Minimal physical model of the cristal Baschet

– The cristal Baschet is a musical instrument created during the 1950 ’ s by Bernard and Francois Baschet. It is composed of a large number of glass rods arranged in a chromatic scale. The sound produced re-sults of vibrations induced by friction between wet ﬁ ngers and the glass rods. Each glass rod is connected to an assembly of threaded shafts and a mass. Mechanical properties of this assembly determine the pitch of the note. Then vibrations are transmitted to large metal panels or cones that act as radiating elements. The manufacturing and tuning of this instrument is based on empirical knowledge and involves many parameters whose effects are not clearly understood. One of the encountered problems is the dif ﬁ culty to produce sound in the high register of the instrument. In an attempt to understand the in ﬂ uences of these parameters on playability, a minimal physical model of the cristal Baschet is developed. It focuses on the interaction between the ﬁ nger and the isolated resonator. The dynamic behavior is described by a set of modes obtained from a ﬁ nite element model or from experimental modal analysis. The musician ’ s gesture is described by two control parameters: the velocity of the ﬁ nger along the glass rod and the normal force applied by the ﬁ nger on the rod. To describe the interaction between the ﬁ nger and the resonator, a friction law is implemented. The in ﬂ uence of different parameters is studied by means of linear stability analysis and time-domain simulations. Speci ﬁ c criteria are developed to highlight the role of design parameters on playability.


Introduction 1.Historical context of cristal Baschet
The cristal Baschet is a musical instrument created by two brothers, Bernard and François Baschet, in 1952 in Paris.This invention is part of a process initiated in 1950, aiming to develop a metal instrument while making use of the richness and diversity of the sounds that it enables.This work led to the creation of a wide range of instruments, called Structures Sonores [1].
F. Baschet and B. Baschet had an original approach in their process of creation and design of musical instruments.They designed instruments by taking into account three major aspects: the visual aspects (particular shapes, sculpture art), the sound aspects (new acoustic sounds) and the social aspects (musical pedagogy, playing accessibility).
The created instruments are modular, i.e. designed as assemblies with interchangeable elements.The patent [2] describes the "Polyphonic, polytone, multi-tone instrument", shown in Figure 1a.Several modes of operation such as impact or friction are suggested and associated with resonators of all kinds: bars (see item 32 in Fig. 1a), flexible rods called whiskers (in French: moustaches) (item 25), ropes (item 18), membranes, etc.The cristal Baschet corresponds to a particular form of this modular instrument.It uses the friction of a glass rod (item 12) (inspired by the research experiments reported by H. Bouasse [3]), connected to a support (item 7) to which bladders (item 44) or a plate acting as a soundboard (item 30) are connected.Over several decades, the cristal has evolved, as shown in Figures 1b and 1c, where the use of bladders and piano soundboard were actually implemented.
The proposed evolutions on the cristal after its invention were oriented on the one hand to improve the playability of the cristal, and on the other hand to improve the efficiency of the acoustic radiation.The Baschet brothers worked with many musicians, such as Yvonne and Jacques Lasry, with whom the orientation of the keyboard was modified.The glass rods, initially vertical (Fig. 1b), were then placed horizontally (Fig. 2a).This has completely changed the musician's gesture and considerably improved the ease of playing.Other evolutions of the cristal, of an exploratory nature, focused on the efficiency of acoustic radiation: the use of bladders (radiating flexible membranes, Fig. 1b) was replaced by plates (piano soundboards or metal cones, see Figs. 1c and 2a).Nowadays, the cristal continues to evolve thanks to the experiments and empirical tests carried out by the association called Structures Sonores Baschet Association (SSB) [4], which continues to keep the work of the Baschet brothers alive, through three types of actions: education, culture and the preservation of the Baschet brothers's legacy.

Presentation of the instrument
In its present form, the cristal Baschet is composed of a large number of glass rods arranged horizontally so as to form a chromatic scale from C1 (65.4 Hz) to C5 (1046.5 Hz).Each glass rod is attached to an assembly consisting of two threaded rods and a mass.This assembly, including the glass rod, is called a resonator.All resonators are attached to a thick metal plate called the collector.Large thin metal panels or cones made of composite materials are connected to the collector to ensure efficient acoustic radiation.
To play cristal, the musician rubs their wet fingers on the glass rods.Three different acoustic functions are thus associated: the excitation function (the friction mechanism), the determination of the playing frequency (resonator constituted by the rods) and the acoustic radiation.This organization follows a functional scheme used in many musical instruments and is well described in the literature [5].The glass harmonica, wine glass and harp glass where the excitation function is similar to the cristal, have been studied [6,7].The cristal, on the other hand, is a little-studied instrument [8] for which no physical model has been proposed.
Cristal players, known as cristalists, often mention a difficulty playing in the high register, in the sense that the sound is harder to produce than in the medium or low registers.This question refers to the analysis of playability, i.e. the ease of production of a sound resulting from friction, and its homogeneity over the whole range.To address this question, we propose in this article to study the conditions of occurrence of self-oscillations induced by friction.Criteria qualifying playability are proposed.This approach is of interest to the instrument maker, since it allows them to analyze the design of resonators.
The paper is organized as follows: Section 2 describes a minimal physical model accounting for the behavior of the cristal Baschet.This toy model contains the essential ingredients to describe the production of musical sound: the interaction between the wet finger and the resonator induces frictional forces that excite the structure, which is described using a modal formalism.In Section 3, the model is used to analyze the appearance of self-sustained oscillations in response to the musician's gesture, in order to propose playability criteria.In Section 4, several parametric studies are carried out to highlight the influence of some design parameters on playability.In particular, the playability of the instrument over its entire range is analyzed to discuss the reduced playability in the high register as expressed by the musicians.

Presentation of the model: assumptions
The physical model representing the cristal is minimal (toy model) and aims to account for friction instabilities, i.e. the generation of self-oscillations.This model is described as "minimal" in the sense that it only describes the interaction between the musician's finger and the isolated resonator.The elements coupled to the resonator (collector and radiating plates) are assumed to have little influence on the generation of self-oscillations.They are therefore not taken into account in the minimal model.
The produced sound is the result of the musician's gesture, described by playing parameters.We make the following assumptions: The rubbing finger is considered to be rigid.Its dynamic behavior is ignored.The interaction between the finger and the glass rod is modeled by a point force.The contact surface is assumed to be very small compared to the length of the rod.The abscissa of the contact point is denoted by x c .The normal force (along the vertical z-axis) applied by the finger on the rod is denoted by F N .The velocity of the finger along the horizontal x-axis is denoted by _ u f .This uni-axial movement describes in a simplified way the musician's gesture.These hypotheses describe the musician's gesture in a simplified way, using three parameters (x c , F N and _ u f ).In a playing situation, there are multiple ways of making a gesture to express musicality.For example, the cristalist can use several fingers, pinch and grab more or less the glass rod. Figure 2b shows the different parameters used to model the interaction between the finger and the glass rod.
Frictional interaction is based on the following assumptions: In a slipping state, the frictional force F T is proportional to the normal force F N and opposes relative motion between the glass rod and the finger.The friction coefficient l is assumed to depend on the relative velocity Á _ u ¼ _ u À _ u f between the glass rod and the finger.In a sticking state, the maximum friction force is described by a static friction coefficient l s .The parameters of the friction law are assumed to be constant over time.In actual playing situation, the amount of water on the glass rod decreases over time, which causes the nature of the friction law to change.The musician must regularly "reload" by dipping their hands into a tank of water.In the minimal model, these variations in friction properties are ignored.
A third group of assumptions concern the resonator: The motion of the resonator is restricted to the (x, z)-plane (see Fig. 2b).
The resonator is considered as a linear and time-invariant structure, so that its dynamics can be described by a set of eigenmodes.The motion is described by an equation of the form: where x(t) = [u(t) w(t)] T is the displacement vector.The components u(t) and w(t) are respectively the axial (along the x-axis) and transversal (along the z-axis) displacements of the interaction point C. M r , C r and K r denote respectively the mass, damping and stiffness matrices.
Damping is considered to be proportional, which implies that the generalized damping matrix is diagonal (see below).
Only the movement along the x-axis of the glass rod is described in Section 3. Indeed, the effect of the finger force F N is not taken into account.The variations of this force are assumed to be slow compared to the dynamics of the resonator, so that they only cause quasi-static deformation of the glass rod along the z-axis.

Modal equations
The displacements x(t) of the interaction point C are expressed using a modal expansion involving N modes, where q(t) and U are respectively the N Â 1 vector of modal coordinates and the 2 Â N modal matrix, whose columns U x,k and U z,k contain the values of mode shapes associated to the degrees of freedom u and w respectively.These N modes can be determined from an equation of motion of the form (1) comprising at least N degrees of freedom.
The orthogonality of modes allows the equation of motion to be written in the modal basis as: k Þ the modal stiffness matrix involving the natural angular frequencies x k .These modal parameters can be obtained experimentally or numerically, as described in the following paragraphs.

Experimental study of the resonator
In the general approach of the study, the modes of the resonator are obtained numerically as this allows useful parametric analyses.However, experimental tests are necessary to validate the model, due to the nature of the connections, which include bolts involving local prestress, and vibrating rods with non-uniform cross-section (threaded rods).This subsection deals with these experimental tests.
The experimental modal analysis is conducted on an isolated resonator, fixed at its base on a stiff and solid support (see Fig. 2c).The resonator is tuned to the note C3# (277.2Hz).A three-axis accelerometer is glued to the upper free edge of the rectangular mass (named "plongeoir" in French).The added mass and damping due to the accelerometer are neglected.An impact hammer with a vinyl tip is used to excite 48 points distributed over the entire structure, in order to achieve a relatively fine representation of the mode shapes.Each point is excited in the three directions when possible.The extraction of modal parameters from the measured transfer functions is performed up to 2000 Hz using the PolyMAX algorithm [9], implemented in the commercial software Simcenter Testlab (Siemens).An example of measured transfer-mobility is shown in Figure 3 and reveals the existence of 9 modes between 20 Hz and 500 Hz.

Numerical study of the resonator
The numerical modal analysis of the resonator is carried out with a finite element model (FEM), using the commercial software COMSOL Multiphysics.The geometry is constructed from the actual dimensions of the resonator previously studied.The bolts are included in the geometry.The threaded rods are modeled by cylindrical rods with uniform cross-section but diameter slightly smaller than the actual outer diameter of the thread.This effective diameter is one of the quantities used for the recalibration of the model.Clamped boundary conditions are considered at the base of the resonator.The mesh is made of tetrahedral elements, whose size is between 3 mm and 40 mm.The material properties considered in the model are shown in Table 1.For each material, the density was directly estimated from measurements.Usual values of Poisson's ratio are considered.The Young's modulus of stainless steel, aluminum and glass are adjusted in order to minimize the frequency deviation between numerical and experimental natural frequencies of the first 9 modes.With the values in Table 1, the frequency deviations df = |(f num À f exp )/ f exp )| have a maximum value of 12.7% (for mode 3) and have an average value of 5% over the first 9 modes, which seems acceptable.The calibration of the FE model is therefore considered valid.
The recalibration of the FE model can be observed in Figure 3, where experimental and numerical transfermobilities are compared.The transfer-mobilities are defined between points A (excitation) and B (response), corresponding respectively to the end of the glass rod and the location of the accelerometer at the end of the upper mass (see Fig. 2).The modes are numbered in increasing order of natural frequency.The mode shapes that exhibit dominant motion in the (x, y)-plane (respectively (x, y)plane) are represented on the first (respectively second) row of mode shapes in Figure 3.
The transfer-mobility along the x-axis mainly presents four resonance peaks.These peaks correspond to modes 1, 4, 7 and 8, whose movements are dominant in the (x, z)-plane.Mode 7 corresponds here to the note played.Its mode shape involves deformation of segment DE of the vibrating shaft (see Fig. 2), behaving like the first mode of a beam clamped at both ends.This motion induces a trans-lation along the x-axis of the glass rod.It can be seen that mode 7 has the lowest modal damping among those identified.In the following, this mode is referred to as the "main mode".The minimal model includes only modes 1, 4, 7, and 8.

Friction Law
The interaction between the finger and the glass rod is described by a friction law that distinguishes sticking and slipping states.This law expresses a relationship between the frictional force F T applied on the glass rod and the relative velocity Á _ u between the glass rod and the finger.It is usually expressed as In the model originally proposed by Coulomb, the coefficient of friction l is constant.Numerous variants were subsequently proposed, including the exponential model [11], or the Derler's model [10].The model used here is a hyperbolic law, described by three parameters: the static friction coefficient l s , the asymptotic dynamic friction coefficient l d , describing the behavior at high relative velocities, and a shape parameter V 0 which controls the curvature of each branch of the friction law.This law is symmetrical, i.e. it is described by an odd function (see Fig. 4a).This expresses that the friction characteristics are independent of the direction of slipping.
The parameters l s , l d and V 0 are adjusted such that the shape of the curve given by equation ( 5) is as close as possible to Derler's law, which is representative of the friction between a wet finger and a glass plate.This semi-empirical power law, F T ¼ 0:25= _ u f 0:37 ; is represented by a dotted line in Figure 4a.The fitted friction law is plotted in red on Figure 4a with the following parameters: V 0 = 0.02 m.s À1 , l s = 2 and l d = 0.3.

Methods of analysis
The analysis of the role of player's control parameters (finger velocity and normal force), as well as the role of design parameters (resonator geometry) is carried out using two complementary approaches: time-domain simulations and Linear Stability Analysis (LSA).

Time-domain simulations
The time-domain simulations aim to describing the occurrence of self-sustained oscillations in a playing situation.We describe here a method from [12] to obtain an explicit numerical scheme to iteratively solve the modal equations (Eq.( 3)).The solution is computed at discrete times t i+1 = t i + Dt, where Dt is the sampling period.When Dt is sufficiently small compared to the characteristic time related to the dynamics of the system, one can consider that the frictional force is constant over the time interval [t i , t i+1 ].The right-hand side of the modal equations (Eq.( 3)) is then constant during this interval.An analytical solution can therefore be used to calculate the state of the system at time t i+1 , described by the set of modal coordinates q(t i+1 ) and their derivatives _ qðt iþ1 Þ, knowing its state at time t i and the constant modal forces f(t i+1 ).This leads to an explicit recurrence relation, (see Appendix A for the derivation of Eq. ( 6) and the expression of the matrices).
The chosen sampling frequency is 48 kHz, so the Dt is approximately equal to 21 ls.A convergence study, not presented in the paper, was carried out to define the sampling frequency.This study showed that increasing the sampling frequency up to 480 kHz does not change the nature of the simulated signals.

Dynamics of the resonator
The glass rod is subject to an external force which is the frictional force F T (t).The corresponding modal forces at time t i+1 can be written as which are related to the relative speed.
Using the recurrence formula from equation ( 6) to replace where the modal forces f(t i+1 ) can be replaced using equation (7).A relationship between the friction force and the relative velocity at time t i+1 is thus obtained, where the coefficients c 1 and c 0 are given by Equation ( 10) captures the dynamics of the resonator as a linear force-velocity relationship.Its graphical representation in the force-velocity plane (Fig. 4b) is a straight line, labeled (D) in Figure 4b.The time-domain simulation consists in finding at each time step the frictional force that simultaneously verifies equations ( 10) and ( 4), which describe the dynamics of the resonator and the friction law, respectively.

Transition between slipping and sticking states
The calculation of the friction force at t i+1 depends on the friction state (sticking or slipping).As this state is not known at t i+1 , a verification is necessary.The conditions to trigger a transition from one state to the other are explained below.
If the system is assumed to be in a sticking state at time t i+1 , the frictional force is obtained by imposing Á _ u t iþ1 ð Þ ¼ 0 in equation (10).If the necessary frictional force is lower that the maximum static friction force, i.e. if |F T (t i+1 )| l s F N , the sticking state at time t i+1 is valid.Otherwise, slipping occurs at time t i+1 .
If the system is assumed to be in a slipping state at time t i+1 , the frictional force takes a value fixed by the friction curve while respecting the dynamics of the resonator.This results in the equality between the expression of the force given by equation ( 4) and the one given by equation (10), which yields Þby its expression from (5) leads to a polynomial equation where the coefficients A, B and C are expressed as The slipping state is possible only if equation ( 13) has real roots.If so, the root chosen, i.e. the value set to Á _ uðt iþ1 Þ, is the one that verifies a hysteresis rule described in [13,14].The corresponding frictional force F T (t i+1 ) is then calculated from equation (10).The transition from slipping to sticking at time t i+1 occurs either if there is no root, or if the determined roots involve a change of sign of the relative velocity.

Occurrence of self-sustained oscillations: typical results
We propose in this paragraph to describe the occurrence of self-sustained oscillations resulting from a typical gesture.This consists in the sudden application of finger velocity _ u f ¼ 0:1 m.s À1 , with a normal force F N = 1 N.These control parameters are assumed to be constant over time and describe a relatively fast and sustained gesture of the musician.The resonator is initially at rest, i.e. u r (0) = 0 and _ u r 0 ð Þ ¼ 0. Figures 5b and 5d show the velocity of the glass rod at the contact point during the transient phase for two different models: a one-mode model including only 1 mode (main mode, i.e. mode 7 in Fig. 3) and a four-mode model (with the modes 1, 4, 7, 8).The corresponding phase diagram are shown in Figures 5a and 5c.
Firstly, similar to the pedagogical model of Akay [15], we consider the results obtained with a one-mode model (see Figs. 5a and 5b).The transient shows a continuous state of slipping, in which an oscillation of the glass rod occurs, with an amplitude that increases over time.In the phase diagram (Fig. 5a), the trajectory follows approximately concentric shapes which become larger and larger, ultimately resulting in an almost circular trajectory.This limit cycle corresponds to a periodic oscillation around a central position, different from the position of the resonator at rest.Particular points in this limit cycle are labeled with letters A, B, B', C, D, E and F which are also shown in Figures 4b, 4c and 4d.Successive phases of the periodic regime are described hereafter.
Portion AB: Point A corresponds to an instant chosen arbitrarily in a slipping phase where the relative velocity Á _ u ¼ _ u À _ u f is increasing.The operating point is defined by the intersection between the line (D) and the friction law (see Fig. 4b).As the slope of (D) is particularly high, the operating point travels along the red arc in the direction indicated by the continuous red arrow, up to point B, very close to the extremum E. The difference between B and E is related to the choice of the time step Dt of the simulation.When the time step is small, this difference is also small.Portion BB': This phase corresponds to a force jump between points B and B'.It is more of a transition than a real phase.Portion B'CD: In this phase, the velocity of the resonator is greater than that of the finger; the relative velocity is slightly positive.To our knowledge, this situation is not encountered in the case of a bowed string, but it exists for other self-oscillating systems involving friction [16] (see Appendix B for a study of parameters involved in this phenomenon).Portion DE: The velocity of the glass rod reaches the velocity of the finger at point D, which initiates a sticking phase.This phase also starts with a jump in frictional force (see Fig. 4d).The frictional force then increases linearly until it reaches the maximum friction force F T = l s F N (point E).Portion EF: After the limit of adherence is reached, a new slipping phase begins, in which the velocity of the resonator is lower than that of the finger.Point F corresponds to the minimum velocity reached during one period.Portion FA: From point F, the relative velocity increases again until it reaches point A, which has been chosen as the starting point to describe the cycle.Now looking at the response of a four-mode model to the same set of control parameters (Figs.5c and 5d), several differences with the one-mode model can be observed.In the phase diagram shown in Figure 5c, the growth towards the periodic regime does not anymore follow a regular spiral as in Figure 5a.The response of several modes, coupled by friction, makes the trajectory in the phase plane more complex.At first glance, Figure 5d seems to indicate that a periodic regime is reached after a time similar to that in Figure 5b, i.e. 0.12 s.However, when the apparent periodic part of the time signal, highlighted in red in Figure 5d, is plotted in the phase plane (Fig. 5c), it does not follow a closed trajectory.Although sticking and slipping phase already alternate at this stage of the solution, they do not do so in a strictly periodic manner.Careful observation of the Figure 5d also reveals an amplitude modulation, visible on the lowest velocities reached by the glass rod.This modulation is due to the response of mode 4 being also unstable for these values of the control parameters, as will be shown later through linear stability analysis (see Sect. 3.2).For this four-mode model, after approximately 1 s, there are no more modulations, the periodic regime is reached.
The time that separates the beginning of the gesture (at t = 0) from the instant when a first sticking phase is triggered is denoted by s, as highlighted by a vertical line in Figures 5b and 5d.It describes the response time of the cristal.It is assumed that the instrument is easier to play if it responds quickly to the musician's gesture.Thus s is a playability criterion: the smaller the s, the greater the playability.In order to estimate this response time for a large number of control or design parameters without having to calculate as many time-domain responses, the following section proposes a less computationally expensive analysis.

Linear stability analysis
Linear Stability Analysis (LSA) is used to describe the behavior of a resonator that undergoes small perturbations around a slipping equilibrium point [17].For this purpose, the equations describing the system are linearized around this equilibrium point, which is fixed by playing parameters.The frictional force thus takes the form where Equation ( 15) corresponds to the first order Taylor expansion of F T (t) around the slipping equilibrium point.The frictional force F T is then written as the sum of a static value " F T and a dynamic value noted here A _ u t ð Þ.The dynamic part of the linearized frictional force is proportional to the velocity of the glass rod.The proportionality factor A depends on the parameters of the friction law, as well as on control parameters of the musician: normal force and finger velocity.The linearized friction law around an equilibrium point can be seen in Figure 4a for _ u f = 0.1 m.s À1 .Inserting the linearized expression of the frictional force, equation (15), into equation (3) yields which can alternatively be written as As seen in equation ( 18), the dynamic part of the linearized frictional force introduces intermodal coupling through damping.
The particular solution of equation ( 18) is a vector of constant modal amplitudes, which corresponds to a static displacement due to the static value F T .The solution of the homogeneous equation has the form Equation ( 20) is a generalized eigenvalue problem, which can be written under the following compact canonical form The eigenvalues k k are the poles of the system.The imaginary part of each pole gives information about the oscillation frequency, whereas the real part carries information about damping.The natural frequency x k and damping ratio n k of each pole can be calculated from The pole is stable if its real part is negative, i.e. n k > 0. In this case, the response of the pole to a small perturbation is an oscillation whose amplitude decreases exponentially.Conversely, the pole is unstable if its real part is positive, i.e. n k < 0. In this case, the pole responds to a small perturbation by an oscillation whose amplitude increases exponentially.This exponential growth is at the origin of selfsustained oscillations.Linear stability analysis can therefore be used to understand which conditions promote the emergence of self-sustained oscillations.These conditions may concern the dynamic parameters of the resonator, contained in the matrices M, C and K in equation ( 21), or the parameters of the musician's gesture or friction law, both encapsulated in the value of A (see Eq. ( 16)).
As an illustration, Figure 6a shows the conditions of existence of an instability in the form of a map with respect to the two control parameters, finger velocity _ u f and normal force F N .To obtain this map, the eigenvalue problem (Eq.( 21)) is solved for different combinations of these two parameters.For each solution, the smallest value of the damping ratio n min = min(n k ) is evaluated.The gray area in Figure 6a indicates the combinations of parameters for which the slipping equilibrium is stable, i.e. n min > 0, while the green area corresponds to an unstable slipping equilibrium, i.e. n min < 0, which may give rise to self-sustained oscillations.The boundary between these two zones, highlighted in red on this graph, thus indicates the minimum normal force to be applied to obtain instability, as a function of finger velocity.This minimal normal force, denoted by F N,min , is a criterion of playability.Indeed, the smaller F N,min , the smaller the force to be applied on the rod to trigger self-sustained oscillations.This minimum force is similar, for a wind instrument, to the minimum mouth pressure required to make the equilibrium solution at constant flow unstable [18].
The map shown in Figure 6a represents only the pole with the lowest damping ratio, but it should be noted that several poles may simultaneously be unstable due to friction.Figure 6b illustrates this by representing in the complex half-plane the evolution of the poles as a function of the normal force F N , for a constant finger velocity _ u f = 0.1 m.s À1 .The conjugate poles (negative imaginary part) are not shown in this figure.The positions of the poles for F N = 0 N correspond to the eigenmodes of the resonator.The numbers shown in Figure 6b thus refer to modes 1, 4, 7, 8 highlighted in Figure 3.As the frictional force increases, the real part of the individual poles increases, i.e. their damping decreases.The different poles become unstable (Rðk k Þ > 0) from a value of normal force which  4b, 4c and 4d.depends on the pole considered.The pole requiring the smallest normal force to become unstable corresponds to the main mode of the resonator (mode 7).This figure also shows that the imaginary part of the poles, i.e. their oscillation frequency, remains almost unchanged when the normal force is increased from 0 to 1 N.

Effects of design parameters on playability 4.1 Effect of glass rod height
As a first parametric study on the geometry of the resonator, we chose a parameter that can reasonably be varied experimentally: the height of the glass rod, denoted by h r in Figure 2b.It is varied here with a regular step of 1 cm.The extreme values that h r can take are chosen such that the bolts of the vibrating shaft (at points E and D in the Fig. 2b) do not touch those of the glass rod hanger.

Effects on modal basis
For each value of h r , the modal basis of the resonator is calculated with the finite element model.The natural frequencies of the modes retained in the minimal model (modes 1, 4, 7 and 8 in the Fig. 3) are plotted in Figure 7a as a function of relative glass rod height h = h r /L r .
The colors are attributed to these modes according to their order of appearance with frequency.The soft green markers correspond to the first flexural mode of the glass rod (mode 1 in Fig. 3).The global flexural mode of the resonator along the x-axis (mode 4 in Fig. 3) is represented by the dark red marker.When their frequencies are well separated, the modes associated with the purple and blue markers correspond respectively to the second flexural mode of the glass rod (mode 8 in Fig. 3) and the main playing mode (mode 7 in Fig. 3).When their frequencies are very close, i.e. for relative rod heights from h = 0.35 to h = 0.6, these two modes are significantly coupled.It is then difficult to distinguish the two coupled modes.A criterion based on the kinetic energy is therefore proposed to identify the main mode because the main mode is the one with the higher kinetic energy in the vibrating rod.The mode that meets this criterion is illustrated with colored crosses + in Figure 7.When the glass rod is fixed near the extremities of the vibrating shaft, the natural frequency of the main mode increases.These frequency variations are mainly due to the stiffness presented by the threaded shaft at the connection point with the glass rod, which is minimal near the middle and maximal at the extremities.
The numerical natural frequencies are validated by those obtained experimentally for some values of h, represented with black crosses Â in Figure 7a.The mode with a natural frequency around 20 Hz could not be identified experimentally due to a poor signal-to-noise ratio at low frequencies.Because of the good agreement between the numerical and experimental natural frequencies, the FE model is then used to calculate the natural frequencies and mode shapes necessary for the analysis of playability, whereas modal damping is set based on experimental data.A damping ratio of 0.09% is attributed to the four modes, for all values of h.This value corresponds to the average of damping ratios identified experimentally.

Effects on minimal normal force
We study the minimum normal force (F N,min ) necessary to make the different poles unstable, for a finger velocity fixed at 0.1 m.s À1 .This minimum force is calculated by LSA for each value of h, as described in Section 3.2, and represented in Figure 7b.
For each rod height, one pole becomes unstable at a lower force than the others.The minimum force associated to this pole can be referred to as the global minimum force.For example, at h = 0.4, the global minimum force is F N,min = 0.06 N, which triggers the instability of the main mode (blue marker).At the same rod height with a higher normal force, e.g.F N = 1 N, two poles are unstable (blue and dark red markers): the main mode and the global flexural mode of the resonator.Up to h = 0.54, the pole carrying the global minimum force corresponds to the main mode (mode 7).For h > 0.54, it is the pole corresponding to the global flexural mode of the resonator (mode 4).Note that the global minimum force is mostly decreasing for increasing h, except for 0.4 < h < 0.54.The global minimum force reaches its lowest value (F N,min % 0.04 N) for the highest glass rod height, which is not a configuration used by instrument makers (h = 0.4).A possible explanation could be that the frequency of the unstable pole, associated with the global flexural mode (f k % 65 Hz), does not correspond to the expected pitch (C3#, 277.2 Hz).This explanation is not sufficient because the resonators could have been designed differently so that the pitch is determined by the frequency of the global flexural mode.To understand the configuration chosen by makers, a second playability criterion is examined hereafter.

Effects on rise time
The rise time is calculated firstly using time domain simulations as described in Section 3.1.3.For each value of h, the response of the resonator to a normal force of 1 N and a finger velocity of 0.1 m.s À1 is simulated.The rise time calculated from the simulated glass rod velocity is illustrated by black square markers in Figure 7c.Note that for h < 0.1, with the chosen playing parameters, all the poles remain stable and therefore no self-sustained oscillation occurs.We notice that s reaches its minimum value around h = 0.4.This result seems to indicate that the glass rod height chosen by the instrument makers, i.e. h = 0.4, corresponds to a compromise between rise time and minimum normal force, which would thus correspond to an optimum of playability.
The rise time calculated from the time-domain simulation is then compared with that predicted by LSA.In order to calculate the rise time associated with each unstable k k pole, it is assumed that the starting transient involves only one pole and takes the form where amplitude C is estimated from the time-domain simulation and corresponds to the amplitude of the first oscillation.The rise time s k is then be calculated from equation ( 23) where the value of The rise time calculated by LSA for each unstable pole is shown in Figure 7c.It can be seen that the shortest rise time obtained by LSA is very close to the one calculated from time-domain simulations, even when several poles are unstable (from h = 0.29 to h = 0.87 for the 1 N force considered here).To illustrate this result, Figures 7d-7g show the response of the resonator for four values of glass rod height: h = 0.2, h = 0.4, h = 0.7 and h = 0.9.These four heights are highlighted in Figures 7a-7c by vertical gray dashed lines.The velocity of the glass rod as a function of time is represented in blue, the velocity of the finger _ u f is in orange.The envelope (see Eq. ( 23)) associated with the pole having the shortest rise time is also shown, using the same color as in Figures 7a-7c.It can be seen that the exponential growth calculated by LSA correlates well with the evolution of amplitude observed on time-domain simulations, at the beginning of the transient.As expected, LSA does not describe accurately the entire transient, which illustrates the limitation of this analysis.However, these results indicate that LSA can be used to provide a rough estimate of the response time of the instrument, which can be useful for anticipating the consequences of certain design parameters., 4.2 From low to high register: playability of the complete instrument So far, the analysis of playability has been carried out on a single resonator (C3#).In this section, playability is characterized over the entire range of the instrument.This is to answer the initial question: whether the weaker playability in the high register can be objectified through criteria accessible with the minimal model.
As seen in Section 2.2.3, the playing frequency corresponds to the natural frequency of the main mode, which essentially involves deformation of DE segment of the threaded rod (see Fig. 2), similar to the first mode shape of a clamped-clamped beam.Therefore, the playing frequency can be calculated as a function of the length of the threaded rod, denoted by L r .The desired playing frequency associated to each length L r is hereafter named the target frequency.

Effects on modal basis
First, only the length of the two threaded rods noted L r is modified.For each target frequency, a modal basis is calculated using the FE model.The natural frequencies are represented in Figure 8a.
It can be seen that the mode labeled by M, which corresponds to the main mode, has a natural frequency almost identical to the target frequency.In the same way as in Section 4.1, this mode can be identified using a criterion based on kinetic energy.Indeed, its mode shape presents a greater kinetic energy of the vibrating rod (DE segment in Fig. 2) than modes B, C and D.
The natural frequency of some modes is barely or not at all affected by the change in length L r .This is the case for modes that do not involve significant deformation of segment DE of the threaded rod, such as modes A, B and C, which correspond to flexural modes of the glass rod.This family of modes was also found using a variant of the FE model where the two threaded shafts were considered rigid, i.e. in a case were the glass rod was not coupled with the rest of the resonator.The corresponding natural frequencies are represented by horizontal dash-dotted lines in Figures 8a and 8d.
There are also modes whose natural frequency evolves in the same way as that of the main mode with respect to the target frequency.The corresponding markers, in Figure 8a, form oblique lines with approximately the same slope.One mode is found to evolve differently with respect to the target frequency.The corresponding markers form a line with a much lower slope.This mode, which is labeled by D and highlighted by the round black markers, involve significant motion of the rectangular mass on (plongeoir).It can also be identified using a criterion based on kinetic energy, since its mode shape presents a higher kinetic energy of the rectangular mass than the other modes.This mode could be described as a local mode of the plongeoir, however its natural frequency seems to be affected by the stiffness of the two threaded rods, which varies in length L r .
The different families of modes, which do not exhibit the same evolution with respect to the target frequency, can couple together when their frequencies get close to each other.A veering phenomenon clearly occurs for some modes, as revealed in Figure 8a by the corresponding branches which repel each other [19].An example of this phenomenon is particularly visible for the first two modes for a target frequency around 100 Hz.The phenomenon also occurs for the main mode in the upper range of the instrument.In the next paragraph, the consequences of intermodal coupling on playability is analyzed.

Effects on minimal normal force
The modal basis obtained at each target frequency is used to calculate the minimum normal force F N,min by LSA, in the same manner as in Section 3.2.The finger velocity is set to _ u f = 0.1 m.s À1 as in the previous analysis.As can be seen in Figure 8b, the main mode is the easiest to play, as it requires the lowest force to become unstable over the whole range of the instrument.Overall, the higher the playing frequency, the higher the minimum force necessary to achieve instability.This observation agrees with the musicians' statement that the notes are more difficult to produce in the high register.In addition to this overall trend, local increases in minimum force can be seen for high playing frequencies.They occur in conjunction with modal coupling between the main mode M and modes C and D. Although less visible in Figure 8b, the phenomenon also occurs when the main mode M couple with mode B. It can be concluded from these observations that when intermodal coupling occurs between the main mode and other modes, the playability is negatively impacted: a higher minimum normal force is necessary to produce the associated notes.Playability is therefore not homogeneous over the whole range of the instrument.

Rectangular mass design modifications
In view of the previous results, a design modification is proposed to avoid coupling between the main mode M and mode D, with the intent to improve playability in the high register.In this part, the length of the rectangular mass, denoted by L m , and the length of the threaded rods L r , is modified as a function of target frequency.More specifically, L m is chosen to be proportional to L r (see Fig. 8f which illustrates the proposed design, in comparison with the original design shown in Fig. 8c).The results are shown in Figures 8d and 8e.It can be observed that the natural frequencies associated with the local mode of the rectangular mass (mode D) no longer approach the natural frequencies of the main mode M (see Fig. 8d), so that intermodal coupling does not occur between these two modes.As a consequence, the minimum force in the high register of the instrument (see Fig. 8e) exhibits less irregularities than the original design (see Fig. 8b).
This example of design modification illustrates how LSA can be used to understand playability issues and propose specific treatments.Preventing the main mode from coupling with other modes seems to be a general rule to follow in order to keep playability homogeneous over the whole range.For instance, modal coupling between modes C (or B) and the main mode M, could be avoided by modifying the design of some resonators close to the corresponding playing frequency.

Conclusion
The cristal Baschet is a friction instrument that allows the production of very expressive and varied sounds.A minimal physical model, or toy model, was developed to describe the sound production of the instrument.It focused on the modeling of the interaction between the finger and a resonator.It takes into account the dynamics of the resonator and the law of friction; it allows the conditions of the occurrence of frictional instability to be described.Parametric studies on the geometry of the resonator have been carried out to understand the influence of the design parameters on the playability.In order to describe the playability, two playability criteria have been proposed, denoted by F N , min and s.These criteria respectively describe the ease of play (minimum normal force to obtain instability) and the reactivity of the instrument (time response of the instrument).The parametric study on the height of the glass rod allows us to understand the empirical choice of the instrument maker.Indeed, the height of the glass rod chosen seems to correspond to a compromise between ease of playing and a short response time of the instrument.The parametric study on the range of the instrument allows us to understand the difficulty for the instrument maker to adjust the instrument in order to make the high notes easy to play.The physical model explains why the ease of playing, which is qualified here by means of appropriate indicators, decreases as the frequency increases.Moreover, the coupling effects between the different modes of the resonator make the instability more complex, which influences and tends to degrade this playability.The model shows indeed that these couplings drastically decrease the playability criteria, to the point of making some notes unplayable.The choice of a different design of the resonator to avoid the coupling between the main mode and a rectangular mass mode has been proposed and allows for better playability.This study shows how a physical model can be used to understand the nature of the mechanisms governing playability.It follows that the analysis can assist instrument makers in the tuning of the instrument, by providing a model of the relationship between resonator design and playability criteria.

Appendix A Derivation of the numerical scheme
In order to facilitate writing and understanding, the procedure to obtain the recurrence relation of equation ( 6) is presented for one mode in particular (mode k).The kth modal coordinate is governed by the ordinary differential equation where the modal force F k is assumed to be constant over the time interval considered.The general solution of equation (A1) is composed of the particular solution and the general solution of the associated homogeneous equation, such that is the damped natural frequency and A k and B k are constants fixed by initial conditions.The time-derivative of q k (t) is At t = 0, following initial conditions are set, which yields Equations (A2) and (A3) can then be rewritten by replacing A k and B k by their expressions from equation (A5).
Knowing the initial values q 0 and _ q 0 as well as the constant force F k , the values of q k and _ q k after a small time interval Dt can then be calculated as The result can be generalized to calculate the solution at any time t i+1 = t i + Dt from the one known at t i by replacing q 0 and _ q 0 with q k (t i ) and _ q k ðt i Þ respectively, which yields These expressions are the basis of equation (6), where A 11 , A 12 , A 21 , A 22 , B 1 and B 2 are diagonal matrices composed of the coefficients defined in equation (A7).These coefficients only depend on modal parameters and chosen sampling time Dt.

Appendix B
Slipping-forward study The results obtained from the time-domain simulation, as described in Section 3.1.3,may appear unconventional when compared to the existing literature, particularly regarding bowed strings.This appendix aims to discuss the key parameters that contribute to this forward-slipping phenomenon, namely mobility, modal mass, and damping.
First, the dynamic behavior of the resonator is compared to that of a violin string.When tuned to the same frequency, the mobility of the resonator is lower than that of the string.For instance, in the case of the G3 note (approximately 200 Hz), the mobility of the resonator is approximately 50 dB lower than that of the string (see Fig. B1.a).As a consequence, the response of the two structures to the same frictional excitation differs significantly, as shown in Figure B1.b.The string has a larger amplitude of motion and exhibits a much longer sticking phase.The existence of harmonic natural frequencies, in the case of the string, also plays a role in these differences in response to the same gesture.In order to investigate an intermediate state between the cristal resonator and a violin string, the mobility of the resonator is increased by numerically reducing its modal mass (see Fig. B1.a and Tab.B1).As the modal mass of the cristal is decreased, the forward slipping effect is progressively reduced until it eventually disappears, as shown in Figure B1.c.
Based on these results, it can be postulated that the higher the mobility of the structure, the larger the amplitude of motion and the longer the duration of the sticking phase.Moreover, forward-slipping is favored by a high modal mass and low mobility.
For the same level mobility at the playing frequency, the forward-slipping phenomenon can be influenced by damping.To illustrate this, three structures are compared: the G3 single-mode cristal and two hybrid cristals created numerically based on this G3 cristal.The damping and modal mass of the latter two structures are simultaneously adjusted using n k = an 0 and m = m 0 /a, ensuring that the maximum mobility at resonance, remains the same (see Fig. B1.d and Tab.B2).The response to the same frictional excitation for the three structures is shown in Fig. B1.e).As a increases, both the amplitude of motion and the duration of the sticking phase increase.Conversely, the forward slipping phenomenon is reduced (see Fig. B1.f).It can be inferred that there exists a critical value of a for which forward slipping no longer occurs.
Table B1.Modal parameters used for three simulations with constant modal damping, corresponding to the results shown in

Figure 1 .
Figure 1.(a) Extracted and freely colored figure of patent of Structures Sonores [2], parts of actual cristal Baschet are highlighted in color; (b) One of the first designs of cristal with vertical keyboard and bladders (from the ImageStudio Ó Royal Museums of Art and History of Bruxelles -Creative Commons CC BY -KMKG/RMAH/MRAH.);(c) cristal Baschet with piano soundboard (photography by Laszlo Ruszka provided by SSB association).

Figure 2 .
Figure 2. (a) Today's cristal Baschet with the elements that compose it (Pictures freely annotated from Collections Musée de la musique (Paris), taken by Jean-Marc Anglès in 1999, referenced as cristal, Bernard et Frangois Baschet, Paris, France, 1980, E.983.14.1);(b) cristal's resonator with the elements that compose it, notations used to describe the model and the geometrical parameters studied; (c) Setup of experimental modal analysis of C3# resonator.

Figure 3 .
Figure 3. Experimental (a) and numerical (b) transfer-mobility between the point A and B.Respectively, the blue, light and dark gray correspond to the transfer-mobility in x-axis, y-axis and z-axis.Modes are numbered in order of appearance.Modal deformations are illustrated and their corresponding experimental eigenfrequencies (and numerical), and experimental modal damping ratios are noted.

Figure 4 .
Figure 4. (a) The red line corresponds to the dynamic friction coefficient, describing the friction law; The dashed black line corresponds to the experimental friction law defined by Derler and Rotaru [10]; the gray line is the linearized friction law at a slipping equilibrium point symbolized by the gray cross; the white background corresponds to the zoom on the friction curve for (b).(b) represents the friction curve.The black line named D corresponds to the linear function that allows the calculation of the time-domain simulation (see Eq. (10)) for the instant A; the seven instants (A-F) describe the stick-slip cycle (refer to the Figs.5a, 4c and 4dfor illustration of these instants).The different steps of the stick-slip cycle are indicated by the different colors (red, cyan, blue and black); the gray line refers to the unused part of the friction law.The arrows show the direction of the cycle, each branch is taken twice during one cycle: the first passage is shown by solid arrows, the second passage by dashed arrows; (c) and (d) correspond respectively to relative velocity and friction force as a function of time; the red and black lines refer to the slipping phase, when the resonator is respectively slower and faster than the finger.The cyan color (for (b)-(c)-(d)) represents the passage between these two parts of the curve.The blue color represents the sticking state.

Figure 5 .
Figure 5. (a) and (b) are the time-response of the one-mode model to frictional excitation ( _ u f = 0.1 m.s À1 and F N = 1 N).(c) and (d) are the time response of the four-mode model (modes 1, 4, 7 and 8) to the same excitation.The orange lines correspond to the speed of the finger, the blue lines correspond to the resonator's movement.(a) and (c) are phase diagrams, corresponding to speed versus displacement of the contact point C of the resonator; the red crosses show the starting point of the simulation: resonator at rest.(b) and (d) are speed versus time graphs, the vertical black lines correspond to the first instant where the speed of both resonator and finger are identical, the red curves correspond to approximately 1 period of the periodic regime.The seven instants (A-F) in (a) are the same as the Figures 4b, 4c and 4d.

7 .
(a) Eigenfrequency versus the relative height of the glass rod, black crosses (Â) refer to experimental frequency, colored crosses (+) correspond to the principal mode (closest to mode 7), colored points correspond to other modes; (b) Minimal applied force to destabilize each pole with _ u f = 0.1 m.s À1 , this playability criteria is calculated by LSA, the colors refer to the color of mode in (a); (c) rise time for each relative height of glass rod with F N = 1 N and _ u f = 0.1 m.s À1 , black squares are calculated from the timedomain simulation; colored dots and crosses correspond to rise time calculated by LSA of each pole in (a); (d), (e), (f) and (g) represent the speed versus time with F N = 1 N and _ u f = 0.1 m.s À1 : for the finger (orange) and for the resonator (blue).The envelope is calculated with LSA, the color corresponds to poles in (a).

Figure 8 .
Figure 8.The left column ((a), (b) and (c)) corresponds to a parametric study of the variation of the length of the shaft L r whose each studied length corresponds to the note of the cristal.In these plots the rectangular mass has always the same length.The right column ((d), (e) and (f)) corresponds to a parametric study where the length of the rectangular mass L m is proportional to the length of the shaft L r ; (a)-(d) variation of eigenfrequencies for each particular resonator pitched on the note of the cristal, the principal mode M is underline by the + marker, the mode of the rectangular mass D is underlined by the o marker.The dash-dotted lines correspond to the eigenfrequencies of the mode A, B and C, these eigenfrequencies are calculated by a specific FEM where the threaded shafts are considered rigid; (b)-(e) is the minimal applied force calculated by LSA for each note and each pole, the color respectively refers to the mode in (a) and (d); (c)-(e) is an illustration of the resonator corresponding to the minimum and maximum note for the parametric study.

FiguresFigure
Figure B1.(a)-(c) refer to three different structures where the playing frequency and the damping rate are invariant (f = 200 Hz and n = 0.29%) in blue line the G3 string of a violin, in black line a G3 resonator of cristal, in yellow line hybrid cristal where the modal mass has been divided by 5; (d)-(f) refer to 1-mode cristal's model with equal maximum mobility, with variation of damping and modal mass by a ratio; (a) and (d) are transfer-mobility functions; (b) and (e) are phase diagram of the periodic regime of the different system responses to the following frictional excitation: sudden application of bowing velocity _ u f = 0.1 m.s À1 , with a normal force F N = 0.3 N; (c) and (d) are the zoomed Figures (b) and (e).

Table 1 .
Material parameters used in FE model of resonator.