Florine Paraz
email: florine.paraz@irphe.univ-mrs.fr

Christophe Eloy
email: christophe.eloy@irphe.univ-mrs.fr

Lionel Schouveiler
email: lionel.schouveiler@irphe.univ-mrs.fr

Experimental study of the response of a flexible plate to a harmonic forcing in a flow

Keywords: Fluid-structure interaction, flexible plate, harmonic forcing, resonance, non-linearity, propulsion Interaction fluide-structure, plaque flexible, forçage harmonique, résonance, non-linarité, propulsion

Most aquatic animals propel themselves by flapping flexible appendages. To gain insight into the effect of flexibility on the swimming performance, we have studied experimentally an idealized system. It consists of a flexible plate whose leading edge is forced into a harmonic heave motion, and which is immersed in a uniform flow. As the forcing frequency is gradually increased, resonance peaks are evidenced on the plate response. In addition to the forcing frequency, the Reynolds number, the plate rigidity and the forcing amplitude have also been varied. In the range of parameters studied, the main effect on the resonance is due to the forcing amplitude, which reveals that non-linearities are essential in this problem.

Introduction

Animal swimming has attracted a lot of attention from hydrodynamicists in the last decades. This interest was partly motivated by the design of bioinspired aquatic propulsion devices. Numerous reviews can be found on the different aquatic propulsion modes, such as the monographs of Lighthill [START_REF] Lighthill | Mathematical biofluiddynamics[END_REF] and Childress [START_REF] Childress | Mechanics of Swimming and Flying[END_REF] or the articles of Lighthill [START_REF] Lighthill | Hydromechanics of aquatic animal propulsion[END_REF], Sfakiotakis et al. [START_REF] Sfakiotakis | Review of fish swimming modes for aquatic locomotion[END_REF] and Triantafyllou et al. [START_REF] Triantafyllou | Hydrodynamics of fishlike swimming[END_REF]. Propulsion using flapping appendages is common for large aquatic animals and, in particular, most of the fast swimmers such as sharks, tuna and dolphins flap their caudal fins to propel themselves. An observation of these appendages shows that they are generally flexible along the chord. It has often been argued (e.g. [START_REF] Bainbridge | Caudal fin and body movement in the propulsion of some fish[END_REF] and [START_REF] Fish | Passive and active flow control by swimming fishes and mammals[END_REF]), that this flexibility enhances the swimming performance. However, the origin of this performance enhancement has proved elusive so far.

Inspired by this caudal-fin swimming mode, different studies have been carried out to quantify the propulsion efficiency of a flapping foil. Propulsion by rigid foils has been first studied theoretically by Lighthill [START_REF] Lighthill | Aquatic animal propulsion of high hydromechanical efficiency[END_REF] and Wu [START_REF] Wu | Hydromechanics of swimming propulsion[END_REF]. It has also been the object of experiments by Triantafyllou et al. [START_REF] Triantafyllou | Wake mechanics of thrust generation in oscillating foils[END_REF], Anderson et al. [START_REF] Anderson | Oscillating foils of high propulsive efficiency[END_REF], Schouveiler et al. [START_REF] Schouveiler | Performance of flapping foil propulsion[END_REF], and Buchholz and Smits [START_REF] Buchholz | The wake structure and thrust performance of a rigid low-aspect-ratio pitching panel[END_REF], among others. These experimental studies have all reported that propulsion performance is maximized for a flapping frequency f corresponding to a Strouhal number St ≈ 0.3, where St = 2f A T E /U with A T E the amplitude of the trailing edge motion and U the swimming speed. The same value of St ≈ 0.3 can be observed in nature for swimmers with moderate aspect ratios (cetaceans, sharks, salmons, trouts and scombrids) as seen in the data recently compiled in [START_REF] Eloy | Optimal Strouhal number for swimming animals[END_REF].

The numerical simulations of Katz and Weihs [START_REF] Katz | Hydrodynamics propulsion by large amplitude oscillation of an airfoil with chordwise flexibility[END_REF] have shown that, using a flapping foil flexible along its chord rather than a rigid one, yields an increase of the propulsive efficiency. In this case, because of the foil flexibility, its natural structural frequencies are introduced in the system, in addition to the flapping frequency. These structural frequencies depend not only on the geometry and the material of the foil but also on the surrounding flow because of added-mass effects. Two-dimensional numerical simulations of Alben [START_REF] Alben | Optimal flexibility of a flapping appendage in an inviscid fluid[END_REF] and Michelin and Llewellyn Smith [START_REF] Michelin | Resonance and propulsion performance of a heaving wing[END_REF] have later revealed peaks in the amplitude of the plate response, when the flapping and structural frequencies are resonant. They showed that efficiency can be maximized at these resonance peaks.

Gain in propulsive performance through the chordwise flexibility has been confirmed by the experimental works of Prempraneerach et al. [START_REF] Prempraneerach | The effect of chordwise flexibility on the thrust and efficiency of a flapping foil[END_REF], Heathcote and Gursul [START_REF] Heathcote | Flexible flapping airfoil propulsion at low Reynolds number[END_REF], or Marais et al. [START_REF] Marais | Stabilizing effect of flexibility in the wake of a flapping foil[END_REF]. Moreover, Dewey et al. [START_REF] Dewey | Scaling laws for the thrust production of flexible pitching panels[END_REF] for a pitching flexible plate, Alben et al. [START_REF] Alben | Dynamics of freely swimming flexible foils[END_REF] and Quinn et al. [START_REF] Quinn | Scaling the propulsive performance of heaving flexible panels[END_REF] for a heaving flexible plate have clearly evidenced the phenomenon of resonance experimentally, as the forcing frequency is varied, and considered its link with propulsive performance, but none of these studies has considered the effect of changing the forcing amplitude.

In this study, to have a better understanding of the dynamics of a flexible fin, we have examined the response of an elastic plate immersed in a uniform flow and forced into a harmonic heave motion at its leading edge. The objective has been to quantify the influence of the different experimental parameters: the amplitude and frequency of forcing, the flow velocity, and the plate bending rigidity. This problem involves complex fluid-structure interactions since the flow load deforms the plate, whose motion in turn affects the flow. We believe that the full complexity of these interactions can only be addressed with experimental studies because, at the present time, accurate numerical simulations are limited to moderate Reynolds numbers and theoretical models usually assume linear problems, i.e. small-amplitude propulsion. This paper is organized as follows: first the experimental set-up and methods are presented. Then the response of flexible plates to harmonic heave forcing is analysed and effects of the different experimental parameters are discussed. Finally, some conclusions are drawn and discussed in the context of swimming.

Experimental set-up

Experiments have been conducted with the set-up schematically illustrated in Fig. 1(a). It consists of a horizontal flexible plate molded out of polysiloxane with a rounded leading edge and a tapered trailing edge (Fig. 1(b)). The plates used have a thickness e = 0.004 m, span s = 0.12 m, and chord c = 0.12 m, giving an aspect ratio s/c = 1. During molding, a rigid axis is inserted inside the plate at the leading edge; this axis has some roughness to prevent the rotation of the plate around it. This set-up reduces the flexible length of the chord to about 0.115 m. The plate is then immersed in a uniform flow, of velocity U , of a free surface water channel. The test section is 0.38 m wide, with a depth of water at rest of 0.45 m.

As shown in Fig. 1(a), the axis is attached to an inverted U-frame such that the plate, at the leading edge, is maintained parallel to channel flow. This inverted U-frame is set in vertical motion by a computer-controlled linear actuator, which allows to impose an arbitrary heave motion to the leading edge. In the present experiments, the leading edge has been forced into a harmonic heave motion: A LE cos(2πf t). The plate is confined between two vertical walls separated from the plate by less than 1 mm. The role of these confinement walls is, first, to minimize the flow around the side edges of the plate and, second, the inverted U-frame being outside of these walls, to avoid the perturbations from the wake of the frame. In addition, free surface effects have been prevented by a rigid horizontal wall placed above the experiment. Channel and confinement walls have been made of transparent material to facilitate visualizations.

To investigate the response of the plate when forced into heave motion, visualizations have been carried out through the side wall of the water channel and recorded with a video camera. The successive shapes and positions of the plate centerline have then been extracted by image analysis of the recordings. A quantitative characterization of the plate response is achieved by measuring the simultaneous displacements of the leading and trailing edges as a function of time by means of two high-accuracy laser sensors (Fig. 1(a)).

In the present study, the variations of four experimental parameters have been considered. First, the frequency f and amplitude A LE of the forcing have been varied in the ranges f = 0.2 -8 Hz and A LE = 0.004 -0.014 m. Then, the flow velocity has been increased up to U = 0.1 m.s -1 . This velocity is expressed in dimensionless form using the Reynolds number based on the chord: Re = U c/ν, where ν is the kinematic viscosity of water. Finally, three plate rigidities have been tested using different materials: B = 0.018, 0.028, 0.053 Nm. For these three plates, the density relative to the water is about 1.2.

A typical experiment consists, for a plate of given bending rigidity, in fixing the Reynolds number Re and the forcing amplitude A LE and in recording the plate response as the forcing frequency f is gradually varied. Experiments have been repeated changing A LE , Re, and using plates of different bending rigidity.

Results

Prior to the experiments with forcing, we have considered the response of the plate to an impulse perturbation of its trailing edge, in water at rest. The damped response to such a perturbation is shown in Fig. 2 for the plate of bending rigidity B = 0.053 Nm. From this signal, we deduce the lowest natural frequency f 0 of the plate in water that will serve as a reference frequency in the following. For the three plates tested in the present study, we find the natural frequencies f 0 = 0.75, 0.99, 1.30 Hz for bending rigidities B = 0.018, 0.028, 0.053 Nm respectively. Note that the ratios of these frequencies scale like the square root of the bending rigidity ratios (as expected from the vibration analysis of a cantilever beam) with an agreement better than 6%. The response of the flexible plates to the harmonic forcing, A LE cos(2πf t), imposed at their leading edge is then considered. We first note that for all the cases tested the plate deforms mainly along the chord and its displacement is harmonic with the same frequency as the forcing. The displacement of the trailing edge can thus be expressed as A T E cos(2πf t + φ) where φ is the phase relative to the leading edge displacement.

The plate response is characterized considering the evolution of the relative amplitude A T E /A LE and the phase φ as the forcing frequency f is varied. This response is illustrated in Fig. 3 for a representative case: A LE = 0.004 m, Re = 6000 and B = 0.018 Nm. In this plot, f is made dimensionless using the natural frequency f 0 of the plate.

Remarkably, the amplitude curve exhibits two distinct peaks. The first peak is sharp and occurs at a frequency f close to the natural frequency f 0 , with a maximum of the trailing edge amplitude more than 2.5 times the forcing amplitude. The second peak, at a frequency f /f 0 between 6.0 and 6.5, is flatter and lower in amplitude. It is recalled that the three first natural vibration modes a clamped plate have dimensionless wavenumbers k 0 c = 1.875, k 1 c = 4.694, and k 2 c = 7.855 and that the ratios of the corresponding natural frequencies f 0 , f 1 and f 2 scale like the square of wavenumber ratios. It results that f 1 /f 0 = (k 1 /k 0 ) 2 ≈ 6.3. We can thus conclude that the two peaks observed at f ≈ f 0 and f 1 in Fig. 3 correspond to resonances of the forcing with the first two natural structural modes. According to this analysis the third resonance peak would be expected at a frequency f ≈ f 2 with f 2 /f 0 = (k 2 /k 0 ) 2 ≈ 17.6, which is outside of the frequency range explored in the present experiments.

The evolution of the trailing edge phase φ relative to the imposed leading edge motion is also shown in Fig. 3. The phase appears to be close to -π far from the two resonance peaks and approaches -π/2 at the first peak which is what is expected from a simple damped oscillator model. In contrast, it is not possible to distinguish a clear trend for φ at the second peak, where it continuously varies between -3 and -5. In Fig. 4, the deformation of the plate at the two resonant peaks is illustrated by plotting superimposed views of the plate centerline during one forcing cycle. It should be noted that direct visualizations of the plate do not reveal significative deformations along the span, such that the plate deflection is well represented by the centerline displacement. It can also be remarked that a weak up-and-down asymmetry in the mode shape is apparent, which is due to the plate material being slightly denser than water.

Figs. 4(a) and (b) show the mode shapes at the two resonance peaks f /f 0 = 1 and 6.3, respectively, in the laboratory frame. Contrary to the first mode, the second mode exhibits a neck close to its trailing edge showing that higher structural modes are involved at high frequencies. The same modes are represented in the frame attached to the plate leading edge in Figs. 4(c) and(d). These views reveal their close similarities with the flutter instability modes of a clamped plate immersed in a uniform flow investigated by Eloy et al. [START_REF] Eloy | Aeroelastic instability of cantilevered flexible plates in uniform flow[END_REF], among others. Note that these deformations are different from the natural modes of a flexible beam in vacuum. In particular, the second mode (Fig. 4(d)) does not exhibit a node but only a pseudo-node or neck, revealing once again that the structural modes and their corresponding frequencies correspond to the system plate + surrounding fluid and not the plate alone, as emphasized by Michelin and Llewellyn Smith [START_REF] Michelin | Resonance and propulsion performance of a heaving wing[END_REF] in their numerical study. So far, we have described the frequency response of a flexible plate for fixed values of the forcing amplitude A LE , the Reynolds number Re, and the bending rigidity B. We will examine now how variations of these parameters affect the frequency response. To do so, the same protocol is used: A LE , Re and B are fixed and the relative response A T E /A LE is plotted as function of the normalized frequency f /f 0 (Fig. 5).

Effect of the plate rigidity

In Fig. 5(a), the response curves are plotted as a function of the dimensional forcing frequency f for three values of the plate rigidity B. We note that the (a) amplitude maxima are not affected by changing B and that the resonance peaks move toward higher frequency as B is increased. However, the natural frequencies f 0 increasing with the plate rigidity, when the response curves are plotted using the dimensionless frequency f /f 0 , they all collapse on a single curve as seen in Fig. 5(b). In conclusion, in the limit of the present study, which considers plates with bending rigidities varying by a factor 3, we could not detect any significant effect of the plate rigidity, as long as the frequency is properly normalized with the natural frequency f 0 .

Effect of the Reynolds number

The effect of the Reynolds number Re = U c/ν on the plate response is illustrated in Fig. 5(c) representing the frequency response for three values of Re. The first value is Re = 0 corresponding to water at rest, the two others are 6000 and 12000, corresponding to U = 0.05 and 0.1 m.s -1 , respectively. We observe both a decrease of the amplitude maxima and a shift of the resonance peaks toward higher frequencies, as Re is increased. This latter observation is due to the normalization performed using the natural frequency f 0 of the plate in water at rest while the natural vibration modes are modified by a surrounding flow as noted in [START_REF] Michelin | Resonance and propulsion performance of a heaving wing[END_REF]. However, these variations in the plate response as Re is changed appear only minor. This is likely due to the large values of the reduced frequency f r = πf c/U , which measures the ratio between the typical time taken by a fluid particle to travel along the chord to the forcing period. For smaller values of f r , one would expect the wake to become more important and to affect the frequency response more significantly.

Effect of the forcing amplitude

Contrary to the others parameters tested, variations of the forcing amplitude A LE have a strong influence on the plate response. This influence is shown in Fig. 5(d) where the frequency response is plotted for three values of A LE . An increase of the forcing amplitude by a factor of 3.5 (from 0.004 to 0.014 m) leads to a decrease of the relative response amplitude of the trailing edge by more than 30% and a slight decrease of the normalized frequency at the resonance peaks. This important effect of the forcing amplitude is a signature of non-linear effects.

These non-linearities likely originate from the large-amplitude plate deformations, which introduce non-linear terms in both the plate and flow dynamical equations, through geometrical and damping terms. For instance, the impermeability condition, which ensures the coupling between the plate and the fluid, has to be applied on a displaced interface, yielding terms that depend non-linearly on the motion amplitude. These terms correspond to cubic nonlinearities [START_REF] Eloy | The origin of hysteresis in the flag instability[END_REF]. Another source of nonlinearity is the drag force exerted on the plate normally to its surface as it moves, also called the resistive force, which corresponds to quadratic non-linearities [START_REF] Ramananarivo | Rather than resonance, flapping wing flyers may play on aerodynamics to improve performance[END_REF].

Calling x the amplitude of the first bending mode of the plate, we expect a weakly nonlinear dynamical equation of the form

ẍ + (2πf 0 ) 2 x + µ ẋ + ν| ẋ| ẋ + δx 3 = γ cos(2πf t), (1)
where the first two terms describe the bending mode of the plate as an harmonic oscillator (with eigenfrequency f 0 ), the third term is related to the internal damping of the plate (µ > 0), the fourth term is the quadratic nonlinearity due to the drag normal to the plate (ν > 0), and the fifth term gathers all the cubic nonlinearities originating from the fluid-structure interaction, the plate dynamics, and the fluid load (in reality there should also be terms of the form x 2 ẍ and x ẋ2 ). The right-hand side corresponds to the forcing due to the heaving motion, where γ > 0 without loss of generality. This equation is the classic Duffing oscillator with an additional quadratic term, ν| ẋ| ẋ. It is beyond the scope of the present paper to calculate the coefficients µ, ν, and δ appearing in this equation, however, it can be formally solved by assuming

x(t) = a cos(2πf t + φ) + h.o.t., (2)
where h.o.t. stands for "higher order terms" and refers here to higher harmonics that can be neglected near resonance. Neglecting also the cubic nonlinearities for simplicity, the dynamical equation ( 1) can be projected onto the main harmonics to give an implicit equation for the amplitude a of the bending mode oscillations

(2π) 4 f 2 0 -f 2 2 a 2 + (2π) 2 µf + 16 3 νf 2 a 2 a 2 = γ 2 . ( 3
)
Solving this implicit equation shows, in particular, that at the resonance (i.e. f = f 0 ), the relative amplitude a/γ is a monotonic decreasing function of γ. This is conform to the experimental observations, where the relative amplitude A T E /A LE is also a decreasing function of the forcing amplitude A LE . In Fig. 6, the relative amplitudes a/γ obtained with equation (3) for different values of the forcing parameter γ are plotted as a function of the normalized forcing frequency f /f 0 . It shows a good qualitative agreement with the experimental measurements reported in Fig. 5(d) near the first resonance. In particular, the value of the relative response at its maximum (around f ≈ f 0 ) is decreasing when the forcing amplitude is increased. The shift of the response peak towards lower frequencies for large forcing amplitude is also reproduced. However, the second peak observed in Fig. 5(d) around f /f 0 ≈ 6 is not reproduced in the model (Fig. 6) since it only accounts for a single natural mode of vibration whereas the flexible plate has an infinite number of structural modes.

The experimental results presented in Fig. 5(d) and the above discussion on the Duffing equation show that an accurate description of the plate response at resonance must include non-linear effects. It is important to stress that these non-linear effects are relevant even if the relative amplitude of the deformation, A T E /c, and the angle between the plate and the flow direction are both relatively small. Yet, these effects are usually neglected in the literature, probably because until now, to our knowledge, there has not been any systematic measurements of the response of a flexible plate when the forcing amplitude is varied (see e.g. [START_REF] Dewey | Scaling laws for the thrust production of flexible pitching panels[END_REF]- [START_REF] Quinn | Scaling the propulsive performance of heaving flexible panels[END_REF]).

Conclusion

In summary, experiments have been conducted to investigate the response of a flexible plate in a uniform flow, when its leading edge is forced into a harmonic heave motion. This study focussed in particular on the influence of the different experimental parameters on the characteristics of the frequency response.

As observed in previous studies, the response curves, representing the relative response amplitude with respect to the forcing frequency, exhibit peaks of resonance for the natural modes of the plate in the fluid flow. The flow velocity and the plate bending rigidity have only minor effects on the response curves as long as the forcing frequency is properly normalized with the natural mode frequency. In contrast, the forcing amplitude strongly affects the plate response indicating the importance of non-linearities in this system.

The present experiments performed with an idealized geometry can help to gain insight into the physical mechanisms of thrust production by flapping flexible appendages. It is known from the seminal work of Lighthill [START_REF] Lighthill | Note on the swimming of slender fish[END_REF] that the thrust scales as the square of the trailing edge deflection amplitude in the linear limit. This study shows that this amplitude can be maximized when the flapping frequency is close to the frequency of one of the natural modes. However, the non-linear effects probably affect Lighthill's scaling of thrust production for large amplitude and there is likely an optimal amplitude at which propulsion efficiency is maximized. Moreover, the present experimental results show that if one wants to model the fluid-structure interactions of a flapping flexible plate in a uniform flow plate, non-linear effects have to be taken into account. Such a model is currently being developed and will be subject of a forthcoming work.

Figure 1 .

 1 Figure 1. Sketch of the experimental set-up (a) and of the flexible plate (b).

Figure 2 .

 2 Figure 2. Response of a plate to an impulse perturbation. The deflection of the trailing edge is plotted as function of time for the plate of bending rigidity B = 0.053 Nm in water at rest.

Figure 3 .

 3 Figure 3. Response of the plate to a harmonic heave forcing. The relative amplitude of the trailing edge displacement, A T E /A LE , and the corresponding phase shift φ are plotted as a function of the normalized forcing frequency f /f 0 , for A LE = 0.004 m, Re = 6000 and B = 0.018 Nm.

Figure 4 .

 4 Figure 4. Mode shapes, at the first f /f 0 = 1 (a, c), and second resonance peak, f /f 0 = 6.3 (b, d). These deformations are shown in the laboratory frame (a, b) and in the frame attached to the leading edge (c, d), A LE = 0.004 m, Re = 6000 (flow from left to right) and B = 0.018 Nm.

Figure 5 .

 5 Figure 5. Effects of the experimental parameters on the frequency response. Response curves as a function of the forcing frequency f (a) and of the normalized frequency f /f 0 (b) for A LE = 0.004 m, Re = 6000 and three values of the bending rigidity B. Frequency response for A LE = 0.004 m, B = 0.028 Nm and three values of the Reynolds number Re (c) and for Re = 6000, B = 0.018 Nm and three values of the forcing amplitude A LE (d).

Figure 6 .

 6 Figure 6. Response curves obtained with equation (3) for different values of the forcing γ. The following parameters have been used: f 0 = 1/2π, µ = 0.1, ν = 3, and γ = 0.04, 0.1, or 0.14.


Professor Caroline Verzat

Fabienne Bornard

Theme: Spaces of Transformation and Transformation of Spaces Symposium: Cradles for imagination and initiative. Title: Imaging the future of entrepreneurship education: scenarios building as shuttles to the future

Introduction

Entrepreneurship Education is fostered by all governments because it is considered as a major source of social and economical value creation. Indeed there has been a boom in the number of programs at university level in the last 20 years. Following that movement, pedagogical research about its final goals, its impact and its methods is now recognized as an important field in entrepreneurship research. [START_REF] Naia | A systematization of the literature on entrepreneurship education,Challenges and emerging solutions in the entrepreneurial classroom[END_REF] conclude from their literature review of the field that most researches produce knowledge about WHAT to teach in entrepreneurship, HOW (with what kind of activities or methods) or FOR WHICH RESULTS (impact studies). But there is a lack of understanding about WHY we should teach entrepreneurship: What are the final goals of entrepreneurship education? Does entrepreneurship education imply a new way of learning? Moreover, although educational practitioners like [START_REF]In pursuit of a new "enterprise" and "entrepreneurship" paradigm for learning: creative destruction, new values, new ways of doing things and new combinations of knowledge[END_REF], [START_REF] Surlemont | Pédagogie et esprit d'entreprendre[END_REF] or Jones (2011) outline that future entrepreneurs need to be socialized within the entrepreneurial ecosystem, we haven't found any clear indication in the literature about WHERE entrepreneurship education should take place: in what kind of educational institution or specific learning environment?

If we consider that there is such a gap between societal needs and available knowledge in entrepreneurship education, we propose to use a creative route to design entrepreneurship education of the future. Drawing on Polak (1973:1) suggestion that society is "pulled forward by its own magnetic images of an idealized future and pushed from behind by its realized past", we tried to investigate "pull-forward" creative visions of the future that are carried out by teachers, researchers, students in entrepreneurship as well as entrepreneurs and coaches in entrepreneurship. Could all these stakeholders reach a common understanding of what is desirable in entrepreneurship education? Do these creative visions give an answer to the WHY and WHERE questions?

For this purpose the scenario building method has been tested as a ...cradle for imagining the future of education. We concretely organized 7 creative workshops with the three populations aiming at letting participants elaborate scenarios for the future of entrepreneurship education. This communication analyses the preliminary results of these scenarios. First we expose the theoretical framework of this explanatory research. The second part of our communication presents the workshop design and the methodology for scenario analysis. The third part displays our results. The conclusion offers a discussion and future paths for research.

1) Theoretical framework

Entrepreneurship education can be defined broadly as "any pedagogical program or process of education for entrepreneurial attitudes and skills" [START_REF] Fayolle A | Assessing the impact of entrepreneurship education programmes: a new methodology[END_REF]. But the absence of a unified, accepted theory or definition of entrepreneurship education is a problem that remains to be solved [START_REF] Sexton | Entrepreneurship education: suggestions for increasing effectiveness[END_REF][START_REF] Hills | Variations in university entrepreneurship education: an empirical study of an evolving field[END_REF][START_REF] Fiet | The theoretical side of teaching entrepreneurship[END_REF]. [START_REF] Fayolle | From craft to science: Teaching models and learning processes in entrepreneurship education[END_REF] argue that there is a lack of a precise definition of entrepreneurship as a teaching field, where philosophical conceptions about teaching, the role of the teacher and the role of the students should be clarified in each course. In fact, the number of entrepreneurship education programs has shown an impressive progression in all countries, and this boom in pedagogical programs was supported by an increase in the diversity of pedagogic approaches, from traditional transmissive approaches in full academic courses to more experiential learning approaches, with case studies, business plans (fictive or real, or in support to a real entrepreneur), interviews with entrepreneurs, guest speakers, business games, internships, development of small businesses or startups in project approaches…These approaches usually meet success with students who appreciate more active pedagogies, enjoy to have the opportunity to speak to "real" entrepreneurs and be challenged with projects…One could be satisfied with this: a lot of work, a lot of energy and imagination to propose those attractive pedagogical sytem. As a matter of fact narrative reviews lead to contradictory findings about the supposed positive link between entrepreneurship education and various entrepreneurship-related human capital assets. Recently some interesting metaanalysis were conducted. For instance [START_REF] Martin | Examining the formation of human capital in entrepreneurship: A meta-analysis of entrepreneurship education outcomes[END_REF] have studied 42 rigorous studies about the impact of Entrepreneurship Education Programs in Higher Education institutions. Their results show that entrepreneurship education has a global positive effect on entrepreneurial knowledge, skills and intentions as well as on the number of start-ups and their performance. Academic courses show a higher impact on firms' numbers and performances but there is no significant impact difference between both categories on human capital criteria. Despite these developments, the lack of precise understanding of what really works in education to reach desirability and feasibility perceptions towards entrepreneurial life and activities is pointed out by many researchers (GEM 1 2010, GEM 2013).

What are the learning and teaching processes that can raise specific self-efficacy perceptions, selfesteem, or entrepreneurial values and attitudes? Which are the achievable learning goals at different ages, for different target groups? What are the adequate contents and activities? Are there more appropriate methods than others? Are there relationships between pre-post satisfaction rates of a particular program, learning outcomes and longer term behavioral transfer? A rather provocative question is raised by Fayolle ( 2013 [START_REF] Fayolle | From craft to science: Teaching models and learning processes in entrepreneurship education[END_REF] to design and implement an education programme: objectives, contents, audience, methodologies and evaluation. Surprisingly, two dimensions appear not to be addressed directly: the objectives (why?) and the audience (for whom ?). The question "why" can be analysed with two levels of learning objectives: individual (or micro) and socio-economical (or macro) objectives. They are related with the development of an entrepreneurial mindset, the transmission of techniques to create new ventures and the improvement of the society mindset concerning the entrepreneurial phenomenon. Finally, we can only list the main insights into best practices for entrepreneurship educators: experiential learning rather than transmission of knowledge, diversity of educational experiences, learner's active participation, etc. So there's a urge in researching the why question.

Facing these difficulties, how can we invent new solutions if we, as professors, designers of educational cursus and researchers in entrepreneurship remain stuck to our routines and mental representations? In fact the amount of collective experience is huge but, at this point of the framing of the field, it's necessary to find means to take some distance with implicit pedagogical models.

1 Global Entrepreneurship Monitor: gemconsortium.org.

How could we create a fertile space of imagination in order to reinvent the entrepreneurship education with its present stakeholders?

For this purpose, we decided to reflect from a prospective point of view in order to create a distance from the present contingencies and to mobilize the imagination resources: as Csikszentmihalyi puts it, the flow of creativity needs forgetting Self, Time, and Surroundings. A few attempts were made in entrepreneurship research to study narrative imaginative resources. For instance [START_REF] Hindle | Skillful Dreaming: Testing a General Model of Entrepreneurial Process with a Specific Narrative of Venture Creation[END_REF], on a proposition of Gartner who launched this original initiative, develops and tests a comprehensive model of entrepreneurial process utilizing content analysis of the book The Republic of tea, How an Idea Becomes a Business? This text presents in detail a venture creation and how the founders dreamed of their future company. [START_REF] Polak | The Image of the Future[END_REF] studies how images of the future has always helped Homo Sapiens to shape the future. The exercise to elaborate images of the future allows to draw boundaries in the thought-realm: it is only by redrawing the boundaries of the unknown that man can increase his knowledge. The entrepreneur, as "a visionary activist" manages to create the future, on the of its vision and holistic guiding images of future possibilities [START_REF] Nyström | Creativity and entrepreneurship in FORD[END_REF].

In order to help developing images of the future, we found that it could be interesting to use scenario building, a collective method traditionally used for strategy [START_REF] Boaventura | Is your vision consistent? A method for checking, based on scenario concepts[END_REF]. Scenario planning consists in trying to imagine the future and its different options: in this way this method implies to fly away from the actual situation. Our analyse of the literature on scenario planning shows that they can be used into different contexts and objectives: planning tools [START_REF] Schoemaker | Scenario planning: A tool for strategic thinking[END_REF], methods to check the consistent of a vision (Boaventura & Fischman, 2008), thought experiments preparing action [START_REF] Aligica | Scenarios and the growth of knowledge: Notes on the epistemic element in scenario building[END_REF], ways to try to control the future [START_REF] Ringland | The role of scenarios in strategic foresight[END_REF][START_REF] Miller | Scenarios, real options and integrated risk management[END_REF]Glykas, 2013) or tools to work on ethics perceptions [START_REF] Nguyen | Ethics Perception: Does Teaching Make a Difference?[END_REF]al. 2008, Persons, 2009). Control an unpredictable future, free imagination...scenario building can also lead, to [START_REF] Csikszentmihalyi | Creativity: Flow and the psychology of discovery and invention[END_REF], Nyström, 1993[START_REF] Fillis | Using creativity to achieve an entrepreneurial future for arts marketing[END_REF], "experimental fumbling" [START_REF] Freinet | La technique Freinet, méthode nouvelle d'éducation populaire, basée sur l'expression libre par l'imprimerie à lécole[END_REF], "Empty Habitable Frameworks" (Fourcade and Krichewsky2 ), "Bricolage" [START_REF] Stinchfield | Learning From Levi-Strauss' Legacy: Art, Craft, Engineering, Bricolage, and Brokerage in Entrepreneurship[END_REF] and Effectual pedagogy (Sarasavathy and Venkataraman, 2011).

2) Methodology

Data were collected during four sessions of this workshop, between January 2013 and February 2014, gathering a total of 176 persons and 65 scenarios (see table 1annexes for more details).

The workshop was structured in 3 parts. The first phase (1h) was dedicated to creative designing of Entrepreneurship Education (EE) in 2044. Participants sitting in groups of 6 were asked to imagine EE in 2044 through a brainwriting procedure. Each person wrote the first sentence of a scenario starting with "Once upon a time in 2044…" and passed this story starting point to his neighbour who was asked to follow the story; The procedure went on until each of the 6 members had added one sentence to the story. Then the group chose his best story and prepared a poster of the scenario.

During the second phase (45 mn) each group presented his poster to all participants of . Thirdly, the researches, as facilitators, helped to formulate a synthesis and a critical reflective analysis on the major common features of the scenarios which had been produced (30 mn). A common discussion took place with participants to explore either philosophical features (learning principles, EE final goals or teacher-students stands), didactic features (specific activities, learning outcomes, audiences or assessment), environmental features (images and organization of learning space, institutional aspects), or characteristics of posters (main messages conveyed, visualization properties).

The 65 scenarios were then systematically analyzed by researchers. This analysis was conducted following the grounded theory methodological principles. A first list of anchors was produced by two independent researchers after reading all 65 scenarios. This list of 15 anchors was then organized in 4 main concepts (philosophical meaning of scenario exploring the WHY questions, didactic features of scenario exploring the HOW questions, environmental features of scenario exploring the WHERE questions, and also narrative features of scenario). These 4 concepts were inspired by different bodies of literature. The 3 educational concepts were borrowed from Fayolle and Gailly (2008) entrepreneurship teaching model, [START_REF] Biggs | Teaching for Quality Learning at University[END_REF][START_REF] Biggs | Aligning Teaching and Assessment to Curriculum Objectives, Imaginative Curriculum Project[END_REF] constructive alignment model, from Carré (1992) 3 macro/meso/micro levels for observing an educational system and from [START_REF] Nunes | An Action Research Model for the Management of Change in Continuing Professional Distance Education, ITALICS, Innovations in Teaching, And Learning in Information and Computer Sciences[END_REF] Educational Management Action Research Model. The fourth concept about narrative features of scenario was inspired by Bremont (1966) semiologic analysis of narratives, emphasizing the importance of the improvement goal and of obstacles and setting 3 possible active or passive roles (agent-hero, allies and adversaries). The third step of our analysis was to refine our coding by systematically applying the 15 anchors and 4 concepts-grid to all scenarios. Two independent researchers coded all scenarios through NVivo, compared and corrected their codings so as to reach a satisfactory Cohen's Kappa coefficient of inter-rater reliability (71% codings above 0.41). 260 codings emerged out of the 65 scenarios. They were grouped in 4 broad categories of nodes, with 15 subnodes. These subnodes help draw out a coherent picture of the most important messages conveyed by creative scenarios about the future of entrepreneurship education.

3) First Results

• Relevant categories to investigate scenarios regarding entrepreneurship education:

Philosophical, Didactical, Environmental and Narrative levels.

Scenario interpretation brought out very detailed categories on most nodes. (see table 2 below)

The philosophical level appears to be the most detailed one, with very clear codings implicitly referring to active pedagogy educational principles (PHI 1) but also to a large view of human development finalities (PHI 2). Not surprisingly a large diversity of activities are described on the didactic level, but with little development of learning goals, assessment or audience. The learning environment is balanced between organizational technological, timeframe and spatial meanings.

Narrative striking characteristics of scenarios are not very much developed: we could only distinguish heroes from obstacles (rarely human) and an ideal tone as opposed to a catastrophic tone.

Table 1: Analytic levels and nodes related

• Dominant nodes: Environmental Level and Philosophical Level

We applied the "Tree Map" analysis to show visually how our coding is distributed (comparison of nodes) and the density within nodes (number of references they contain). The greener the node is, the more significant it is.

Results presented below indicate that the future of entrepreneurship education is related to a

Learning Space where learners can experiment with a particular relationship between learner and teacher. Those three themes dominate the scenarios content:

-"Learning Space" -Environmental Level: 62% of scenarios / 103 encoded references. This node describes where the learning may emerge. It deals with either concrete spaces (garden, under the sea, on a boat, a big entrepreneurship house, on the beach) or with virtual spaces (bubble spaces, spaceship, flying structure, through dreams, intelligent clothes). (We will detail the content of this node in the following paragraph) -"Experimenting-Acting and Trying"-Philosophical Level: 51% of scenarios / 68 encoded references. This node belongs to the "Experiential Learning" node which is centered on real activities (projects, building schools, practical workshops, laboratory) and approaches (practicing, applying, learning by doing, entrepreneurial effectuation method) that can be useful to learn entrepreneurship. Participants insist on the possibility to try and to fail by transforming directly ideas into concrete business: "There will be no content anymore, just exchanging and reflecting between people. They will use an ideas modeling room which leads to an action room that helps to launch ideas."

-"Learner and Teacher Relationship"-Philosophical Level: 51% of scenarios / 67 encoded references. This node expresses the relationship that exists between the learner and the teacher: the roles (learning facilitators, actors of their own learning, openers of possibilities, help to reveal talents, explore learner's mind, adapt to learners), who the teachers are (robots, serial entrepreneurs, experts, wise people) , the nature of interaction (cooperation, exchanges, mentoring, tutorship, meeting in various places), the frequency of the interaction (sometimes, just when it is needed, new teachers appearing).

• Encoded level by population: Action (Coaches and Entrepreneurs) versus Reflection (Researchers and Students)

We wanted to compare the encoded references depending on the population profile: who talks about what ? The table below highlights the levels encoded by points of view. We apply a smoothing factor to ensure that each population is well represented whatever its size during the data collection.

Populations who are daily in action mode (Entrepreneurs and Coaches in Entrepreneurship) use less easily the narrative level than those who are more in reflection mode (Students in Entrepreneurship and Researchers). We observe that Researchers and Students in entrepreneurship can easily project into the Learning Space and can use a narrative level (heroes, overcome obstacles, emotions, particular tone) to describe their scenarios compared to the others populations. However there is little difference concerning the didactic level. From the philosophical level, we observe that researchers and entrepreneurs seem to be more visionary.

Table 2: Encoded references by population and broad category of code (level 1)..

• Learning Space : a facilitator for experimenting, dreaming, achieving finalities

This request helps us to underline the elements (the nodes) with which the Learning Space is often associated. The results below show that Learning Space (Environmental Level) is mainly linked to the Philosophical Level (50% of references), then to the Didactic Level (38% of references) and finally to the Environment Level (12% of references).

• Learning Space : Main images used within the scenarios

The Learning Space node includes 103 encoded references among the 40 scenarios. We highlight here the main images which have been used by the participants to describe the Learning Space. Four main images emerge from data:

-"Ephemeral sites". The Learning Space helps participants (learners, teachers…) to travel from one place to another one (spaceship, flying machines, teletransportation): "the students built a spaceship to go to another planet and create a school". It encourages also to learn by travelling where the activity happens: "Students travel from third-space to others in order to meet various groups of people and they refused to settle a base-camp". The learning space has no frontier to help developing freedom of movements.

-"Harmony with Nature". The Learning space is frequently associated to the nature (ocean, beach, mountains, countryside, garden) and to its participants: "All the building suspend in the sky, move around and fly. The school will look like a garden". They live, they change together: "Children can be on a mountain or closed to the sea. Depending on their biotic environment, children can create the place in which they feel like small fishes".

-"Technological tools". The Learning Space is equipped with different technologies that should help to improve learning ("Massive Open Course on the web": "intelligent cloth"), travelling ("flying carpets"; "a transport mean has been invented to explore the solar system"), experimenting ("This is not a school, but a laboratory, a workshop"; "experiment spaces thanks to the instantaneous transfer")

-"Caring and stimulating environment". The Learning Space is described as a place where people can learn comfortably ("a soothing and relaxing place") and experiment freely new ideas being protected by different systems : "soap-bubbles; bubbles-city; entrepreneurship's sanctuary". Or they can create their own space that will be shared : "Everyone creates a life-space in which everyone can look for original and fantastic ideas which have been cooked by the owner (...). No one is gonna judge, we give, we exchange, serving the idea production".

4) Conclusion

We trace a path from critical reflective analysis [START_REF] Mezirow | Transformative dimensions in adult learning[END_REF] of existing higher education institutions and their educational practices to pedagogical refoundation [START_REF] Fayolle | Personal views on the future of entrepreneurship education[END_REF].

The scenario building method appeared to be an efficient shuttle towards the future of entrepreneurship education with different stakeholders and different contexts. The results were quite surprising in the sense that scenarios were not as divergent as we could have expected: a few categories emerged quite easily from the first reading as relevant to the grounded theory. When a deeper analyse was conducted, we found that, as [START_REF] Naia | A systematization of the literature on entrepreneurship education,Challenges and emerging solutions in the entrepreneurial classroom[END_REF] demonstrated, the pedagogical activities were described with precision and variety but little development was noted on the subject of learning goals, assessment or audience.

The main result relies on the fact that we finally managed to explicit the controversial most of the time implicit "WHY" debate about the final goals and educational principles associated with entrepreneurship education (see the "Philosophical Level" node).

The second main result concerns the description of the "Learning Space" that has been imagined by participants, which was deeply linked to the future of entrepreneurship education. This Learning

  ): Do we really know what we are doing when we teach or educate people in entrepreneurship? The recent research of Naïa and al. (2014) seems to give him reason. They studied 60 articles of peer-reviewed articles over the period 2000-2011 using the theoretical framework proposed by Mialaret (2005/1976), Béchard and Grégoire (2005) and Fayolle

  the point of view of certain researchers, to change organizational culture (Korte and al. 2007): this proactive tool has the potential to help developing alternative possible cultures. Organizational culture is usually viewed as set of underlying assumptions, beliefs and norms governing how individuals should act in a specific context. Understanding shared cognition helps to initiate change in these deep representations. We launched a research program based on the design and animation of a creative workshop open to educators, researchers and entrepreneurs involved in educational entrepreneurship programs. The participants were proposed to explore new ways by imagining the future of entrepreneurship education. The workshop design is based on prospective thinking, creative imagination

Figure :

 : Figure: Tree Map Analysis -comparing number of references by encoded nodes.

Table 3 :

 3 Crossing Learning Space node with its mains linked nodes This table reveals that the Learning Space helps to achieve the Educational Principles (Experiential Learning, Personal Development Learning) and Entrepreneurship Education final goals (Human and Knowledge Development, Sustainable Development…). It also provides a space where activities can be led (projects, journeys, leisure -recreation, meeting others) by learners (students, diversity) who live adventures (Narrative Level: ideal tone and heroes) in a techno-pedagogic environment (intelligent clothes, virtual class, big data).

To be published.


Olivier Derrien
email: derrien@lma.cnrs-mrs.fr

A VERY LOW LATENCY PITCH TRACKER FOR AUDIO TO MIDI CONVERSION

An algorithm for estimating the fundamental frequency of a singlepitch audio signal is described, for application to audio-to-MIDI conversion. In order to minimize latency, this method is based on the ESPRIT algorithm, together with a statistical model for partials frequencies. It is tested on real guitar recordings and compared to the YIN estimator. We show that, in this particular context, both methods exhibit a similar accuracy but the periodicity measure, used for note segmentation, is much more stable with the ESPRITbased algorithm. This allows to significantly reduce ghost notes. This method is also able to get very close to the theoretical minimum latency, i.e. the fundamental period of the lowest observable pitch. Furthermore, it appears that fast implementations can reach a reasonable complexity and could be compatible with real-time, although this is not tested is this study.

INTRODUCTION

MIDI (Musical Interface for Digital Instruments) is the most widely used standard for connecting digital instruments. It specifies both the hardware interface and the data transmission protocol. It allows for instance to encode a melody as a collection of notes (note starting points, durations, pitches...) and to control a compatible synthesizer with an external interface, for instance a digital keyboard or a "wind controller" which mimics a wind instrument. However, for some instruments like guitars, designing an appropriate digital controller is difficult. Then, the original acoustic instrument can be used as a MIDI controller by adding an audio-to-MIDI converter. Such a device basically consists of a microphone that captures the acoustic signal produced by the instrument and a pitchtracker which estimates the evolution of pitch during time. For guitars, one usually uses an under-saddle pickup for each string, connected to a series of monophonic pitch trackers, one for each string [START_REF] Zölzer | DAFX, Digital Audio Effects[END_REF]. Such MIDI converters have been marketed for a few decades, but suffer from many flaws: latency, ghost notes, octave errors... The performance constantly improve, but latency still remains an issue: With an up-to-date Roland GR55 (built-in audio-to-MIDI converter and synthesizer) connected to a compatible acoustic guitar (Godin Multiac), we measured an average latency of 50 ms between the output of the under-saddle pickups and the audio output of the synthesizer (constant over the guitar frequency-range). Thus, playing a guitar synth is not easy and requires to develop specific skills.

In this study, we focus on the issue of latency for monophonic pitch tracking. We assume that pitch estimation is similar to fundamental frequency detection (noted f0), and that the observed signal is harmonic. It appears that latency has several sources that add up. First, the "algorithmic delay", which is inherent to the pitch-detection algorithm. It corresponds to the length of the time-interval that is required for the algorithm to give an accurate estimation. This delay has a fundamental lower bound which related to the minimum f0 value than can be detected. For a "Spanish" guitar1 , the minimum f0 value is approximately 80 Hz, which corresponds to a minimum delay of 12.5 ms. Then, the "computational delay", which is the time required by the digital signal processor (DSP) to perform the pitch estimation. This delay can be reduced by increasing the speed of the DSP.

The issue of pitch detection is a classical problem and many algorithms have been proposed in the past decades. These methods can be roughly classified in two categories: time-domain and frequency-domain. Time-domain methods usually consist of finding a maximum of the auto-correlation function (or another similar function), while frequency-domain methods rely on a spectral analysis stage followed by a peak-picking stage. It was proved that time-domain methods are usually more efficient for real-time estimation of single-pitch [START_REF] Dem Knesebeck | Comparison of pitch trachers for real-time guitar effects[END_REF]. Especially, the YIN algorithm, proposed by de Cheveigné et al. [START_REF] De Cheveigné | Yin, a fundamental fre-quency estimator for speech and music[END_REF], can be considered as a reference f0 estimator. It is based on the observation of a "cumulative mean normalized difference function", which is characterized by dips at the time-lags corresponding to the periodicity. This method is accurate, has a moderate complexity and a relatively low algorithmic delay. In [START_REF] Dem Knesebeck | Comparison of pitch trachers for real-time guitar effects[END_REF], the delay of the full method was estimated around 30 ms for a "Spanish" guitar. However, this value is still approximately twice the theoretical minimum delay.

In this paper, we consider a new approach to reduce the algorithmic delay. Most f0 estimators are non-parametric methods in the sense that they do not use a priori information about the signal. In contrast, parametric methods, which rely on a signal model, are known to be more precise when the observed signal correctly fits the model, but usually fail in the opposite case. For that reason, non-parametric methods are often considered more robust. However, audio signals coming from an under-saddle guitar pickup usually produce a quasi-harmonic sound with a very low noise, which justifies the use of a parametric method based on a sinusoidal signal model. In this study, we choose the Exponentially Damped Sinusoidal (EDS) model. The model parameters are estimated with a method derived from the ESPRIT algorithm [START_REF] Badeau | EDS parametric modeling and tracking of audio signals[END_REF]. This phase is similar to a spectral analysis and a peak-picking stage. To fulfill the pitch estimation, we use a spectral f0 estimator inspired by the one proposed by Doval et al. [START_REF] Doval | Estimation of fundamental frequency of musical sound signals[END_REF]. Algorithms derived from ESPRIT are known for their good frequency resolution, but also have the reputation to require high computation time. However, fast algorithms have been proposed in the last decade [6, Chapter V] which exhibit a complexity not much higher than a FFT. Thus, this method is theoretically suitable for real-time implementation, although this is not tested is this study.

This paper is organized as follows: In a first part, we consider more precisely the issue of algorithmic delay in a f0 estimator. In a second part, we describe the proposed method. In a third part, we give results obtained from real guitar sounds both for our algorithm and for the YIN estimator, concerning pitch accuracy and delay. In the last part, we draw conclusions.

THE ISSUE OF ALGORITHMIC DELAY

Most f0 estimators are frame-based methods. An input buffer of N samples is used, and the estimation of f0 is made every a samples (a ∈ N {0}). In other words, a sliding analysis window of N samples is used, with a hop-size a. The f0 estimator should ideally be associated to the estimation of a "periodicity measure", i.e. whether the signal is pitched or not. A common periodicity measure is obtained by computing the energy of the periodic components in the signal, called "voiced" components in the case of a speech signal [START_REF] Richard | Analysis/synthesis and modification of the speech aperiodic component[END_REF]. Such a periodicity measure influences the accuracy of the note segmentation process: A simple way to detect notes is to threshold the periodicity measure.

It is often believed that the algorithmic delay is equal to the window length, but this more complex. As exemplified on figure 1, the algorithmic delay corresponds to the time-interval between the beginning of a note and the last sample of the first window for which f0 estimate is accurate (and eventually the periodicity measure is higher than the threshold). Sometimes, the algorithm returns the accurate f0 even if the pitched signal does not "fill" the window (plotted case). Then, the delay can be shorter that N samples. Sometimes, the estimator takes some time to return the accurate f0 and the delay can be longer that N samples. As explained in [START_REF] De Cheveigné | Yin, a fundamental fre-quency estimator for speech and music[END_REF], performing a correct f0 estimation requires that the window length is no shorter than the largest expected period. But, according to the well known rule of thumb, a correct estimation requires enough signal to cover twice the largest period. Thus, the minimum delay is equal to 1/f min 0 , f min 0 being the lowest f0 value than can be detected. As explained previously, this corresponds to approximately 12.5 ms for a "Spanish" guitar. As a consequence, the window length N must be higher that fs/f min 0 where fs is the sampling frequency. But practically, we expect a minimum delay of 25 ms.

THE PROPOSED METHOD

This method is divided in two stages: in the first one, the most significant sinusoidal components are extracted according to a signal model. Then, in the second stage, the most probable fundamental frequency is estimated using a statistical model.

Sinusoidal modeling

In this part, we describe the signal model and the estimation algorithm. Both have been extensively discussed in the literature. We choose to reproduce this description from a previous work [START_REF] Sirdey | Modal analysis of impact sounds with esprit in gabor frames[END_REF] in order to render the paper self-contained.

In the EDS model, the signal to be analyzed is written:

x[n] = s[n] + w[n], (1)
where the deterministic part s[n] is a sum of K damped sinusoids:

s[n] = K-1 k=0 α k z n k . (2)
Complex amplitudes are defined as α k = a k e iφ k (containing initial amplitude a k and phase φ k ), and poles are defined as z k = e -d k +2iπν k (containing damping d k and normalized frequency ν k ). The stochastic part w[n] is a gaussian white noise.

The estimation algorithm consists in finding the best values of K, α k and z k for a given signal in the least square sense. In this study, an estimation algorithm proposed by Badeau et al. [START_REF] Badeau | EDS parametric modeling and tracking of audio signals[END_REF] is used, which is derived from the ESPRIT algorithm. The principle consists of performing an SVD on an estimate of the signal correlation matrix. The eigenvectors corresponding to the K highest eigenvalues correspond to the so-called signal space, while the remaining vectors correspond to the so-called noise space. The shift invariance property of the signal space allows a simple solution for the optimal poles values z k . Then, the amplitudes α k can be recovered by solving a standard least square problem. The algorithm can be described as follows:

We define the signal vector:

x = x[0] x[1] . . . x[N -1] T , ( 3
)
where N is the length of the analysis window. We assume that N is even. The Hankel signal matrix is defined as:

X =      x[0] x[1] . . . x[Q -1] x[1] x[2] . . . x[Q] . . . . . . . . . x[R -1] x[R] . . . x[N -1],      , ( 4
)
where We also define the amplitude vector:

Q, R > K and Q + R -1 = N . Q ≈ R
α = α0 α1 . . . αK-1 T , (5)
and the Vandermonde matrix of the poles:

Z N =      1 1 . . . 1 z0 z1 . . . zK-1 . . . . . . . . . . . . z N -1 0 z N -1 1 . . . z N -1 K-1      . ( 6
)
Performing a SVD on X leads to:

X = [U1U2] Σ1 0 0 Σ2 V1 V2 , (7)
where Σ1 and Σ2 are diagonal matrices containing respectively the K largest singular values and the remaining singular values.

[U1U2] and [V1V2] are respectively the corresponding left and right singular vectors. The shift-invariance property of the signal space yields to:

U ↓ 1 Φ1 = U ↑ 1 , V ↓ 1 Φ2 = V ↑ 1 , (8)
where the poles are eigenvalues of matrix Φ1 and Φ2. (.) ↑ and (.) ↓ respectively stand for the operators that discard the first line and the last line of a matrix. Here, we estimate:

Φ1 = (U ↓ 1 ) † U ↑ 1 , (9)
where (.) † denotes the pseudoinverse operator. The estimates of z k are obtained by diagonalization of Φ1. The associated Vandermonde matrix Z N is computed. Finally, the estimates of amplitudes with respect to the least square criterion are obtained by:

α = (Z N ) † x. (10)
Badeau et al. also proposed a criterion (ESTER) which measures the adequacy between the signal and the model [START_REF] Badeau | Selecting the modeling order for the esprit high resolution method: an alternative approach[END_REF]. It is based on the fact that equations ( 8) are strictly verified only when the signal exactly follows the EDS model defined by equation ( 2) without noise. In the general case, a distance between U ↑ 1 and U ↓ 1 Φ1 can be used to measure the model error. Is was that the original ESTER criterion naturally favors low values for the model order K. In order to minimize this effect, we propose a modified version of this criterion:

J = (K -1) 2 U ↑ 1 -U ↓ 1 Φ1 2 . ( 11
)
The numerator simply performs a normalization of the denominator by the size of the matrix inside the norm. A high value for J means a good match between the signal and the model. This criterion can be used to automatically determine the best model order K, or in our case, to derive a periodicity measure.

Fundamental frequency estimation

It is assumed that each damped sinusoid in the EDS decomposition corresponds to a partial. Its frequency is related to the pole estimate by f k = fsν k = fs 2π arg(z k ). Doval et al. proposed a statistical method that allows estimating the most probable fundamental frequency of a harmonic signal given a set of partials [START_REF] Doval | Estimation of fundamental frequency of musical sound signals[END_REF]. The main idea is to compute a likelihood function of the fundamental frequency based on a probabilistic model of the observed partials. The best estimate for the fundamental frequency is the global maximum of this function. In the original method, the statistical model is elaborated and has many parameters. Estimating these parameters requires a learning database of recorded notes. Furthermore, the computation of the likelihood function can be time-consuming.

With our application, a low-complexity algorithm is desirable. We also wish that our method does not depend on a learning database. Thus, we modify the model in order to reduce the complexity and to minimize the number of parameters. In particular, the distribution of energy between partials is not modeled. This probably degrades the efficiency of the f0 estimation compared to the original method, but it happens to be sufficient for this application.

For a given value of f0, we define a set of frequency intervals Im centered on mf0:

Im = m - 1 2 f0, m + 1 2 f0 , m ∈ N {0}, (12)
which define a partition of the frequency scale. The partials are dispatched in these intervals according to their frequency f k . Some intervals can contain several partials, and some others can be empty.

In each non-empty interval, we define the most probable "harmonic partial" as the one which frequency is closer to mf0, noted f h m . The others are called "supplementary partials" (see figure 2). The likelihood function is written as:

L(f0) = m∈M g f h m f0 -m PS(f0) PE(f0), (13)
where M is the set of indices m corresponding to non-empty intervals Im. The first term is the a posteriori probability to observe the set of harmonic partials. The second and third terms, PS(f0) and PE(f0), are respectively the a posteriori probability to observe the set of supplementary partials and empty intervals. g is a probability function that models the frequencies of harmonic partials, which is assumed to be gaussian:

g f h m f0 -m ∝ e -1 σ 2 f h m f 0 -m 2 . ( 14
)
σ 2 represents the variance of the reduced frequencies f h m /f0 around the mean value m. PS(f0) and PE(f0) are estimated by:

Ps(f0) = 1 - NS K α S , PE(f0) = 1 - NE M α E , ( 15
)
where NS is the number of supplementary partials, NE the number of empty intervals and M the total number of intervals. αS and αE are constants that allow adjusting the influence of NS and NE on the likelihood function. Thus, when the frequencies of the harmonic-partials are close to mf0, the likelihood increases. When the number of supplementary partials or empty intervals increases, the likelihood decreases. This method naturally avoids octave errors: A lower (resp. higher) octave generates supplementary partials (resp. empty intervals), which lowers the probability PS(f0) (resp. PE(f0)) and finally lowers the likelihood. However, this requires a fine tuning on αS and αE.

Implementation details

The EDS estimation algorithm is applied on analysis segments without weighting function (also called "rectangular analysis window" in the literature). The model order K can be set to a constant, or one can define maximum and minimum values and use the ES-TER criterion to select the optimal order. From our experiments on isolated guitar notes, it appears that K = 6 is good choice for a constant. Otherwise, K may vary between 4 and 12. Choosing a constant K saves execution time but this is sub-optimal. More precisely, bass notes usually have more harmonics than treble notes. Observing a larger set of harmonics on bass notes is desirable because these notes are more difficult to detect (there are fewer fundamental periods in the analysis window), and a larger set of harmonics gives a more robust estimation of f0.

A periodicity measure ideally measures the energy of the periodic component in the signal. The criterion J defined in equation ( 11) is simply a ratio (without dimension) that measures the signalto-model adequation. Thus, we derive our periodicity measure by multiplying J by the energy of the signal in the analysis window.

The estimation of f0 implies computing the likelihood for all possible values of f0. This can be accelerated by testing only discrete values, for instance on the tempered scale. If a finer estimation is required, a refinement stage can be added [START_REF] Doval | Estimation of fundamental frequency of musical sound signals[END_REF]. We set σ = 1/8, αS = 8 and αE = 4. This set of parameters appears to give a robust estimation over all the guitar frequency range. However, it is possible to choose a different set of parameters for each f0 which could improve the detection accuracy.

Concerning complexity, ESPRIT is obviously the most critical part. A non-optimized version of ESPRIT has a complexity in O(N 3 ) which is hardly suitable for real-time implementation. But a fast implementation of ESPRIT [START_REF] Badeau | High resolution methods for estimating and tracking modulated sinusoids[END_REF] has a complex-ity in O(KN (K + log(N ))). When K is small, this reduces to O(KN log(N )), which is not much more than a FFT. When the overlap between adjacent analysis windows is high, using adaptive algorithms allow to reduce again the complexity [START_REF] Badeau | High resolution methods for estimating and tracking modulated sinusoids[END_REF].

RESULTS AND DISCUSSION

In this section, we report test results for our algorithm and the YIN estimator on the same audio excerpts. The signal is the output of an under-saddle piezo-pickup on a solid-body acoustic guitar. The original signal is sampled at 44.1 kHz, downsampled at 11.025 kHz to reduce complexity. This appears to be sufficient for estimating the highest pitch on a Spanish guitar (between 930 and 1200 Hz). The implementation is in Matlab, and thus the estimation is an offline process. According to the results given in section 2, the minimum buffer length is N = 138 for f min 0 = 80 Hz. We choose for both methods a hop-size of a = 8 samples, which corresponds to 0.72 ms.

For the YIN estimator, the author's implementation [START_REF]The YIN algorithm documentation[END_REF] is used. For the proposed method, the implementation of the ESPRITbased estimator relies on the DESAM Toolbox [START_REF]The DESAM toolbox[END_REF], which is nonoptimized. The minimum f0 is set to 80 Hz for both methods. Buffer length was adjusted so that a correct estimation of f0 is obtained for the whole guitar range. N = 300 is the minimal value for the YIN estimator, and N = 260 is the minimal value for the ESPRIT-based method. The YIN estimator gives a continuous frequency estimation that we round to the tempered scale. The new method is implemented only for discrete f0 values corresponding to the tempered scale. Frequencies are then converted into MIDI note index for both methods. The periodicity measure in the case of YIN is the inverse of the so-called "aperiodicity mea- sure" returned by the algorithm [START_REF]The YIN algorithm documentation[END_REF]. For both methods, the note segmentation is obtained by thresholding the periodicity measure. The thresholds were adjusted empirically in order to get accurate segmentation on several test recordings. The threshold was set to 100 for ESPRIT and to 60 for YIN. Although, in a finely tuned application, a different threshold could be set for each string.

On figure 3, we plot the results for both algorithms on a sequence of high-pitched notes played on the same string, with a pitch-bend during the third note. The estimated MIDI note is accurate and stable for both methods when the signal is stationary. As expected, both return insignificant pitch values between the notes. The case of the periodicity measure is more contrasted: With the YIN algorithm, the periodicity evolves sharply in time, but is not very stable. One can not define a threshold on the periodicity measure that avoids ghost notes. With ESPRIT, the periodicity is more stable and an accurate thresholding can avoid most ghost notes, but it evolves more slowly in time.

On figure 4, we plot the results for a low-pitch single note (E2), which is the lowest note on a Spanish guitar, and zoom around the onset. With both methods, the estimated MIDI note is accurate and stable after a transition phase. As regards the algorithmic delay (we do not consider the computational delay in this section), the ESPRIT-based method returns the correct raw MIDI note after 12 ms, which is approximately one period of the signal (i.e. the theoretical minimum value), whereas YIN returns the correct raw MIDI note after 20 ms. However, one must take into account the periodicity measure to evaluate the actual delay. Both methods exhibit a raising front on the periodicity which allows a precise thresholding, approximately 12 ms after the onset for the ESPRITbased method and 35 ms after the onset for the YIN algorithm. This is close to the value obtained by Knesebeck et al. in [START_REF] Dem Knesebeck | Comparison of pitch trachers for real-time guitar effects[END_REF].

On figure 5, the results for a high-pitch single note (E4) are plotted. As expected, the results are globally similar to the previous case because the analysis parameters (especially the window size) did not change. However, one can see that the periodicity measure is less sharp with ESPRIT: there is a shelf between 12 and 24 ms that could extend the delay, or even generate ghost notes, depending on the threshold value. This can be explained by the fact that the signal exhibits a pseudo-periodicity before the onset that might come from the interaction between the string and the pick. The periodicity onset is sharper with YIN.

CONCLUSION

In this paper, an algorithm for estimating the fundamental frequency of single-pitch notes was described. The application to audio-to-MIDI conversion for guitar was especially considered. This application requires very-low algorithmic delay, which is still an issue with state-of-the-art pitch trackers. In order to minimize this delay, a new method was proposed. The first stage, equivalent to a spectral peak-picking algorithm, uses an algorithm from the literature derived form the ESPRIT method. The second stage is a fundamental frequency estimator inspired by the method proposed by Doval et al., which consists in maximizing a likelihood function. The new method was tested on real guitar recordings and was compared to the YIN estimator proposed by de Cheveigné et al. which can be considered as a reference method. It was showed that, on this test material, both methods exhibit a similar accuracy, but it is important to notice that only the closest MIDI note was considered, and not the continuous fundamental frequency estimation. Concerning the periodicity measure which is used for note segmentation, the new method was found more stable than the YIN estimator. This allows to significantly reduce ghost notes that are commonly observed in audio-to-MIDI conversion. It was also showed that the ESPRIT-based method is able to provide note tracking with an algorithmic delay that is very close to the theoretical limit, i.e. the fundamental period of the lowest observable pitch, which is not the case with the YIN method. For that reason, this new estimator may allow to significantly reduce the latency of audio-to-MIDI conversion. However, the issue of computational cost is crucial. The YIN estimator is a fast method, well suited for real-time implementation. In this preliminary study, our method was only tested off-line using a non-optimized implementation in Matlab. But theoretical studies have showed that fast implementations of the ESPRIT algorithm can reach a reasonable complexity in O(KN log(N )) where K is the number of partials to be observed (typically 6) and N is the length of the analysis window (here less than 300 points). This is not much more than a FFT, which means that an optimized version would be theoretically compatible with real-time. This point will be investigated in the future.

Figure 1 :

 1 Figure 1: Measurement of the algorithmic delay.

  was proved to be an efficient solution, thus we choose Q = N/2 and R = N/2 + 1.

Figure 2 :

 2 Figure 2: Classification of partials for a given f0.

Figure 3 :

 3 Figure 3: Outputs of the estimation process for a short sequence of notes, ESPRIT (left column) and YIN (right column). First row: Signal waveform. Second row: MIDI note returned by the algorithm. Third row: Periodicity measure (blue) and threshold (red), log scale. Fourth row: Final MIDI note after thresholding the periodicity measure.

Figure 4 :

 4 Figure 4: Outputs of the estimation process for E2 note, ESPRIT (left column) and YIN (right column). First row: Signal waveform. Second row: MIDI note returned by the algorithm (blue) and theoretical value (red). Third row: Periodicity measure (blue) and threshold (red), log scale. Fourth row: Final MIDI note after thresholding the periodicity measure (blue) and reference value (red). Time origin is manually aligned with the onset.

Figure 5 :

 5 Figure 5: Outputs of the estimation process for E4 note, ESPRIT (left column) and YIN (right column). First row: Signal waveform. Second row: MIDI note returned by the algorithm (blue) and theoretical value (red). Third row: Periodicity measure (blue) and threshold (red), log scale. Fourth row: Final MIDI note after thresholding the periodicity measure (blue) and reference value (red). Time origin is manually aligned with the onset.

A "Spanish" guitar means a 6 string instrument tuned to the standard scale E-A-D-G-B-E. Thus, the lowest note is E2, corresponding to a fundamental frequency of 82.41 Hz.


F Paraz

C Eloy

L Schouveiler

Keywords: Flexible plate, Flapping plate, Vortical structures, Thrust

We present dye visualizations of the wake behind a flexible plate flow when its leading edge is forced into harmonic heave motion. We show that, when the forcing amplitude is sufficiently large, a reverse Bénard-von Kármán vortex street is formed in the wake, indicative of the production of positive thrust.

Introduction

Flapping fins are commonly used by swimming animals to produce locomotive forces. These fins have inspired a vast amount of studies motivated, for a large part, by the design of novel propulsion devices. These studies have generally considered a foil or a plate harmonically actuated in heave motion (translation), in pitch motion (rotation), or in a combination of both motions. Interestingly, the propulsive performances have been shown to increase when a flexible plate, rather than a rigid one, is used. The thrust produced by flapping fins is an increasing function of the amplitude of the trailing edge deflection [START_REF] Lighthill | Note on the swimming of slender fish[END_REF]. To assess the propulsive performances of such systems, the knowledge of the deformation kinematics is thus of primary importance.

In the present work, we consider the flow generated by a flexible plate whose leading edge is forced into harmonic heave motion. It follows a previous study in which we have investigated the plate response to this forcing [START_REF] Paraz | Experimental study of the response of a flexible plate to a harmonic forcing in a flow[END_REF]. In particular, we have evidenced that the deflection amplitude of the trailing edge was maximal when the forcing frequency was resonant with a structural eigenfrequency of the flexible plate. We now turn to the visualization of the vortical structures generated in the wake of this model swimmer, as it is known to be a good indicator of the thrust produced [START_REF] Dabiri | On the estimation of swimming and flying forces from wake measurements[END_REF].

The experimental setup, shown in Fig. 1a, consists of a horizontal flexible plate, of bending rigidity B, immersed in a uniform horizontal flow of velocity U generated by free-surface water channel. The plate is of thickness e = 0.004 m, span s = 0.12 m, and chord c = 0.12 m. It is molded out of polysiloxane with a tapered trailing edge and with a rigid axis inserted at the rounded leading edge. This rigid axis is attached to a U-frame that can be actuated with a linear motor into harmonic motion such that the leading elevation is imposed: A LE cos(2πf t). The flow around the plate is confined between two vertical parallel walls to avoid the perturbations due to the U-frame.

The explored ranges of the experimental parameters A LE , f , B and U are given in Fig. 1b. To visualize the vortical structures generated by the plate motion, the plate is coated with a solution of fluorescent dye prior immersion. The patterns created by the dye entrained in the flow are illuminated at mid-span with a vertical sheet of light generated with an Argon Ion laser. Images are captured, through the transparent channel sidewall, with a camera aligned perpendicular to the light sheet.

Results and perspectives

In Fig. 2, the evolution of the vortex patterns in the wake of the heaving flexible plate are visualized with dye over one period. The experimental parameters are A LE = 0.004 m, f = 0.8 Hz and U = 0.05 m s -1 , corresponding to a resonant case (i.e. a maximum of the trailing edge deflection amplitude). In Fig. 2, it can be observed that a vortex forms at the leading edge on the upper side during the plate downstroke (Figs 2a-c) and a second one on the lower side during the upstroke (Figs 2d-f). These vortices are likely due to a flow separation at the leading edge. This separation occurs because the combination of the incoming flow and of the heave motion results in a non-zero effective angle of attack at the leading edge. For the parameters used here, this angle is arctan(2πf A LE /U ) = 23 • . The leading edge vortices can be followed on the image sequence as they travel along the plate chord up to the trailing edge where additional vortices are generated. Both vortices combine into large scale vortices shed twice per cycle: the shedding of a counter-clockwise vortex is seen in Fig. 2a and of a clockwise one in Fig. 2d. As they are advected downstream, they form a reverse Bénard-von Kármán vortex street, similar to the wakes observed behind bluff bodies but with reversed sense of rotation of the vortices. This reverse Bénard-von Kármán wake corresponds to a jet-like average velocity profile and is thus associated with thrust production. We are currently trying to measure this thrust directly to confirm quantitatively the present results of qualitative nature.

Fig. 1 (

 1 Fig. 1 (a) Sketch of the experimental setup and (b) ranges of the experimental parameters explored.

Fig. 2

 2 Fig. 2 Dye visualizations of the vortices generated by the heaving flexible plate over one period for A LE = 0.004 m, f = 0.8 Hz (period T = 1.25 s), B = 0.018 Nm and U = 0.05 m.s -1 (flow from left to right). The motion of the plate leading edge is indicated by an arrow.

Acknowledgements We warmly thank Thomas Leweke for his help with the visualizations.