Page  00000216 Friction and Application to Real-time Physical Modeling of a Violin Stefania Serafin* Christophe Vergez and Xavier Rodet,, Ircam - Centre Georges Pompidou 1, Place Igor Stravinsky 75004- Paris - France Abstract In this paper we focus on two bow-string interaction models, which are included in a physical model of a bowed string instrument. While the first model is based on friction measurements, the second one is an actual model of adhesion between two bodies in contact. I Introduction One key point in physical modeling of bowed string instruments is to find a suitable equation that describes the friction force between the bow and the string. This force is a highly nonlinear and eventually discontinuous function of bow parameters and dynamics of the model. The friction mechanism gives rise to the well known stick-slip phase alternation, characteristic of the so-called Helmholtz motion. In this paper, we focus on two models of bowstring interaction. The first one is based on simplifications of friction measurements. The bow-string interaction is then represented by an hyperbola, which allows us to solve analytically the coupling with the linear vibrational behaviour of the string. Since friction appears almost everywhere, it is interesting to analyse how models that describe this phenomenum in other contexts behave, when applied to a bowed string model. Thus, the second model of bowstring interaction is an actual model of adhesion between two bodies in contact. It was first proposed by Dahl [1] and then improved by Hayward and Armstrong [7] to synthesize friction in the context of haptic rendering. The main characteristics of this model are the fact that it is autonomous (i.e. independent of time), easy to be discretized and with parameters that have a simple physical interpretation. To evaluate the influence of the friction models on the quality of the synthesis, we built a complete violin model, as described in the next section. 2 Structure of a Violin Model We adapted a general bowed string model (cf. [2], [6]) that has been used for a long time for its ability of repro*Current Address: CCRMA, Stanford University, CA. USA - serafin fb Vbb Figure 1: General structure of a bowed string model ducing, despite its simplicity, many of the phenomena observed in bowed string instruments. The structure of the model is represented in figure 1. This model supposes that the bow is applied to a single point of the string, which is at the normalized distance Pb from the bridge (Pb = 0.5 represents the middle of the string). The bow velocity and the bow force are noted Vb and fb respectively. At the contact point, two variables are considered, which are the friction force f and the transverse velocity of the string v. Friction and velocity are, to a good approximation, related by a highly non-linear function. When the two body stick together, the velocity of the bow Vb is equal to the one of the string v, otherwise they are sliding. The alternance of stick and slide phases gives rise to the Helmholtz motion. In our model strings are represented by fractional delay lines, and losses along the string and at each extremity are lumped into low-pass filters Hn(z) and Hb(z) (where n stands for nut and b for bridge), whose coefficients are estimated according to impulse response of violin's strings. The velocity v at the contact point results from the contribution of the waves vi. -216 - ICMC Proceedings 1999

Page  00000217 and vi~ coming from the nut and the bridge respectively. Two travelling waves v,, and Vo, resulting from the bow-string interaction propagate toward the nut and the bridge, in such a way that we can write: V = von + vi, = Vo, + Vi, -- rb + ib The contribution of the reflected waves vi, and vib are summed at the contact point: Vh = vi,+ Nib Bow string interaction is represented by: 4100 4200 4300 4400 4500 t (samples) f = 2 Z(v - vh) (la) f = 7(v-vb) (lb) (1) where 7 is a function that represents the bow-string interaction. Once this coupling has been solved, the new outgoing waves o,, and vob are calculated by equations (2a) and (2b): %f Figure 2: Velocity of the string v at the contact point 0.3. 0........ ' - -; I0.1 5 -................,.................................. -..... 005................................... -0.0 -0.05...... vto = i, + "t (2a) (2b) (2) 3 An Hyperbolic Friction Model In this section, we are mainly interested in modelling the friction mechanism. 3.1 Description of the Model The first approach consists in considering that, as in equation (lb), friction is simply a function of the relative velocity v-Vb of the bow and the string. In this section, this function is first approximated by fitting an hyperbola to data measured in a real instrument (as seen, for example, in [3]). This leads to: (V - vb)= 0.3+ 0.2 + Iv - vb.1 (3) v - vb < 0 In this case, we have shown that the coupling between the bow and the string, i.e. system (1), can be solved analytically. In fact, equations (3) and (la) lead to a second order polynomial equation, which is easily solved. Since the vertical portion of the curve (cf. figure 3) gives a third solution, the choice between multiple solutions is made by the so-called hysteresis rule (cf. [2]). This phenomenological rule stipulates that if the system is in a stick state, it remains in a stick state. However the actual solution is never the one which lies between two other solutions. 3.2 Simulation Results The bow-string interaction just seen above, is included in the model described in section 2. I S -0.8 *0.6 -0.4 -0.2 0 0.2 0.4 0.8 0.0 V Figure 3: Friction f versus string velocity v Figure 2 shows the velocity of the string v at the contact point. Simulations have been performed with fb = 0.3N, vb = 0.7m/s, pb = 0.15. The corresponding friction f versus string velocity v is shown in figure 3. Note that the set of all the crosses, obtained by solving the coupling at each time step, allows to reconstruct, as expected, the hyperbola branch chosen as friction curve, displayed for comparisions with the one obtained in the model introduced in the next section. Note also that the points that lie on the vertical portion of the curve correspond to a stick state (v = vb), whereas the others correspond to a slip state (v < Vb). This model runs in real time on a G3 Macintosh computer, using the platform Max/MSP and is controlled by a tablet-based playing interface, as described in a companion paper [4). Despite its simplicity, the perceptive results are satisfatory. It is therefore interesting to explore how more complex models influence the quality of the synthesis. 4 A New Friction Model It has recently been observed ([5]) that considering friction as a function of relative bow-string velocity, as supposed in the previous section, is just an approximation. Actually at the contact point, the friction force versus ICMC Proceedings 1999 - 217 -

Page  00000218 c 80 0 c; 75 0 70 70.~.~.~...... ~..; ~......~..... ~. i................'~~..........................................,....... - -............-......-...... -......-........ 7.~~~~I~~~~~~~~~~~~*~ 7~r ~ ~~v a ~~~~ ~~~~~ ~ ~~r 1.2 0..................................................... *..................., I........................................................ I................ 400 450 500 550 800 650 700 750 400 450 500 550 800 650 700 750 800 time Figure 4: Left picture: positions of the moving object x (straigth line) and of the contact point w versus time (in samples). Right picture: distance z = x - w versus time. velocity exhibits an hysteresis loop which is not represented in the "classic" friction curve. In order to reproduce the friction mechanism with more accuracy, we focus on a model first proposed by Dahl ([1]). This model emerges from the observation that bodies in contact exhibit a presliding displacement, which consists on a surface deformation of the contact. One of the main differences compared to the previous model is the fact that a microscopic displacement can arise even when there is no macroscopic motion. 4.1 Description of the Dahl Model Let's consider two objects in contact where x is the position of the moving object and w the position of the adhesion point. A micro-displacement between the two objects is measured by z = x - w (with jzi < Zmax). Dahl model supposes that the friction force is proportional to the distance z, as if the two objects where attached by a spring. As detailed in [7], the most general form proposed by Dahl is: d- f= ov (1 - a(z) z sgn(v)|' sgn (1 - a(z) z sgn(u)) dt (4) where v = -, f represents the friction force and a is a nonlinear function of z which accounts for adhesion properties of the specific materials. When i = 1, an algorithm is proposed in [7] to solve numerically the Dahl model. 4.2 Discretization of the Dahl Model As proposed in [7], equation (4) can be solved in terms of displacement as follows: t (samples) Figure 5: Friction f versus time (in samples) in the physical model of violin including the Dahl model for friction. * otherwise Wk = Wk-1 + 1 Xk - Xk-.1 a(Xk - Wk-1) (Xk - Wk-i) The shape of a can vary to better fit the adhesion properties of the two materials. However we choose, as proposed by [7] formula (5) which seems suitable for our needs: 1 -8 -(nk) =.8 -mar cstick + Zk (5) where:maz is the maximal distance between positions of the moving object xk and the adhesion point Wk, and Zstick is a parameter related to the adhesion properties. The way the model behaves is better understood looking at figure 4: we see how the contact point wk moves when it is driven by the moving object zk, and the oscillation of the corresponding distance z.. * if a(k - Wk-1) |Ik - Wk-i > 1 wk = Xk ~ zmar 4.3 Application to a Bowed String Model The friction model described in section 4 has been included in the physical model described in section 2. ICMC Proceedings 1999 - 218 -

Page  00000219 1v - V6 Figure 6: Friction f versus relative velocity u - ub between the string and the bow in the physical model of violin including the Dahl model for friction. In this new model, since microdisplacements are taken into account, it is possible to distinguish between Xk, which represents the position of the string related to the position of the bow at time k, and wk which is, as before, the contact point between the bow and the string at time k. The linear part of the violin's model, i.e. the propagation of the waves in the string remains unchanged. At the bow contact, friction is obtained by solving the differential equation (4) (with i = 1) together with equation (la),.which leads to the following algorithm: * The position of the string at time k is first calculated: zk = (vk - vb) x (tk - tk-1) + Xk-i * Calling zk = Xk - Wk-., c(zk) is calculated according to relation (5). * According to the distance zk, and taking into account the adhesion properties (i.e. a(zk)) the new position of the adhesion point is calculated: - if Izk xa(zk) > 1: notice the difference between the hyperbolic model and the Dahl model, comparing figures 3 and 6. In the first one, friction only depends on v-vb, while in the second one an hysteresis loop is present, as in the measures described in [5], which show that velocity is not the only state variable which determines the friction force. 5 Conclusion Two models of bow-string interaction have been proposed. The "hyperbolic model" has been included in a real-time model of violin and gives satisfactory perceptive results. However, another model of friction has been tested to simulate bow-string interaction. The friction characteristics obtained with this last model shows an hysteresis loop, as recently been measured in stickslip oscillations. For this reason we believe that the friction mechanism itself is better reproduced with this last model. However, further investigations are necessary to evaluate the characteristics of this new model in a realtime musical context. References [1] P. R. Dahl. Solid friction damping of mechanical vibrations. AIAA, 40(3):1675-82, 1976. [2] M. E. McIntyre, R. T. Schumacher, and J. Woodhouse. On the oscillations of musical instruments. 74:1325-1345, 1983. [3] R. T. Schumacher. Measurements of some parameters of bowing. JASA, October 94. [4] S. Serafin, R. Dudas, M. Wanderley, and X. Rodet. Gestural control of a real-time physical model of a bowed string instrument. In ICMC'99 Proceedings, Beijing, October 1999. Icma. [5] J. H. Smith. Stick-slip vibrations and its constitutive laws. PhD thesis, University of Cambridge, 1990. [6] J. O. Smith. Synthesis of bowed strings. Venice, Italy, 1982. Computer Music Association. [7] B. Armstrong V. Hayward. A new computational model of friction applied to haptic rendering. In Preprints of ISER'99 (6th Int. Symp. on Experimental Robotics), march 1999. ({ Wk f = k - X k mar = o"0 zmax where ( = sgn(zk x a(:k)) - iflzk xa(:k) < 1: f k= Wk(.-l + k 1 k - Xk.k-1 X C(zk) X Zk { = O~O(Xk-Wk 1. S= -(Ok) (xT - Wk-1) k - k* New outgoing velocities vo,, and v,, traveling towards the nut and the bridge respectively are finally calculated with system (2). 4.4 Simulation Results In figure 5,. we can see the friction force versus time, whereas in figure 6 is represented the friction force f versus the relative string-bow velocity v - vb. We can ICMC Proceedings 1999 - 219 -