtained with the FEM for an outside impact at different time (t =0.228ms (a); 1.2ms (b); 2.15ms (c) and 5ms (d))

Figure 3 .

 3 Figure 3.11: the experimental set-up. Plate dimension is 1200x800x0.1 mm 3 .

  Figure shows a good agreement between predictions and experimental results on the transient stage. At ms t 28 1  , the reflections on the plate boundaries make the analytical model diverging from other responses. In a first approximation, this is not a blocking point because it is considered that reflections induce a nil wave flow through a closed surface (Figure 3.11 abcd). So we can consider here that only transient response (for t<t 1 ) gives information about wave flow.

Figure 3 . 12 :3. 6 Conclusion

 3126 Figure 3.12: Experimental results and analytical and FEM prediction for a force applied at location (0, 0)

Figure 4 . 1 .

 41 Figure 4.1. (a) detection of energy flow direction for 1D propagation; (b) impact detection ( Inside or Outside) for 1D propagation. When force is applied in the area I, energy loss from sensor 2 to 1 and 3 to 4 are positive. When force is applied in the area O, energy loss from sensor 2 to 1 is negative, sensor 3 to 4 is positive.

Figure 4 .Figure 4 . 2 :

 442 Figure 4.2: Integrated surface-1

( 4 . 8 ) 9 )

 489 The ingoing energy minus the outgoing energy equals the total energy difference:The right term of Equation (4.9) has two components; clarity reasons, only the first one will be investigated. It is possible because the 2 components are independent and the second one can be obtained by a similar analysis.As Figure4.3 shows, Equation (4.10) can be obtained from Equation (4.9) when inclination angle equals to 0.

Figure 4 . 3 :

 43 Figure 4.3: Integrated surface for an incident angle 0  

Figure 4 . 4 .

 44 Figure 4.4. The sensor centers are placed in parallel on a-b and c-d lines.

Figure 4 . 4 :

 44 Figure 4.4: Integrated surface with 2 piezoelectric sensors.

  half width w and the distance between two sensors l ad are small enough compared to the wavelength of bending wave, the two piezoelectric sensor voltages

  .4 which is the surface of the abcd parallelogram.

Chapter 4

 4 Force detection using energy flow direction estimator with piezoelectric it is therefore possible to obtain the derivative of the first term of the right member of Equation (4.12) from Equations (4.16) and(4

Figure 4 . 5 :

 45 Figure 4.5: Integrated surface with 4 piezoelectric sensors-four outputs

Figure 4 . 6 :

 46 Figure 4.6: Integrated surface with 4 piezoelectric sensors-two output Let us consider now the specific arrangement using eight sensors and electrically connected as shown in Figure 4.7.

Figure 4 . 7 )

 47 is that it needs four sensor outputs. If the object is only to know if the force location is inside or outside the integrated surface, it is possible to reduce computational requirements. Two vertical directions are considered at the same time. They are independent from each other because of the far field assumption. When the inner surface width l ad (Figure4.3) is small enough compared to radial distance r, it is possible to define the energy density difference estimator 2 1

Figure 4 . 7 :,

 47 Figure 4.7: Integrated surface with eight piezoelectric sensors-four outputs

Figure 4 . 8 : 2 4. 7

 4827 Figure 4.8: Integrated surface with 8 piezoelectric sensors-2

Figure 4 . 9 :

 49 Figure 4.9: Schematics of the considered structure-infinite plate

Figure 4 . 10 :Figure 4 . 11 :Figure 4 . 12 :Figure 4 . 13 :Figure 4 . 14 :Figure 4 .

 4104114124134144 Figure 4.10: Impact force applied to the considered-infinite plate

Chapter 4 Figure 4 .Figure 4 . 1

 4441 Figure 4.16: Schematics of different location inside

Figure 4 .

 4 Figure 4.19: Experimental structure

Figure 4 . 20 :Figure 4 . 21 :

 420421 Figure 4.20: Impact force to the considered structure-finite plate-1

Chapter 4 Figure 4 . 22 : 2 -

 44222 Figure 4.22: Impact force to the considered structure-finite plate-2

Figure 4 .Figure 4 .Figure 4 .

 444 Figure 4.23: q value response to the pulse force -force outside at (0.25, 0.3)-force inside at (0, 0.01)

Figure 5 .-Figure 5 . 1 :

 551 Figure 5. 1: principle of the energy flux expressed by Poynting vector in an infinite structure (a) without consideration of losses. Sum of energy flux trough the surface (S)

3 )

 3 We are now going to transform the equation to introduce the Poynting vector.

Figure 5 . 2 .Chapter 5 Figure 5 . 2 :

 52552 Figure 5.2: Schematics of the considered structure-infinite plate

Figure 5 . 3 :Figure 5 . 4 :-Figure 5 . 5 :

 535455 Figure 5.3: Impact force to the considered structure-infinite plate

Chapter 5 -Figure 5 . 6 :

 556 Figure 5. 6: Energy for various force locations outside (damping coefficient is 0.1)

Figure 5 . 7 :Figure 5 . 8 :

 5758 Figure 5. 7: Energy for various damping coefficient -force locations at the origin (0, 0).In a similar way, Figure5.8 shows that the overall magnitudes of Q curves would be increased if we increase the force peak as shown in Figure5.3. The reason is that more energy flows through the contour. The value of Q is four times greater since the energy depends on the squared force magnitude. Needless to say, Figure5.8 shows Q

1

 1

  Because  equals to 0.05~0.1, that, the first right term of Equation (5.20) is negligible compare to the second term. Equation (5.20) turns to:

Chapter 5

 5 Force detection using energy flow estimator with piezoelectric sensors -100 ----

Figure 5 . 9 :

 59 Figure 5. 9: Integrated surface with 2 piezoelectric sensors

  value, hence, the signs of x P do not depend of this term. Based on the above results, we can simplify the energy flow as:

Figure 5 . 10 : 2 Chapter 5

 51025 Figure 5.10: Integrated surface with 4 piezoelectric sensor-1

Figure 5 . 11 :

 511 Figure 5.11: Integrated surface with 4 piezoelectric sensor-2

Figure 5 . 13 :

 513 Figure 5.13: Integrated surface with 8 piezoelectric sensors-2It is therefore possible to know the impact location whether inside or outside with the sign of Q simply from two piezoelectric sensors output. When the impact is applied

Figure 5 . 14 :

 514 Figure 5.14: Schematics of the considered structure-infinite plate with surface-bonded piezoelectric element

Figure 5 .Figure 5 . 15 :-Figure 5 .

 55155 Figure 5.16 and Figure 5.19. In both of these figures, two locations of impact force in side and outside are given. The inside force location is at (0, 0). The outside force location is at (0.2, 0.2).These figures show good agreements between previous theory and approximation based on simulation and experiment data from piezoelectric sensors.

Chapter 5 -Figure 5 .

 55 Figure 5.17: approximation energy response to different peak pulse force

Figure 5 . 18 :-Figure 5 .-Figure 5 .

 51855 Figure 5.18: approximation energy for various damping coefficient -force location at the origin (0,0)

Figure 5 . 21 -

 521 Figure 5.23 show the results. These results clearly demonstrate that the impact location area can be efficiently detected from the value of the proposed criterion.

Figure 5 .-Figure 5 .Figure 5 . 23 :

 55523 Figure 5.21: energy response to a pulse force -1 (experimental, plastic tip of hammer)

Figure 5 .Figure 5 .

 55 Figure 5. 24 energy response to different pulse force

Chapter 5 -Figure 5 .

 55 Figure 5.26 experiment set-up for composite plate-1

Chapter 5 -Figure 5 .

 55 Figure 5.27 energy response to a pulse force -1(composite plate)

Figure 5 . 2 Chapter 5 -Figure 5 .

 5255 Figure 5.28 experiment set-up for composite plate-2

Figure 1 . 1 .

 11 Figure 1.1. Beam acceleration response under impulse force. ................................................ Figure 1.2 sensor network for the detection of the energy flow direction in a 1D structure. ..Figure 1.3: principle of the energy flux expressed by Poynting vector in an infinite structure (a) without consideration of losses. Sum of energy flux trough the surface (S) is null for an outside impact (O-point) and not null for an inside impact (I-point). For a finite structure (b), reflection on the boundaries induce secondary sources (s1..si) with a null associated energy flow. ................................................................................................... Figure 1.4. Piezoelectric element used as bending sensor. ...................................................... Figure 1.5. primary forms of piezoelectric material ................................................................ Figure 1.6: Massive ceramic: industrial standard materials. ................................................... Figure 1.7. piezoelectric fibers (a) and an example of integration (b) ....................................Figure 2.1: The studied structure. The PZT elements are uniformly bound on the beam with a

Figure 2 . 1 :

 21 Figure 1.1. Beam acceleration response under impulse force. ................................................ Figure 1.2 sensor network for the detection of the energy flow direction in a 1D structure. .. Figure 1.3: principle of the energy flux expressed by Poynting vector in an infinite structure (a) without consideration of losses. Sum of energy flux trough the surface (S) is null for an outside impact (O-point) and not null for an inside impact (I-point). For a finite structure (b), reflection on the boundaries induce secondary sources (s1..si) with a null associated energy flow. ................................................................................................... Figure 1.4. Piezoelectric element used as bending sensor. ...................................................... Figure 1.5. primary forms of piezoelectric material ................................................................ Figure 1.6: Massive ceramic: industrial standard materials. ................................................... Figure 1.7. piezoelectric fibers (a) and an example of integration (b) ....................................Figure 2.1: The studied structure. The PZT elements are uniformly bound on the beam with a xstep. E in is the stored energy in the beam due to the impact at the point A. Epi is the energy extracted by the i-element and dissipated in the resistor. Eb is the mechanical energy in the beam. ..................................................................................................................................

Figure 2 . 2 :

 22 Figure 2. 2: Spring-Mass representation of an infinite beam. .................................................Figure 2.3 Energy flow in the beam, differentiating the energy over sensor 1 to 2 is negative

Figure 2 . 3

 23 Figure 2. 2: Spring-Mass representation of an infinite beam. .................................................Figure 2.3 Energy flow in the beam, differentiating the energy over sensor 1 to 2 is negative ( 0 12   p E ), sensor 2 to 3 is null ( 0 23   p E ), 3 to 4 and 4 to 5 is positive

Figure 2 . 4

 24 Figure 2.4 Normalized evolutions of the extracted energy (dotted) and the proposed criterion (plain)-force location at x 0 =7.5. ................................................................................... Figure 2.5: The simulated model. Dimensions are in mm ......................................................Figure 2.6: Meshing of the structure (Zoomed area)...............................................................Figure 2.7: Example of a sensor (PZT2) response for an impact in part 0. .............................

Figure 2 . 6 :

 26 Figure 2.4 Normalized evolutions of the extracted energy (dotted) and the proposed criterion (plain)-force location at x 0 =7.5. ................................................................................... Figure 2.5: The simulated model. Dimensions are in mm ......................................................Figure 2.6: Meshing of the structure (Zoomed area)...............................................................Figure 2.7: Example of a sensor (PZT2) response for an impact in part 0. .............................

Figure 2 . 7 :

 27 Figure 2.4 Normalized evolutions of the extracted energy (dotted) and the proposed criterion (plain)-force location at x 0 =7.5. ................................................................................... Figure 2.5: The simulated model. Dimensions are in mm ......................................................Figure 2.6: Meshing of the structure (Zoomed area)...............................................................Figure 2.7: Example of a sensor (PZT2) response for an impact in part 0. .............................

Figure 2

 2 Figure 2.8: simulation results: 1-2 means

Figure 2 . 9

 29 Figure 2.9 Experimental set-up ............................................................................................Figure 2.10: Experimental (plain) and results for the proposed criterion using a-530KHz piezoelectric moveable source. ....................................................................................... Figure 3.1: Schematics of the considered infinite plate ..........................................................

Figure 2 . 10 :

 210 Figure 2.9 Experimental set-up ............................................................................................Figure 2.10: Experimental (plain) and results for the proposed criterion using a-530KHz piezoelectric moveable source. ....................................................................................... Figure 3.1: Schematics of the considered infinite plate ..........................................................

Figure 3 . 2 :

 32 Figure 3.2: Impulse response at the point (0.2, 0.2) ................................................................ Figure 3.3: Impulse response at the point (0.4, 0.4) ................................................................ Figure 3.4: Schematics of the considered structure with surface-bonded piezoelectric element

Figure 3 . 3 :

 33 Figure 3.2: Impulse response at the point (0.2, 0.2) ................................................................ Figure 3.3: Impulse response at the point (0.4, 0.4) ................................................................ Figure 3.4: Schematics of the considered structure with surface-bonded piezoelectric element

Figure 3 . 4 :

 34 Figure 3.2: Impulse response at the point (0.2, 0.2) ................................................................ Figure 3.3: Impulse response at the point (0.4, 0.4) ................................................................ Figure 3.4: Schematics of the considered structure with surface-bonded piezoelectric element

Figure 4 .

 4 1. (a) detection of energy flow direction for 1D propagation; ..................................

Figure 4 . 2 :

 42 Figure 4.2: Integrated surface-1 ..............................................................................................Figure 4.3: Integrated surface for an incident angle

Figure 4 . 3 :

 43 Figure 4.2: Integrated surface-1 ..............................................................................................Figure 4.3: Integrated surface for an incident angle 0   ...................................................Figure 4.4: Integrated surface with 2 piezoelectric sensors. ...................................................Figure 4.5: Integrated surface with 4 piezoelectric sensors-four outputs ................................Figure 4.6: Integrated surface with 4 piezoelectric sensors-two output ..................................Figure 4.7: Integrated surface with eight piezoelectric sensors-four outputs ..........................Figure 4.8: Integrated surface with 8 piezoelectric sensors-2 .................................................Figure 4.9: Schematics of the considered structure-infinite plate ...........................................Figure 4.10: Impact force applied to the considered-infinite plate .........................................Figure 4.11: q value response to an impact excitation ............................................................Figure 4.12: q value response to a pulse force excitation .......................................................Figure 4.13: q value response to a pulse force excitation .......................................................Figure 4.14: q value response to a pulse force excitation .......................................................Figure 4.15: q value response to a pulse force excitation .......................................................Figure 4.16: Schematics of different location inside ...............................................................Figure 4.17: peak of q value response to different location inside from point o to a ..............Figure 4.18: peak of q value response to different location inside from point o to b ..............Figure 4.19: Experimental structure ........................................................................................Figure 4.20: Impact force to the considered structure-finite plate-1 .......................................Figure 4.21: q value response to the pulse force outside and inside-1 ....................................Figure 4.22: Impact force to the considered structure-finite plate-2 .......................................Figure 4.23: q value response to the pulse force -force outside .............................................Figure 4.24: q value response to the pulse force -force outside .............................................Figure 4.25: q value response to the pulse force -force outside .............................................Figure 5. 1: principle of the energy flux expressed by Poynting vector in an infinite structure

Figure 4 . 4 :

 44 Integrated surface with 2 piezoelectric sensors. ...................................................

Figure 4 . 5 :

 45 Integrated surface with 4 piezoelectric sensors-four outputs ................................

Figure 4 . 6 :

 46 Integrated surface with 4 piezoelectric sensors-two output ..................................

Figure 4 . 7 :

 47 Integrated surface with eight piezoelectric sensors-four outputs ..........................

Figure 4 . 8 :

 48 Integrated surface with 8 piezoelectric sensors-2 .................................................

Figure 4 . 9 :

 49 Schematics of the considered structure-infinite plate ...........................................

Figure 4 . 10 :

 410 Impact force applied to the considered-infinite plate .........................................

Figure 4 .

 4 11: q value response to an impact excitation ............................................................

Figure 4 .

 4 12: q value response to a pulse force excitation .......................................................

Figure 4 .

 4 13: q value response to a pulse force excitation .......................................................

Figure 4 .

 4 14: q value response to a pulse force excitation .......................................................

Figure 4 .

 4 15: q value response to a pulse force excitation .......................................................

Figure 4 .

 4 16: Schematics of different location inside ...............................................................

Figure 4 .

 4 17: peak of q value response to different location inside from point o to a ..............

Figure 4 .

 4 18: peak of q value response to different location inside from point o to b ..............

Figure 4 . 19 :

 419 Experimental structure ........................................................................................

Figure 4 . 20 :

 420 Impact force to the considered structure-finite plate-1 .......................................

Figure 4 .

 4 21: q value response to the pulse force outside and inside-1 ....................................

Figure 4 . 22 :

 422 Impact force to the considered structure-finite plate-2 .......................................

Figure 4 .

 4 23: q value response to the pulse force -force outside .............................................

Figure 4 .

 4 24: q value response to the pulse force -force outside .............................................

Figure 4 .

 4 25: q value response to the pulse force -force outside .............................................

Figure 5 . 1 :

 51 principle of the energy flux expressed by Poynting vector in an infinite structure (a) without consideration of losses. Sum of energy flux trough the surface (S) is null for an outside impact (O-point) and not null for an inside impact (I-point). For a finite structure (b), reflexion phenomena induce secondary sources (s 1 ..s i ) which are the List of figure -123 ----associated null energy flux total .....................................................................................

Figure 5 . 2 :Figure 5 . 3 :Figure 5 . 6 :Figure 5 . 7 :Figure 5 . 9 :Figure 5 . 10 :Figure 5 . 11 :Figure 5 . 12 :Figure 5 . 13 :

 5253565759510511512513 Figure 5.2: Schematics of the considered structure-infinite plate ........................................... Figure 5.3: Impact force to the considered structure-infinite plate ......................................... Figure 5.4: energy response to a pulse force excitation .......................................................... Figure 5.5: energy response to a pulse force excitation .......................................................... Figure 5. 6: Energy for various force locations outside ..........................................................Figure 5. 7: Energy for various damping coefficient ..............................................................Figure 5.8: Energy for various force curve peak-force locations ............................................Figure 5. 9: Integrated surface with 2 piezoelectric sensors ................................................. Figure 5.10: Integrated surface with 4 piezoelectric sensor-1 ............................................... Figure 5.11: Integrated surface with 4 piezoelectric sensor-2 ............................................... Figure 5.12: Integrated surface with 8 piezoelectric sensors-1 ............................................. Figure 5.13: Integrated surface with 8 piezoelectric sensors-2 ............................................. Figure 5.14: Schematics of the considered structure-infinite plate with surface-bonded piezoelectric element ..................................................................................................... Figure 5.15: Impact force to the structure (with piezoelectric sensor) .................................. Figure 5.16: approximation energy response to a pulse force (simulation) .......................... Figure 5.17: approximation energy response to different peak pulse force .......................... Figure 5.18: approximation energy for various damping coefficient .................................... Figure 5.19: approximation energy response to a pulse force-1 (experimental, steel tip of hammer) ........................................................................................................................Figure5.20: energy response to a pulse force -2 (experimental, steel tip of hammer) ......... Figure5.21: energy response to a pulse force -1 (experimental, plastic tip of hammer) ...... Figure5.22: energy response to a pulse force -2 (experimental, plastic tip of hammer) ...... Figure5.23: energy response to a pulse force -3 (experimental, plastic tip of hammer) ...... Figure 5. 24 energy response to different pulse force ........................................................... Figure 5.25 energy response to different amplitude pulse force (force inside of frame) ...... Figure 5.26 experiment set-up for composite plate-1............................................................ Figure 5.27 energy response to a pulse force -1(composite plate) ........................................ Figure 5.28 experiment set-up for composite plate-2............................................................ Figure 5. 29 energy response to a pulse force -2(composite plate) .......................................

Figure 5 .

 5 Figure 5.2: Schematics of the considered structure-infinite plate ........................................... Figure 5.3: Impact force to the considered structure-infinite plate ......................................... Figure 5.4: energy response to a pulse force excitation .......................................................... Figure 5.5: energy response to a pulse force excitation .......................................................... Figure 5. 6: Energy for various force locations outside ..........................................................Figure 5. 7: Energy for various damping coefficient ..............................................................Figure 5.8: Energy for various force curve peak-force locations ............................................Figure 5. 9: Integrated surface with 2 piezoelectric sensors ................................................. Figure 5.10: Integrated surface with 4 piezoelectric sensor-1 ............................................... Figure 5.11: Integrated surface with 4 piezoelectric sensor-2 ............................................... Figure 5.12: Integrated surface with 8 piezoelectric sensors-1 ............................................. Figure 5.13: Integrated surface with 8 piezoelectric sensors-2 ............................................. Figure 5.14: Schematics of the considered structure-infinite plate with surface-bonded piezoelectric element ..................................................................................................... Figure 5.15: Impact force to the structure (with piezoelectric sensor) .................................. Figure 5.16: approximation energy response to a pulse force (simulation) .......................... Figure 5.17: approximation energy response to different peak pulse force .......................... Figure 5.18: approximation energy for various damping coefficient .................................... Figure 5.19: approximation energy response to a pulse force-1 (experimental, steel tip of hammer) ........................................................................................................................Figure5.20: energy response to a pulse force -2 (experimental, steel tip of hammer) ......... Figure5.21: energy response to a pulse force -1 (experimental, plastic tip of hammer) ...... Figure5.22: energy response to a pulse force -2 (experimental, plastic tip of hammer) ...... Figure5.23: energy response to a pulse force -3 (experimental, plastic tip of hammer) ...... Figure 5. 24 energy response to different pulse force ........................................................... Figure 5.25 energy response to different amplitude pulse force (force inside of frame) ...... Figure 5.26 experiment set-up for composite plate-1............................................................ Figure 5.27 energy response to a pulse force -1(composite plate) ........................................ Figure 5.28 experiment set-up for composite plate-2............................................................ Figure 5. 29 energy response to a pulse force -2(composite plate) .......................................

Figure 5 .

 5 Figure 5.2: Schematics of the considered structure-infinite plate ........................................... Figure 5.3: Impact force to the considered structure-infinite plate ......................................... Figure 5.4: energy response to a pulse force excitation .......................................................... Figure 5.5: energy response to a pulse force excitation .......................................................... Figure 5. 6: Energy for various force locations outside ..........................................................Figure 5. 7: Energy for various damping coefficient ..............................................................Figure 5.8: Energy for various force curve peak-force locations ............................................Figure 5. 9: Integrated surface with 2 piezoelectric sensors ................................................. Figure 5.10: Integrated surface with 4 piezoelectric sensor-1 ............................................... Figure 5.11: Integrated surface with 4 piezoelectric sensor-2 ............................................... Figure 5.12: Integrated surface with 8 piezoelectric sensors-1 ............................................. Figure 5.13: Integrated surface with 8 piezoelectric sensors-2 ............................................. Figure 5.14: Schematics of the considered structure-infinite plate with surface-bonded piezoelectric element ..................................................................................................... Figure 5.15: Impact force to the structure (with piezoelectric sensor) .................................. Figure 5.16: approximation energy response to a pulse force (simulation) .......................... Figure 5.17: approximation energy response to different peak pulse force .......................... Figure 5.18: approximation energy for various damping coefficient .................................... Figure 5.19: approximation energy response to a pulse force-1 (experimental, steel tip of hammer) ........................................................................................................................Figure5.20: energy response to a pulse force -2 (experimental, steel tip of hammer) ......... Figure5.21: energy response to a pulse force -1 (experimental, plastic tip of hammer) ...... Figure5.22: energy response to a pulse force -2 (experimental, plastic tip of hammer) ...... Figure5.23: energy response to a pulse force -3 (experimental, plastic tip of hammer) ...... Figure 5. 24 energy response to different pulse force ........................................................... Figure 5.25 energy response to different amplitude pulse force (force inside of frame) ...... Figure 5.26 experiment set-up for composite plate-1............................................................ Figure 5.27 energy response to a pulse force -1(composite plate) ........................................ Figure 5.28 experiment set-up for composite plate-2............................................................ Figure 5. 29 energy response to a pulse force -2(composite plate) .......................................

Figure 5 .

 5 Figure 5.2: Schematics of the considered structure-infinite plate ........................................... Figure 5.3: Impact force to the considered structure-infinite plate ......................................... Figure 5.4: energy response to a pulse force excitation .......................................................... Figure 5.5: energy response to a pulse force excitation .......................................................... Figure 5. 6: Energy for various force locations outside ..........................................................Figure 5. 7: Energy for various damping coefficient ..............................................................Figure 5.8: Energy for various force curve peak-force locations ............................................Figure 5. 9: Integrated surface with 2 piezoelectric sensors ................................................. Figure 5.10: Integrated surface with 4 piezoelectric sensor-1 ............................................... Figure 5.11: Integrated surface with 4 piezoelectric sensor-2 ............................................... Figure 5.12: Integrated surface with 8 piezoelectric sensors-1 ............................................. Figure 5.13: Integrated surface with 8 piezoelectric sensors-2 ............................................. Figure 5.14: Schematics of the considered structure-infinite plate with surface-bonded piezoelectric element ..................................................................................................... Figure 5.15: Impact force to the structure (with piezoelectric sensor) .................................. Figure 5.16: approximation energy response to a pulse force (simulation) .......................... Figure 5.17: approximation energy response to different peak pulse force .......................... Figure 5.18: approximation energy for various damping coefficient .................................... Figure 5.19: approximation energy response to a pulse force-1 (experimental, steel tip of hammer) ........................................................................................................................Figure5.20: energy response to a pulse force -2 (experimental, steel tip of hammer) ......... Figure5.21: energy response to a pulse force -1 (experimental, plastic tip of hammer) ...... Figure5.22: energy response to a pulse force -2 (experimental, plastic tip of hammer) ...... Figure5.23: energy response to a pulse force -3 (experimental, plastic tip of hammer) ...... Figure 5. 24 energy response to different pulse force ........................................................... Figure 5.25 energy response to different amplitude pulse force (force inside of frame) ...... Figure 5.26 experiment set-up for composite plate-1............................................................ Figure 5.27 energy response to a pulse force -1(composite plate) ........................................ Figure 5.28 experiment set-up for composite plate-2............................................................ Figure 5. 29 energy response to a pulse force -2(composite plate) .......................................

Figure 5 .

 5 Figure 5.2: Schematics of the considered structure-infinite plate ........................................... Figure 5.3: Impact force to the considered structure-infinite plate ......................................... Figure 5.4: energy response to a pulse force excitation .......................................................... Figure 5.5: energy response to a pulse force excitation .......................................................... Figure 5. 6: Energy for various force locations outside ..........................................................Figure 5. 7: Energy for various damping coefficient ..............................................................Figure 5.8: Energy for various force curve peak-force locations ............................................Figure 5. 9: Integrated surface with 2 piezoelectric sensors ................................................. Figure 5.10: Integrated surface with 4 piezoelectric sensor-1 ............................................... Figure 5.11: Integrated surface with 4 piezoelectric sensor-2 ............................................... Figure 5.12: Integrated surface with 8 piezoelectric sensors-1 ............................................. Figure 5.13: Integrated surface with 8 piezoelectric sensors-2 ............................................. Figure 5.14: Schematics of the considered structure-infinite plate with surface-bonded piezoelectric element ..................................................................................................... Figure 5.15: Impact force to the structure (with piezoelectric sensor) .................................. Figure 5.16: approximation energy response to a pulse force (simulation) .......................... Figure 5.17: approximation energy response to different peak pulse force .......................... Figure 5.18: approximation energy for various damping coefficient .................................... Figure 5.19: approximation energy response to a pulse force-1 (experimental, steel tip of hammer) ........................................................................................................................Figure5.20: energy response to a pulse force -2 (experimental, steel tip of hammer) ......... Figure5.21: energy response to a pulse force -1 (experimental, plastic tip of hammer) ...... Figure5.22: energy response to a pulse force -2 (experimental, plastic tip of hammer) ...... Figure5.23: energy response to a pulse force -3 (experimental, plastic tip of hammer) ...... Figure 5. 24 energy response to different pulse force ........................................................... Figure 5.25 energy response to different amplitude pulse force (force inside of frame) ...... Figure 5.26 experiment set-up for composite plate-1............................................................ Figure 5.27 energy response to a pulse force -1(composite plate) ........................................ Figure 5.28 experiment set-up for composite plate-2............................................................ Figure 5. 29 energy response to a pulse force -2(composite plate) .......................................

Figure 5 .

 5 Figure 5.2: Schematics of the considered structure-infinite plate ........................................... Figure 5.3: Impact force to the considered structure-infinite plate ......................................... Figure 5.4: energy response to a pulse force excitation .......................................................... Figure 5.5: energy response to a pulse force excitation .......................................................... Figure 5. 6: Energy for various force locations outside ..........................................................Figure 5. 7: Energy for various damping coefficient ..............................................................Figure 5.8: Energy for various force curve peak-force locations ............................................Figure 5. 9: Integrated surface with 2 piezoelectric sensors ................................................. Figure 5.10: Integrated surface with 4 piezoelectric sensor-1 ............................................... Figure 5.11: Integrated surface with 4 piezoelectric sensor-2 ............................................... Figure 5.12: Integrated surface with 8 piezoelectric sensors-1 ............................................. Figure 5.13: Integrated surface with 8 piezoelectric sensors-2 ............................................. Figure 5.14: Schematics of the considered structure-infinite plate with surface-bonded piezoelectric element ..................................................................................................... Figure 5.15: Impact force to the structure (with piezoelectric sensor) .................................. Figure 5.16: approximation energy response to a pulse force (simulation) .......................... Figure 5.17: approximation energy response to different peak pulse force .......................... Figure 5.18: approximation energy for various damping coefficient .................................... Figure 5.19: approximation energy response to a pulse force-1 (experimental, steel tip of hammer) ........................................................................................................................Figure5.20: energy response to a pulse force -2 (experimental, steel tip of hammer) ......... Figure5.21: energy response to a pulse force -1 (experimental, plastic tip of hammer) ...... Figure5.22: energy response to a pulse force -2 (experimental, plastic tip of hammer) ...... Figure5.23: energy response to a pulse force -3 (experimental, plastic tip of hammer) ...... Figure 5. 24 energy response to different pulse force ........................................................... Figure 5.25 energy response to different amplitude pulse force (force inside of frame) ...... Figure 5.26 experiment set-up for composite plate-1............................................................ Figure 5.27 energy response to a pulse force -1(composite plate) ........................................ Figure 5.28 experiment set-up for composite plate-2............................................................ Figure 5. 29 energy response to a pulse force -2(composite plate) .......................................

Figure 5 .

 5 Figure 5.2: Schematics of the considered structure-infinite plate ........................................... Figure 5.3: Impact force to the considered structure-infinite plate ......................................... Figure 5.4: energy response to a pulse force excitation .......................................................... Figure 5.5: energy response to a pulse force excitation .......................................................... Figure 5. 6: Energy for various force locations outside ..........................................................Figure 5. 7: Energy for various damping coefficient ..............................................................Figure 5.8: Energy for various force curve peak-force locations ............................................Figure 5. 9: Integrated surface with 2 piezoelectric sensors ................................................. Figure 5.10: Integrated surface with 4 piezoelectric sensor-1 ............................................... Figure 5.11: Integrated surface with 4 piezoelectric sensor-2 ............................................... Figure 5.12: Integrated surface with 8 piezoelectric sensors-1 ............................................. Figure 5.13: Integrated surface with 8 piezoelectric sensors-2 ............................................. Figure 5.14: Schematics of the considered structure-infinite plate with surface-bonded piezoelectric element ..................................................................................................... Figure 5.15: Impact force to the structure (with piezoelectric sensor) .................................. Figure 5.16: approximation energy response to a pulse force (simulation) .......................... Figure 5.17: approximation energy response to different peak pulse force .......................... Figure 5.18: approximation energy for various damping coefficient .................................... Figure 5.19: approximation energy response to a pulse force-1 (experimental, steel tip of hammer) ........................................................................................................................Figure5.20: energy response to a pulse force -2 (experimental, steel tip of hammer) ......... Figure5.21: energy response to a pulse force -1 (experimental, plastic tip of hammer) ...... Figure5.22: energy response to a pulse force -2 (experimental, plastic tip of hammer) ...... Figure5.23: energy response to a pulse force -3 (experimental, plastic tip of hammer) ...... Figure 5. 24 energy response to different pulse force ........................................................... Figure 5.25 energy response to different amplitude pulse force (force inside of frame) ...... Figure 5.26 experiment set-up for composite plate-1............................................................ Figure 5.27 energy response to a pulse force -1(composite plate) ........................................ Figure 5.28 experiment set-up for composite plate-2............................................................ Figure 5. 29 energy response to a pulse force -2(composite plate) .......................................

Figure 5 .

 5 Figure 5.2: Schematics of the considered structure-infinite plate ........................................... Figure 5.3: Impact force to the considered structure-infinite plate ......................................... Figure 5.4: energy response to a pulse force excitation .......................................................... Figure 5.5: energy response to a pulse force excitation .......................................................... Figure 5. 6: Energy for various force locations outside ..........................................................Figure 5. 7: Energy for various damping coefficient ..............................................................Figure 5.8: Energy for various force curve peak-force locations ............................................Figure 5. 9: Integrated surface with 2 piezoelectric sensors ................................................. Figure 5.10: Integrated surface with 4 piezoelectric sensor-1 ............................................... Figure 5.11: Integrated surface with 4 piezoelectric sensor-2 ............................................... Figure 5.12: Integrated surface with 8 piezoelectric sensors-1 ............................................. Figure 5.13: Integrated surface with 8 piezoelectric sensors-2 ............................................. Figure 5.14: Schematics of the considered structure-infinite plate with surface-bonded piezoelectric element ..................................................................................................... Figure 5.15: Impact force to the structure (with piezoelectric sensor) .................................. Figure 5.16: approximation energy response to a pulse force (simulation) .......................... Figure 5.17: approximation energy response to different peak pulse force .......................... Figure 5.18: approximation energy for various damping coefficient .................................... Figure 5.19: approximation energy response to a pulse force-1 (experimental, steel tip of hammer) ........................................................................................................................Figure5.20: energy response to a pulse force -2 (experimental, steel tip of hammer) ......... Figure5.21: energy response to a pulse force -1 (experimental, plastic tip of hammer) ...... Figure5.22: energy response to a pulse force -2 (experimental, plastic tip of hammer) ...... Figure5.23: energy response to a pulse force -3 (experimental, plastic tip of hammer) ...... Figure 5. 24 energy response to different pulse force ........................................................... Figure 5.25 energy response to different amplitude pulse force (force inside of frame) ...... Figure 5.26 experiment set-up for composite plate-1............................................................ Figure 5.27 energy response to a pulse force -1(composite plate) ........................................ Figure 5.28 experiment set-up for composite plate-2............................................................ Figure 5. 29 energy response to a pulse force -2(composite plate) .......................................

Figure 5 .

 5 Figure 5.2: Schematics of the considered structure-infinite plate ........................................... Figure 5.3: Impact force to the considered structure-infinite plate ......................................... Figure 5.4: energy response to a pulse force excitation .......................................................... Figure 5.5: energy response to a pulse force excitation .......................................................... Figure 5. 6: Energy for various force locations outside ..........................................................Figure 5. 7: Energy for various damping coefficient ..............................................................Figure 5.8: Energy for various force curve peak-force locations ............................................Figure 5. 9: Integrated surface with 2 piezoelectric sensors ................................................. Figure 5.10: Integrated surface with 4 piezoelectric sensor-1 ............................................... Figure 5.11: Integrated surface with 4 piezoelectric sensor-2 ............................................... Figure 5.12: Integrated surface with 8 piezoelectric sensors-1 ............................................. Figure 5.13: Integrated surface with 8 piezoelectric sensors-2 ............................................. Figure 5.14: Schematics of the considered structure-infinite plate with surface-bonded piezoelectric element ..................................................................................................... Figure 5.15: Impact force to the structure (with piezoelectric sensor) .................................. Figure 5.16: approximation energy response to a pulse force (simulation) .......................... Figure 5.17: approximation energy response to different peak pulse force .......................... Figure 5.18: approximation energy for various damping coefficient .................................... Figure 5.19: approximation energy response to a pulse force-1 (experimental, steel tip of hammer) ........................................................................................................................Figure5.20: energy response to a pulse force -2 (experimental, steel tip of hammer) ......... Figure5.21: energy response to a pulse force -1 (experimental, plastic tip of hammer) ...... Figure5.22: energy response to a pulse force -2 (experimental, plastic tip of hammer) ...... Figure5.23: energy response to a pulse force -3 (experimental, plastic tip of hammer) ...... Figure 5. 24 energy response to different pulse force ........................................................... Figure 5.25 energy response to different amplitude pulse force (force inside of frame) ...... Figure 5.26 experiment set-up for composite plate-1............................................................ Figure 5.27 energy response to a pulse force -1(composite plate) ........................................ Figure 5.28 experiment set-up for composite plate-2............................................................ Figure 5. 29 energy response to a pulse force -2(composite plate) .......................................

  Guyomar, Professor and Pro. Lionel Petit. I owe profound debt of gratitude to Pro.Daniel Guyomar who gives me a chance to study this thesis, for his sense of responsibility, enlightening guidance, and incredible patience during the whole course of my studying. I would like to thank to Pro. Lionel Petit for his kindness support during this work.

I would like to thank to Kaori Yuse who give me many advises on writing science paper.

I would also like to express my heartfelt gratefulness to my teammate Lallart

Mickael for his kindness support during this work.

The experimental work could not have progressed as quickly without the support of Defromerie Frederic.

I would like to thank to Evelyne Dorieux, secretary of Laboratory, for her supporting.

  [START_REF] Jordan | Remote impact analysis by use of propagated acceleration signals II: Comparison between theory and experiment[END_REF] , Bateman et al29 Kim and Lyon 30 , Lin and Bapat 31 , McCarthy and Lyon 32

Table 1

 1

	.1. Only

Table 2 .

 2

			1: Nomenclature of the lumped model
	M	lineic mass
	K	stiffness
	u	x	longitudinal displacement
	V	piezovoltage
			lineic mass where a piezoelement is
	M
		p	bonded
			stiffness where a piezoelement is
	K
		p	bonded
	Considered that the piezoelectric inserts does not significantly modify the local
	stiffness and mass of the structure, one can assume that	M and K	p are equals to	M
				p
	and

Table 2 .

 2

		2: FEM simulation parameters
	Parameter	Value (SI)	Parameter	Value (SI)
	Resistivity		1	Density	7850
	Length of the piezoelements	2 2  10 .	Length of the beam	74 . 4
	Thickness of the piezoelements	4 3  10 .	Thickness of the beam	2 3  10 .
	Relative permittivity	1142	Elasticity	10 . 195	9
	Voltage coefficient		18 . 6

Table 3 .

 3

	1: Simulation parameters of the plate
	Parameter		Value
	Surface density m	6.4	Kg	2 .  m
	Young modulus Y	195 GPa
	Thickness h	0.8 mm
	Poisson ration 		0.33
	Damping coefficient 		0.08

-53 ----

Table 3 .

 3 2: Simulation parameters of the piezoelectric element

	Parameter	Value
	Length L	180 mm
	Width W	20 mm
	Thickness l	1 mm
	Voltage coefficient 31

Table 3 .

 3

	3 Dimensions (in mm) of the FEM model
	plate (Length X Width X Thickness H)	2500 X 2000 X 1
	piezoelement square size (side L X Width W X	40 X 40 X 1 X 0.5
	Thickness l)

[START_REF] Ventsel | Thin Plates and Shells: Theory, Analysis and Applications[END_REF]

, to respect thin plate conditions (Kirchhoff 's plate theory). In this way, the vertical shear strains  xy and  yz are negligible. The dimensions of the plate FEM model are summarized in table

Table 3.3

Table 4 .

 4

						1: Matlab simulation parameters
		Parameter			Value		Parameter	Value
	Plate	surface	6.4 / Kg m	2	Outside sensor surface length	m 14 . 0
	density m					L2
	Plate	elastic	11 1.95 10 Pa	Outside sensor surface width	m 003 . 0
	modules	E					W2
	Plate	thickness	4 8.10 m 		Inside sensor surface length	m 132 . 0
	H						L1
	Poisson ratio 			0.33		Inside sensor surface width	m 003 . 0
								W1
	Damping ratio		n	0.1		Relative permittivity	33 /  s	0	1142
	Voltage coefficient	1 Cm  4.3 . 
	e	31
	Sensor thickness

Table 5 .

 5

			1: Matlab simulation parameters
	Parameter	Value		Parameter	Value
	Plate	surface	6.4 / Kg m	2	Integral surface	m 18 . 0
	density m			length L
	Plate	elastic	11 1.95 10 Pa	Integral surface	m 18 . 0
	modules E			width W
	Plate	thickness	4 8.10 m
	H
			0.33
	Damping ratio n 	0.1



Poisson ratio 

Table 5 .

 5 2: Matlab simulation parameters of the piezoelectric element

		Parameter			Value		Parameter	Value
	Plate	surface	6.4 / Kg m	2	Integral surface length L2	m 14 . 0
	density m
	Plate	elastic	11 1.95 10 Pa	Integral surface width W2	m 003 . 0
	modules	E
	Plate thickness	h	4 8.10 m 		Integral surface length L1	m 138 . 0
	Poisson ratio 			0.33		Integral surface width W1	m 003 . 0
	Damping ratio		n	0.1		Relative permittivity	33 /  s	0	1142
	Voltage coefficient	1 Cm  4.3 . 
	e	31

Table 2 .

 2 1: Nomenclature of the lumped model ....................................................................... Table 2.2: FEM simulation parameters ................................................................................... Table 2.3: Experimental parameters ........................................................................................ Table 3.1: Simulation parameters of the plate .........................................................................

Table 3 .

 3 2: Simulation parameters of the piezoelectric element .............................................. Table 3.3 Dimensions (in mm) of the FEM model ............................................................... Table 4.1: Matlab simulation parameters ................................................................................ Table 5. 1: Matlab simulation parameters ............................................................................... Table 5.2: Matlab simulation parameters of the piezoelectric element .................................

-60 ----

-79 ----

-95 ----

Doyle JF, Experimentally determining the contact force during the transverse impact of an orthotropic plate, Journal ofSound and Vibration, 118,441-448, 1987

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Acknowledgements

Acknowledgements