Chaotic behavior of the piccolo

Transcription

1 Buens Aires 5 t 9 September, 2016 Acustics fr the 21 st Century PROCEEDINGS f the 22 nd Internatinal Cngress n Acustics Numerical Cmputatin in Musical Acustics: Paper ICA Chatic behavir f the piccl Nichlas Girdan Auburn University, Auburn, Alabama, United States, njg0003@auburn.edu Abstract A direct numerical slutin f the Navier-Stkes equatins in three dimensins has been used t cmpute the sund pressure prduced by a piccl as a functin f time p(t). Fr mderate blwing speeds u, a pure tne is prduced, but as u is increased p(t) exhibits an increasingly cmplex behavir. The behavir f p(t) is cnsistent with a psitive Lyapunv expnent at high values f u. Detailed results fr the pwer spectrum reveal a simple pure tne dminated by a single frequency at lw u, as expected. As u is increased additinal frequencies appear in the spectrum alng with bradband nise in certain spectral regins. The results suggest that the piccl is, under certain blwing cnditins, a chatic system. Keywrds: piccl, instrument mdelling, wind instruments, chas

2 22 nd Internatinal Cngress n Acustics, ICA 2016 Buens Aires 5 t 9 September, 2016 Acustics fr the 21 st Century 1 Intrductin Chatic behavir f the piccl Much prgess has been made in recent years in the mdelling f musical instruments at a first principles level. By this we mean mdelling based n the fundamental laws f mechanics. We nw have mdels f this kind fr many string instruments such as the pian and guitar (see, e.g., [1-4]). Wind instruments present a bigger challenge than string instruments, since a detailed descriptin f a wind instrument requires applicatin f the Navier-Stkes equatins. These are a set f nnlinear partial differential equatins that describe the behaviur f the air flwing thrugh an instrument. Fr realistic instrument gemetries slutin f the Navier-Stkes equatins requires a numerical treatment with a high perfrmance (parallel) cmputer, and in the past several years available cmputers have becme pwerful enugh t yield useful results fr wind instruments, althugh early wrk f this kind was first given mre than a decade ag (e.g., [5,6]). In this paper we describe such a simulatin study f the piccl. We present results fr the acustic pressure as a functin f time p(t) utside the instrument fr different values f the blwing speed. The results suggest that at high blwing speeds the behaviur f p(t) can be chatic. 2 The mdel The Navier-Stkes equatins in three dimensins are given in Eq. 1. Here u, v, and w are the cmpnents f the air velcity alng the x, y, and z, directins and ρ is the density, c is the speed f sund and ν is the viscsity. Nte that we must assume a cmpressible fluid in rder t calculate the sund pressure. Fr an ideal gas such as air, the sund pressure p (which is the variatin f the pressure frm its equilibrium value) is prprtinal t variatins f the density frm its backgrund value. ρ t + (ρu) x + (ρv) y + (ρw) = 0 z u t + u u u +v x y + w u z + c 2 ρ ρ x ν 2 u = 0 v t + u v v +v x y + w v z + c 2 ρ ρ y ν 2 v = 0 w t + u w w +v x y + w w z + c 2 ρ ρ z ν 2 w = 0 (1) Our mdel is shwn in Figure 1. This is a simplified mdel f a piccl, in which the bdy f the instrument is a cylindrical tube f ttal length 270 mm and diameter 12.5 mm, with a rectangular 2

3 22 nd Internatinal Cngress n Acustics, ICA 2016 Buens Aires 5 t 9 September, 2016 Acustics fr the 21 st Century embuchure pening f length 10 mm (alng the axis) and width 7 mm. The distance frm the center f the embuchure t the clsed end was 11 mm. There are n tne hles and the embuchure is a straight hle thrugh the tube with n lip plate. The lips f the player were mdelled as a slid channel clsely adjacent t the embuchure as shwn in Figure 1. T blw the flute, we impsed a cnstant air flw at the end f the lip channel farthest frm the embuchure and parallel t the channel. The channel curves as it appraches the embuchure s as t direct its air jet at an angle with respect t the directin tangent t the tp f the tube. Fr all f the results presented in this paper that angle was 20º. This blwing angle was fund t give a well behaved result fr p(t) fr a wide range f blwing speeds. Sme ther blwing angles were als investigated and thse results will be presented elsewhere. These dimensins and the verall design f ur mdel are similar thse f a real piccl. We expect the behaviur we calculate t als be fund in the transverse flute. lips embrchure hle bdy f instrument Figure 1. Mdel piccl shwing the lcatin and rientatin f the channel that acts as the lips. The end f the tube at the lwer left is clsed while the end at the upper right is pen. The dimensins are given in the text. The Navier-Stkes equatins were slved in a nn-unifrm Cartesian grid using an explicit finite difference time dmain algrithm described elsewhere [7]. Inside and near the piccl the grid spacing was 0.2 mm perpendicular t the axis and 0.5 mm alng the axis, and the time step was 0.2 µs. The piccl was cntained in a clsed rectangular bx with dimensins 60x60x360 mm 3 and with a ttal f 2x10 7 grid pints. 3

4 22 nd Internatinal Cngress n Acustics, ICA 2016 Buens Aires 5 t 9 September, 2016 Acustics fr the 21 st Century 3 Results 3.1 Time dependence f the acustic pressure Figure 2 shws typical results fr the acustic pressure as a functin f time p(t), as calculated fr a pint utside the piccl. In Fig. 2a the blwing pressure was u = 14 m/s, and the behaviur f p(t) was regular and peridic. As a musical tne, it was nearly a pure tne with nly a small amunt f pwer at the secnd harmnic (the spectrum will be shwn belw). Figure 2a actually shws the behaviur with u = 14 m/s with tw slightly different blwing cnditins. The slid curve shws results with the blwing speed ramped up linearly with time starting frm zer and reaching the final value f u = 14 m/s at t = 5.00 ms (a ramp-up time that seems typical fr wind instruments.) The dtted curve in Fig. 2a shws results with a ramp-up time f 5.05 ms. It is seen that this 1% change in the ramp-up time prduced very little change in the resulting p(t) t ramp = 5.00 m/s t ramp = 5.05 m/s (a) t ramp = 5.00 m/s t ramp = 5.05 m/s (b) p (Pa) u = 14 m/s t (ms) p (Pa) u = 20 m/s t (ms) Figure 2: Results fr the sund pressure as a functin f time, p(t), at a lcatin utside the tube and away frm the tube axis in Fig. 1. Left: Fr a blwing speed u = 14 m/s. Right: Fr a blwing speed u = 20 m/s. The slid and dtted curve shw results fr tw different ramp-up times as explained in the text. Figure 2b shws results fr a larger blwing speed u = 20 m/s. Fr this blwing speed p(t) has a mre cmplex spectrum, and deviates frm a pure tne in ways that will be described in detail belw. In additin, changing the ramp-up time frm 5.00 ms t 5.05 ms prduces a very significant change in p(t). This sensitivity t a small change in the blwing cnditins seen in Fig. 2b suggests that when blwn at u = 20 m/s led us t cnsider if this mdel piccl is a chatic dynamical system [8,9]. A central prperty f a chatic dynamical system is an extreme sensitivity t changes in the initial cnditins. Fr ur mdel piccl, the mvement f the state f the piccl thrugh phase space can be described by p(t). If we then cnsider tw phase space trajectries p 1 (t) and p 2 (t) that riginate frm tw slightly different initial cnditins, then fr a chatic system the tw trajectries diverge expnentially with time, with an expnent λ called the Lyapunv expnent 4

5 22 nd Internatinal Cngress n Acustics, ICA 2016 Buens Aires 5 t 9 September, 2016 Acustics fr the 21 st Century [10]. The Lyapunv expnent is psitive fr a system in a chatic state, and zer r negative fr a nn-chatic system. An analysis f ur mdel piccl suggests that λ underges a transitin frm a negative t a psitive value fr blwing speeds arund 15 m/s [11]. The piccl wuld nt be the first musical instrument t exhibit chatic behaviur. Previus wrk n the clarinet [12-14] and the cymbal [15] has fund that these tw instruments can als be chatic. In the case f the clarinet the nnlinearity f the reed is respnsible fr the chatic behaviur, while fr the cymbal the nnlinear equatin f mtin f the cymbal is the direct cause. Fr the piccl, the chatic behaviur seems t be due t the inherent prperties (i.e., nnlinearities) f the Navier-Stkes equatins. This is nt surprising in view f the wrk f Lrenz [5] and many ther analyses f fluid systems described by the Navier-Stkes equatins. 3.2 Spectral prperties Our results suggest that there is a transitin t a chatic state as the blwing speed is increased. It is believed that there are a relatively small number f different ways that this transitin can ccur. Well knwn rutes t chas include perid dubling, intermittancy, and a rute invling Hpf bifurcatins [10]. These different rutes t chas can be distinguished thrugh the spectrum f the trajectry thrugh phase space; in ur system this trajectry is specified by p(t). Figure 3 shws results fr the pwer spectrum f p(t) at tw blwing speeds, u = 14 and 20 m/s, the same blwing speeds cnsidered in Fig. 2. Examinatin f these spectra and the spectra fr ther blwing speeds reveals the fllwing. At lw blwing speeds p(t) is apprximately a pure tne, dminated by a cmpnent with a fundamental frequency f 1 and with a much smaller cmpnent at f 2, where f 2 = 2 f 1. The cmpnent at f 2 is thus simply the secnd harmnic f f 1, which is what wuld be expected fr a typical piccl tne. This behavir is seen in Fig. 3a with u = 14 m/s (a) u = 14 m/s 1000 (b) u = 20 m/s Pwer (arb. units) f 1 f 2 Pwer (arb. units) f 1 f 2 f f (Hz) f (Hz) Figure 3: Results fr the spectrum f p(t) at the tw blwing speeds cnsidered in Fig. 2.The apprximate lcatins f the fundamental frequency f 1, the secnd harmnic f 2, and a third frequency cmpnent f 3 that is nt harmnically related t f 1 are shwn. While the vertical scale is arbitrary, it is the same in parts (a) and (b). 5

6 22 nd Internatinal Cngress n Acustics, ICA 2016 Buens Aires 5 t 9 September, 2016 Acustics fr the 21 st Century As the blwing speed is increased, the cmpnent at f 1 bradens and the cmpnent at f 2 als bradens and reduces in amplitude, becming a shulder n a smth backgrund. This is illustrated in Fig. 3b with u = 20 m/s. In additin, a third cmpnent appears at a higher frequency f 3. frequency (Hz) f f 2 f u (m/s) Pwer (arb. units) 1000 P P 2 P u (m/s) Figure 4: Left: Frequencies f 1, f 2, and f 3 f the cmpnents identified in the pwer spectra in Fig. 3 as a functin f blwing speed u. Right: Peak pwer f the cmpnents at frequencies f 1, f 2, and f 3. as a functin f blwing speed. Figure 4 shws hw the frequencies f these three cmpnents and their peak pwers vary as a functin f blwing speed. The secnd harmnic at f 2 disappears int the backgrund arund u = 20 m/s while the cmpnent at f 3 becmes visible nly at a slightly lwer blwing speed. Nte that we are shwing the peak pwer in Fig. 4; the ttal integrated pwer under these peaks wuld exhibit a slightly different behaviur, since the peak widths change cnsiderably with u. Hwever, allwing fr variatins f the width wuld nt change the qualitative behaviur seen in Fig Discussin The current understanding f chatic dynamical systems is that there are a relatively small number f rutes t chas [10]. One f thse rutes invlves a Hpf bifurcatin in the system s trajectry in phase space, t a state in which an scillatin with a particular frequency is fund. As the system is driven harder, i.e., by varying a parameter such as the blwing speed in ur case, the system may then underg a series f Hpf bifurcatins with each crrespnding t the appearance f a new mde with its wn frequency. Theretical arguments [9,10] suggest that after tw r three bifurcatins the system enters a chatic state with tw r mre independent frequencies. This picture is bradly cnsistent with the behavir seen in the spectra in Fig. 3 and the behavir f the mdes fund in Fig. 4. 6

7 22 nd Internatinal Cngress n Acustics, ICA 2016 Buens Aires 5 t 9 September, 2016 Acustics fr the 21 st Century 5 Cnclusins The results fr the spectra suggest that ur mdel piccl is a chatic dynamical system in which the transitin t the chatic state is gverned by a Hpf bifurcatin. Our results are bradly similar t that fund in ther systems described by the Navier-Stkes equatins [16]. We shuld nte that an analysis f the sensitivity f p(t) t changes in the initial cnditins and the assciated Lyapunv expnent [11] suggests the transitin t chas ccurs at a blwing speed f abut u = 15 m/s, which is near where the fundamental cmpnent exhibits a peak pwer in Fig. 4. There is ne aspect f ur mdel piccl that is a bit unexpected. The variatin f f 1 with blwing speed in Fig. 4a is much strnger that expected r fund fr a real piccl [17]. Fr a real instrument ne finds that f 1 increases nly a little with blwing speed, usually much less than a semitne, befre jumping an ctave. Our mdel piccl shws a much strnger variatin f f 1. The reasn fr this is nt entirely clear, but sme preliminary measurements n piccls with different shaped embuchures reveal that embuchures with shaper edges, like thse in ur mdel (Fig. 1) can exhibit a quite sizeable variatin f f 1 befre jumping t a higher ctave. This will require further study, as will the pssibility f chatic behaviur f p(t) prduced by a real instrument. Acknwledgments The cmputatins described in this paper were perfrmed at the Rsen Center fr Advanced Cmputing at Purdue University and the Auburn University Center fr High Perfrmance Cmputing. This wrk was supprted by U.S. Natinal Science Fundatin grant PHY References [1] Girdan, N.; Jiang, M. Physical mdeling f the pian, Eurpean Jurnal f Applied Signal Prcessing, special issue, Vl , 2004, [2] Chabassier, J.; Chaigne, A., Jly, P. Mdeling and simulatin f a grand pian, Jurnal f the Acustical Sciety f America, Vl 134 (1), 2013, [3] Derveaux, G.; Chaigne, A.; Béchache, E.; Jly, P. Time-dmain simulatin f a guitar. I. Mdel and methd, Jurnal f the Acustical Sciety f America, Vl 114 (6), 2003, [4] Bader, R. Cmputatinal mechanics f the classical guitar. Springer, Berlin (Germany), [5] Skrds, P. A. Mdeling f flue pipes: Subsnic flw, lattice Bltzmann and parallel distributed cmputers, Ph.D. thesis, MIT, Cambridge, MA (USA), [6] Kuhnelt, H. Simulating the mechanism f sund generatin in flutes using the lattice Bltzmann methd, Prceedings f the Stckhlm Musical Acustics Cnference (SMAC 03), (2003). [7] Girdan, N.: Simulatin studies f a recrder in three dimensins, Jurnal f the Acustical Sciety f America, Vl 135 (2), 2014, [8] Lrenz, E. N. Deterministic nnperidic flw, Jurnal f Atmspheric Science, Vl 20 (2), 1963, [9] Schuster, H. B., Deterministic Chas. Physik-Verlag, Deerfield Beach, FL (USA),

8 22 nd Internatinal Cngress n Acustics, ICA 2016 Buens Aires 5 t 9 September, 2016 Acustics fr the 21 st Century [10] Eckmann, J-P.; Ruelle, D. Ergdic thery f chas and strange attractrs, Reviews f Mdern Physics, Vl 57 (3), 1985, [11] N. Girdan, t be published. [12] Maganza, C.; Caussé, R.; Lalë, F. Bifurcatins, perid dublings, and chas in clarinet-like systems, Eurphysics Letters,Vl 1 (6), 1986, [13] Takahashi, K.; Kdama, H.; Nakajima, A. Numerical study n multi-stable scillatins f wdwind single-reed instruments, Acustica-Acta Acustica, Vl 95 (6), 2009, [14] Taillard, P. A.; Kergmard, J.; Lalë, F. Iterated maps fr clarinet-like systems, Nnlinear Dynamics, 62 (1-2), 2010, [15] Chaigne, A.; Tuzé C. Lyapunv expnents frm experimental time series: Applicatin t cymbal vibratins, Acustica-Acta Acustica, 86 (3), (2000) [16] Gllub, J. P.; Swinney, H. L. Onset f turbulence in a rtating fluid, Physical Review Letters, Vl 35 (14), 1975, [17] N. Girdan and C. Laird, unpublished. 8