The Dynamics of Tibetan Singing Bowls

Transcription

1 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls he Dyamics of ibeta Sigig Bowls Octávio Iácio & Luís L. Herique Istituto Politécico do Porto, Escola Superior de Música e Artes do Espectáculo, Musical Acoustics Laboratory, Rua da Alegria, 503, Porto, Portugal José Atues Istituto ecológico e uclear, Applied Dyamics Laboratory I/ADL, Estrada acioal 0, Sacavém Codex, Portugal Summary ibeta bowls have bee traditioally used for ceremoial ad meditatio purposes, but are also icreasigly beig used i cotemporary music-makig. hey are hadcrafted usig alloys of several metals ad produce differet toes, depedig o the alloy compositio, their shape, size ad weight. Most importat is the soudproducig techique used either impactig or rubbig, or both simultaeously as well as the excitatio locatio, the hardess ad frictio characteristics of the excitig stick (called puja). Recetly, researchers became iterested i the physical modellig of sigig bowls, usig waveguide sythesis techiques for performig umerical simulatios. heir efforts aimed particularly at achievig real-time sythesis ad, as a cosequece, several aspects of the physics of these istrumets do ot appear to be clarified i the published umerical formulatios ad results. I the preset paper, we exted to axi-symmetrical shells subjected to impact ad frictio-iduced excitatios our modal techiques of physical modellig, which were already used i previous papers cocerig plucked ad bowed strigs as well as impacted ad bowed bars. We start by a experimetal modal idetificatio of three differet ibeta bowls, ad the develop a modellig approach for these systems. Extesive oliear umerical simulatios were performed, for both impacted ad rubbed bowls, which i particular highlight importat aspects cocerig the spatial patters of the frictioiduced bowl vibratios. Our results are i good agreemet with prelimiary qualitative experimets. PACS o Kk. Itroductio Several frictio-excited idiophoes are familiar to wester musical culture, such as bowed vibraphoe ad marimba bars, the ail violi, the musical saw, musical glasses ad the glass harmoica, as well as some atural objects rubbed agaist each other, like sea shells, boes, stoes or pie-coes. I a iterestig tutorial paper, Akay [] presets a overview of the acoustics his paper is a elarged versio of work preseted at the 34 th Spaish atioal Acoustics Cogress ad EEA Symposium (eciacústica 2003) ad at the Iteratioal Symposium o Musical Acoustics (ISMA 2004), Japa. pheomea related to frictio, which is the mai soudgeeratig mechaism for such systems. Some of these musical istrumets have bee experimetally studied, i particular by Rossig ad co-workers, a accout of which will be foud i [2]. evertheless, the aalysis of idiophoes excited by frictio is comparatively rare i the literature ad mostly recet see Frech [3], Rossig [4], Chapuis [5] ad Essl & Cook [6]. Amog these studies, oly [3] ad [6] aim at physical modellig, respectively of rubbed glasses ad bowed bars. I our previous work Iácio et al. [7-9] we also ivestigated the stick-slip behaviour of bowed bars uder differet playig

2 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls coditios, usig a modal approach ad a simplified frictio model for the bow/bar iteractio. Recetly, some researchers became iterested i the physical modellig of sigig bowls, usig waveguide sythesis techiques for performig umerical simulatios [0-2]. heir efforts aimed particularly at achievig real-time sythesis. herefore, uderstadably, several aspects of the physics of these istrumets do ot appear to be clarified i the published formulatios ad results. For istace, to our best kowledge, a accout of the radial ad tagetial vibratory motio compoets of the bowl shell ad their dyamical couplig has bee igored i the published literature. Also, how these motio compoets relate to the travellig positio of the puja cotact poit is ot clear at the preset time. Details of the cotact/frictio iteractio models used i simulatios have bee seldom provided, ad the sigificace of the various model parameters has ot bee asserted. O the other had, experimets clearly show that beatig pheomea arises eve for ear-perfectly symmetrical bowls, a importat aspect which the published modellig techiques seem to miss (although beatig from closely mistued modes has bee addressed ot without some difficulty [2] but this is a quite differet aspect). herefore, it appears that several importat aspects of the excitatio mechaism i sigig bowls still lack clarificatio. I this paper, we report ad exted our recet studies [3,4] by applyig the modal physical modellig techiques to axi-symmetrical shells subjected to impact ad/or frictio-iduced excitatios. hese techiques were already used i previous papers cocerig plucked ad bowed strigs [5-8] as well as impacted [9] ad bowed bars [7-9]. Our approach is based o a modal represetatio of the ucostraied system here cosistig o two orthogoal families of modes of similar (or ear-similar) frequecies ad shapes. he bowl modeshapes have radial ad tagetial motio compoets, which are proe to be excited by the ormal ad frictioal cotact forces betwee the bowl ad the impact/slidig puja. At each time step, the geeralized (modal) excitatios are computed by projectig the ormal ad tagetial iteractio forces o the modal basis. he, time-step itegratio of the modal differetial equatios is performed usig a explicit algorithm. he physical motios at the cotact locatio (ad ay other selected poits) are obtaied by modal superpositio. his eables the computatio of the motio-depedet iteractio forces, ad the itegratio proceeds. Details o the specificities of the cotact ad frictioal models used i our simulatios are give. A detailed experimetal modal idetificatio has bee performed for three differet ibeta bowls. he, we produce a extesive series of oliear umerical simulatios, for both impacted ad rubbed bowls, demostratig the effectiveess of the proposed computatioal techiques ad highlightig the mai features of the physics of sigig bowls. We discuss extesively the ifluece of the cotact/frictio ad playig parameters the ormal cotact force the tagetial velocity F ad of V of the exciter o the produced souds. May aspects of the bowl resposes displayed by our umerical simulatios have bee observed i prelimiary qualitative experimets. Our simulatio results highlight the existece of several motio regimes, both steady ad usteady, with either permaet or itermittet bowl/puja cotact. Furthermore, the ustable modes spi at the agular velocity of the puja. As a cosequece, for the listeer, sigig bowls behave as rotatig quadropoles. he soud will always be perceived as beatig pheomea, eve if usig perfectly symmetrical bowls. From our computatios, souds ad aimatios have bee produced, which appear to agree with qualitative experimets. Some of the computed souds are appeded to this paper. 2. ibeta sigig bowls ad their use Sigig bowls, also desigated by Himalaya or epalese sigig bowls [20] are traditioally made i 2

3 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls ibet, epal, Bhuta, Mogolia, Idia, Chia ad Japa. Although the ame qig has bee applied to lithophoes sice the Ha Chiese Cofucia rituals, more recetly it also desigates the bowls used i Buddhist temples [2]. I the Himalaya there is a very aciet traditio of metal maufacture, ad bowls have bee hadcrafted usig alloys of several metals maily copper ad ti, but also other metals such as gold, silver, iro, lead, etc. each oe believed to possess particular spiritual powers. here are may distict bowls, which produce differet toes, depedig o the alloy compositio, their shape, size ad weight. Most importat is the soud producig techique used either impactig or rubbig, or both simultaeously as well as the excitatio locatio, the hardess ad frictio characteristics of the excitig stick (called puja, frequetly made of wood ad evetually covered with a soft ski) see [22]. he origi of these bowls is t still well kow, but they are kow to have bee used also as eatig vessels for moks. he sigig bowls dates from the Bo civilizatio, log before the Buddhism [23]. ibeta bowls have bee used essetially for ceremoial ad meditatio purposes. evertheless, these amazig istrumets are icreasigly beig used i relaxatio, meditatio [23], music therapy [20, 24, 25] ad cotemporary music. he musical use of ibeta sigig bowls i cotemporary music is a cosequece of a broad artistic movemet. I fact, i the past decades the umber of percussio istrumets used i Wester music has greatly icreased with a ivasio of may istrumets from Africa, Easter, South-America ad other coutries. May Wester composers have icluded such istrumets i their music i a acculturatio pheomeo. he ibeta bowls ad other related istrumets used i cotemporary music are referred to, i scores, by several ames: temple bells, campaa di templo, japoese temple bell, Buddhist bell, cup bell, dobaci Buddha temple bell. Several examples of the use of these istrumets ca be foud i cotemporary music: Philippe Leroux, Les Us (200); Joh Cage/Lou Harriso, Double Music (94) percussio quartet, a work with a remarkable Easter ifluece; Olivier Messiae, Oiseaux Exotiques (955/6); Joh Keeth aveer, catata otal Eclipse (999) for vocal soloist, boys choir, baroque istrumets, brass, ibeta bowls, ad timpai; a Du Opera Marco Polo (995) with ibeta bells ad ibeta sigig bowls; Joyce Bee ua Koh, Lè (997) for choir ad ibeta bowls. Figure. hree sigig bowls used i the experimets: Bowl (φ = 80 mm); Bowl 2 (φ = 52 mm); Bowl 3 (φ = 40 mm). Figure 2. Large sigig bowl: Bowl 4 (φ = 262 mm), ad two pujas used i the experimets. 3. Experimetal modal idetificatio Figures ad 2 show the four bowls ad two pujas used for the experimetal work i this paper. I order to estimate the atural frequecies ω, dampig values ς, modal masses m ad modeshapes ϕ ( θ, z) to be used i our umerical simulatios, a detailed experimetal modal idetificatio based o impact testig was performed for Bowls, 2 ad 3. A mesh of 20 test locatios was defied for each istrumet (e.g., 24 poits regularly spaced azimuthally, at 5 differet heights). Impact excitatio was performed o all of the poits ad 3

4 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls the radial resposes were measured by two accelerometers attached to ier side of the bowl at two positios, located at the same horizotal plae (ear the rim) with a relative agle of 55º betwee them, as ca be see i Figure 3. Modal idetificatio was achieved by developig a simple MDOF algorithm i the frequecy domai [26]. he modal parameters were optimized i order to miimize the error ε ( ω, ς, m, ϕ ) betwee the measured trasfer fuctios H ( ω ) = Y ( ω) F ( ω) ad er r e the fitted modal model Hˆ ( ω ; ω, ς, m, ϕ ), for all er measuremets ( P e excitatio ad P respose locatios), r i a give frequecy rage [, ] modes. Hece: e r m ax e= r = ω m i ω ω ecompassig ε ( ω, ς, m, ϕ ) = P P ω () = H ( ) ˆ er ω H er ( ω ; ω, ς, m, ϕ ) d ω with: Hˆ ( ω ; ω, ς, m, ϕ ) = er + er A = ω C + C = mi max 2 2 ω 2 2 ω ω + 2iω ω ς 2 (2) where the modal amplitude coefficiets are give as er e e r r A = ϕ ( θ, z ) ϕ ( θ, z ) m ad the two last terms i (2) accout for modes located out of the idetified frequecy-rage. he values of the modal masses obviously deped o how modeshapes are ormalized (we used ϕ( θ, z) = ). ote that the idetificatio is max er oliear i ω ad ς but liear i A. Results from the experimets o the three bowls show the existece of 5 to 7 promiet resoaces with very low modal dampig values up to frequecies about 4 ~ 6 khz. For these well-defied experimetal modes, the simple idetificatio scheme used proved adequate. As a illustratio, Figure 3 depicts the modulus of a frequecy respose fuctio obtaied from Bowl 2, relatig the acceleratio measured at poit (ear the bowl rim) to the force applied at the same poit. he shapes of the idetified bowl modes are maily due to bedig waves that propagate azimuthally, resultig i patters similar to some modeshapes of bells [2]. Followig Rossig, otatio ( j, k ) represets here the umber of complete odal meridias extedig over the top of the bowl (half the umber of odes observed alog a circumferece), ad the umber of odal circles, respectively. Despite the high maufacturig quality of these hadcrafted istrumets, perfect axi-symmetry is early impossible to achieve. As will be explaied i sectio 4, these slight geometric imperfectios lead to the existece of two orthogoal modes (hereby called modal families A ad B), with slightly differet atural frequecies. Although this is ot apparet i Figure 3, by zoomig the aalysis frequecy-rage, a apparetly sigle resoace ofte reveals two closely spaced peaks. Figure 4 shows the perspective ad top views of the two orthogoal families of the first 7 soudig (radial) modeshapes (rigid-body modes are ot show) for Bowl 2, as idetified from experimets. I the frequecy-rage explored, all the idetified modes are of the ( j, 0) type, due to the low value of the height to diameter ratio ( Z / φ ) for ibeta bowls, i cotrast to most bells. he modal amplitudes represeted are ormalised to the maximum amplitude of both modes, which complicates the perceptio of some modeshapes. However, the spatial phase differece ( π / 2 j ) betwee each modal family (see sectio 4) is clearly see. Although modal frequecies ad dampig values were obtaied from the modal idetificatio routie, it was soo realized that the accelerometers ad their cables had a o-egligible ifluece o the bowl modal parameters, due to the very low dampig of these systems, which was particularly affected by the istrumetatio. Ideed, aalysis of the ear-field soud pressure timehistories, radiated by impacted bowls, showed slightly higher values for the atural frequecies ad much loger decay times, whe compared to those displayed after trasducers were istalled. Hece, we decided to use some modal parameters idetified from the acoustic resposes of o-istrumeted impacted bowls. Modal frequecies were extracted from the soud pressure spectra ad dampig values were computed from the logarithm-decremet of bad-pass filtered (at each mode) soud pressure decays. 4

5 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls Abs ( H ( f ) ) [(m/s2)/] Frequecy [Hz] Figure 3. Experimetal modal idetificatio of Bowl 2: Picture showig the measuremet grid ad accelerometer locatios; Modulus of the accelerace frequecy respose fuctio. (2,0) (3,0) (4,0) (5,0) (6,0) (7,0) (8,0) Modal family B Modal family A (j,k) Figure 4. Perspective ad top view of experimetally idetified modeshapes (j,k) of the first 7 elastic mode-pairs of Bowl 2 (j relates to the umber of odal meridias ad k to the umber of odal circles see text). able I Modal frequecies ad frequecy ratios of bowls, 2 ad 3 (as well as their total masses M ad rim diameters φ). Bowl M = 934 g φ = 80 mm otal Mass Diameter Bowl 2 M = 563 g φ = 52 mm Bowl 3 M = 557 g φ = 40 mm Mode f A [Hz] f B [Hz] f A B f AB f A [Hz] f B [Hz] f A B f AB f A [Hz] f B [Hz] f A B f AB (2,0) (3,0) (4,0) (5,0) (6,0) (7,0) (8,0)

6 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls able I shows the values of the double modal B frequecies ( f ad f ) of the most promiet modes A of the three bowls tested, together with their ratios to the AB fudametal mode (2,0) where f represets the average frequecy betwee the two modal frequecies B f ad f. hese values are etirely i agreemet with A the results obtaied by Rossig [2]. Iterestigly, these ratios are rather similar, i spite of the differet bowl shapes, sizes ad wall thickess. As rightly poited by Rossig, these modal frequecies are roughly 2 proportioal to j, as i cylidrical shells, ad iversely 2 proportioal to φ. Rossig explais this i simple terms, somethig that ca be also grasped from the theoretical solutio for i-plae modes for rigs [27]: ω = j 2 ( ) j j 2 j + EI ρ AR 4, with j =,2,..., (3) where E ad ρ are the Youg Modulus ad desity of the rig material, I the area momet of iertia, A the rig cross sectio area ad R the rig radius. It ca be see that as j takes higher values, the first term of equatio 3 2 teds to j, while the depedecy o the rig diameter is embedded i the secod term. he frequecy relatioships are mildly iharmoic, which does ot affect the defiite pitch of this istrumet, maily domiated by the first (2,0) shell mode. As stated, dissipatio is very low, with modal dampig ratios typically i the rage ς = 0.002~0.05 % (higher values pertaiig to higher-order modes). However, ote that these values may icrease oe order of magitude, or more, depedig o how the bowls are actually supported or hadled. Further experimets were performed o the larger bowl show i Figure 2 (Bowl 4), with φ = 262 mm, a total mass of 533 g ad a fudametal frequecy of 86.7 Hz. A full modal idetificatio was ot pursued for this istrumet, but te atural frequecies were idetified from measuremets of the soud pressure resultig from impact tests. hese modal frequecies are preseted i able II, which show a similar relatio to the fudametal as the first three bowls preseted i this study. For this istrumet all these modes were assumed to be of the (j,0) type. able II Modal frequecies ad frequecy ratios of Bowl 4. Mode (j,k) f [Hz] f /f (2,0) (3,0) (4,0) (5,0) (6,0) (7,0) (8,0) (9,0) (0,0) (,0) Formulatio of the dyamical system 4.. Dyamical formulatio of the bowl i modal coordiates Perfectly axi-symmetrical structures exhibit double vibratioal modes, occurrig i orthogoal pairs with A B idetical frequecies ( ω = ω ) [4]. However, if a slight alteratio of this symmetry is itroduced, the atural frequecies of these two degeerate modal families deviate from idetical values by a certai amout ω. he use of these modal pairs is essetial for the correct dyamical descriptio of axi-symmetric bodies, uder geeral excitatio coditios. Furthermore, shell modeshapes preset both radial ad tagetial compoets. Figure 5 displays a represetatio of the first four modeshape pairs, ear the bowl rim, where the excitatios are usually exerted (e.g., ze Z ). Both the radial (gree) ad tagetial (red) motio compoets are plotted, which for geometrically perfect bowls ca be formulated as: with ϕ ϕ Ar At ( θ ) = ( θ ) r + ( θ ) t (4) ( θ ) = ( θ ) r + ( θ ) t A Ar At ϕ ϕ ϕ B Br Bt ϕ ϕ ϕ = cos ( θ ) ( θ ) = si ( θ ) ( θ ) ; Br ϕ Bt ϕ = si ( θ ) ( θ ) = cos ( θ ) ( θ ) (5,6) Ar where ϕ ( θ ) correspods to the radial compoet of the At A family th modeshape, ϕ ( θ ) to the tagetial 6

7 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls compoet of the A family th mode shape, etc. Figure 5 shows that spatial phase agles betwee orthogoal mode pairs are π / 2 j. Oe immediate coclusio ca be draw from the polar diagrams show ad equatios (5,6): the amplitude of the tagetial modal compoet decreases relatively to the amplitude of the radial compoet as the mode umber icreases. his suggests that oly the lower-order modes are proe to egage i self-sustaied motio due to tagetial rubbig excitatio by the puja. If liear dissipatio is assumed, the motio of the system ca be described i terms of the bowl s two families of modal parameters: modal masses m, modal circular frequecies ω, modal dampig ζ, ad mode shapes ϕ ( θ ) (at the assumed excitatio level ze Z ), with =,2,,, where stads for the modal family A or B. he order of the modal trucatio is problem-depedet ad should be asserted by physical reasoig, supported by the covergece of computatioal results. he maximum modal frequecy to be icluded, ω, mostly depeds o the short time-scales iduced by the cotact parameters all modes sigificatly excited by impact ad/or frictio pheomea should be icluded i the computatioal modal basis. he forced respose of the damped bowl ca the be formulated as a set of 2 ordiary secod-order differetial equatios { A } { B } M 0 A Q ( t) + 0 M B Q ( t) { A } { B } C 0 A Q ( t) C B Q ( t) { } { } { Ξ } { Ξ } K 0 A Q ( t) ( t) A A + = 0 K B Q ( t) ( t) B B (7) (2,0) A (3,0) A (4,0) A (5,0) A f = 34 Hz f 2 = 836 Hz f 3 = 59 Hz f 4 = 2360 Hz (2,0) B (3,0) B (4,0) B (5,0) B Figure 5. Mode shapes at the bowl rim of the first four orthogoal mode pairs (Blue: Udeformed; Gree: Radial compoet; Red: agetial compoet). 7

8 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls where: [ ] M = Diag( m,, m ), [ ] ω ζ C = Diag(2 m,, 2 m ω ζ ), [ ] K = m m, 2 2 Diag( ( ω ),, ( ω ) ) are the matrices of the modal parameters (where stads for A or B), for each of the two orthogoal mode families, while { } { } Q ( t) = q ( t),, q ( t) ad Ξ ( t) = I ( t),, I ( t) are the vectors of the modal resposes ad of the geeralized forces, respectively. ote that, although equatios (7) obviously pertai to a liear formulatio, othig prevets us from icludig i I ( t) all the oliear effects which arise from the cotact/frictio iteractio betwee the bowl ad the puja. Accordigly, the system modes become coupled by such oliear effects. he modal forces I ( t) are obtaied by projectig the exteral force field o the modal basis: 2π r t I ( t) = (, ) ( ) (, ) ( ) F θ t ϕ θ F θ t ϕ θ dθ 0 + r t (8) =, 2,, where F ( θ, t) ad F ( θ, t) are the radial (impact) ad r t tagetial (frictio) force fields applied by the puja e.g., a localised impact F ( θ, t) ad/or a travellig rub r c F ( θ ( t ), t ). he radial ad tagetial physical motios r, t c ca be the computed at ay locatio θ from the modal amplitudes q ( t ) by superpositio: 4.2. Dyamics of the puja As metioed before, the excitatio of these musical istrumets ca be performed i two basic differet ways: by impact or by rubbig aroud the rim of the bowl with the puja (these two types of excitatio ca obviously be mixed, resultig i musically iterestig effects). he dyamics of the puja will be formulated simply i terms of a mass m P subjected to a ormal (e.g. radial) force F ( t ) ad a imposed tagetial rubbig velocity V ( t ) which will be assumed costat i time for all our exploratory simulatios as well as to a iitial impact velocity i the radial directio V ( t ). hese three 0 parameters are the most relevat factors which allow the musicia to play the istrumet ad cotrol the mechaism of soud geeratio. May distict souds may be obtaied by chagig them: i particular, V ( t ) with F = V = 0 will be pure impact, ad 0 0 F ( t) 0, V ( t) 0 with V ( t ) = 0 will be pure 0 sigig (see sectio 4). he radial motio of the puja, resultig from the exteral force applied ad the impact/frictio iteractio with the bowl is give by: m y = F ( t) + F ( θ, t) () P P r where F ( θ, t) is the dyamical bowl/puja cotact force. r 4.3. Cotact iteractio formulatio he radial cotact force resultig from the iteractio betwee the puja ad the bowl is simply modelled as a cotact stiffess, evetually associated with a cotact dampig term: ( θ ) = ( θ, ) ( θ, ) F K y t C y t (2) r c c r c c r c Ar A Br B y ( t) = ( ) ( ) ( ) ( ) r ϕ θ q t + ϕ θ q t (9) = = At A Bt B y ( t) = ( ) ( ) ( ) ( ) t ϕ θ q t + ϕ θ q t (0) ad similarly cocerig the velocities ad acceleratios. where ỹ r ad ẏ are respectively the bowl/puja relative r radial displacemet ad velocity, at the (fixed or travellig) cotact locatio θ ( t ), K ad c c C are the c cotact stiffess ad dampig coefficiets, directly related to the puja material. Other ad more refied cotact models for istace of the hertzia type, evetually with hysteretic behaviour could easily be 8

9 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls implemeted istead of (2). Such refiemets are however ot a priority here Frictio iteractio formulatio I previous papers we have show the effectiveess of a frictio model used for the simulatio of bowed bars ad bowed strigs [7-9, 5-8]. Such model eabled a clear distictio betwee slidig ad adherece states, slidig frictio forces beig computed from the Coulomb model Ft = Fr µ d ( y t ) sg( y t ), where ẏ is the t bowl/puja relative tagetial velocity, ad the adherece state beig modelled essetially i terms of a local adherece stiffess K ad some dampig. We were a thus able to emulate true frictio stickig of the cotactig surfaces, wheever Ft < F r µ s, however at the expese of a loger computatioal time, as smaller itegratio time-steps seem to be imposed by the trasitios from velocity-cotrolled slidig states to displacemet-cotrolled adherece states. I this paper, a simpler approach is take to model frictio iteractio, which allows for faster computatio times, although it lacks the capability to emulate true frictio stickig. he frictio force will be formulated as: ( y ) ( y ) F ( θ, ) (, ) (, ) (, ) t c t = F θ r c t µ d t θc t sg t θc t, if y t ( θc, t) F ( θ, ) (, ) (, ), if (, ) t c t = F θ r c t µ y s t θc t ε y t θc t < ε ε (3) where µ is a static frictio coefficiet ad µ ( y ) is a s dyamic frictio coefficiet, which depeds o the puja/bowl relative surface velocity followig model: ẏ t. We will use the ( ) µ ( y ) = µ + ( µ µ )exp C y ( θ, t) (4) d t s t c where, 0 µ µ is a asymptotic lower limit of the frictio coefficiet whe s y, ad parameter C cotrols the decay rate of the frictio coefficiet with the relative slidig velocity, as show i the typical plot of Figure 6. his model ca be fitted to the available t d t experimetal frictio data (obtaied uder the assumptio of istataeous velocity-depedece), by adjustig the empirical costats µ, µ s ad C. otice that both equatios (3) correspod to velocitycotrolled frictio forces. For values of ẏ t outside the iterval [ ε, ε ], the first equatio simply states Coulomb s model for slidig. Iside the iterval [ ε, ε ], the secod equatio models a state of pseudo-adherece at very low tagetial velocities. Obviously, ε acts as a regularizatio parameter for the frictio force law, replacig the zero-velocity discotiuity (which reders the adherece state umerically tricky), as show i Figure 6. his regularizatio method, extesively developed i [28], has bee ofte used as a pragmatic way to deal with frictio pheomea i the cotext of dyamic problems. However, usig this model, the frictio force will always be zero at zero slidig velocity, iducig a relaxatio o the adherece state (depedet o the magitude of ε ), ad therefore disablig a true stickig behaviour. How pericious this effect may be is problem-depedet systems ivolvig a prologed adherece will obviously suffer more from the relaxatio effect tha systems which are slidig most of the time. For the problem addressed here, we have obtaied realistic results usig formulatio (3), for small eough values of the regularizatio domai (we used 4 - ± ε 0 ms ) results which do ot seem to critically deped o ε, withi reasoable limits ime-step itegratio For give exteral excitatio ad iitial coditios, the previous system of equatios is umerically itegrated usig a adequate time-step algorithm. Explicit itegratio methods are well suited for the cotact/frictio model developed here. I our implemetatio, we used a simple Velocity-Verlet itegratio algorithm [29], which is a low-order explicit scheme. 9

10 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls a) b) Figure 6. Frictio coefficiet as a fuctio of the cotact relative tagetial velocity ( µ = 0.2, µ = 0.4, s C = 0): For -< ẏ t <; For -0.0< ẏ t < umerical simulatios he umerical simulatios preseted here are based o the modal data of two differet sized istrumets: Bowl 2 ad Bowl 4, which were idetified i sectio 2. he simulatios based o the smaller istrumet data will be used to highlight the mai features of the dyamics of these istrumets, while the larger istrumet simulatios will serve the purpose of studyig the ifluece of the cotact/frictio parameters o the oscillatio regimes. he puja is modeled as a simple mass of 20 g, movig at tagetial velocity V, ad subjected to a exteral ormal force F as well as to the bowl/puja oliear iteractio force. We explore a sigificat rage of rubbig parameters: F = ~ 9 ad V = 0. ~ 0.5 m/s, which ecompass the usual playig coditios, although calculatios were made also usig impact excitatio oly. For clarity, the ormal force ad tagetial velocity will be assumed time-costat, i the preset simulatios. However, othig would prevet us from imposig ay time-varyig fuctios F ( t ) ad V ( t ), or eve as musicias would do to couple the geeratio of F ( t ) ad V ( t ) with the oliear bowl/puja dyamical simulatio, through well-desiged cotrol strategies, i order to achieve a suitable respose regime. he cotact model used i all rubbig simulatios of Bowl 2 has a cotact stiffess of cotact dissipatio of K = c 6 0 /m ad a C = 50 s/m, which appear c adequate for the preset system. However, cocerig impact simulatios of this istrumet, cotact parameters te times higher ad lower were also explored. he frictio parameters used i umerical simulatios of this istrumet are µ s = 0.4, µ = 0.2 ad C = 0 (see Figure 6). I relatio to the umerical simulatios of Bowl 4, differet cotact/frictio parameters were used to simulated frictio by pujas made of differet materials, amely rubber ad wood. Its values will be described i sectio 5.4. I sectio 4. a few geeral remarks were produced cocerig the order of the modal basis to use. With respect to the preset system, the choice of the modal basis order of trucatio is ot difficult ad certaily ot critical, as oly a few modes are excited (i cotrast with bowed strigs). For easily uderstadable physical reasos, modes with modal stiffess much higher tha the cotact stiffess are ot sigificatly excited, so a reasoable criterio to choose a miimum order of trucatio is to compare K c with the successive K of the modal series. I the preset study a maximum value of 6 K c = 0 /m is used, the it is reasoable to 7 assume that modes with K much higher tha 0 /m will be useless. For Bowl 2, we decided to use seve mode pairs, the maximum value K beig of the order 8 ~ 0 /m. Ideed, this is a geerous modal basis, K 2 7 ad four mode pairs would do equally well, as K ~ 0 /m. However it is poitless to discuss o such a detail, whe the umber of modes is low. I relatio to the larger Bowl 4, te mode pairs were used, followig the same reasoig. Both computatioal ad experimetal results cofirm that the trucatio criterio adopted is adequate. 0

11 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls As discussed before, assumig a perfectly symmetrical bowl, simulatios were performed usig idetical A B frequecies for each mode-pair ( ω = ω ). However, a few computatios were also performed for less-thaperfect systems, asymmetry beig the modelled itroducig a differece (or split ) ω betwee the frequecies of each mode pair. A average value of 0.005% was used for all modal dampig coefficiets. I order to cope with the large settlig times that arise with sigig bowls, 20 secods of computed data were geerated (eough to accommodate trasiets for all rubbig coditios), at a samplig frequecy of Hz. 5.. Impact resposes Figures 7(a, b) display the simulated resposes of a perfectly symmetrical bowl to a impact excitatio ( 0 V ( t ) = m/s ), assumig differet values for the cotact model parameters. he time-histories of the respose displacemets pertai to the impact locatio. he spectrograms are based o the correspodig velocity resposes. ypically, as the cotact stiffess icreases from 0 5 /m to 0 7 /m, higher-order modes become icreasigly excited ad resoate loger. he correspodig simulated souds become progressively brighter, deotig the metallic bell-like toe which is clearly heard whe impactig real bowls usig wood or metal pujas Frictio-excited resposes Figure 8 shows the results obtaied whe rubbig a perfectly symmetrical bowl ear the rim, usig fairly stadard rubbig coditios: F = 3 ad V = 0.3 m/s. he plots show pertai to the followig respose locatios: the travellig cotact poit betwee the bowl ad the puja; a fixed poit i the bowl s rim. Depicted are the time-histories ad correspodig spectra of the radial (gree) ad tagetial (red) displacemet resposes, as well as the spectrograms of the radial velocity resposes. As ca be see, a istability of the first "elastic" shell mode (with 4 azimuthal odes) arises, with a expoetial icrease of the vibratio amplitude util saturatio by oliear effects is reached (at about 7.5 s), after which the self-excited vibratory motio of the bowl becomes steady. he respose spectra show that most of the eergy lays i the first mode, the others beig margially excited. otice the dramatic differeces betwee the resposes at the travellig cotact poit ad at a fixed locatio. At the movig cotact poit, the motio amplitude does ot fluctuate ad the tagetial compoet of the motio is sigificatly higher tha the radial compoet. O the cotrary, at a fixed locatio, the motio amplitude fluctuates at a frequecy which ca be idetified as beig four times the spiig frequecy of the puja: 4 4( 2V φ ) Ω = Ω =. Furthermore, at fluct puja a fixed locatio, the amplitude of the radial motio compoet is higher tha the tagetial compoet. he aimatios of the bowl ad puja motios eable a iterpretatio of these results. After sychroisatio of the self-excited regime, the combied resposes of the first mode-pair result i a vibratory motio accordig to the 4-ode modeshape, which however spis, followig the revolvig puja. Furthermore, sychroisatio settles with the puja iteractig ear a ode of the radial compoet, correspodig to a ati-odal regio of the tagetial compoet see Figure 5 ad Equatios (5,6). I retrospect, this appears to make sese ideed, because of the frictio excitatio mechaism i sigig bowls, the system modes self-orgaize i such way that a high tagetial motio-compoet will arise at the cotact poit, where eergy is iputted. At ay fixed locatio, a trasducer will see the vibratory respose of the bowl modulated i amplitude, as the 2j alterate odal ad ati-odal regios of the sigig modeshape revolve. For a listeer, the rubbed bowl behaves as a spiig quadropole or, i geeral, a 2j-pole (depedig o the self-excited mode j) ad the radiated soud will always be perceived with beatig pheomea, eve for a perfectly symmetrical bowl. herefore the soud files available were geerated from the velocity time-history at a fixed poit i the bowl rim.

12 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls Figure 7. Displacemet time histories (top) ad spectrograms (bottom) of the respose of Bowl 2 to impact excitatio with two differet values of the bowl/puja cotact stiffess: 0 5 /m (soud file available); 0 7 /m (soud file available). 2

13 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls Figure 8. ime-histories, spectra ad spectrograms of the dyamical respose of Bowl 2 to frictio excitatio whe F = 3, V = 0.3 m/s: at the bowl/puja travellig cotact poit; at a fixed poit of the bowl rim (soud file available). Figure 9. ime-histories, spectra ad spectrograms of the dyamical respose of Bowl 2 to frictio excitatio whe F = 7, V = 0.5 m/s: at the bowl/puja travellig cotact poit; at a fixed poit of the bowl rim (soud file available). 3

14 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls Figure 0. ime-histories, spectra ad spectrograms of the dyamical respose of Bowl 2 to frictio excitatio whe F =, V = 0.5 m/s: at the bowl/puja travellig cotact poit; at a fixed poit of the bowl rim (soud file available). (c) Figure. Radial (gree) ad tagetial (red) iteractio forces betwee the bowl ad the travellig puja: F = 3, V = 0.3 m/s; F = 7, V = 0.5 m/s; (c) F =, V = 0.5 m/s. 4

15 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls Followig the previous remarks, the out-of-phase evelope modulatios of the radial ad tagetial motio compoets at a fixed locatio, as well as their amplitudes, ca be uderstood. Ideed, all ecessary isight stems from Equatios (5,6) ad the first plot of Figure 5. I order to cofirm the rotatioal behaviour of the selfexcited mode we performed a simple experimet uder ormal playig (rubbig) coditios o Bowl 2. he ear-field soud pressure radiated by the istrumet was recorded by a microphoe at a fixed poit, approximately 5 cm from the bowl s rim. While a musicia played the istrumet, givig rise to a self-sustaied oscillatio of the first shell mode (j = 2, see Figures 4 ad 5), the positio of the rotatig puja was moitored by a observer which emitted a short impulse at the puja passage by the microphoe positio. Sice soud radiatio is maily due to the radial motio of the bowl, the experimet proves the existece of a radial vibratioal odal regio at the travellig poit of excitatio. Betwee each two passages of the puja by this poit (i.e. oe revolutio), 4 soud pressure maxima are recorded, corroboratig our previous commets that the listeer hears a beatig pheomea (or pseudo-beatig) origiatig from a rotatig 2j-pole source, whose beatig-frequecy is proportioal to the revolvig frequecy of the puja. Such behaviour will be experimetally documeted i sectio 5.4. It should be oted that our results basically support the qualitative remarks provided by Rossig, whe discussig frictio-excited musical glass-istrumets (see [4] or his book [2] pp , the oly refereces, to our kowledge, where some attetio has bee paid to these issues). However, his mai poit he locatio of the maximum motio follows the movig figer aroud the glass may ow be further clarified: the maximum motio followig the exciter should refer i fact to the maximum tagetial motio compoet (ad ot the radial compoet, as might be assumed). Before leavig this example, otice i Figure the behaviour of the radial ad tagetial compoets of the bowl/puja cotact force, o several cycles of the steady motio. he radial compoet oscillates betwee almost zero ad the double of the value F imposed to the puja, ad cotact is ever disrupted. he plot of the frictio force shows that the bowl/puja iterface is slidig durig most of the time. his behaviour is quite similar to what we observed i simulatios of bowed bars, ad is i clear cotrast to bowed strigs, which adhere to the bow durig most of the time see [9], for a detailed discussio. he fact that stickig oly occurs durig a short fractio of the motio, justifies the simplified frictio model which has bee used for the preset computatios. Figure 9 shows the results for a slightly differet regime, correspodig to rubbig coditios: F = 7 ad V = 0.5 m/s. he trasiet duratio is smaller tha i the previous case (about 5 s). Also, because of the higher tagetial puja velocity, beatig of the vibratory respose at the fixed locatio also displays a higher frequecy. his motio regime seems qualitatively similar to the previous example, however otice that the respose spectra display more eergy at higher frequecies, ad that is because the cotact betwee the exciter ad the bowl is periodically disrupted, as show i the cotact force plots of Figure. Oe ca see that, durig about 25% of the time, the cotact force is zero. Also, because of moderate impactig, the maxima of the radial compoet reach almost 3 F. Both the radial ad frictio force compoets are much less regular tha i the previous example, but this does ot prevet the motio from beig early-periodic. Figure 0 shows a quite differet behaviour, whe F = ad V = 0.5 m/s. Here, a steady motio is ever reached, as the bowl/puja cotact is disrupted wheever the vibratio amplitude reaches a certai level. As show i Figure (c), severe chaotic impactig arises (the amplitude of the radial compoet reaches almost 7 F ), which breaks the mechaism of eergy trasfer, leadig to a sudde decrease of the motio amplitude. he, the motio build-up starts agai util the saturatio level is reached, ad so o. As ca be expected, this itermittet respose regime results i 5

16 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls curious souds, which iterplay the aerial characteristics of sigig with a distict rigig respose due to chaotic chatterig. Ayoe who ever attempted to play a ibeta bowl is well aware of this soorous saturatio effect, which ca be musically iterestig, or a vicious uisace, depedig o the cotext. o get a clearer picture of the global dyamics of this system, Figures 2 ad 3 preset the domais covered by the three basic motio regimes (typified i Figures 8-0), as a fuctio of F ad V : () Steady self-excited vibratio with permaet cotact betwee the puja ad the bowl (gree data); (2) Steady self-excited vibratios with periodic cotact disruptio (yellow data); (3) Usteady self-excited vibratios with itermittet amplitude icreasig followed by atteuatio after chaotic chatterig (orage data). ote that, uder differet coditios, the self-excitatio of a differet mode may be triggered for istace, by startig the vibratio with a impact followed by rubbig. Such issue will be discussed later o this paper. Figure 2 shows how the iitial trasiet duratio depeds o F ad V. I every case, trasiets are shorter for icreasig ormal forces, though such depedece becomes almost egligible at higher tagetial velocities. At costat ormal force, the ifluece of V strogly depeds o the motio regime. Figure 2 shows the fractio of time with motio disruptio. It is obviously zero for regime (), ad growig up to 30 % at very high excitatio velocities. It is clear that the rigig regime (3) is more proe to arise at low excitatio forces ad higher velocities. Figures 3 ad show the root-mea-square vibratory amplitudes at the travelig cotact poit, as a fuctio of F ad V. otice that the levels of the radial compoets are much lower tha the correspodig levels of the tagetial compoet, i agreemet with the previous commets. hese plots show some depedece of the vibratory level o the respose regime. Overall, the vibratio amplitude icreases with V for regime () ad decreases for regime (3). O the other had, it is almost idepedet of F for regime (), while it icreases with F for regime (3). 4,0 30,0% 2,0 25,0% 3 ormal Force [] , 0,2 0,3 0,4 0,5 agetial Velocity [m/s] 6,0 4,0 2,0 0,0 0,0 8,0 Iitial rasiet [s] 3 ormal Force [] , 0,2 0,3 0,4 0,5 agetial Velocity [m/s] 20,0% 5,0% 0,0% 5,0% 0,0% Figure 2. Iitial trasiet duratio ad percetage of time with o bowl/puja cotact, as a fuctio of F ad V. 6

17 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls 8,0E-05 2,0E-04 Radial Displacemet Amplitude RMS [m] 6,0E-05 4,0E-05 2,0E-05 0,0E+00 0,5 0,4 0,3 agetial Velocity [m/s] 0,2 0, ormal Force [] agetial Displacemet Amplitude RMS [m],6e-04,2e-04 8,0E-05 4,0E-05 0,0E+00 0,5 0,4 0,3 agetial Velocity [m/s] 0,2 0, ormal Force [] Figure 3. Displacemet amplitude (RMS) at the bowl/puja travellig cotact poit, as a fuctio of Radial motio compoet; agetial motio compoet. F ad V : 5.3. o-symmetrical bowls Figures 4 ad eable a compariso betwee the impact resposes of perfectly symmetrical ad a osymmetrical bowls. Here, the lack of symmetry has bee simulated by itroducig a frequecy split of 2% betwee the frequecies of each mode-pair (e.g. ω = 0.02ω ), all other aspects remaiig idetical such crude approach is adequate for illustratio purposes. otice that the symmetrical bowl oly displays radial motio at the impact poit (as it should), while the usymmetrical bowl displays both radial ad tagetial motio compoets due to the differet propagatio velocities of the travellig waves excited. O the other had, oe ca otice i the respose spectra of the usymmetrical system the frequecy-split of the various mode-pairs. his leads to beatig of the vibratory respose, as clearly see o the correspodig spectrogram. Figure 5 shows the self-excited respose of the symmetrical bowl, whe rubbed at F = 3 ad V = 0.3 m/s. otice that soud beatig due to the spiig of the respose modeshape domiates, whe compared to effect of modal frequecy-split. Iterestigly, the slight chage i the modal frequecies was eough to modify the ature of the self-excited regime, which wet from type () to type (3). his fact shows the difficulties i masterig these apparetly simple istrumets Ifluece of the cotact/frictio parameters Playig experiece shows that rubbig with pujas made of differet materials may trigger self-excited motios at differet fudametal frequecies. his suggests that frictio ad cotact parameters have a importat ifluece o the dyamics of the bowl regimes. Although this behaviour was preset i all the bowls used i this study, it was clearly easier to establish these differet regimes o a larger bowl. herefore we illustrate the differet behaviours that ca be obtaied, by usig Bowl 4 ad parameters correspodig to two pujas, respectively covered with rubber ad made of aked wood. As the frequecy separatio betwee mode-pairs was relatively small for this bowl, we assume a perfectly symmetrical bowl, ad performed simulatios usig 0 A B mode-pairs with idetical frequecies ( ω = ω ) see able II. A average value of 0.005% was used for all modal dampig coefficiets. I order to cope with the large settlig times that arise with sigig bowls, 30 secods of computed data were geerated (eough to accommodate trasiets for all rubbig coditios). Figure 6 shows a computed respose obtaied whe usig a soft puja with relatively high frictio. Here a cotact stiffess K c = 0 5 /m was used, assumig frictio parameters µ = 0. 8, µ = 0. 4 ad C = 0, s uder playig coditios F = 5 ad V = 0.3 m/s. 7

18 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls Figure 4. Dyamical resposes of a impacted bowl, at the impact locatio: Axi-symmetrical bowl (0% frequecy split); o-symmetrical bowl with 2% frequecy split (soud file available). Figure 5. Dyamical respose of a rubbed bowl with 2% frequecy split whe F = 3, V = 0.3 m/s: at the bowl/puja travellig cotact poit; at a fixed poit of the bowl rim (soud file available). 8

19 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls Figure 6. ime-histories, spectra ad spectrograms of the dyamical respose of Bowl 4 excited by a rubber-covered puja for F = 5 ad V = 0.3 m/s: at the bowl/puja travellig cotact poit; at a fixed poit of the bowl rim (soud file available). Figure 7. ime-histories, spectra ad spectrograms of the dyamical respose of Bowl 4 excited by a woode puja for F = 5 ad V = 0.3 m/s: at the bowl/puja travellig cotact poit; at a fixed poit of the bowl rim (soud file available). 9

20 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls he plot show i a) displays the radial (gree) ad tagetial (red) bowl motios at the travellig cotact poit with the puja. hese are of about the same magitude, ad perfectly steady as soo as the selfexcited motio locks-i. I cotrast, plot b) shows that the radial motio clearly domiates whe lookig at a fixed locatio i the bowl, with maximum amplitudes exceedig those of the travellig cotact poit by a factor two. Most importat, beatig pheomea is observed at a frequecy related to the puja spiig frequecy Ω = 2V φ, as also observed i relatio to Bowl 2. he p spectrum show i plot c) presets the highest eergy ear the first modal frequecy, while the spectrogram d) shows that the motio settles after about 7 secods of expoetial divergece. Ideed, our computed aimatios show that the ustable first bowl mode ( 87Hz) spis, followig the puja motio, with the cotact poit located ear oe of the four odes of the excited modeshape (see Figure 4). he bowl radiates as a quadro-pole spiig with frequecy Ω p, ad beatig is perceived with frequecy Ω = 4Ω. beat p Figure 7 shows a computed respose obtaied whe usig a harder puja with lower frictio, assumig 6 K = 0 /m, µ = 0. 4 ad µ = 0. 2, uder the same c playig coditios as before. s he self-excited motio takes loger to emerge ad is proe to qualitative chages. However, vibratio is essetially domiated by the secod modal frequecy ( 253Hz), with a sigificat cotributio of the first mode durig the iitial 25 secods. his leads to more complex beatig pheomea, except durig the fial 5 secods of the simulatio, where oe ca otice that, i spite of the similar value of V used, beatig is at a higher frequecy tha i Figure 6. Ideed, because the secod elastic mode is ow ustable (see Figure 4), the bowl radiates as a hexa-pole spiig with frequecy Ω p, ad beatig is perceived with frequecy Ω beat = 6Ω p. Figure 8 shows the experimetal results recorded by a microphoe placed ear the bowl rim, while playig with a rubber-covered puja. As described before, timig pulses were geerated at each cosecutive revolutio, whe the puja ad microphoe were earby. Vibratio was domiated by a istability of the first mode (2,0) ad, i spite of mildly-cotrolled huma playig, it is clear that radiatio is miimal ear the cotact poit ad that four beats per revolutio are perceived. Whe a harder aked wood puja was used, the iitial trasiet became loger, before a istability of the secod mode (3,0) settled. he bowl resposes teded to be less regular, as show i Figure 8, however six beats per revolutio are clearly perceived. All these features support the simulatio results preseted i Figures 6 ad 7, as well as the physical discussio preseted i sectio 5.2. he preset results stress the importace of the cotact/frictio parameters, if oe wishes a bowl to sig i differet modes such behaviour is easier to obtai i larger bowls. As a cocludig remark, we stress that a soorous bowl/puja rattlig cotact ca easily arise, i particular at higher tagetial velocities ad lower ormal forces, a feature which was equally displayed by may experimets ad umerical simulatios, as discussed before. a) ime [s] b) ime [s] Figure 8 ear-field soud pressure waveform (blue) due to frictio excitatio by: a) a rubber-covered puja ad b) a woode puja o Bowl 4, ad electrical impulses (red) sychroized with the passage of the puja by the microphoe positio (soud files available). 6. Coclusios I this paper we have developed a modellig techique based o the modal approach, which ca achieve accurate 20

21 Iácio, Herique & Atues: he Dyamics of ibeta Sigig Bowls time-domai simulatios of impacted ad/or rubbed axisymmetrical structures such as the ibeta sigig bowl. o substatiate the umerical simulatios, we performed a experimetal modal aalysis o three bowls. Results show the existece of 5 to 7 promiet vibratioal mode-pairs up to frequecies about 6 khz, with very low modal dampig values. he umerical simulatios preseted i this paper show some light o the soud-producig mechaisms of ibeta sigig bowls. Both impact ad frictio excitatios have bee addressed, as well as perfectly-symmetrical ad less-thaperfect bowls (a very commo occurrece). For suitable frictio parameters ad for adequate rages of the ormal cotact force F ad tagetial rubbig velocity V of the puja, istability of a shell mode (typically the first "elastic" mode, with 4 azimuthal odes) arises, with a expoetial icrease of the vibratio amplitude followed by saturatio due to oliear effects. Because of the itimate couplig betwee the radial ad tagetial shell motios, the effective bowl/puja cotact force is ot costat, but oscillates. After vibratory motios settle, the excitatio poit teds to locate ear a odal regio of the radial motio of the ustable mode, which correspods to a ati-odal regio of the frictio-excited tagetial motio (this effect is somewhat relaxed for softer pujas). his meas that ustable modes spi at the same agular velocity of the puja. As a cosequece, for the listeer, souds will always be perceived with beatig pheomea. However, for a perfectly symmetrical bowl, o beatig at all is geerated at the movig excitatio poit. ypically, the trasiet duratio icreases with V ad decreases for higher values of F. he way vibratory amplitudes deped o V ad F chages for differet respose regimes. hree basic motio regimes were obtaied i the preset computatios, depedig o ad V : () Steady self-excited vibratio with permaet cotact betwee the puja ad the bowl; (2) Steady selfexcited vibratios with periodic cotact disruptio; (3) Usteady self-excited vibratios with itermittet amplitude icreasig followed by atteuatio after chaotic chatterig. F It was demostrated through computatios ad experimets that the order j of the mode triggered by frictio excitatio is heavily depedet o the cotact/frictio parameters. I our computatios ad experimets o a large bowl, the first mode respoded easily whe usig a soft high-frictio puja, while istability of the secod mode was triggered by usig a harder lower-frictio woode puja. he first motio regime offers the purest bowl sigig. Our results suggest that higher values of F should eable a better cotrol of the produced souds, as they lead to shorter trasiets ad also reder the system less proe to chatterig. As a cocludig ote, the computatioal methods preseted i this paper ca be easily adapted for the dyamical simulatio of glass harmoicas, by simply chagig the modes of the computed system, as well as the cotact ad frictio parameters. Ackowledgmets his work has bee edorsed by the Portuguese Fudação para a Ciêcia e ecologia uder grat SFRH/BD/2806/2003. Refereces [] A. Akay, Acoustics of Frictio, Joural of the Acoustical Society of America, pp (2002). [2]. D. Rossig, he Sciece of Percussio Istrumets, Sigapore, World Scietific, (2000). [3] A. P. Frech, I Vio Veritas: A Study of Wieglass Acoustics, America Joural of Physics 5, pp (983). [4]. D. Rossig, Acoustics of the Glass Harmoica, J. Acoust. Soc. Am. 95, pp. 06- (994). [5] J.-C. Chapuis, Ces Si Délicats Istrumets de Verre, Pour la Sciece 272, pp , (2000). [6] G. Essl ad P. Cook, Measuremet ad Efficiet Simulatios of Bowed Bars, Joural of the Acoustical Society of America 08, pp (2000) [7] O. Iácio, L. Herique, J. Atues, Dyamical Aalysis of Bowed Bars, Proceedigs of the 8th 2