Numerical simulations of piano strings. I. A physical model for a struck string using finite difference methods AntoineChaigne SignalDepartment, CNRS UIL4 820, TelecomParis, 46 rue Barrault, 75634Paris Cedex13, France Anders Askenfelt Departmentof SpeechCommunication and Music Acoustics;Royal Institute of Technology(KTH), P.O. Box 700 14, S-100 44 Stockholm, Sweden (Received 8 March 1993;accepted for publication26 October 1993) The first attempt to generatemusical sounds by solvingthe equationsof vibratingstrings by meansof finitedifference methods (FDM) wasmade by Hiller and Ruiz [J. Audio Eng. Soc.19, 462472 (1971)]. It is shownhere how this numericalapproach and the underlyingphysical modelcan be improvedin order to simulatethe motion of the piano stringwith a high degree of realism.Starting from the fundamentalequations of a damped,stiff stringinteracting with a nonlinear hammer, a numerical finite differencescheme is derived, from which the time histories of stringdisplacement and velocityfor eachpoint of the stringare computedin the timedomain. The interactingforce between hammer and string,as well as the forceacting on the bridge,are givenby the samescheme. The performanceof the model is illustratedby a few examplesof simulated string waveforms. A brief discussionof the aspectsof numerical stability and dispersionwith referenceto the properchoice of samplingparameters is alsoincluded. PACS numbers: 43.75.Mn LIST OF SYMBOLS N numberof stringsegments coefficientsin the discretewave equation p stiffnessnonlinear exponent all(t) hammer acceleration s cross-sectional area of the core bl ,b3 dampingcoefficients T string tension c = transversewave velocity of string v(x,t) transversestring velocity E Young's modulusof string hammer velocity f(x,Xo,t) force density VHO initial hammer velocity (t = 0) fl fundamentalfrequency Xo distanceof hammerfrom agrafie L samplingfrequency y(x,t) transversedisplacement of string Fs(t) bridge force a =xo/L relativehammer striking position (RHSP) F/•(t) hammer force at=l/fe time step g( x,Xo) spatialwindow Ax= L/N spatialstep i spatialindex string stiffnessparameter K coefficient of hammer stiffness •/(t) hammer displacement L string length K radius of gyration of string Ms=pL string mass linear massdensity of string M•r hammer mass decay rate M•r/Ms hammer-stringmass ratio (HSMR) •= decaytime time index angular frequency INTRODUCTION from sucha set of equations.Consequently, it is necessary to use numerical methodswhen testing the validity of a The vibrationalproperties of a musicalinstrument-- physicalmodel of a musicalinstrument. like any other vibrating structure•can be describedby a Once the numerical difficulties have been mastered, a set of differentialand partial differentialequations derived simulationof a traditional instrumentby a physicalmodel from the generallaws of physics.Such a set of equations, meansthat the influenceof step-by-stepvariations of sig- which definethe instrumentwith a higher or lesserdegree nificant designparameters like string properties,plate res- of perfection,is oftenreferred to as a physicalmodel. Due onances,and others, can be evaluated. Such a systematic to the complexdesign of the traditionalinstruments, which researchmethod could hardly be achievedwhen working in most cases also include a nonlinear excitation mecha- with real instruments, not even with the assistance of nism, no analytical solutionscan, however,be expected skilled instrumentmakers. In the future, it is hoped that 1112 d. Acoust.Soc. Am. 95 (2), February1994 0001-4966/94/95(2)/1112/7/$6.00 ¸ 1994 AcousticalSociety of Annefica 1112 advancedphysical models, which reproducethe perfor- with a nonlinearcompression characteristic. He investi- manceof traditionalinstruments with high fidelity,can be gated, in particular,some details of the hammer-string usedas a tool for computer-aided-lutherie(CAL). interaction,and the efficiencyin the energytransmission Variousnumerical methods have been used extensively from hammerto string.The effectof stringinharmonicity for manyyears in otherbranches of acoustics,for example was taken into accountin a simplifiedmanner by slightly in underwateracoustics where the goalis to solvethe elas- modifyingthe values of thelumped string compliances? tic waveequation in a fluid.I In musicalacoustics, it is of In a recentpaper, Hall madeuse of anotherapproach greatvalue to obtaina solutiondirectly in the time domain, for simulatinga stiffstring excited by a nonlinearhammer, sinceit allowsus to listen to the computedwaveform di- which he named a standing-wavemodel. His method can rectly,and judge the realismof the simulation.Among the be regardedas a seminumericalapproach, since it partially large numberof numericaltechniques available, finite dif- makesuse of analyticalresults. By this method, he inves- ferencemethods (FDM) are particularlywell suitedfor tigatedsystematically the effectsof stepby stepvariations solvinghyperbolic equations in the time domain. 2 For sys- of hammer nonlinearityand stiffnessparameters, among tems in one dimension, like the transverse motion of a otherthings? vibrating string, the use of FDM leads to a recurrence In comparison with the earlier studies mentioned equationthat simulates the propagation along the string. 3 above,the presentmodel has the featureof a detailedmod- The generalityof FDM makesit possibleto alsouse them eling of the piano stringand hammer as closelyas possible for solvingproblems in two and three dimensions.The to the basicphysical relations: Our modelis entirelybased main practicallimit then is set by the rapidly increasing on finite differenceapproximations of the continuousequa- computingtime. tions for the transversevibrations of a dampedstiff string Historically,Hiller and Ruiz werethe first to solvethe struckby a nonlinearhammer. The blow of the hammeris equationsof the vibratingstring numericallyin order to representedby a forcedensity term in the waveequation, simulatemusical sounds? The model of thepiano string distributedin time and space,and the dampingis fre- and hammerused by thesepioneers was, however,rather quencydependent. crudein view of the improvementsin pianomodeling over The presentationis organizedas follows.In Sec.I, the thelast two decades? For example, the crucial value of the continuousmodel for the dampedstiff string is briefly re- contact duration between hammer and string, in reality viewed,with regardto the waveequation, and to the equa- being a result of the complexhammer-string interaction, tions governingthe hammer-stringinteraction. In Sec. If, was set beforehandas a known parameter. it is shownhow this theoreticalbackground can be put into Someyears later, Baconand Bowsherdeveloped a dis- a discreteform for time-domainsimulations. Some impor- crete model for the struck string where the hammer was tant aspectsof numerical stability, dispersion,and accu- definedby its mass and its initial velocity. 6 Displacement racy are briefly discussed here, in particularthe selectionof waveformswere computed for both hammerand stringat the appropriatenumber N of spatialsteps as a functionof the contactpoint. Their modelcan be regardedas the first the fundamentalfrequency f• of the string, for a given seriousattempt to achievea realistic descriptionof the samplingfrequency f•. A detailedtreatment of the numer- hammer-stringinteraction in the time domain.However, ical aspectscan be found in a previouspaper by the first severaleffects were not modeled in detail. The damping author.• In Sec.III, thestructure of the computer program was includedas a single fluid (dashpot) term, and the is presented,and a few examplesof the capabilitiesof the stiffnessof the string was neglected.The model assumed modelfor representingthe wavepropagation on the string further a linear compressionlaw of the felt. From a nu- are given. mericalpoint of view, no attemptswere made to investigate A thorough evaluation of the model by systematic stability,dispersion, and accuracyproblems. comparisonsbetween simulated and measuredwaveforms More recently,Boutilion made use of finite differences and spectrawas left as a separatestudy. That work will for modelinga piano string without stiffness,assuming a alsoinclude a systematicexploration of the influenceof the nonlinearcompression law and the presenceof a hysteresis hammer-stringparameters on the piano tone. in the felt. He investigated,in particular, the hammer- I. THEORETICAL BACKGROUND stringinteraction for two notes,in the bassand mid range, respectively.7 A. Wave propagation on a damped stiff string In all three papers mentioned, the numerical velocity, The presentmodel describesthe transversemotion of a i.e., the ratio betweenthe discretespatial and time steps, piano string in a plane perpendicularto the soundboard. was set equal to the physicaltransverse velocity of the The vibrationsare governedby the followingequation: string. It has been shown that this particularchoice is possiblefor an ideal string only, and that the numerical 2-2 oy scheme becomesunstable if stiffness,or nonlinear effects -- •+2b3 •---•+f(x,xo,t), due to largevibration amplitudes, are takeninto accountin (1) the model.3 in which stiffnessand damping terms are included. The At aboutthe sametime, Suzukipresented an alterna- stiffnessparameter is given by tive for simulatingthe motionof hammerand string,using a string model with lumped elementsstruck by a hammer e=•(ES/TL2). (2) 1113 J. Acoust.Soc. Am., Vol. 95, No. 2, February1994 A. Chaigneand A. Askenfelt:Simulations of pianostring 1113 It