Simon Benacchio

Adrien Mamou-Mani

Baptiste Chomette

Victor Finel

Active control and sound synthesistwo different ways to investigate the influence of the modal parameters of a guitar on its sound

The vibrational behavior of musical instruments is usually studied using physical modeling and simulations. Recently, active control has proven its efficiency to experimentally modify the dynamical behavior of musical instruments. This approach could also be used as an experimental tool to systematically study fine physical phenomena. This paper proposes to use modal active control as an alternative to sound simulation in order to study the complex case of the coupling between classical guitar strings and soundboard. A comparison between modal active control and sound simulation investigates the advantages, the drawbacks and the limits of these two approaches.

I. INTRODUCTION

Musical instruments are finely tuned mechanical structures involving complex phenomena. The relation between their vibrational properties and sound attributes remains a major question for musical acousticians. Jansson [START_REF] Jansson | Admittance Measurements of 25 High Quality Violins[END_REF] compares the bridge admittance of 25 violins.

His aim is to find overall characteristics giving indications of the quality of musical instruments. Results from this kind of measurement campaign are difficult to formalize, as using several instruments involves many uncontrolled physical parameters. Other approaches allow researchers to get around this problem. For instance, Gough [START_REF] Gough | The Resonant Response of a Violin G-string and the Excitation of the Wolf-Note[END_REF][START_REF] Gough | The Theory of String Resonances on Musical Instruments[END_REF] compares theoretical predictions based on an analytical model and the measurements of the mechanical admittance of two G-strings mounted on a violin. His aim is to study the coupling between the string and the body of a string instrument. Results about the wolf-note phenomenon and about the two independent modes of transverse vibration of the string are given thanks to the modification of different parameters of both experimental setup and modeling. However, the modal parameters of the soundboard cannot be modified while they have a decisive influence in the coupling between the string and the body. In [4,[START_REF] Woodhouse | Plucked Guitar Transients: Comparison of Measurements and Synthesis[END_REF], Woodhouse proposes several sound synthesis methods and validates them comparing with real guitar sounds. One of these methods is then used to find thresholds of perception thanks to modifications of the main parameters of the model [START_REF] Woodhouse | Perceptual Thresholds for Acoustical Guitar Models[END_REF]. Virtual changes of the modal parameters of the guitar body and of the mechanical parameters of the string give the so-called just-noticeable differences. Fritz et al. [START_REF] Fritz | Perceptual Threshols for Detecting Modifications Applied to the Acoustical Properties of a Violin[END_REF] use the same approach to explore the relationships between acoustical characteristics of violins and their perceived qualities. Wright [START_REF] Wright | The Acoustics and Psychoacoustics of the Guitar[END_REF] uses also a sound synthesis method and psychoacoustics tests in order to determine the relative importance of the different acoustical parameters of the guitar body in the tone quality of the notes played with an instrument.

He also says that, thanks to computer synthesis, "the construction parameters can be varied individually in a manner that would be impossible on real instruments". This assertion reveals that active control of vibration was not well-known in the field of musical acoustics in 1996.

Few applications of active control to musical instruments are actually published at this time. Griffin [START_REF] Griffin | Acoustic Replication in Smart Structurs Using Active Structural/Acoustic Control[END_REF] uses active control methods to change the sound of a guitar modifying certain modal properties. He noticeably controls the frequency and the damping of the first three modes of the soundboard, but he does not give their effects on the sound. Later, he also uses active control to cancel the feedback phenomenon when amplified acoustic guitars are played [START_REF] Griffin | The Application of Smart Structures Toward Feedback Suppression in Amplified Acoustic Guitars[END_REF]. At the same time, Besnainou [START_REF] Besnainou | Modal Stimulation: A Sound Synthesis New Approach[END_REF] modifies the sound of a xylophone bar controlling the phase and the amplitude of its displacement. More recently, Boutin improves this control using a sum of second order band-pass filters in order to modify the amplitudes and the frequencies of the first resonances of a xylophone bar [START_REF] Boutin | Modifying the resonances of a xylophone bar using active control[END_REF]. Using simulations, he also investigates the possible effects of the Proportional-Integral-Derivative control method on a model of a violin "bridge hill" [START_REF] Boutin | Physical Parameters of the Violin Bridge Changed by Active Control[END_REF]. In [START_REF] Berdahl | Active damping of a vibrating string[END_REF], Berdahl studies the effect of an active damping applied on the modes of the string of an electric guitar. He succeeds in controlling independently the two transverse modes of the string. These studies reveal that the in situ modifications of the mechanical parameters of musical instruments are possible. To the knowledge of the authors no comparison have been made between the active control method, and the well-known sound synthesis method. This paper deals with the comparison between these two methods in the study of the coupling between a classical guitar strings and soundboard. First, the frequency domain sound synthesis approach is presented, as well as the modal active control method. The sounding effects of the modal frequency and damping modifications from both methods are compared and discussed. Finally, a short discussion gives the advantages, the drawbacks and the limitations of these two different methods.

II. MOTIVATIONS AND APPROACHES DESCRIPTION A. The Guitar

The classical guitar chosen for this study is presented in Figure 1. This musical instrument is made up of a lot of parts whose the basic are the strings, the bridge and the body made up of the soundboard, the back, the sides and the air cavity. The small radiating surface of the strings does not allow them to radiate sound directly. The air cavity and the large surface of the soundboard allow for an efficient radiated sound. The design of the soundboard is then crucial to obtain certain final sound attributes. Selection of soundboard material, shape and assembly with ribs are examples of stages mastered by the guitar maker to target these final sound attributes. This maker know-how implies certain vibrational properties, well characterized in the low frequency region (approximately under 1500 Hz) by its modes of vibration. The fact that the modal parameters of guitars are used for example to discriminate them in different classes [START_REF] Elie | Macro Parameters Describing the Mechanical Behavior of Classical Guitar[END_REF] motivates the choice of modal approach to compare simulation and active control.

B. Sound Synthesis

The first approach presented in this paper is the sound synthesis method called the frequency domain synthesis. It is based on the model of the string/body coupling in the frequency domain. Then, the inverse Fast Fourier Transform (FFT) is used to find the timevarying transient response of the model. This method proposed by Woodhouse is quickly described here and fully detailed in [4,[START_REF] Woodhouse | Plucked Guitar Transients: Comparison of Measurements and Synthesis[END_REF] where all the following equations can be found. This method uses the fact that, when two systems are coupled together at a single point, the admittance Y of the coupled system defined at this point is given by

1 Y = 1 Y string + 1 Y body , (1)
where Y string and Y body are the input admittances of the string and the body respectively and defined at the coupling point chosen at the bridge in this case. Using the inverse FFT, this equation gives the time-varying velocity response of the body at the coupling point to an impulse applied to the string. This transfer function is the same as the acceleration response of the body to a step function force applied to the string. The step function force is a good model of the force applied by a musician on the string when playing. Then a drastic approximation is made to give the sound pressure in the far field of the guitar. According to [START_REF] Chaigne | Acoustique des Instruments de Musique (Acoustics of Musical Intsruments)[END_REF], the sound pressure radiated by a spherical body of radius a moving in the symmetric breathing mode with a uniform velocity V is given by

p(r) = V ρ air c a r jka 1 + jka e -jk(r-a) , (2)
with k = ω/c. c is the speed of the sound and ρ air the density of the air. r is the distance between the center of the spherical body and the point of measurement of the sound pressure. Considering that the acceleration of the sphere is equal to jωV , the sound pressure is calculated filtering the computed bridge acceleration by

R(r) = 1 jω ρ air c a r jka 1 + jka e -jk(r-a) . (3)
This basic model is used to compute the radiated sound of the synthesized guitar regarding its body as a sphere of radius a = 20 cm and with r = 50 cm.

The string admittance at the bridge is calculated using

1 Y string = - iT L 1 ω + ∞ n=1 2ω -iω n η n ω 2 -iωω n η n -ω 2 n , (4)
with L and T the string length and tension respectively. ω n and ν n are the angular frequency and the loss factor of the n th string mode respectively defined by

ω n = iπc s L 1 + B 2T nπ L 2 (5)
and

η n = T (η F + η A /ω n ) + Bη B (nπ/L) 2 T + B(nπ/L) 2 , (6)
where c s = T /ρ s is the speed of the wave in the string, B the bending stiffness and η F , η A , η B the coefficients determining friction, air and bending damping respectively. The values of these parameters used for the sound synthesis are extracted from [START_REF] Woodhouse | Plucked Guitar Transients: Comparison of Measurements and Synthesis[END_REF] and given in Tables 3 and4 in Appendix A. However, (4) gives the impedance of the string at the bridge. As the point where the string is plucked is very important for the sound production, the transfer function between the string displacement at the plucking point y x and the string displacement at the bridge y L must be added to the model. This transfer function is given by

y x y L = x L + c L ∞ n=1 (-1) n 2ωsin(nπx/L) ω 2 -iωω n η n -ω 2 n , (7)
with x the distance between the guitar head and the plucking point. The admittance of the guitar taking into account this point is given by

Y pluck = y x y L Y. (8)
The body admittance Y body is given by

Y body = N k=1 jω m k (ω 2 k -ω 2 + 2jωω k ξ k ) (9)
with N the number of modeled modes. ω k , ξ k and m k are the modal frequency, damping factor and effective mass of the k th mode of the body respectively. It is important to note that the effective mass includes the modal mass of the corresponding mode but also the characteristics of the position of the admittance on the guitar. This input admittance is measured at the bridge using an impulse hammer and an accelerometer. Then, modal parameters are identified from this measurement using the Rational Fraction Polynomial (RFP) algorithm [START_REF] Richardson | Parameter Estimation from Frequency Response Measurements using Rational Fraction Polynomials[END_REF].

The total transfer function Y total between the plucking point and the sound pressure is given using the relation (1) between the string admittance (4) and the body admittance [START_REF] Griffin | Acoustic Replication in Smart Structurs Using Active Structural/Acoustic Control[END_REF], the transfer function between the plucking point and the bridge [START_REF] Fritz | Perceptual Threshols for Detecting Modifications Applied to the Acoustical Properties of a Violin[END_REF] and the transfer function between the bridge acceleration and the sound pressure [START_REF] Gough | The Theory of String Resonances on Musical Instruments[END_REF]. The final equation is given by

Y total = y x y L Y R(r). (10)
It should be noticed that all parts of the model are time-invariant. Finally, the time-varying transient response of the synthesized guitar is given using the inverse fast Fourier transform (FFT) of the frequency response function given by [START_REF] Griffin | The Application of Smart Structures Toward Feedback Suppression in Amplified Acoustic Guitars[END_REF]. To sum up, this equation is the transfer function between the string displacement at the plucked position and the radiated sound pressure of the guitar built using the string and the body admittances. As the latters are dependent on frequency, the inverse FFT of ( 10) is dependent on time and gives the sound of the synthesized guitar. Results of this sound synthesis method are given in section III.

C. Active Control

The second approach presented in this paper is the active control of the guitar soundboard.

The technique chosen to control its modal parameters is called the modal active control method [START_REF] Preumont | Vibration Control of Active Structures, An Introduction[END_REF][START_REF] Gawronski | Dynamics and Control of Structures: A Modal Approach[END_REF] and is based on the description of the structure in the modal state space. This method has already proven its efficiency controlling the modal parameters of several musical instruments in different frequency ranges [START_REF] Benacchio | Mode Tuning of a Simplified String Instrument Using Time-Dimensionless State-Derivative Control[END_REF][START_REF] Meurisse | Experimental demonstration of the Modification of the Resonances of a Simplified Self-Sustained Wind Instrument Through Modal Active Control[END_REF]. Its main advantage is to target the energy of the control on the modes of interest using a minimal number of transducers. To ensure the control of all the modal parameters of the soundboard, a proportional and derivative modal state control is used. This technique is quickly described here and fully detailed in [START_REF] Benacchio | Mode Tuning of a Simplified String Instrument Using Time-Dimensionless State-Derivative Control[END_REF]. Figure 2 presents the general principle of the modal active control. The vibration of the guitar is measured using a sensor bound on the soundboard. This measured signal y(t)

is used in the control system to compute the command signal u(t). This signal is added to the disturbance signal w(t) using an actuator and is chosen to modify the parameters of the controlled structure. In this case, the disturbance signal w(t) is created by the string. In the state space, the dynamic of a linear system disturbed and controlled with a proportional and derivative modal state control method can be written

                   ẋ(t) = Ax(t) + Bu(t) + Gw(t) y(t) = Cx(t) u(t) = -K 1 x(t) -K 2 ẋ(t) , (11)
where x t (t) = [q t (t) qt (t)] is the state vector with q(t) the vector of the modal displacement.

A, B, C and G are the matrices of the structure, the input, the output and the disturbance signal respectively. For N modes, one actuator and one sensor, the state matrices can be written

A =     0 N,N Id N,N -Ω 2 -2ΞΩ     , (12)
B =     0 N,1 Π a     , G =     0 N,1 Π w     , C =     Π c 0 N,1     t , (13)
with Ω = diag(ω k ) and Ξ = diag(ξ k ). The parameters ω k and ξ k are the same as in [START_REF] Griffin | Acoustic Replication in Smart Structurs Using Active Structural/Acoustic Control[END_REF]. Π a , Π c and Π w are the modal vector of the actuator, the sensor and the disturbance respectively.

The effective masses presented in [START_REF] Griffin | Acoustic Replication in Smart Structurs Using Active Structural/Acoustic Control[END_REF] are included in the modal vector of the actuator. These parameters are identified from the measured transfer function between the sensor and the actuator using a RFP algorithm.

According to the first equation of [START_REF] Besnainou | Modal Stimulation: A Sound Synthesis New Approach[END_REF], the dynamic of the disturbed structure can be modified using the command signal u(t). Thanks to the proportional and derivative modal state control, the modal frequencies, damping factors and effective masses of the structure can be modified. The gain vectors of the controller K 1 and K 2 are chosen using pole placement algorithms adapted for the state [START_REF] Kautsky | Robust Pole Assignement in Linear State Feedback[END_REF] and the state-derivative [START_REF] Abdelaziz | Robust Pole Assignment for Linear Time-invariant Systems using State-derivative Feedback[END_REF] control.

According to the control system described in Figure 2, a Luenberger observer is used [START_REF] Luenberger | Observing the State of a Linear System[END_REF]. Indeed, the third equation of [START_REF] Besnainou | Modal Stimulation: A Sound Synthesis New Approach[END_REF] shows that the command signal is built with the state and the derivative state of the system. As it is not possible to directly measure these two vectors, they are estimated with an observer using the model of the structure. To make sure that the control system is stable, the real part of the poles of the observer are chosen from twice to six times lower than the corresponding poles of the controlled structure. As the observer works with a fixed model of the structure the quality of the estimates and the accuracy of the control depend on the changes in the nominal system. The model of the structure can be quickly identified before each measurement to improve the efficiency of the control system. Although the observer brings robustness issues, it estimates each modal displacement, velocity and acceleration during control. The control system then targets the energy on modes of interest modifying their modal parameters simultaneously and separately.

The dynamic and the design step of the observer are given in [START_REF] Luenberger | Observing the State of a Linear System[END_REF].

III. COMPARISON BETWEEN SOUND SYNTHESIS AND ACTIVE CONTROL

In order to compare these two approaches the modal frequency and the damping factor of the second mode of the soundboard of the guitar are modified. The frequency of this mode is 213 Hz. The effects of the control in the sound are studied investigating the coupling between this mode and the second partial of the A string of the guitar since its frequency is 220 Hz and roughly matches the frequency of the controlled mode. This mode is chosen for its important role in the sound of the instrument. Its large amplitude is significant for the mastering of the amplitude and the decay of several fundamental and second partials of the guitar. The study focuses on this frequency range and on the coupling phenomenon to achieve an accurate and detailed comparison of the control and synthesis approaches. The proposed illustration is a specific case of coupling. Other modifications of the modal parameters of the soundboard should be performed in other frequency ranges to generalize the results regarding the coupling phenomenon between a string and the modes of a soundboard. In this paper, several modifications of the damping factor and of the frequency are studied and compared when applied using the synthesis method or the active control.

• In the case of sound synthesis, the modal parameters are directly modified in the impedance of the soundboard changing the parameters of [START_REF] Griffin | Acoustic Replication in Smart Structurs Using Active Structural/Acoustic Control[END_REF]. The initial values of these modal parameters are the same that those of the control system described in the next paragraph. Then the synthesized radiated sound of the guitar is calculated using the method presented in section II.

• In the case of active control, the control system is designed to match the targeted modifications using a pole placement algorithm. To modify the sound of the guitar, the real part of the poles can be placed close to zero (for example to decrease the damping factor of a mode). Thus, the stability margins of the control system are sometimes decreased but always kept large enough to maintain its stability. The modal model of the soundboard is built using 23 modes identified using the measured impedance.

The large number of modes used in the model avoids strong spillover effects in the studied range of frequency. The identified parameters of the first ten modes and the corresponding values of B and C are given in Table 5 in Appendix A. The bridge impedance is measured using an integrated signal from an accelerometer and an impact hammer. The radiated sound is measured 50 cm in front of the soundboard using a microphone. The string excitation is produced using thin brass wire which is attached to the string 15 cm from the bridge. Pulling this wire until it breaks gives a repeatable plucked string excitation. The experimental setup is presented in Figure 1.

A. Modification of the Damping Factor

The first comparison between the two proposed approaches is done on the damping factor of the second mode of the soundboard. Seven cases are studied. The damping factor of the targeted mode is altered by +400%, +200%, +100%, 0%, -10%, -50% and -80%. The case with no modification is called the reference case. Figure 3 gives the synthesized and the measured bridge impedance of the guitar for these different cases. These two figures focus only on the targeted mode to make them more readable. Inserts show the studied frequency range in larger views of the transfer functions. These figures show that the two approaches are efficient. The active control method seems to be slightly less accurate than the synthesis method. Indeed, Figure 3(b) shows that the control of the damping factor slightly modifies the frequency of the controlled mode. This might be due to an approximate identification of this mode for the model used to design the control system. The values of the damping factor after the modifications are given in Table 1 for the control and the synthesis methods.

To further investigate the control efficiency, Figure 4 gives the upper part of the targeted and actual pole locations of the controlled mode. The non-circled symbols give the targeted locations and match the nearly horizontal dotted line which is the locus of equal frequency locations. The circled symbols give the actual pole locations of the controlled mode. As they don't match the nearly horizontal line, the frequency of the controlled mode is also modified. The system is stable since the real parts of the poles are always under zero. If one looks at the other poles of the system this stability criterion is also observed. However, if the damping factor of the mode decreases more than -80%, the real part of the pole could be too close to zero and the system could be unstable.

Then, the sounds produced using the two methods are studied. The spectrum of the synthesized sound and the Fourier Transform of the measured radiated sound are presented in Figure 5. Again, these two figures focus on the second partial of the played note. Once again the two approaches give similar results. The coupling between the string and the guitar body is clearly modified when the damping factor of the body mode is shifted. The body/string coupling phenomenon is fully described by Gough in [START_REF] Gough | The Resonant Response of a Violin G-string and the Excitation of the Wolf-Note[END_REF]. When decreasing, the damping factor increases the coupling between the body and the string modes. According to Gough, these two modes can be either strongly or weakly coupled. For a strong coupling, the decrease of the damping factor of one of these modes creates two peaks. Their frequencies move away and their damping factors decrease together. This is clearly the case of the -50% and -80% shifts of the damping factor. The reference case and the -10% shift are also cases of strong coupling since the damping factor of the visible peaks in Figure 5 decreases when the damping factor of the body mode decreases. The +100%, +200% and +400% shifts are cases of weak coupling. Indeed, according to Gough [START_REF] Gough | The Resonant Response of a Violin G-string and the Excitation of the Wolf-Note[END_REF], when the damping factor of the body mode lower in frequency than the string partial decreases, the frequency and the damping factor of the partial increase. This is clearly the case of these three last shifts of the damping factor as shown in Figure 5.

To support these observations, the temporal envelope for the second partial of the string is given in Figure 6 for the seven modifications of the damping factor. These temporal envelopes are extracted from the spectrograms of the synthesized and measured sounds.

They represent the time evolution of the energy of the targeted partial. This figure shows a good similarity between the two approaches. Moreover, it can be seen that the +100% shift is the limit between the strong and the weak coupling since the slope of the corresponding time evolution is the largest. Once again, Figure 6(b) shows that the control approach is more variable than the synthesis approach. The temporal envelope of the measured sound is much more complex than the time evolution of the synthesized sound. For example, the curves in Figure 6(a) are straight while some curves in Figure 6(b) present bounces. This might be due to the polarization of the string, which is not included in the model used in sound synthesis. This phenomenon is described in [4].

The comparison between the controlled and synthesized temporal envelopes given in Fig- Gough's model. Once again the results are similar. Figure 7(a) reveals that the stronger is the coupling between the body and the string modes the higher is the amplitude of the coupled partial. Figure 7(b) supports the fact that the +100% shift is the limit between the strong and the weak coupling since the slope is the largest for this case. This observation reveals also that the coupling in the reference case is strong since all the points to the left

B. Modification of the Modal Frequency

The second illustration of the two proposed approaches is done for the modal frequency of the second mode of the soundboard. The principle is the same as in the previous section.

In addition to the reference case, the frequency of the mode is altered by ±5%, ±10% and ±15%. 2 for the control and the synthesis methods. It should be noticed that the frequency shifts in Hertz depends on the location of the controlled mode across the range of frequencies. To control the frequency of another mode the shifts in percent should be adapted to the frequency range of work. The synthesis approach gives expected results while those of the control approach are less precise. The targets in percent are not perfectly reached for the active control method. This might be due to computation errors during the design of the control system, the use of a non perfect model or the influence of the residual modes. The curves matching the +10% and +15% shifts show that the small mode located at a frequency of 240 Hz has not a negligible influence on the results. Indeed, its amplitude increases when the modified mode is shifted to higher frequencies. In active control applications this usual phenomenon is known as the spillover effect. It is due to the imprecise modeling of the modes which are not much excited during the identification step. Then, a part of the energy used to modify the controlled mode is passed to these modes unintentionally increasing their amplitudes. However, even if the shift to high frequencies is disturbed by the mode located at 240 Hz, the amplitude increase makes the controlled impedance not so far from the synthesized one. The digital damping is also a usual constraint of active control systems.

As this changes the damping factor of the controlled mode, the control system is designed to keep the modal damping factor constant.

The upper part of the targeted (non-circled symbols) and actual (circled symbols) pole locations of the controlled mode are given in Figure 9. The nearly vertical line gives the This might be due to the spillover effect. However, this effect is accentuated by the coupling between the two modes. Indeed, the modification of the damping factor of the small mode when the controlled mode is shifted in low frequencies reveals a coupling between these two modes. The spillover effect also limits the frequency shift of the controlled mode since the real part of the pole of the small mode could be shifted too close to zero and could make the system unstable.

Then, the sounds produced using the two proposed approaches are studied. Figure 10(a) gives the spectrum of the synthesized sounds. second partial of the played note. The effects of the frequency shift on the partial of the string are similar for the synthesized and the measured sounds. The results observed in Figure 10 indicate that the body and the string modes are strongly coupled. As described by Gough [START_REF] Gough | The Resonant Response of a Violin G-string and the Excitation of the Wolf-Note[END_REF], when the frequency of the body mode is shifted far away from the frequency of the string partial, the latter is slightly shifted from its original frequency. When the string partial is lower than the body mode, its frequency slightly decreases. When the string partial is higher than the body mode, its frequency slightly increases. When the frequency of the body mode roughly matches the frequency of the string partial, the coupling becomes stronger. In this case, it is more difficult to discriminate between the two modes. This is the case in Figure 10 for the reference sound and for the +5% shift of the frequency.

Then, the effects of these modifications in the temporal envelope of the second partial of the A string are studied. lower for the control approach.

Figure 12 gives the maximum amplitude and the slope of the temporal envelope of the second partial of the string for each frequency shift. Figure 12(a) shows that the maximum amplitude of the partial increases when the frequency of the body mode is moved closer to the frequency of the partial. The maximum amplitude is reached for the +5% shift while shows that the slope of the partial also increases when the frequency of the body mode is moved closer to its frequency. Figure 11(b) shows that this slope is modified by more complex phenomena after few seconds. However, it seems that the closer are the frequencies of the body and the string modes, the quicker is the decrease of the partial. According to the conclusion of Section III.A, the coupling is strong in the reference case and decreases when the frequency of the mode is moved away from the frequency of the partial. However it is not possible to give the split between strong and weak coupling since no specific features is observed. Once again, it should be noticed that the modification of the coupling between the string and the mode of the soundboard depends on the initial coupling conditions. The results would have been different if the coupling of the reference case has not been strong.

IV. DISCUSSION

The two proposed approaches seem to be efficient ways to study complex phenomena like the sound production of musical instruments. The previous sections reveal that each approach has its own advantages and drawbacks. The choice of the method depends on the targeted goal.

On a real instrument, a lot of parameters like the temperature or the humidity can modify the mechanical properties of the studied structure. In the model used in the synthesis approach, these parameters are fixed by the user as it is shown in section II.B. If the influence of one of these parameters has to be studied, it is possible to modify it without changing the others. This is not always possible with active control. For example, Figure 8 reveals that the modal frequency can be modified without changing the damping factor for the synthesis approach but not for the control approach. Synthesis is also really convenient to study the influence of a wide range of parameters. In section II.B, a lot of parameters can be modified, while the model of the control approach is focused on the properties of the soundboard.

Another advantage of synthesis is to enable the modification of the model parameters until extreme limits that are impossible to reach with modal active control, which is limited by the quality of the computation, the energy needed to reach a control target, and by the real-time requirements.

However, it is not possible to play or to listen the sound of the synthesis method as the sound of a real instrument. Controllers (like computers or MIDI keyboards) and sound emission systems (like loudspeakers or headphones) are one of the limits of the sound synthesis. To be used in psychoacoustical tests, the sound synthesis must be realistic, thus needing a very complex physical model (plucked string, soundboard, sound radiation, etc).

Results from sound synthesis must be compared to more reliable ones, coming generally from experimental data. As the modal active control only needs the knowledge of the modes of vibration of the soundboard, it does not need a physical model of a complete instrument.

For the modal active control method, the modeling step is much easier and less complicated (although limited to linear control theory). The realism is not an issue for the active control approach, since the modifications are made in situ, thus the sound is radiated directly by the instrument, and the instrument is still playable. Modal active control enables the study of very complex situations (multiphysics or involving human player for instance), while needing only simple models of the sub-system that is controlled (like the soundboard).

V. CONCLUSION

This paper proposes to compare two different approaches in order to modify the modal parameters of a guitar soundboard. The first one is a frequency domain synthesis method using the bridge impedance and the inverse Fast Fourier Transform. A model is used to compute the radiated sound of a plucked string fixed on a modal model of the guitar soundboard. The second one is an active control method using a linear modal state model of the controlled structure and enabling the in situ modifications of the modal parameters of the guitar soundboard. This paper compares these two approaches modifying the modal damping factor and the modal frequency of the second vibration mode of the soundboard. Different modifications are studied in each case. The effects of these modifications are studied in the bridge impedance of the soundboard and in the radiated sound of the synthesized and of the controlled guitar.

The close results obtained for the two approaches are the first significant result of this study. These results are not exactly the same but they lead to the same conclusion about the influence of the modal parameters in the sound of musical instruments. Indeed, the effects of the modal modifications are similar for the two proposed approaches both in the bridge impedance and in the radiated sounds.

The second important result concerns the choice of one of these two methods according to the targeted goal of the user and knowing the characteristics of both approaches. The synthesis approach is more convenient to reach precisely the targeted modal properties, while modal active control is more realistic for studies where the whole instrument, listening and playing experience are important.

Figure 1 :

 1 Figure 1: Measurement on the guitar in an anechoic chamber. ➀ Guitar soundboard, ➁ A

Figure 2 :

 2 Figure 2: General principle of the modal active control.

  Measured bridge impedances -Active control approach.

Figure 3 :

 3 Figure 3: Modifications of the damping factor of the second mode in the soundboard

Figure 4 :

 4 Figure 4: Pole locations of the controlled mode. Reference case •, damping factor altered by

  Fourier Transform of the measured radiated sound of the controlled guitar.

Figure 5 :

 5 Figure 5: Comparison between the radiated sound for the synthesis and the active control.

  Measured temporal envelope -Active control approach.

Figure 6 :

 6 Figure 6: Temporal envelope of the second partial of the A string. Reference case [ ],

ure 6

 6 is difficult. The main characteristics of these curves are detailed in Figure7in order to better observe the coupling modification as a function of the modal parameter shifts. Figure 7 gives the maximum amplitude and the slope of the temporal envelope of the second partial of the string in the different studied cases. The maximum amplitudes are normalized between 0 and 1 since the scales of the spectrum of the synthesized sound and the Fourier Transform of the measured sound are not the same. The slopes are calculated using 0.5 second of the temporal envelope after their maximum amplitudes. Only the transient of the partials is studied since it is known to be very important regarding the sound attributes of musical instruments. Arrows indicate the observed coupling behavior also predicted by

  Slope calculated using 0.5 second of the temporal envelope after its maximum amplitude.

Figure 7 :

 7 Figure 7: Maximum amplitude and slope of the temporal envelopes given in Figure 6 for

Figure 8 (

 8 a) gives the seven impedances used in the synthesis model and Figure 8(b) gives the controlled impedances measured at the bridge. For clarity, these two figures focus only on the targeted mode. Inserts show the studied frequency range in larger views of the transfer functions. These figures show that the synthesis and the active control give close results. The values of the frequency after the modifications and the actual shifts in Hertz are given in Table

  Measured bridge impedances -Active control approach.

Figure 8 :

 8 Figure 8: Modifications of the frequency of the second mode in the soundboard impedance.

Table 2 :

 2 Values of the frequency of the soundboard mode around 213 Hz for the synthesis and the control methods. locus of equal damping factor locations. The poles of the shifts to low frequencies and to high frequencies respectively stay close to and move away from this line. The damping factor is clearly modified for the cases where the frequency is increased. Figure9also gives the unexpected pole modifications of the small mode located at 240 Hz. The damping factor of this mode decreases when the damping factor of the controlled mode increases and vice versa.

Figure 9 :

 9 Figure 9: Pole locations of the controlled mode. Reference case •, frequency altered by -15%

  Fourier Transform of the measured radiated sound of the controlled guitar.

Figure 10 :

 10 Figure 10: Comparison between the radiated sound for the synthesis and the active control.

Figure 11

 11 presents the first three seconds of the synthesized and measured temporal envelopes in the seven cases presented in Figure8. Again, the behavior of the temporal envelopes are close. As in Figure8, the effects of the frequency shift are Synthesized temporal envelopes -Synthesis approach. Measured temporal envelopes -Active control approach.

Figure 11 :

 11 Figure 11: Temporal envelopes of the second partial of the A string. Reference case [ ],

  Slopes calculated using 0.5 second of the temporal envelope after its maximum amplitude.

Figure 12 :

 12 Figure 12: Maximum amplitude and slope of the temporal envelopes given in Figure 11 for

f

  string [Hz] T [N] ρ s [kg.m -1 ] c s [m.s -1 ] B [Nm 2 ] η F [] η B [] η A [s -1 ]

Table 1 :

 1 Values of the damping factor of the soundboard mode for the synthesis and the control methods.

	Targeted shift [%]	-80	-50	-10	0	+100 +200 +400
	Synthesis 0.005 0.013 0.023 0.025 0.05 0.075 0.126
	Damping factor [ ]
	Control 0.013 0.017 0.024 0.025 0.047 0.057 0.072

Table 3 :

 3 Parameters of the synthesized string. c [m.s -1 ] ρ air [kg.m -3 ]

	340	1.184

Table 4 :

 4 Air parameters. ten modes identified using the impedance measured at the bridge of the soundboard.Mode numberf k [Hz] ξ k [] Π a k [] Π c k []

	1	87	0.007 0.073	1
	2	113	0.029 0.166	1
	3	213	0.025 1.254	1
	4	232	0.009 0.105	1
	5	271	0.007 0.074	1
	6	378	0.008 0.078	1
	7	409	0.019 0.031	1
	8	418	0.014 0.218	1
	9	504	0.011 0.141	1
	10	563	0.013 0.091	1

Table 5 :

 5 Modal parameters of the first ten modes of the soundboard.

Acknowledgements

This work has been done during the PhD of Simon Benacchio, funded by the Agence National de la Recherche (ANR 11 PDOC 010 01).

Appendix A

Tables 3 and4 give the parameters of the string and of the air used in the model of the synthesis method. Comparison between the radiated sound for the synthesis and the active control. 17


1

The farmers of the associative movement "Terre de liens" in France: reshaping farming representations and identities.

Pibou Elsa 1

Abstract -This article explores the ruptures and continuities in the farmer's ways of being in a contemporary and activist organization called "Terre de liens". This uncommon French structure, which buys and leases land to farmers, tries to raise public awareness of agricultural issues and to integrate members who are traditionally far away from land tenure issues.

Even if the main purposes of this organization is to fight against land speculation, provide support to the setting-up of farmers and develop sustainable activities, the action of actually and concretely setting-up farmers raises many questions, among which the place and representations of the farmers in this process. This article will study the case of the farmer profile within "Terre de liens", in order to highlight what being farmer in this structure changes or does not change. This study will try to show how the socioprofessional identities of the farmers could be transformed. 1

TDL FARMERS: A SOCIOLOGICAL APPROACH
The strategies developed by project initiators in agriculture in order to access land tenure face several obstacles, especially when they lack sufficient financial, informational and relational resources.

Since the end of World War II, the French government established various policies meant to help farmers bypass these difficulties and preserve the agricultural use of the land (Sencébé et al., 2013). But given the persistent problems and the context of multifaceted crisis that our societies face, some citizens-both farmers and civil society actors-have been involved, since 2003, in the creation of an associative initiative called "Terre de liens" (TDL)literally "Land of links". TDL was initially created as an association aiming at helping farmers set-up their agriculture projects and providing advice and information on the best way to collaborate with other actors in several juridical forms. The association thus reminded the citizens and the regional authorities about land tenure issues and federated the various branches of the association through the French regions, allocating them resources. After this initial phase, TDL developed its own financial tools in order to provide operational help to farmers. These tools, called La foncière and La fondation, enable TDL to collect funds through public offerings. Thanks to these donations from citizens and private businesses, TDL can able buy land and lease it to new farmers.

1 University of Toulouse II Le Mirail, Laboratoire Dynamiques Rurales, Toulouse, France (elsa.pibou@univ-tlse2.fr).

In return, the tenants must relinquish part of their independence. Indeed, they have to renounce entirely or partially their desire of becoming land owners. Furthermore, TDL implemented more or less systematized devices to control the proper use of land: a tenancy agreement with specific environmental clauses, an agronomic inventory of the farms, as well as several meetings taking place in the farms between TDL members, volunteers, donors, shareholders and farmers. From a sociological perspective, I want to understand the reasons that led these farmers to become members of this organization (was it because of opportunities, reflective choices, pragmatism?). I also wish to grasp the impacts and the consequences of their commitment in the organization and their influence on the process of construction and reconstruction of their socio-professional identities. I will investigate the relations, the connections or the breaks between the objectives set by TDL at a broad scale, and the individual representations of the farmers who lease TDL land. Is there a significant gap between the collective ambition and the actual representations that the farmers finally have? For that matter, is it relevant to analyse the situation in terms of vertical or horizontal relationships between the farmers and the organization? Do TDL farmers knowingly waive their freedom of farming as they intend to, as well as their desire to own and to pass down land? What about their autonomy? Does the collective nature of ownership effectively preclude the farmers from developing an intense emotional connection with land (Hervieu, 2005: 95)?

A POLYMORPHIC DATA COLLECTION PLAN

To provide some elements to respond to these questions, my study was built on several modalities of data collection. On the one hand, I organized a quantitative survey to analyse the profiles and the personal evolution of TDL farmers at a national level and I conducted long interactive interviews, with TDL farmers in one region ("Midi-Pyrénées"), to better approach and understand their representations and values. On the other hand, through participant-observation, I attended several meetings at different levels in order to grasp the implications and the real place(s) of the farmers within this structure.

RUPTURES AND BOUNDARIES IN TDL FARMERS POSITION-

ING

My analytic work shows that the organization highlights paradoxes inherent to the understanding of the farmer's place and work. On the one hand, one can observe similar patterns in the personal path of TDL farmers (education, work experience…) and in their practices. Most of them chose the profession, turned themselves into farmers and studied to that end. Furthermore, they developed practices in accordance with TDL's wishes: organic agriculture, often mixed farming, and products commercialised locally. My initial hypothesis-that TDL farmers do not opt for this organization by chance-was thus corroborated. They all chose first a profession before choosing a place where they would eventually practise this occupation. Generally, their farm research was not focused on one single region. This pattern breaks with the family and territorial continuum, which is a typical feature of French peasantry [START_REF] Mendras | Sociétés paysannes[END_REF][START_REF] Mendras | La fin des paysans, 1 ère ed[END_REF]Hervieu, 2001;[START_REF] Rémy | Les mondes agricoles en politique[END_REF]. Many farmers collaborated with TDL in order to ensure that they would pass down their farms to farmers who would respect the social and environmental principles supported by the organization. Nevertheless, these remarks should not lead to minimise their sense of belonging to one specific place: their roots and their territorial registration remains an important marker of their identity (Jean, 1997). All these elements suggest the development of the profile of the intermediary farmer [START_REF] Dufour | Reconstruction des identités professionnelles autour de la mutlifonctionnalité de l'agriculture. L'exemple des coteaux du Lyonnais[END_REF][START_REF] Papy | Nouveaux rapports à la nature dans les campagnes[END_REF])-farmers who get into a territory and whose speeches reveal some "peasant" characteristics, as described by rural sociologists [START_REF] Mendras | La fin des paysans, 1 ère ed[END_REF][START_REF] Rémy | Les mondes agricoles en politique[END_REF].

On the other hand, I did not expect to observe such diversity in their attempts, in their positioning and in the reasons encouraging them to call on this organization. The diversity that we traditionally find in the French farmers' profile-their values, representations and desires-is no exception and can be found within TDL. For some farmers, it is their militant path and commitment that led them to turn to the organization and not just their desire to settle. As a matter of fact, most of them wanted to access land in order to farm. They generally hoped to access land through a communitarian way and they do not make the property a priority. Actually, they lost the ability of owning their land but they have the right to be involved in the meetings of the organization and they have the possibility to sit on the decisionmaking structures. Nevertheless, this possibility as well as involvement and opening towards the entire society-such as TDL wants to experiment-did not necessarily correspond to the farmer's expectations. This could be observed in their relations with TDL: some of them seized the opportunity to commit themselves within the structure, to get their voices heard, but a lot of farmers did not want to make these commitments a daily priority. We can observe various degrees of farmer involvement: this depends on their ability to convert their systems of norms, to grasp the societal representations of their profession, and to adopt the linguistic and organizational code of TDL. These elements especially resurface in conflict situations and in power struggles that such conflicts cause.

Thus, what can be observed in the structure leads to another question -that of the farmer's place. Actually, are we not witnessing the development of new ways of controlling farms, farming and agricultural practices? With this kind of structure, is not it this profession, which is generally described as considering autonomy as an important asset, becoming a new "object class" to quote Pierre [START_REF] Bourdieu | Une classe objet[END_REF]?

  Finally and contrary to what I expected to witness, the study of TDL and TDL farmers showed that power struggles between farmers and other rural actors do not disappear within this kind of structure, no matter how open it may be, but seem to crystallize on various topics that raise the question of the farmer's place in society, through a new point of view. At this stage of development, the organization that I observed does not seem to recreate new farming modalities but it recreates new types of solidarity and dialogue that are likely to transform the positioning of farmers in the public debate. Moreover, it recreates a situation of test of their desire of autonomy as well as their capacity of being involved in this debate. This kind of structure, the claims it defends and what it reveals are part of an international dynamic that takes the issue of access to land and the practice of farming profession and echo back other social movements or structures such as the MST in Brazil, the Via Campesina movement, the marches of access to land in India.

July -1 August 2013 in Florence, Italy


Guillaume Lévy

On uniqueness for a rough transport-diffusion equation

In this Note, we study a transport-diffusion equation with rough coefficients and we prove that solutions are unique in a low-regularity class.

Introduction

In this note, we address the problem of uniqueness for a transport-diffusion equation with rough coefficients. Our primary interest and motivation is a uniqueness result for an equation obeyed by the vorticity of a Leray-type solution of the Navier-Stokes equation in the full, three dimensional space. The main theorem of this note is the following. Theorem 1.1 : Let v be a divergence free vector field in L 2 (R + , Ḣ1 (R 3 )) and a a function in L 2 (R + ×R 3 ). Assume that a is a distributional solution of the Cauchy problem

(C)    ∂ t a + ∇ • (av) -∆a = 0 a(0) = 0, (1)
where the initial condition is understood in the distributional sense. Then a is identically zero on R + ×R 3 .

Email address: levy@ljll.math.upmc.fr (Guillaume Lévy).

As a preliminary remark, the assumptions on both v and a entail that ∂ t a belongs to L 1 loc (R + , H -2 (R 3 )) and thus, in particular, a is also in C(R + , D ′ (R 3 )). In Theorem 1.1, a is to be thought of as a scalar component of the vorticity of v, which is in the original problem a Leray solution of the Navier-Stokes equation. In particular, we only know that a belongs to L 2 (R + × R 3 ) and L ∞ (R + , Ḣ-1 (R 3 )), though we will not use the second assumption. The reader accustomed to three-dimensional fluid mechanics will notice that, comparing the above equation with the actual vorticity equations in 3D, a term of the type a∂ i v is missing. In the original problem, where this Theorem first appeared, we actually rely on a double application of Theorem 1.1. For some technical reasons, only the second application of Theorem 1.1 takes in account the abovementioned term.

As opposed to the standard DiPerna-Lions theory, we cannot assume that a is in L ∞ (R + , L p (R 3 )) for some p ≥ 1. However, our proof does bear a resemblance to the work of DiPerna and Lions; our result may thus be viewed as a generalization of their techniques. Because of the low regularity of both the vector field v and the scalar field a, the use of energy-type estimates seems difficult. This is the main reason why we rely instead on a duality argument, embodied by the following theorem.

Theorem 1.2 : Given v a divergence free vector field in L 2 (R + , Ḣ1 (R 3 )) and a smooth ϕ 0 in D(R 3 ), there exists a distributional solution of the Cauchy problem

(C ′ )    ∂ t ϕ -v • ∇ϕ -∆ϕ = 0 ϕ(0) = ϕ 0 (2) with the bounds ϕ(t) L ∞ (R 3 ) ≤ ϕ 0 L ∞ (R 3 ) (3)
and

∂ j ϕ(t) 2 L 2 (R 3 ) + t 0 ∇∂ j ϕ(s) 2 L 2 (R 3 ) ds ≤ ∂ j ϕ 0 2 L 2 (R 3 ) + ϕ 0 2 L ∞ (R 3 ) ∂ j v 2 L 2 (R+×R 3 ) (4)
for j = 1, 2, 3 and any positive time t. By reversing the arrow of time, this amounts to build, for any strictly positive T , a solution on [0, T ] × R 3 of the Cauchy problem

(-C ′ )    -∂ t ϕ -v • ∇ϕ -∆ϕ = 0 ϕ(T ) = ϕ T , (5)
where we have set ϕ T := ϕ 0 for the reader's convenience.

Proofs

We begin with the dual existence result. Proof (of Theorem 1.2.) : Let us choose some mollifying kernel ρ = ρ(t, x) and denote v δ := ρ δ * v, where ρ δ (t, x) := δ -4 ρ( t δ , x δ ). Let (C ′ δ ) be the Cauchy problem (C ′ ) where we replaced v by v δ . The existence of a (smooth) solution ϕ δ to (C ′ δ ) is then easily obtained thanks to, for instance, a Friedrichs method combined with heat kernel estimates. We now turn to estimates uniform in the regularization parameter δ. The first one is a sequence of energy estimates done in L p with p ≥ 2, which yields the maximum principle in the limit. Multiplying the equation on ϕ δ by ϕ δ |ϕ δ | p-2 and integrating in space and time, we get

1 p ϕ δ (t) p L p (R 3 ) + (p -1) t 0 ∇ϕ δ (s)|ϕ δ (s)| p-2 2 2 L 2 (R 3 ) ds = 1 p ϕ 0 p L p (R 3 ) . (6)
Discarding the gradient term, taking p-th root in both sides and letting p go to infinity gives

ϕ δ (t) L ∞ (R 3 ) ≤ ϕ 0 L ∞ (R 3 ) . (7)
To obtain the last estimate, let us derive for 1 ≤ j ≤ 3 the equation satisfied by ∂ j ϕ δ . We have

∂ t ∂ j ϕ δ -v δ • ∇∂ j ϕ δ -∆∂ j ϕ δ = ∂ j v δ • ∇ϕ δ . ( 8
)
Multiplying this new equation by ∂ j ϕ δ and integrating in space and time gives

1 2 ∂ j ϕ δ (t) 2 L 2 (R 3 ) + t 0 ∇∂ j ϕ δ (s) 2 L 2 (R 3 ) ds = 1 2 ∂ j ϕ 0 2 L 2 (R 3 ) + t 0 R 3 ∂ j ϕ δ (s, x)∂ j v δ (s, x) • ∇ϕ δ (s, x)dxds. ( 9
)
Since v is divergence free, the gradient term in the left-hand side does not contribute to Equation (9). Denote by I(t) the last integral written above. Integrating by parts and recalling that v is divergence free, we have

I(t) = - t 0 R 3 ϕ δ (s, x)∂ j v δ (s, x) • ∇∂ j ϕ δ (s, x)dxds ≤ ϕ 0 L ∞ (R 3 ) t 0 ∂ j v δ (s) L 2 (R 3 ) ∇∂ j ϕ δ (s) L 2 (R 3 ) ds ≤ 1 2 t 0 ∇∂ j ϕ δ (s) 2 L 2 (R 3 ) ds + 1 2 ϕ 0 2 L ∞ (R 3 ) t 0 ∂ j v δ (s) 2 L 2 (R 3 ) ds.
And finally, the energy estimate on ∂ j ϕ δ reads

∂ j ϕ δ (t) 2 L 2 (R 3 ) + t 0 ∇∂ j ϕ δ (s) 2 L 2 (R 3 ) ds ≤ ∂ j ϕ 0 2 L 2 (R 3 ) + ϕ 0 2 L ∞ (R 3 ) ∂ j v 2 L 2 (R+,×R 3 ) . (10)
Thus, the family (ϕ

δ ) δ is bounded in L ∞ (R + , H 1 (R 3 )) ∩ L 2 (R + , Ḣ2 (R 3 )) ∩ L ∞ (R + × R 3 ).
Up to some extraction, we have the weak convergence of (ϕ δ ) δ in L 2 (R + , Ḣ2 (R 3 )) and its weak- * convergence in

L ∞ (R + , H 1 (R 3 )) ∩ L ∞ (R + × R 3 ) to some function ϕ.
By interpolation, we also have ∇ϕ δ ⇀ ∇ϕ weakly in L 4 (R + , Ḣ 1 2 (R 3 )) as δ → 0. As a consequence, because v δ → v strongly in L 2 (R + , Ḣ1 (R 3 )) as δ → 0, the following convergences hold :

∆ϕ δ ⇀ ∆ϕ in L 2 (R + × R 3 ); v δ • ∇ϕ δ , ∂ t ϕ δ ⇀ v • ∇ϕ , ∂ t ϕ in L 4 3 (R + , L 2 (R 3 )).
In particular, such a ϕ is a distributional solution of (C ′ ) with the desired regularity. We now state a Lemma which will be useful in the final proof. Lemma 2.1 : Let v be a fixed, divergence free vector field in L 2 (R + , Ḣ1 (R 3 )). Let (ϕ δ ) δ be a bounded family in L ∞ (R + × R 3 ). Let ρ = ρ(x) be some smooth function supported inside the unit ball of R 3 and define

ρ ε := ε -3 ρ • ε . Define the commutator C ε,δ by C ε,δ (s, x) := v(s, x) • (∇ρ ε * ϕ δ (s))(x) -(∇ρ ε * (v(s)ϕ δ (s)))(x). Then C ε,δ L 2 (R+×R 3 ) ≤ ∇ρ L 1 (R 3 ) ∇v L 2 (R+, Ḣ1 (R 3 )) ϕ δ L ∞ (R+,H 1 (R 3 )) . (11)
This type of lemma is absolutely not new. Actually, it is strongly reminiscent of Lemma II.1 in [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF] and serves the same purpose. We are now in position to prove the main theorem of this note.

Proof (of Theorem 1.1.) : Let ρ = ρ(x) be a radial mollifying kernel and define ρ ε (x) := ε -3 ρ( x ε ). Convolving the equation on a by ρ ε gives, denoting a ε := ρ ε * a,

(C ε ) ∂ t a ε + ∇ • (a ε v) -∆a ε = ∇ • (a ε v) -ρ ε * ∇ • (av). ( 12
)
Notice that even without any smoothing in time, a ε ,

∂ t a ε are in L ∞ (R + , C ∞ (R 3 )) and L 1 (R + , C ∞ (R 3 ))
respectively, which is enough to make the upcoming computations rigorous. In what follows, we let ϕ δ be a solution of the Cauchy problem (-C ′ δ ), with (-C ′ δ ) being (-C ′ ) with v replaced by v δ . Let us now multiply, for δ, ε > 0 the equation (C ε ) by ϕ δ and integrate in space and time. After integrating by parts (which is justified by the high regularity of the terms we have written), we get

T 0 R 3 ∂ t a ε (s, x)ϕ δ (s, x)dxds = a ε (T ), ϕ T D ′ (R 3 ),D(R 3 ) - T 0 R 3 a ε (s, x)∂ t ϕ δ (s, x)dxds and T 0 R 3 [∇ • (v(s, x)a ε (s, x)) -ρ ε (x) * ∇ • (v(s, x)a(s, x))] ϕ δ (s, x)dxds = T 0 R 3 a(s, x)C ε,δ (s, x)dxds,
where the commutator C ε,δ has been defined in the Lemma. From these two identities, it follows that

a ε (T ), ϕ T D ′ (R 3 ),D(R 3 ) = T 0 R 3 a(s, x)C ε,δ (s, x)dxds - T 0 R 3 a ε (s, x) -∂ t ϕ δ (s, x) -v(s, x) • ∇ϕ δ (s, x) -∆ϕ δ (s, x) dxds.
From the Lemma, we know that (

C ε,δ ) ε,δ is bounded in L 2 (R + × R 3 ). Because v • ∇ϕ δ → v • ∇ϕ in L 4 3 (R + , L 2 (R 3
)) as δ → 0, the only weak limit point in L 2 (R + × R 3 ) of the family (C ε,δ ) ε,δ as δ → 0 is C ε,0 . Thanks to the smoothness of a ε for each fixed ε, we can take the limit δ → 0 in the last equation, which leads to a ε (T ), ϕ T D ′ (R 3 ),D(R 3 ) = T 0 R 3 a(s, x)C ε,0 (s, x)dxds.

Again, the family (C ε,0 ) ε is bounded in L 2 (R + × R 3 ) and its only limit point as ε → 0 is 0, simply because v • ∇ϕ ερ ε * (v • ∇ϕ) → 0 in L 4 3 (R + , L 2 (R 3 )). Taking the limit ε → 0, we finally obtain a(T ), ϕ T D ′ (R 3 ),D(R 3 ) = 0.

This being true for any test function ϕ T , a(T ) is the zero distribution and finally a ≡ 0.


Manuel Pozo
email: manuel.pozo@isep.fr

Raja Chiky
email: raja.chiky@isep.fr

AN IMPLEMENTATION OF A DISTRIBUTED STOCHASTIC GRADIENT DESCENT FOR RECOMMENDER SYSTEMS BASED ON MAP-REDUCE

Keywords: recommender system, collaborative filtering, matrix factorization, gradient descent, parallelization

This work presents an implementation of a Distributed Stochastic Gradient Descent (DSGD) for Recommender Systems based on Hadoop/MapReduce. Recommender Systems aim at presenting first the information in which users may be more interested. To do this, they analyse a great volume of data that represent the users preferences (e.g. ratings). Thus, this stirs up the need of load-balancing. DSGD is a Matrix Factorization technique that has demonstrated high accuracy and scalability. In this work we expose this algorithm and modify it to improve its accuracy and adaptability to a hadoop cluster. The experimentation phase uses Movie-Lens datasets. Comparisons with other algorithms are given. Results show the good performance of the implementation.

INTRODUCTION

The web content has become so vast that users hardly find the information they are looking for. Recommender Systems aim at selecting and presenting first the information in which users may be more interested. Nowadays, enterprises use these systems to personalize items for users, i.e. recommendations, by analysing their preferences [START_REF] Paul B Kantor | Recommender systems handbook[END_REF]. For instance, Amazon and Youtube propose new products/videos regarding what is already bought/watched. A very interesting datum has been offered by the video media service Netflix in 2014 [START_REF] Hunt | Recsys 2014. the value of better recommendations[END_REF]: improving the accuracy of recommendations by 10% may lead in around $500M per month of outcomes. Hence, these systems are becoming more and more important in the past decade.

The interest of users in items can be explicit (e.g. giving a rating to a movie) or implicit (e.g. tracking navigational behaviour). Recommender Systems analyze this feedback in order to generate recommendations [START_REF] Paul B Kantor | Recommender systems handbook[END_REF]. Furthermore, the number of users and items uses to be high. For instance, Netflix has around 20 millions customers, 80 thousands movies and 5 billions ratings [START_REF] Makari | Shared-memory and shared-nothing stochastic gradient descent algorithms for matrix completion[END_REF]. Normally users just rate some of these items. This yields in a sparse dataset that contains a huge volume of missing/unknown ratings to predict. Thus, these predictions are computationally expensive. Novel recommenders should be accurate to fit users interests, and scalable to alleviate time processing. Matrix Factorization (MF) is a recommendation technique that has demonstrated great accuracy and scalability for very large datasets [START_REF] Koren | The bellkor solution to the netflix grand prize[END_REF][START_REF] Koren | Matrix factorization techniques for recommender systems[END_REF].

This work focuses on a particular Matrix Factorization technique called Distributed Stochastic Gradient Descent (DSGD). The contribution in this paper are: (1) we detail a MapReduce implementation of this technique in a real Hadoop environment, (2) we did a modification in the paradigm distribution to achieve better accuracy and better adaptability to the hadoop cluster. The experimentation phase uses the publicly available MovieLens dataset. Main techniques are compared in terms of recommendation accuracy and time-consumption. The results indicate the good performance of the implementation.

This article is structured as follows: section 2 presents a brief state of the art in Matrix Factorization. In sections 3 and 4, we explain the distribution paradigm for Stochastic Gradient Descent techniques and the proposed modification. The experimentation phase is given in section 5. Finally, in section 6 conclusions and future work are exposed.

RELATED WORK

This state of the art mainly focuses on Matrix Factorization since it has demonstrated high accuracy predicting recommendations [START_REF] Koren | The bellkor solution to the netflix grand prize[END_REF]. It relies on collaborative filtering assumptions: users who agreed in past tend to agree in future. Hence, it groups people of similar tastes, and it recommends past liked items from people with the same preferences.

The Matrix Factorization technique decomposes a matrix R into two random matrices, P and Q, in such a way that the multiplication of both matrices gives approximately the original one [START_REF] Koren | Matrix factorization techniques for recommender systems[END_REF]. This concept is used in recommender systems to predict the missing ratings or unknown values of a sparse rating matrix R. Typically, known ratings in the matrix are de- noted by r u,i , where u stands for a user-row and i for an itemcolumn. Thus, r u,i can be obtained by the multiplication of a vector p u from matrix P and a vector q i from matrix Q. The dimensions of these vectors are latent factors, a.k.a. latent features, without any special semantic meaning used to represent information. Figure 1 illustrates this technique.

As a consequence, the goal is to find the matrices P and Q that best approximate known entries in R. That is, to look for the best vectors so that a prediction ru,i is close to its real value r u,i . This creates a basic baseline predictor ru,i = p u • q t i . Thus, other missing values might be discovered by minimizing the quadratic error of the difference between real and predicted values: min ru,i∈R (r u,i -ru,i ) 2 .

Indeed, this problem can be solved by using optimization algorithms. The two most known optimization techniques that may find out accurate predictions are based on alternating minimization and gradient descent [START_REF] Koren | Advances in collaborative filtering[END_REF].

On the one hand, alternating minimization techniques has simple algebraic resolution. Popularized by the Alternating Least Square (ALS) [START_REF] Schafer | Collaborative filtering recommender systems[END_REF][START_REF] Zhou | Large-scale parallel collaborative filtering for the netflix prize[END_REF][START_REF] Pilászy | Fast als-based matrix factorization for explicit and implicit feedback datasets[END_REF][START_REF] Takács | Alternating least squares for personalized ranking[END_REF][START_REF] Jain | Low-rank matrix completion using alternating minimization[END_REF], this technique decomposes the problem into two simple optimization problems represented in P and Q. Then, by fixing one matrix, the other one has to be guessed. Iterating the fixed matrix in order to guess the other one yields in an approximated result for R. In order to minimize the error, all ratings are consider in each iteration. On the other hand, the gradient descent optimization technique includes learning-parameters that study the ratings patterns to improve the results of the algorithm. This technique was popularized as Gradient Descent (GD) and Stochastic Gradient Descent (SGD) [START_REF] Funk | Netflix update: Try this at home (3rd place)[END_REF][START_REF] Takács | Major components of the gravity recommendation system[END_REF][START_REF] Koren | Factorization meets the neighborhood: a multifaceted collaborative filtering model[END_REF]. In order to minimize the error, it iterates among each single entry in R looking for a global minimum. After each iteration, the parameters are updated taking the negative gradient of the function into account, what improves the accuracy by making little steps per iteration. In [START_REF] Koren | Advances in collaborative filtering[END_REF][START_REF] Koren | Factor in the neighbors: Scalable and accurate collaborative filtering[END_REF][START_REF] Koren | Factor in the neighbors: Scalable and accurate collaborative filtering[END_REF] authors detail the evolution of this technique to exploit both explicit and implicit feedback.

SGD techniques are more accurate than ALS techniques [START_REF] Makari | Shared-memory and shared-nothing stochastic gradient descent algorithms for matrix completion[END_REF][START_REF] Koren | Advances in collaborative filtering[END_REF]. However, it is difficult to implement in a parallel way. As a consequence, there is a tendency to use ALS techniques due to their good accuracy and easy scalability [START_REF] Jain | Low-rank matrix completion using alternating minimization[END_REF][START_REF] Gemulla | Large-scale matrix factorization with distributed stochastic gradient descent[END_REF]. Yet, a highly paralellizable version of SGD has been recently proposed in [START_REF] Makari | Shared-memory and shared-nothing stochastic gradient descent algorithms for matrix completion[END_REF][START_REF] Gemulla | Large-scale matrix factorization with distributed stochastic gradient descent[END_REF], which has demonstrated that SGD may overcome ALS in both accuracy and scalability. In this work we offer an implementation of the Distributed SGD algorithm explained in [START_REF] Makari | Shared-memory and shared-nothing stochastic gradient descent algorithms for matrix completion[END_REF]. In addition, we have slightly changed this algorithm to achieve better accuracy and better adaptability to the cluster. The following sections will explain the paradigm in the distribution of SGD and our contribution.

,i = µ + b u + b i + p u • q t i ,
where µ is the average of the ratings in the matrix R, and b u , b i are biases for the user u and the item i respectively, which represent a deviation from the average. In order to learn this parameters and to find the best P and Q, the goal is to minimize the λ-regularized squared error of:

min bu,bi,pu,qi ru,i∈R (r u,i -r u,i ) 2 +λ•(b 2 u +b 2 i + p u 2 + q i 2 )
This technique iterates among ratings one by one evaluating the error in the prediction, e u,i = r u,i -ru,i . Stochastic Gradient Descent proposes to iterate over a batch of ratings instead, what allows a faster convergence in very large datasets. Then, they update users and items parameters and matrices by taking the negative gradient, where a learning rate γ is introduced:

• b u := b u + γ • (e u,i -λ • b u ) • b i := b i + γ • (e u,i -λ • b i ) • p u := p u + γ • (e u,i • p u -λ • q i ) • q i := q i + γ • (e u,i • q i -λ • p u )
However, either simple iteration or batch iterations, one can highlight the interdependency of the new updated values and previous values [START_REF] Makari | Shared-memory and shared-nothing stochastic gradient descent algorithms for matrix completion[END_REF]. For instance, new value for b u depends on last computed value of b u for this user u. In a single machine, new updated values after one iteration are ready for a new iteration. On the contrary, parallelizing the algorithm may lead in miss-updated values or lack of synchronization.

Avoiding this dependency is possible by analysing the matrix R. In fact, independent entries can be found, and thus, can be computed in parallel. Indeed, independent entries do not share rows and/or columns (users and/or items). As a consequence, they have no common parameters; hence, there is no dependency. Then, computing only independent entries at a time may allow to safely update parameters for future iterations. Based on this idea, authors in [START_REF] Makari | Shared-memory and shared-nothing stochastic gradient descent algorithms for matrix completion[END_REF][START_REF] Gemulla | Large-scale matrix factorization with distributed stochastic gradient descent[END_REF] have developed a fully distributed stochastic gradient descent, which is described below.

Distribution: paradigm and flexibility.

The main idea behind the parallelization of this algorithm is the division of the matrix into blocks and stratums. A block is a batch of ratings that should be computed. It is computed by only one processor, and hence, parameters updates are available within the same processor and there is no dependency problems inside. However, only independent blocks can be computed in parallel. In fact, parallelizable blocks do not share rows neither columns, as a consequence, ratings of different blocks are independent. A stratum is defined as a set of independent blocks. Then, inside a stratum there are no dependency problems. Thus, this creates a stratum schedule to follow, where stratums can be computed in sequence or arbitrary sequence [START_REF] Makari | Shared-memory and shared-nothing stochastic gradient descent algorithms for matrix completion[END_REF]. Then, SGD is stratified and safely parallelizable. Figure 2 represents this paradigm.

This method allows to distribute the process of blocks into machines in a cluster. In order to distribute the algorithm, the number of nodes w and thread processors per node t are taken into account. Thus, the number of single process units is b = w • t (supposing similar node machines). Looking for b independent blocks in the matrix maximizes the efficiency of the distribution in the cluster. However, this may cause a problem of flexibility to the cluster. Imagine a matrix of dimension n u x n i , the key point in the block decomposition is the divisibility of the dimensions in integer parts. That is, dimensions might be divisible by b in order to properly create blocks (n u %b = 0 and n i %b = 0). This integer divisibility constraint creates a conflict while decomposing matrices in blocks, since dimensions may not verify this "condition".

FLEXIBLE DISTRIBUTED STOCHASTIC GRADIENT DESCENT

Our work focuses on coping this divisibility condition. In [START_REF] Makari | Shared-memory and shared-nothing stochastic gradient descent algorithms for matrix completion[END_REF][START_REF] Gemulla | Large-scale matrix factorization with distributed stochastic gradient descent[END_REF], the deletion of rows/column (i.e. users/items) is suggested to achieve it. However, this may yield in a loss of accuracy: if a significant number of data is deleted the system may loose important information. In fact, this shows an accuracy/cluster flexibility trade-off: more clusters might lead in more deletions, hence an accuracy loss.

Our contribution is flexible in the condition accomplishment. On the one hand, deleting dimensions loses data. On the other hand, adding dimensions may increase the timeconsumption of the algorithm. However, this does not yields in a loss of data, since inserting empty dimension does not add any noisy information.

Based on these ideas, we have developed a Flexible Distributed Stochastic Gradient Descent. It uses three modes of usage regarding the available processing nodes and the matrix dimensions: (1) under dimensionality, (2) upper dimensionality, and (3) flexible dimensionality. The under dimensionality is the proposition given by [START_REF] Makari | Shared-memory and shared-nothing stochastic gradient descent algorithms for matrix completion[END_REF]. This mode resize the matrix by deleting dimensions, and thus resized dimension are equal or minor to the real one to accomplish the decomposing condition. The upper dimensionality resizes the matrix by inserting empty dimensions until accomplishing the decomposing condition. Finally, the flexible dimensionality technique is a hybrid of both techniques. It looks for the closest dimension (above or below) to the real one. This technique balances the trade-off. Any of these techniques return new dimensions, n u and n i , that fits properly with a matrix block decomposition.

DSGD Hadoop/MapReduce

In this section, we present the steps of our implementation in MapReduce. Three parameters have to be set: row original matrix dimension n r , column original matrix dimension n c , and the number of units b. There are three main points in our implementation:

(1) Matrix block decomposition: it adapts the dataset to the number of units of the cluster. To accomplish the decomposition condition and make the rating matrix decomposable in blocks, this method will delete entries and/or add empty entries. Once the decomposing condition is solved, block sizes can be defined. The row step size of a block is given by rowStep = n r /b and the column step size is given by columnStep = n c /b, where n r and n c are the new dimensions. Hence, the blocks can be formed by moving through the matrix.

(2) Stratum assignment: it groups independent blocks in the same stratum. Note that, the number of the stratums is actually the number of nodes b, and thus, up to b blocks are in the same stratum and can be run in parallel.

(3) Stratum execution: This step computes SGD in stratums blocks and then updates parameters at the end of their execution. This guarantees the integrity and independence of parameters for next stratums. The stratum are sequentially executed, yet this sequence may be randomly created. Finally, the stratums sequence is run a defined number of iterations or until convergence/achieved accuracy.

Our approach uses a SGDJob in MapReduce that analyses multiple entries. These are the blocks of a stratum, which contain the ratings to execute Gradient Descent optimization. The output of the job consist of vectors containing the last updated parameters for users and items. This process is done with a single M ap-f unction. The input key is a pair of val-ues which represent the entries of the matrix, the input value is the rating itself. After computation and a clean-up map process, the map function can return the results. The output key are the row/column of the matrix and a vector representing the updated parameters.

EXPERIMENTATION

We assess the performance of our implementation by using the publicly available MovieLens 10M dataset 1 , which has 10 million ratings applied by 69878 users to 10667 movies. We create a matrix of rows (users) and columns (items) that contains available ratings. The evaluation and comparisons are carried in terms of accuracy and scalability. On the one hand, the dataset is split up into a training set and a test set, containing respectively the 90% and the 10% of the ratings. These tests allow to evaluate the accuracy by using Root Mean Square Error (RMSE). This metric computes the square error in the difference between predicted values and real-observed values, the lower the better. On the other hand, the time-consuming and scalability parameters depend on machines and clusters. We vary the number of nodes in the cluster. That is, we use b = 1, 2, 5, 7, 15. In this paper we run the experimentation in single node cluster: a MacOS 4Go RAM with 2 cores (2.53GHz). This makes mappers in mapreduce to be computed sequentially, thus time-consumption is higher than in multi-cluster. Amazon EMR2 instances are used for real cluster experimentation, yet results are still on work and are not shown in this paper.

Three methods are compared: under dimensionality (un-derDSGD), upper dimensionality (upperDSGD), and flexible dimensionality (flexDSGD). These techniques have been equally trained: the overfitting parameter and learning rate were obtained by cross-validation: λ = 0.025 and γ = 0.0075. In addition, the algorithms run for 30 iterations and use 30 latent features. Henceforth, it is expected that upperDSGD obtains slightly better results in terms of RMSE, since it does not delete data. Yet, underDSGD might get slightly better time performance. FlexDSGD will show intermediary results in both comparisons. Moreover, we offer a comparison to ALS that shows the superiority of DSGD. The technique uses λ = 0.025, value obtained by cross-validation. 30 iterations over 30 latent features were performed as well.

Table 1 shows the comparisons and results. The performance in RMSE and the time achieved in computing one iteration in a single node cluster are presented. Besides, it shows the number of rows/columns (users/items) that has been added or removed (denoted by "+" or "-", respectively) in order to achieve the divisibility "condition": n u %b = 0 and n i %b = 0.

As it was expected, DSGD techniques are more accurate (with a lower RMSE) and faster than ALS. In addition, up- perDSGD and flexDSGD slightly overcome to underDSGD in accuracy. Indeed, these two modes deal with more ratings, what allows achieving (tiny 1%) better results. This fact explains as well the little extra time taken in computation. Nevertheless, less or none information is deleted. For instance, focus in the case of 15 number of units, which has 15 independent blocks per stratum. In order to achieve this decomposition, underDSGD has deleted 8 rows (users) and 2 columns (items), yielding in a loss of data. In addition, the deletion is not controlled, and thus one may delete very informativeness rows and columns.

CONCLUSION

Recommender Systems aim at selecting and presenting first the information in which users may be more interested. Matrix Factorization recommendation technique achieves high accuracy and scalability in very large datasets. In this paper we focused on a Distributed Stochastic Gradient Descent (DSGD) algorithm based on this technique. It has higher performance compared to other algorithms; yet its distribution paradigm does not fit properly the cluster, what may cause light loss in accuracy. This paper details an implementation of Distributed Stochastic Gradient Descent in a real Hadoop environment. In addition, we have proposed a slight modification to (1) adapt the distribution to the cluster set up, and (2) avoid the minor loss of accuracy of the current distribution paradigm. The experimentation phase uses the public MovieLens dataset. The evaluations show the good performance of the approach in terms of accuracy and scalability. Future work focuses on enhancing and improving this implementation by incorporating more heterogeneous data to the process.

Fig. 1 .

 1 Fig. 1. Matrix Factorization technique. Decomposition in matrices P and Q.

1 .

 1 Stochastic Gradient Descent (SGD)Gradient Descent (GD) is based on Matrix Factorization. It goals to explain the ratings in a matrix R by using two matrices P and Q. The baseline predictor is defined as ru

Fig. 2 .

 2 Fig. 2. Stratification and Block decomposition. Blocks 1,5, 9 do not share rows neither columns, thus, they form a stratum to run in parallel. Once computed, another stratum (2,6,7 or 3,4,8) can run.

Table 1 .

 1 Experimentation results.

	Technique	RMSE	Time (min)	N. Units (b) Rows Columns
	ALS [7, 8]	0.79603	3.019	1	=	=
	Under DSGD [3]	0.77571	1.303	1	=	=
	Flex DSGD	0.77571	1.303	1	=	=
	Upper DSGD	0.77571	1.303	1	=	=
	Under DSGD [3]	0.77611	1.850	2	=	-1
	Flex DSGD	0.77555	1.852	2	=	+1
	Upper DSGD	0.77559	1.885	2	=	+1
	Under DSGD [3]	0.77626	5.518	5	-3	-2
	Flex DSGD	0.77617	5.524	5	+2	-2
	Upper DSGD	0.77597	5.525	5	+2	+3
	Under DSGD [3]	0.77586	8.978	7	-4	-6
	Flex DSGD	0.77548	8.985	7	+3	+1
	Upper DSGD	0.77565	9.024	7	+3	+1
	Under DSGD [3]	0.77593	28.506	15	-8	-2
	Flex DSGD	0.77596	28.526	15	+7	-2
	Upper DSGD	0.77555	28.536	15	+7	+13

http://grouplens.org/datasets/movielens/

http://aws.amazon.com/elasticmapreduce/

Acknowledgments

This work has been supported by FIORA project, and funded by "DGCIS" and "Conseil Regional de l' Île de France".


Hardness Evaluation of Porous Hydroxyapatite Coating

I. INTRODUCTION

The hardness test, because of its simplicity, is commonly used to estimate the mechanical properties of coatings. There are several approaches to measuring coating hardness. For thin coatings, a modeling approach can be used to determine the coating hardness and indentation size effect (ISE) behavior. For the Vickers pyramid indenter, the hardness is defined as the load divided by the surface area of the impression. The Vickers hardness number (VHN) is defined as the ratio of load to the total area of impression. According to the usual practice VHN is expressed in kgfmm -2 and the units are omitted.

Two main problems arise when this technique is applied to coated materials: a) The coating thickness varies typically between 2000Af or an ion-implanted substrate to 20 micron for the hard coatings used to improve the wear resistance of tool steels. To obtain the bulk hardness value for the film it is necessary to satisfy requirement that the film thickness should be ten times greater than the penetration depth [1]. As the indentation depth is D=d /7, the substrate hardness generally influences the measured hardness values. b) In the load range commonly used in microhardness characterization (typically 5-1000 gf) the hardness number varies with the applied load. This situation is called the indentation size effect (ISE) and has been attributed to a wide variety of mechanisms [1,2]. ISE can be modeled in two ways. The first is empirical and corresponds to a power law. P = a d n .

(

) 1
This equation is called Meyer relationship. The second is a linear relationship between hardness and the reciprocal length of the impression [3]:

VHN = H 0 + B/d. ( 2
)
Where H 0 is the absolute hardness and B is a constant. This equation corresponds to a variation of the indentation diagonal with the applied load [4]:

P = a 1 d + a 2 d 2 .
(

With a 2 = H 0 and B=1.8544 a 1. Some attempts have been made to give a physical significance to Eq. ( 3) [5]. In our opinion, the size effect in hardness is due to the formation of a pile-up at the immediate edge of the impression [6]. This is confirmed by the investigation of Chaudhri and Winter [7], which show that the pile-up supports a part of the indenter load. Many attempts have been made to establish a quantitative correlation between the hardness of composite material and that both of coating and substrate. The main models developed are based on the area law-of-mixture [8][9][10][11][12][13][14][15] or the volume law-of-mixture [16][17][18]. Jönsson et al. proposed that the composite hardness H C can be expressed as a weighted mean of the film hardness H f and the substrate one H S:

H C = (A f /A) Hf + (A s /A) Hs. (4) A f /A = 2kt/d-(kt/d) 2 .
Where A is total projected area of the impression, equal to the sum of the projected areas of the rim, A f, and the interior, A S.

The objective of this study was to evaluate the hardness of porous hydroxyapatite coated onto Ti6A14V substrate.

II. EXPERIMENTAL PROCEDURE

Plasma spraying is used to produce the composite of hydroxyapatite coating on Ti6A14V substrate. Before spraying, the samples were sandblasted in order to obtain a favorable random surface roughness for mechanical bonding in the coating/substrate interface. Surface roughness (R a ) was 2.29 micrometers and coating thickness varies between 100 and 120 micrometers. Indentation were made using a Vickers diamond indenter at peak contact loads between 50 gf and 2000 gf on the surface prepared by the way of polishing. The indentations were made and measured in ambient air and approximately 1 minute elapsed between indenting and measuring.

III. RESULTS AND DISCUSSION

The hardness values of substrate (H S ) and of composite (HV C ) were determined according to general equation of Vickers hardness. The particular points of model are shown in the table 1.

The measured hardness values were to be dependent on coating thickness. The hardness values are plotted in Fig. 1 as a function of the reciprocal indentation depth 1/d. At each load, more than six experimental data are recorded to calculate the hardness value.Interpretation of the model based on the area law-of-mixture:

Iost and Bigot [19] consider the relation (4) without the Jönsson and Hogmark simplifications. Taking into account the second-and third-order 1/d terms, composite hardness can be expressed using the following expansions:

H c = H 0s + A 1 /d + A 2 /d 2 + A 3 /d 3 .
(

) with A 1 = 2kt (H 0f -H 0s ) +B S A 2 = 2kt (B f -B S )-k 2 t 2 (H 0f -H 0s ) A 3 = -k 2 t 2 (B f -B S ) 5
A mathematic analysis shows that the form of equation ( 5) depends on the product of = (H 0f -H 0s )/ (B f -B S ) and kt.

For our experimental data, = 0.03 and k = 0.5 since cracks formation occur at the corners of the indentation print. This product is less than 0.5.The substrate hardness measured at different applied load after removal of the coating is:

H s = 233 + 4871/d (6)
Application of Eq. 5 and 6 with the experimental results of the composite hardness aim to obtain the film hardness and its variation with the applied load :

H f = 209 + 4061/d (7)
It is worth noting that a hardness intrinsic value of 484 VHN has been measured for dense HA.

IV. CONCLUSIONS

One of the most important particularities of hardness measurement is that they are load dependent. In order to obtain an intrinsic value of hardness for the coating/substrate systems, it is possible to use the Jönsson et al. Model. The model based on the area law-of-mixture is suitable at condition that Kt/d is less than 1. Values higher than 1 for kt/d correspond to an indent diagonal length less than the coating thickness, and consequently the measured hardness equal the coating one. From the above discussion it is not so difficult to obtain an absolute hardness because the influence of substrate has been found effective only for the highest load. It is also to be noticed that the hydroxyapatite hardness of the coating is less than the bulk hydroxyapatite and substrate ones. This result is a consequence of the presence of porosity in the plasmasprayed hydroxyapatite coating.

A1

  . Behnamghader 1 , B. Farsadzadeh 2 , D. Najjar 3 , A. Iost 3 Materials and Energy Research center, Tehran, P.O. Box 14155-4777, Iran 2 Biomedical Department, Azad Islamic University, P.O. Box 14665-1445, Tehran, Iran 3 ENSAM/LMPGM CNRS UMR 8517, 8 Bd . Louis XIV, 59046 Lille Cedex, France Abstract--The extensive use of appropriate coatings to improve wear resistance, friction coefficient, electrical properties, corrosion resistance and biomedical application has stimulated a growing interest in their mechanical properties and especially hardness testing that is routinely used for coating evaluation. In this study Jönsson and Hogmark model is applied for the porous hydroxyapatite produced by plasma spraying on Ti6A14V substrate. Firstly, the effect of indentation load on hardness values of coating and substrate are studied. The modified Jönsson and Hogmark model is used to explain the composite hardness behavior and the effect of coating porosity. Keywords-Hardness, Coating, Hydroxyapatite, Plasma-spray, porosity

Fig. 1 .

 1 Fig. 1. A Schematic representation of the composite hardness versus 1/d.Tangent at H C , Model, H f , H s.

Table 1 .

 1 Some particular points of composite, substrate and coating hardness.

	d	Kt/d	A f /A	H c	H s	H f
	(mm)			(Kgf/mm 2 )	(Kgf/mm 2 )	(Kgf/mm 2 )
	High	0	0	233	233	209
	60	1	1	277	314	277
	30	2	0	395	395	344