Non-Smooth Model of the Grand Piano Action Anders Thorin

Non-smooth model of the grand piano action Anders Thorin

To cite this version:

Anders Thorin. Non-smooth model of the grand piano action. Solid mechanics [physics.class-ph]. Ecole Polytechnique X, 2013. English. pastel-00939493

HAL Id: pastel-00939493 https://pastel.archives-ouvertes.fr/pastel-00939493 Submitted on 30 Jan 2014

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE DE DOCTORAT DE L’ÉCOLE POLYTECHNIQUE

Présentée par Anders THORIN

Pour obtenir le grade de DOCTEUR DE L’ÉCOLE POLYTECHNIQUE

Spécialité : Mécanique

Non-smooth model of the grand piano action

Préparée au Laboratoire de Mécanique des Solides et au CEA List

Soutenue le 22 octobre 2013 devant le jury composé de :

M. Bernard BROGLIATO Rapporteur INRIA Grenoble - Rhône-Alpes M. René CAUSSÉ Rapporteur IRCAM, Paris M. Paul FISETTE Président Université catholique de Louvain M. Stephen PAULELLO Invité Atelier Paulello, Villethierry M. José LOZADA Co-Encadrant CEA List, Paris M. Xavier MERLHIOT Co-Encadrant CEA List, Paris M. Xavier BOUTILLON Directeur École polytechnique, Paris

Imagine, maintenant : un piano. Les touches ont un début. Et les touches ont une fin. Toi, tu sais qu’il y en a quatre-vingt-huit, là-dessus personne peut te rouler. Elles ne sont pas infinies, elles. Mais toi, tu es infini, et sur ces touches, la musique que tu peux jouer est infinie. Elles, elles sont “quatre-vingt-huit. Toi, tu es infini. Voilà ce qui me plaît. C’est quelque chose qu’on peut vivre. Mais si tu / Mais si je monte sur cette passerelle, et que devant moi / Mais si je monte sur cette passerelle, et que devant moi se déroule un clavier de millions de touches, des millions et des milliards/ Des millions et des milliards de touches, qui ne finissent jamais, c’est la vérité vraie qu’elles ne finissent jamais, et ce clavier-là, il est infini / Et si ce clavier-là, il est infini, alors / Sur ce clavier-là, il n’y a aucune musique que tu puisses jouer. Tu n’es pas assis sur le bon tabouret : ce piano-là, c’est Dieu qui y joue / Nom d’un chien, mais tu les as seulement vues, ces rues? Rien qu’en rues, il y en avait des milliers, comment vous faites là-bas pour en choisir une / Pour choisir une femme / Une maison, une terre qui soit la vôtre, un paysage à regarder, une manière de mourir / Tout ce monde, là / Ce monde collé à toi, et tu ne sais même pas où il finit / Jusqu’où il y en a / Vous n’avez pas peur, vous, d’exploser, rien que d’y penser, à toute cette énormité, rien que d’y penser? D’y vivre... / Moi, j’y suis né, sur ce bateau. Et le monde y passait, mais par deux mille personnes à la fois. Et des désirs, il y en avait aussi, mais pas plus que ce qui pouvait tenir entre la proue et la poupe. Tu jouais ton bonheur, sur un clavier qui n’était pas infini. C’est ça que j’ai appris, moi. La terre, c’est un bateau trop grand pour moi. C ’est un trop long voyage. Une femme trop belle. Un parfum trop fort. Une musique que je ne sais pas jouer. Pardonnez-moi. Mais je ne descendrai pas.

Alessandro Baricco – Novecento : pianiste”

À Daniel Yvon

Remerciements

Encore indécis à m’engager sur un doctorat à l’époque, je me souviens lire le descriptif de la thèse proposée par le LMS et le CEA LIST sur internet. Mes hésitations se sont instantanément levées : il s’agissait d’une conciliation entre ma passion pour la musique et celle des sciences. Je suis très reconnaissant envers le LMS et le CEA LIST ainsi que des membres qui les composent, d’avoir rendu possible la réalisation de la thèse dans un environnement si favorable et fructueux. Mes premiers remerciements s’adressent à Xavier Boutillon, avec qui j’ai pris plaisir à collaborer de fao¸n symbiotique tout au long de la thèse et grâce à qui j’ai beaucoup appris. Je lui suis d’autant plus reconnaissant qu’il s’est toujours montré attentif quand il le fallait, compréhensif et humain. Merci également à José Lozada et Xavier Merlhiot pour leurs nombreux conseils avisés et leur initiation aux domaines de la mécanique analytique, de la géométrie différentielle ou de la mécatronique. Merci à Jean-Christophe Eytard, Vincent de Greef, Hakim Gharbi, Erik Guimbretière, Ali Bozetine et William Gilbert pour leur aide sur la partie expérimentale. Je souhaite également remercier Bernard Brogliato et René Caussé d’avoir accepté la charge de rapporteur, Paul Fisette celle d’examinateur et de président, et Stephen Paulello de s’être joint au jury. Puis, il m’est très important de remercier ceux qui contribuent à ce qui me semble être l’un des plus précieux attributs de la recherche : le partage inconditionnel de savoir. Je tiens en effet à souligner la générosité de ceux qui donnent de leur temps pour transmettre leurs connaissances, même à des inconnus, aussi je prends le risque d’offusquer ceux que j’omets en ne citant que certains altruistes m’ayant aidé directement pour ma thèse. Merci donc à Patrick Ballard (accumulations d’impact), Alain Ehrlacher (énumérer les objets serait trop long !), Tony Lelièvre (mesures de Stieljes), Antonin Chambolle (décomposition des mesures), Vincent Acary (dynamique non-régulière) et particulièrement Mathieu Gautier et Jean Sreng (C++ et Python pour XDE). Que les autres sachent qu’ils ne sont pas pour autant oubliés ! Enﬁn, je remercie les acteurs du LMS, du CEA LIST et du LadHyx pour leur contribution à la si bonne ambiance qui y règne. Cela inclut en particulier Benoît Theckes et Loïc Tadrist avec chacun desquels j’ai passé deux tiers de thèse très riches en discussions fructueuses, mais aussi des collègues qui auront plus marqués le bureau 65 20 08 par leur caractère que par la durée de leur passage, tels Thomas Barois et ses sacs, Pascal Khalife (alias 110Ch210) et son tornado976 ou encore Eddy Malou – ça, c’est clair !

Résumé

Le mécanisme de la touche de piano à queue sert à propulser le marteau vers les cordes. Ce mécanisme permet au pianiste de contrôler avec précision la vitesse et l’instant d’impact du marteau sur la corde. Il est raisonnable de penser que c’est le comportement dynamique de la touche qui permet cette contrôlabilité. Avec pour perspective l’amélioration du rendu haptique des claviers numériques, cette thèse propose une méthode de simulation d’un modèle complet du mécanisme. Le son généré par la vibration qui suit l’impact du marteau sur les cordes n’entre pas dans le cadre de l’analyse. Des modèles du mécanisme comportant plusieurs degrés de liberté, des frottements et des contacts intermittents, ont été proposés depuis une quinzaine d’années. Notre approche se distingue de celles suivies jusqu’ici par un changement du point de vue adopté pour valider et pour simuler le modèle. En se fondant sur l’étude approfondie d’un modèle à un degré de liberté, il est en effet montré que la simulation d’un modèle dynamique complet doit se faire à l’aide d’un pilotage en déplacement, tandis que les travaux récents et anciens présentent des simulations pilotées en force. Une analyse des problèmes numériques liés aux discontinuités de vitesses survenant au sein du mécanisme durant l’enfoncement de la touche est présentée. Ils sont résolus par des méthodes de dynamique non-régulière implémentées dans le logiciel XDE. Les résultats sont présentés sous forme de comparaison avec les mesures expérimentales. La plupart des irrégularités des forces mesurées se retrouvent dans les forces simulées, en jeu piano comme en jeu forte. Les simulations rendent également bien compte de la ciné- matique de chaque élément du mécanisme. Une analyse de sensibilité du comportement dynamique aux paramètres du modèle est enﬁn exposée. Mots-clefs : piano, dynamique multi-corps, dynamique non régulière, modélisation, simulation, haptique.

Abstract

The grand piano action aims at propelling the hammer up to the strings. This mechanism provides the pianist with a high-controllability of the time of impact of the hammer with the strings and the hammer’s velocity at the impact. This controllability is believed to be due to the dynamic behaviour of the piano action. The present thesis proposes a simulation method of a complete model of the mechanism, which opens doors to improvements of the haptic rendering of digital keyboards. The sound following the impact of the hammer on the strings is not analysed. In the last ﬁfteen years, various models of the piano action including several degrees of freedom, friction and intermittent contacts, have been proposed. Our approach differs from existing work in that it is based on a new viewpoint for model validation and simulation. Indeed, using a in-depth study of a model with a single degree of freedom, it is shown that the simulation of a complete dynamic model must be driven with a displacement whilst, until now, only force driven simulations have been presented. Velocity discontinuities, occurring during the descent of the key, raise numerical issues which are analysed. They are overcome by non-smooth numerical methods that have been implemented in the computer program XDE. The results of the simulation are presented and compared to experimental measurements. For both piano and forte keystrokes, most of the irregularities in the measured force are reﬂected in the simulated force. The kinematics of the bodies is also correctly predicted. Eventually, a sensitivity analysis of the dynamic behaviour to the model’s parameters is proposed. Keywords : piano, multibody dynamics, non-smooth dynamics, modelling, simulation, haptic.

Contents

General introduction 1

1. Introduction 3 1.1. Presentationofthegrandpiano...... 4 1.2. Description of the grand piano’s action ...... 4 1.3. History ...... 7 1.4. Motivationsandobjectives ...... 8 1.5. Literaturereview ...... 12 1.6. Studiedaction...... 15

2. Modelling the dynamics of the piano action: is apparent success real? 17 2.1. Introduction...... 18 2.2. Experiments ...... 20 2.3. Simplemodels ...... 22 2.4. Simulations ...... 25 2.5. Discussion ...... 27 2.6. Conclusion...... 30

3. Experimental set-up 33 3.1. Kinematics...... 35 3.2. Force ...... 39 3.3. Actuationofthekey ...... 41 3.4. Triggerandacquisition ...... 45 3.5. Summary of measurement errors and uncertainties ...... 45

4. Model 47 4.1. Characteristics of the piano action ...... 48 4.2. Generalities ...... 49 4.3. Physicalelements...... 52 4.4. Classiﬁcationofparameters ...... 56 4.5. Dynamics:descriptionandvalues ...... 57 4.6. Matrixformulationofthedynamics ...... 68

5. Simulation methods 75 5.1. Limitations of the regularising approach ...... 76 5.2. Formulation of the piano action as a non-smooth dynamical multibody system 85 5.3. eXtendedDynamicEngine(XDE)...... 90 5.4. Adjustments of the model for its implementation ...... 91 5.5. AdjustmentsofXDEforthesimulation ...... 93 5.6. Simulatinginpractice ...... 95

xi Contents

6. Results and discussions 97 6.1. Regulationofthevirtualpianoaction ...... 98 6.2. Position-drivensimulations...... 98 6.3. Force-drivensimulations ...... 106 6.4. Discussion ...... 107 6.5. Asensitivityanalysis...... 113

Conclusion 123

A. Adjustment procedure of the grand piano action I

B. Application of the Lagrangian to a double pendulum V

C. Example of the calculation of ∂xδKH IX

xii List of Figures

1.1. Moderngrandpiano[Blackham,1965]...... 4 1.2. Sideviewofthegrandpiano ...... 5 1.3. Terminologyofthegrandpianoaction ...... 6 1.4. Pianoactionatsuccessivestages ...... 6 1.5. Actionoftheclavichord...... 7 1.6. Cristofori’spianoforte,1726 ...... 7 1.7. Today’s state-of-art actions for digital keyboards ...... 11 1.8. Lozada’shapticdevice(2007)...... 11

2.1. Experimentalset-up ...... 20 2.2. Measured position y, measured acceleration ¨y, measured force F in a mezzo forte keystroke...... 21 2.3. Grandpianoaction(withoutdamper)...... 22 2.4. Simpliﬁedschemeofthegrandpianoaction...... 22 2.5. Simple model of the piano action: the simplistic key ...... 23 2.6. Position y of the end of the key during various strokes ...... 26 2.7. Force F applied on the end of the key during various strokes ...... 27

3.1. Completeexperimentalset-up...... 34 3.2. Schemeofthemeasurements ...... 34 3.3. Set-upforactuatingthekey ...... 35 3.4. Comparison of the integration of the acceleration and the differentiation of the position, piano keystroke ...... 36 3.5. TestedpatternsforKLTtracking ...... 38 3.6. Illustration of KLT tracking with the selected patterns ...... 38 3.7. Angles of the bodies tracked by KLT algorithm ...... 39 3.8. Correction of the 50 Hz signal captured in the force measurement ...... 40 3.9. Typical measurements, piano ...... 40 3.10.Typical measurements, forte ...... 41 3.11.Scheme of the piston used to press on the silicone ...... 42 3.12.Typical ﬁt of the linear viscoelastic behaviour of the piston ...... 43 3.13.Ten identiﬁcations of the linear viscoelastic parameters of the piston . . . . . 44 3.14.Comparison of the force measurement and its estimation using the vis- coelasticpiston ...... 44 3.15.Schemeoftheset-up ...... 45 3.16.Photography of the piano action equipped for measurements ...... 45

4.1. Approximation of each rigid body as a clamped beam ...... 50 4.2. Location of the felts of the grand piano action ...... 52 4.3. Photograph of the felt bushing on the jack ...... 53 4.4. Zones where Coulomb friction is considered ...... 54

xiii List of Figures

4.5. Locationofthecouplingsprings ...... 55 4.6. Adjustable geometrical parameters, on a real grand piano action...... 56 4.7. Examplesofnon-measurableparameters ...... 57 4.8. Parameters for the description of the geometry for the contact zone centred in P ...... 58 4.9. Keyframe,measurements ...... 60 4.10.Contactsdescription,key ...... 60 4.11.Whippenframe,measurements ...... 61 4.12.Contactsdescription,whippen ...... 61 4.13.Jackframe,measurements ...... 62 4.14.Contactsdescription,jack ...... 62 4.15.Leverframe,measurements ...... 63 4.16.Contactsdescription,lever ...... 63 4.17.Hammerframe,measurements ...... 64 4.18.Contactsdescription,hammer...... 64 4.19.Damperframe,measurements...... 65 4.20.Contactsdescription,damper ...... 65 4.21.Groundframe,measurements ...... 66 4.22.Contactsdescription,ground ...... 66 4.23.Initialpositionsoftheframes ...... 67

5.1. Plots of multi-valued sign and single-valued sign functions...... 77 5.2. Scheme of the hammer studied as a freely oscillating pendulum ...... 79 5.3. Example of instability of the explicit scheme ...... 83 5.4. Example of non-physical convergence due toﬁ regularisation ...... 84 5.5. Example of non-convergence due to too small Θ ...... 84 5.6. Chatteringofregularisedschemes ...... 84 5.7. Convergenceoftheimplicitscheme ...... 85 5.8. Convergence of the non-smooth scheme ...... 89 5.9. Comparison of the regularised and non-smooth approaches ...... 89 5.10.Convergence of the regularised scheme and the non-smooth scheme for several Υ ...... 90 5.11.Scheme of the Kelvin-Voigt model added for the simulation ...... 91 5.12.Description of the contact with prismatic joint and felt law, illustrated with thehammer-stringcontact ...... 93 5.13.Illustration of the implementation of the felt law as a linear model with adjustableparameters...... 94 5.14.Synchronized connexions of Orocos agents ...... 95 5.15.Simulationsprinciple ...... 95

6.1. Scheme of the measured and simulated physical quantities (position-driven) 99 6.2. Simulation results for a piano keystroke (input: displacement) ...... 100 6.3. Simulation results for a forte keystroke (input: displacement) ...... 101 6.4. Comparison of the position of the bodies (tracking vs simulation) for a piano keystroke...... 103 6.5. Film and simulation screenshots comparison for notable events (see Fig- ure6.9)...... 105 6.6. Scheme of the measured and simulated physical quantities (force-driven) . . 106

xiv List of Figures

6.7. Simulation results for a piano keystroke (input: force) ...... 106 6.8. Simulation results for a forte keystroke (input: force) ...... 106 6.9. Notable events during a piano keystroke ...... 107

6.10.Computed force Fsimu fordifferenttimesteps ...... 111 6.11.Simulation results for a piano keystroke without felts (input: force) ...... 111 6.12.Simulation results for a piano keystroke without felts (input: displacement) 112

6.13.Sensitivity of J with respect to SK+y ...... 114 6.14.Sensitivity of J with respect to KWx ...... 114 6.15.Sensitivity of J with respect to KWy ...... 114 6.16.Sensitivity of J with respect to SJy ...... 115 6.17.Sensitivity of J with respect to JWx ...... 115 6.18.Sensitivity of J with respect to SLy ...... 115 6.19.Sensitivity of J with respect to LWy ...... 115 6.20.Sensitivity of J with respect to KHx ...... 116 6.21.Comparison of simulation results with measured and optimised values of

LWy for a piano keystroke(input: displacement) ...... 116 6.22.Comparison of simulation results with measured and optimised values of

LWy for a forte keystroke(input: displacement)...... 116 6.23.Comparison of the position of the bodies (tracking vs simulation) for a piano

keystroke with the optimised value of LWy ...... 121

A.1. Terminologyofthegrandpianoaction ...... I A.2. Locationoftheregulationsettings ...... II A.3. Appropriate adjustment of the jack regulating screw ...... II

B.1. Non-inertial frame of the lever and the jack connected to the whippen with hingesjoints...... V B.2. Scheme and parametrisation of the double pendulum ...... V

C.1. Scheme of the key compressing the front rail punching ...... IX

List of Tables

2.1. Parameter values of the grand piano action used in experiments according toLozada[2007]ormeasuredbyus...... 24 2.2. Parameters of the SK and BH models, according to Eqs. (2.1), (2.2) and (2.4) 25 2.3. Sensitivity of the results of the simulation ...... 28

3.1. Summaryofmeasurements ...... 43 3.2. Summaryofallmeasurement’serrors ...... 46 3.3. Detailsofkinematicerrors ...... 46

4.1. Order of magnitude of the inﬂuence of ﬂexibility in statics anddynamics . . 50 4.2. Description of the contact geometries ...... 51 4.3. The three cases of dry friction (M: external moment, D: dryfriction) . . . . . 54 4.4. The three cases of Coulomb friction (F: external force, T: tangential friction force) ...... 55 4.5. Conventions for the notations of the parameters ...... 58 4.6. Elementsinvolvedinthekey’sdynamics ...... 60 4.7. Elementsinvolvedin the whippen’s dynamics ...... 61 4.8. Elementsinvolvedinthejack’sdynamics ...... 62 4.9. Elementsinvolvedinthelever’sdynamics ...... 63 4.10.Elements involved in the hammer’s dynamics ...... 64 4.11.Elementsinvolvedin the damper’s dynamics ...... 65 4.12.Parameters of the linear elastic law of the coupling springs ...... 68

5.1. Summary of the three dimensionless parameters ...... 81 5.2. Value of Υ foreachbody ...... 81

6.1. Escapement velocities, measured and simulated ...... 110 6.2. Sensitivity of the cost function with respect to parameters of Category II . . . 118 6.3. Sensitivity of the cost function with respect to parameters of Category III . . 119

xvii

General introduction

The grand piano allows a tremendous variety of musical expressions, which is a reason for its success. This instrument is made of two uncoupled systems: the action, which transfers the energy given by the pianist to the hammer, and the vibrating ensemble, which produces the sound after the hammer has hit the strings. The action of the pianist, during a keystroke, results in a precise instant and a precise velocity of escapement of the hammer. The expressive potential of the piano results from the fact that the action allows pianists to control these two physical quantities with precision. This property is believed to originate in the tactile feedback of the key, hence in its dynamical behaviour. The piano action is therefore a haptic system which differs from many other ones by the highly accurate control that it offers. This concept is referred to as human-controllability. The grand action is a planar mechanism composed of several wooden pieces, joined by hinges with felt bushings around their axes. The possible contacts between pieces are ensured by felts. The mechanism also includes springs, screws and some metallic parts. Its overall dynamics is complex and highly non-linear. Moreover, it is very sensitive to small geometrical adjustments. Because touch is so important, manufacturers of digital pianos try to design keyboards that are haptically realistic. Old or cheap keyboards fail to reproduce the behaviour of the grand piano action. Nowadays, the best keyboards tend to imitate the original action by using some of its elements, if not all. However, these passive systems are still not fully satisfactory. This thesis is part of a research project led at LMS and CEA List, which aims at reproducing the dynamic behaviour of the grand piano action by means of an active or semi-active system controlled in real time. This approach is expected to provide cheaper, lighter, realistic and haptically-adjustable keyboards, and also an interface which allows to carry out series of experiments in view of improving the understanding of human-controllability in haptics. The reproduction of the dynamical behaviour of the grand piano action, by means of an active or semi-active solution, requires a valid model of the action and its real-time simulation. The literature reports many models, more and more complex. Complete models, where all the pieces of the grand piano action are considered, started to appear around 2000. All dynamical models, even the simple ones, were claimed to be good models in that their simulation exhibited a high degree of similarity with recorded positions of the key or the hammer, in response to an applied force. However, the fact remains that no satisfactory haptic device has been proposed yet.

1 List of Tables

This thesis has two main purposes. The first one is to examine why simple models look as good as complex ones. The second one is to propose a complete model and efficient numerical methods to simulate it, with the requirement of real time applicability. The dissertation is divided into six chapters. In Chapter 1, the piano and its action are presented and described. The motivations of our research project are detailed. Then, the literature on piano action models and simulations is reviewed, with an insight on the haptic aspects. A single degree-of-freedom model, similar to the models proposed until approximately 2000, is studied in Chapter 2 in view of understanding why its predictions look so realistic. The conclusion is that a model of the dynamics of the piano action should be validated by a comparison of measured and calculated forces, in response to a displacement instead of a force. The experimental set-up is described in Chapter 3. It includes dynamic measurements of the key and kinematic measurements of the whole. Typical measurements are presented for piano and forte keystrokes. A complete model, widely inspired of [Lozada, 2007], is given in Chapter 4. It consists in a multibody system made of six rigid bodies. All its parameters have a physical interpreta- tion and are measurable. They are categorised in view of a sensitivity analysis. All their values are given in this chapter. Simulating this model raises difficulties because of its non-smooth laws. In Chapter 5, we discuss their different possible treatments. We chose to apply a non-smooth method, implemented in the computer program XDE. A pendulum with dry friction is used to illustrate a numerical method for non-smooth systems and quantify its efficiency compared with a method based on a regularised model. Eventually, results are presented as a comparison with measurements in Chapter 6 with a displacement input, for both piano and forte dynamics. Results are also presented for the same model, using a force as an input. A sensitivity analysis of the reaction force of the key on the finger to the model’s parameters is carried out.

2 Contents 1 ..Ltrtr eiw...... review . Literature . . . . . 1.5...... objectives . . and . Motivations . . . . . 1.4...... History . action piano’s . 1.3. grand . the of . Description . piano 1.2. grand the of Presentation 1.1. ..Suidato ...... action Studied 1.6. ...Dgtlkyors...... keyboards . Digital . 1.4.2. view of point Haptical 1.4.1. ...Hpia eie ...... devices . Haptical . . . . . 1.5.4...... models Complete . . . . 1.5.3. . . . models Simple action 1.5.2. piano the of Description 1.5.1. Introduction hpe 1 Chapter 4 ...... 8 . . . . . 4 . . . . . 12 . . . . 12 . . . 15 . . 9 . 9 . . . 14 . . 7 . 13 12 3

55 ms CHAPTER 1. INTRODUCTION

1.1. Presentation of the grand piano A grand piano is a musical instrument played by means of a keyboard, see Figure 1.1. The modern keyboard is typically made of 88 keys. Each key is part of an action which transfers the energy given by the pianist to its hammer. The hammer strikes the strings which then vibrate. The strings transmit their vibrations to a bridge, and to a large wooden plate called soundboard. Eventually, the vibrations of the soundboard produces acoustic waves in the air: the sound of the piano.

strings

cast-iron frame

soundboard pin block bridge case action

Figure 1.1 – Modern grand piano [Blackham, 1965].

Once the hammer has escaped from the action, it moves freely until it strikes the strings: the pianist has no more control on its motion. Therefore, the keyboard and the strings of the grand piano are uncoupled. The instrument can therefore be described as two separate mechanisms: one which propels the hammers, and one (strings, bridge, soundboard, frame) which produces the sound. For this reason, it is possible to study these two elements independently. Here, we focus on the study of the grand piano action. It is shown in [Repp, 1999] that once a piece of music has been learnt, the playing of the musician is not coupled to the sound feedback of the piano. This justiﬁes why the piano action can be studied from of mechanical point of view, independently of the sound it produces.

1.2. Description of the grand piano’s action The location of the action is shown in Figure 1.2. The purpose of the piano action is to propel the hammer up towards the string and to catch it back after its escapement. The terminology of the mechanism is given in Figure 1.3 and the way it works is explained in the following sequence of events, the main stages of which are displayed in Figure 1.4.

4 hni scekd ti lce ytebccekﬁe nteky(Figure key the on ﬁxed backcheck the by blocked is it checked, is it When hspaelssbetween lasts phase This h ueosflspa ao oei h oc u r humidity-sen are but touch the in role major a play felts numerous The eoetehme istestring the hits hammer the Before pe ftehme,telte sete hce,o lce tteh the at blocked or checked, either is latter string the hammer, the the hit of speed has hammer the After keystroke. the of dynamics the h hl ehns oe ihtm ota h cin hc sv is is which procedure action, regulation the classical A that so regulated. periodically time be with moves mechanism whole the otesae4 ftepae rse h e gi,tepaoact piano retu the mechanism again, the key all key, the gravity. the piano of presses releases The because player he the If released. If completely stages. following been 4. not se state has to the ready key Th to position, the hammer. resting the if its releases even to backcheck string returns the jack up, the go compres and to and hammer begins downwards key the lever as repetition soon the pushes knuckle hammer 1 fteato spoel adjusted. properly is action the If iue1.2 Figure .Tejc tp tteltofbto n ttesm time same the at and button let-off the at stops damper. jack the The lift to 3. begins also key the asse through, Half-way whippen-lever-jack the 2. lifting rotate, to starts key The 1. .Tekycniust ittewipnlvrjc sebyadt and assembly whippen-lever-jack the lift to continues key The 4. .Poeldb h ak h amrecpsfo h cina giv a at action the from escapes hammer the jack, the by Propelled 5. .I t eoiyi iheog,tehme istestring. the hits hammer the enough, high is velocity its stops. If and punching 7. rail front the compresses key The 6. ttedo ce Fgr 1.4(b)). (Figure screw drop the at eeiinlvrrtt eaieyt h hpe.Tehammer The jack. the whippen. by the only to relatively rotate lever repetition r 1.4(c)). ure frame – ieve ftegadpao[fe lkRms iiei Com Wikimedia / Remesz Olek [after piano grand the of view Side bridge soundboard 20 sand ms string 200 ..DSRPINO H RN IN’ ACTION PIANO’S GRAND THE OF DESCRIPTION 1.2. ea ehns,rods mechanism, pedal s[sefl n aso,19]dpnigon depending 1990] Jansson, and [Askenfelt ms eedn ntepsto ftekyadthe and key the of position the on Depending otnt rail sostenuto ie nApni A. Appendix in given aobar capo 1 n oisrsigposition, resting its to rns dtehme pt the to up hammer the nd mbly. h eeiinlvrstops lever repetition the o otne ihthe with continues ion snwbigpushed being now is me knuckle. ammer r estv,hsto has sensitive, ery cini hnback then is action e t pig As spring. its ses iie Moreover, sitive. ejc n the and jack he piglfsthe lifts spring e nsed(Fig- speed en frame .().The 1.4(d)). pedals f iue1.3 Figure cf. mons]. 5

54 ms CHAPTER 1. INTRODUCTION

damper head string tuning pin

repetition lever hammer shank knuckle hammer drop screw

jack backcheck let-off button let-off button punching damper lever whippen

back rail cloth spoon capstan screw key front rail punching Figure 1.3 – Terminology of the grand piano action [after Olek Remesz / Wikimedia Commons].

(a) rest (b) contact jack / let-off button

Figure 1.4 – Piano action at successive stages [personal transformations of Figure 1.3].

6 16-70 slkl ohv ealdtewd xrsinpossi expression wide the recalled have to likely is (1669-1750) ihsalhmesta tukte,cetn hth called he what creating them, struck that hammers small with chara its of control the for allow not did but sound, loud a produced which a mrvdb rsooihmefadms fteeeet ftdysp today’s of elements the of most and himself Cristofori by improved was h lvcod oeunro h in,apae tteedo the of end the at appeared piano, the of forerunner clavichord, The otibShöe rpsdasmlraction. similar a proposed Schröter Gottlieb used were keys Its century. 16th the in appeared harpsichord The amrddlie tmao)i nisrmn hc ae bac dates which instrument an is (tympanon) dulcimer hammered A emnkyor ntuet ae otre ibra ul nacti an built Silberman Gottfried maker instruments keyboard German tuksrn ntuetwt ebadadhdalreexpress large sound. a feeble very had a and produced keyboard it that a was with drawback instrument string struck a enhl,aPrsa ntuetmkrcle aisadthe and Marius called maker instrument Parisian a Meanwhile, w forte jacks harpsichord the replaced He 1709. Ferdi in prop for one working were proposed repairer mechanisms Florence, and tuner three century, piano a 18th Cristofori, the Bartolomeo of beginning the In clavichord. the to opposed osssi tiigsrnswt n alti ahhn.Isim Its hand. each in mallet one with strings striking in consists ovneta lvcodo apihr ebad,weeec ﬁnger each st where be keyboards, to harpsichord note. hand or whole clavichord one as required convenient string Each instruments. string ae ieisnm othe to name its give later oisadShöe CosnadAe,17] n ul h rtpi ﬁrst the built and 1977], Ames, and inve [Closson Schröter their and from fori’s anything gained ever inventors three these of None ..History 1.3. led netd e iue1.6. Figure see invented, ke already same the using while loudnesses, different with notes audible hc en sf n odhrscod.Ti aewl become will name This harpsichord". loud and "soft means which " damper iue1.6 Figure hammer iue1.5 Figure ealcprusv device percussive metallic piano string string – h aninvto sta talwdtepouto of production the allowed it that is innovation main The . – rsooispaoot,1726 pianoforte, Cristofori’s whippen cino h clavichord the of Action jack spring c

Thorin. emncmoe Christoph composer German c rvmn yHebenstreit by provement

Thorin. " iiisofrdb struck by offered bilities bad h mechanism The yboard. key lvcmaoclpaoe piano col clavicembalo nfreoeya after year one anoforte ukwihwsntas not was which ruck ad I e eiiin Medici de’ III nando sdt tiestrings. strike to osed tos n12,the 1726, In ntions. key ideAe.I was It Ages. Middle v oe.Ismain Its power. ive ihpukdstrings plucked hich opuktestrings the pluck to ncpe nCristo- on copied on oAtqiyand Antiquity to k pianoforte aoatoswere actions iano a rdc one produce can ..HISTORY 1.3. trsis as cteristics, n will and 7

53 ms CHAPTER 1. INTRODUCTION

Cristofori’s death in 1731. he Frederick the Great of Prussia ordered seven pianofortes at once. Silberman had also exchanges with Bach who first criticised the instrument, but then gave his approval two years later of the improved pianoforte. The real success of the fortepiano began in the second half of the 18th century, mainly in Germany, France and England. Bach’s son Johann Christian played the pianoforte publicly and taught it to Queen Charlotte, while Mozart was composing specifically for the pianoforte. Meantime the square piano, smaller and cheaper, had an increasing success especially in England. In 1777, a skilled craftsman by the name of Sébastien Érard built his first pianoforte. He managed to get along with Louis XVI, and had to flee to London during the French revolution, in 1789. He came back to Paris from 1796 to 1808, then to London again until 1815. During this period he made many inventions, especially one for which he is still famous: in 1823, he invented the double escapement, allowing to propel the hammer up to the string several times in a row. Pleyel made some criticism about its fragility and preferred to use the simple escapement, but eventually had to face the success of the double escapement and lined up with the other piano makers. Many small changes were added to the piano and its action in the 19th century. The upright piano is an example of an instrument which originates from the piano. Henri Pape joined the Pleyel factory in Paris in 1811 and then continued on his own. He applied for 137 patents, most of which were not used, but including some inventions which have become today’s standards such as the addition of felts on the hammers (1826) or the crossing of the strings (1839). Pleyel also innovated, with for instance the addition of a cast iron frame in 1826. His factory prospered and he became an important rival of Érard. Many French factories were created in the 19th century but the small ones disappeared. Germany and England also produced large quantities of pianos. At the end of the century, Pleyel and Érard had both build their hundred-thousandth piano [Closson and Ames, 1977]. The pianos also spread to America in the 19th century, especially as many Germans emigrated over there. A famous example is that of Steinweg and his sons who fled from Europe to avoid the revolutions of 1848 and settled in New York in 1850, where they created Steinway & Sons in 1853. It became a major factory within a few years. Square pianos had a large success although they had developed from reconditioned clavichords and had been surpassed by grand and upright pianos from a technical point of view. In 1904, the American piano makers and dealers bought two hundreds of them, and burnt them in a giant bonfire at Atlantic City [Times, 1904] in order to inform the populace about the obsolescence of square pianos, and revive grand and upright pianos.

1.4. Motivations and objectives The present thesis is part of a larger double-faceted research project. The ﬁrst facet concerns the research on the mechanical properties of haptic devices which allow them to perform highly accurate dynamical tasks. The other one is to develop digital keyboards with realistic touch, using active or semi-active systems.

8 natv ebad erdcn h oc ftegadpaoato,w action, piano grand the of touch the reproducing keyboard, active An h eodsnec rvsta orpoueteietao h gr the of inertia the reproduce to that proves sentence second The h oino oc a enetnieysuidadcmetdfrth for commented and studied extensively been has touch of notion The hsifrainoiiae rmtecmlxdnmc ftemechan the of dynamics in complex necessary the gives from piano the originates of information touch the This system, haptic any in As o h V-0,Ymh eindec e ifrnl.Terad Their differently. key each designed Yamaha CVP-609, the For them. imitates manufacturer nteYmh L-9 case: CLP-990 Yamaha the In CLP the between similarities claiming: the by e pianos’ praises is Yamaha This instance, websi 1.7. For manufacturers’ on Figure found see examples action, numerous debatable, the piano to results real according the a are reproduce only to Not pian tends actions. digital piano of grand manufacturers of behaviour today’s matter, does touch Because keyboards Digital 1.4.2. t of model dynamic a of time. simulation real the in requires ma keyboard to a doors such open would Building it particular, In accurately. dynamics, so hammer action the piano the in what, understanding for interest lospoesoa insst control to pianists professional allows hi oc aebe tde n[rnieu n otlo,200 Acc Boutillon, important. and [Principeaud very in is studied touch been that have feel touch pianists min their that in fact has a already is he interesting It that the sound conveys given also a It to corresponding pianists. touch for the is touch [Lutring crucial in how reported is pianists professional with interviews oeta w centuries two than more h finger. the ...Hpia on fview of point Haptical 1.4.1. t e 2 n10,Ceet ulse oko o opa h pianofor the play to how on book a published Clementi 1803, In 1803]. n h velocity the and • • oe esaehayi oc n h ihrky r ih,wt l t all with light, are keys between." higher in the and gradations touch in in action heavy are authentic keys an lower provides Hammer Graded Linear "Moreover, dista the long." extremely because is base, fulcrum the the to to key the down of touch tip pleasing the from a syst provide hammering keys same "The the piano." is grand touch a as superior mechanism this spring-less behind secret "The v t e e scnrle ytepae within player the by controlled is a erpoue iharltv ro around error relative a with reproduced be can v e ftehme tecpmn.Tecnlso sta h action the that is conclusion The escapement. at hammer the of 2 eyitrsigaayi ftecneto oc,involving touch, of concept the of analysis interesting very A . t e and 2 ms; v e ihrmral accuracy: remarkable with ..MTVTOSADOBJECTIVES AND MOTIVATIONS 1.4. e oue nfigr n ad [Clementi, hands and fingers on focused te, 2 %. rFehr 02.I highlights It 2002]. er-Flecher, rc n erdcblt of reproducibility and uracy lostepaitt control to pianist the allows 45sato n grand and action -465’s etsmn states: vertisement omto o t control. its for formation yrlvn experiments. relevant ny ] osdrn h time the considering 8], data ins finds pianist a that idea e e eln point, selling key a ven styt erdc the reproduce to try os d. tes. n in es the keys, piano and u h conception the but s,atae by actuated ism, udb fgreat of be ould in,adfor and piano, e enatural he epaoaction, piano he hc the which mand em nce 9

52 ms CHAPTER 1. INTRODUCTION

"For even more realism, this new keyboard also uses a new hammer action that lets you actually feel the hammers and cushions as you play. [. . . ] The result is the same feel and response as a grand piano’s keyboard when playing fast, delicate passages." This last quote underlines that low dynamics, such as for piano keystrokes, are harder to be haptically rendered. The peak of this tendency is reached with the Yamaha AvantGrand, which includes a complete grand action (except hammer felts). Instead of reproducing the piano action’s dynamic behaviour by copying its complexity, the present research aims at doing so with cheap electronic active or semi-active3 components. They would be controlled in order to reproduce a faithful force feedback, computed by real-time simulations. Such a solution is likely to be much cheaper, lighter, smaller, but also to enable adjustable touch. A haptic device was proposed in 2007 by Lozada, see Figure 1.8. When a controlled magnetic field B is generated from the solenoid, the ferromagnetic particles inside the magnetorheological fluid react by creating aggregates, so that the apparent viscosity of the fluid is raised. Adjusting B results in adjusting the reaction force of the key, so that it matches the dynamical behaviour of the key. For numerical reasons related to the complexity of the piano action, Lozada did not succeed to simulate the model. The realisation of a haptic device which reproduces the dynamical behaviour of the grand piano action requires a valid model of mechanism, as well as efficient simulation tech- niques. Both are presented in the present thesis.

3Such as magnetorheological ﬂuids which have a short time response.

10 ftekey. the of iue1.8 Figure – oaashpi eie(07 iiga erdcn h dyna the reproducing at aiming (2007) device haptic Lozada’s iue1.7 Figure a NtrlKyor" aaac Yamaha Keyboard", "Natural (a) – c M,Kwi(00 c (2010) Kawai RM3, (c) o a’ tt-fatatosfrdgtlkeyboards. digital for actions state-of-art Today’s b H I,Rln 20)c (2009) Roland III, PHA (b)

musicpromusic

..MTVTOSADOBJECTIVES AND MOTIVATIONS 1.4.

Roland pianoworld antrelgclﬂuid magnetorheological key rod connecting oeodfrcontrollable for solenoid ii plate rigid ia behaviour mical B 11

51 ms CHAPTER 1. INTRODUCTION

1.5. Literature review

1.5.1. Description of the piano action The piano action, from Cristofori’s action (e.g. [Pollens, 2002]) to modern actions, has been extensively described, from several point of views. Some are focused on its history, some are intended to piano technicians, others are haptically-oriented. In [Pfeiffer, 1962], most of the actions developed in history are presented. Pfeiffer made elementary models of the hammer and basic experimental testing by means of calibration weights. He mentioned the friction force occurring at escapement4. The intermittent contacts occurring within the grand piano action, often referred to as changing kinematic constraints, have been studied in [Askenfelt and Jansson, 1990] where their timing is experimentally quantified. The touch, the motion of the mechanism and the hammer-string contact were characterised experimentally in [Askenfelt and Jansson, 1991]. The vibrations of the hammer shank were already discussed (first mode frequency: 50 Hz). Many studies starting in the 90s aimed at contributing to the domain of haptics. [Gillespie, 1996] noted that contrary to pianos’, synthesisers’ actions are not subject to changing kinematic constraints, even though keyboardists rely upon them to develop and exert fine control. [Hayashi et al., 1999] explained that producing a stable soft tone (piano keystroke) is difficult, even for experienced pianists. [Goebl et al., 2005] investigated the temporal behaviour of grand piano actions from different manufacturers under different touch conditions and dynamic levels.

1.5.2. Simple models After World War I, the piano manufacturer Pfeiffer applied scientiﬁc methods to the piano action [Pfeiffer, 1950, 1962; Pfeiffer et al., 1967]5. Several elementary models of the action were proposed. They became more and more complex. A unidimensional model was given in [Rimski-Korsakov and Maveev, 1938]6. It inspired some improvements [Oledzki, 1973] where two masses (one for the hammer and one for all the other parts) were connected by a spring representing the internal ﬂexibility of the action. In order to improve the results, the mass of the hammer were then made time-varying. A frictionless model was also proposed in [Dijksterhuis, 1965] where the key, the whippen-jack assembly and the hammer were superimposed masses. The last two models used forces as inputs. Later, [Gillespie and Cutkosky, 1992] presented a model made of four bodies (key, whippen, jack and hammer), for the motion from the beginning until the hammer-string contact. It included inertia and weight but neglected damping, compliance and friction. Gillespie mentioned that friction did matter at let-off. In order to simplify the simulation for real-time, a 2-DOF model, not good for escapement but allowing the solving, was

4He concluded that because the upright action has less friction, it is better than the grand piano action. This is questionable from the haptical point of view, and does not correspond to the common preference of pianists. 5Wolfgang Pfeiffer passed away in 1960; the posthumous dates actually corresponds to the translations. 6A. Rimski-Korsakov, relative to the composer Nikolaï Rimski-Korsakov, studied string instruments (for example [Rimski-Korsakov, 1937] and [Rimski-Korsakov and Samoilenko, 1937]).

12 egt.Teatoshglgtdteiflec fteregulati the of influence the highlighted authors The weights. a osdrdfrec e fknmtccntans hc e,ov led, which constraints, kinematic of set each for considered was hsmdlwste oie yteadto fa lsi tig[Vya string calcul elastic on an focused were of simulations addition The the 2009]. by al., et modified Vyasarayani then was model This 08 dsteflxblt ftehme hn ntemdl swel as model, the in shank hammer the of flexibility the adds 2008] c very law a with modelled are felts The 2004]. Hirschkorn, 2006; h oprsno h esrdadsmltdknmtc scorrec is kinematics simulated and measured the of comparison The ilsi mrvdhsmdlb digterptto ee [Gil lever repetition the adding by model his improved Gillespie opeemdli ie n[a e egee l,19] where 1995], al., et Berghe den [Van in given is model complete A oehrwt h nraeo opttoa osblte,tepi the possibilities, computational of increase the with Together eoiiso osataclrtoso h e.Tersligmoti resulting The key. the of accelerations constant or i velocities consisted It 1999]. al., et [Hayashi in proposed was model Another h amrshank. hammer the eiiiyddntafc h ehns eoeteipc fteha obse the was of impact It the force. before mechanism input the given affect a not for did keystroke, flexibility one of kinematics simulate opnn oteatclto fteky tnadP required PC standard A key. the of articulation the to ke the component of 200 displacements [Izadbakhsh, the Waterloo, of on University focused the is at th Hirschkorn discussion Again, Following the regularised. and is force, friction a dry and 1975] Crossley, and ishonpooe opeemdlwt esrdprmtr [Hi parameters measured with model complete a bu measured proposed not Hirschkorn are used parameters The ones. c possible are the constraints all kinematic cover of sets Several haptically-oriented. ipe iersdmdli loconsidered. also is model linearised simpler osrit.Teipti oc n h iuainrslsare results required simulation model the complete and The force key. for a the account and is cannot model hammer input This The whippen. the constraints. to fixed are lever tition na46Xprocessor, 486DX a on ...Cmlt models Complete 1.5.3. comp of qualified are models Such considered. wher is models action references section piano following grand The complex. more and more svldtdfis ycmaigtesmltdadtemaue d key. the measured th on of input the displacement force the and given Then, a input. simulated for position the key given the comparing a c for and by hammer is key damping first the nor by friction validated No constituted is model modelled. not a therefore uses is author escapement the 2006], [Oboe, In simulat the discussed. of inputs The hammer. the representing mass free a h e epyatr t yais n htrcvr fteorigin the of recovery that non-linearities. t and mechanism noteworthy of dynamics, is because its It alters hammer. deeply the key of the velocity escapement the on ularly rpsdi GlepeadCtok,19]adrue n[Gillespi in reused and 1993] Cutkosky, and [Gillespie in proposed oiue ndmninlmdl eysmlrt h n fDijkster of from one forces the experiments, to similar the very model, unidimensional a used Mori ODEs. continuous 33 H.B oa tnad,ra-iecudntb osbe A possible. be not could real-time standards, today By MHz. 1 to N 10 eeipsdt h e ymaso calibrated of means by key the to imposed were N 4 fsmlto for simulation of h no h ehns,partic- mechanism, the of on nu ftesmlto is simulation the of input e n cinmdl became models action ano niee u hyd not do they but onsidered epe 96.Hssuyis study His 1996]. lespie, ..LTRTR REVIEW LITERATURE 1.5. oswr ihrconstant either were ions tn h eetosof deflections the ating ldnmc sdifficult, is dynamics al h ieaiso the of kinematics the niee.Temodel The onsidered. -O oe with model 2-DOF a n amri observed is hammer e h akadterepe- the and jack the ,19] n model One 1994]. e, t. no h amris hammer the of on mradtestring. the and mmer a digwih to weight adding hat us[oi 97.In 1997]. [Mori, huis oet hto [Hunt of that to lose eemodels. lete slcmnso the of isplacements ;Iabks tal., et Izadbakhsh 6; hnigkinematic changing 75 n h hammer. the and y rl,t piecewise to erall, ahbd fthe of body each e satranslation a as l aulyfitted. manually t i nodrto order in min aaai 2009; sarayani, shone al., et rschkorn 1 amr The hammer. vdta the that rved nra life real in s 13

50 ms CHAPTER 1. INTRODUCTION

A micromechanical model of piano felts based on interactions between fibres is proposed in [Masoudi, 2012]. The model was implemented in an upright piano action model, based on Hirschkorn’s [Masoudi and McPhee, 2012]. The simulations were driven with a force and the kinematics were observed. The reported running time for one simulation was 87 min. Another a complete model with measured parameters is proposed in [Lozada, 2007]. Friction laws were regularised but still, numerical difficulties were encountered when simulating the multibody system with input positions, in order to calculate the reaction forces of the key on the pianist’s finger. [Mamou-Mani and Maniguet, 2009] claimed to have characterised differences in the mechanical behaviour of Érard and Broadwood’s pianos. No details on the model were given. In [Gillespie et al., 2011], the piano action is modified using rubber, in order to "linearise" its behaviour. The authors call the model a hybrid dynamical model because it combines continuous variables, and discrete variables to describe the state of the kinematic constraints. The input is still a force, and the discussion is focused on the position of the key. Another model where Coulomb friction is smoothed is proposed in [Links, 2011]. The position of the key in correctly simulated, except at the beginning of the motion. The correlations between the measured and simulated hammer positions is good. A study, aimed at understanding the historical evolution of the piano action, is given in [Bokiau et al., 2012]. The kinematics are simulated from an input idealised smooth force profile. To sum up, many different complete models have been proposed. In all of them, friction has been regularised to make the simulation easier. Also, none of them calculated the reaction force of the key: nearly all of them used it as an input and made kinematic observations.

1.5.4. Haptical devices

We give here a brief insight of a few haptically-oriented studies. Details on the musculoskeletal system of the ﬁnger are given in [Dennerlein et al., 1998]. In 1990, Cadoz et al. created a keyboard with tactile feedback [Cadoz et al., 1990]. It was also a modular device: the feedback could be changed. [Gillespie and Cutkosky, 1992] relate the importance of touch and the idea of replacing the piano action by actuators or programmable passive devices coupled with a control system. Voice-coil motors are employed in an electro-mechanical apparatus in [Gillespie and Cutkosky, 1993] and [Gillespie, 1994]. In [Gillespie, 1996], the author explains that real-time feedback is possible during events which last longer than 200 ms. Otherwise, the control requires learning and practise. [Chu, 1996] exposes the project of building a force-feedback based MIDI controller. An- other haptic device aimed at reproducing the piano action behaviour using voice-coil motors is proposed in [Oboe and Poli, 2002] and [Oboe, 2006]. The model used for the real time simulations is made of two bodies (key and hammer) without friction, damping or escapement.

14 Lns 01 eot oeitrsigodr fmgiue h s the magnitude: of orders interesting some reports 2011] [Links, ob ttsta [i nig]sgetta esr informati sensory that suggest findings] "[his that states Goebl eea oe speetdi h hss h aiaino the of validation The thesis. the in p presented grand the is of model enough general representative be A to presumed is action This h xeiet r efre nasnl in e ehns ma mechanism key piano single a on performed are experiments The niain ntedsg fhpi ebad r ie n[ovt an [Horváth in given are keyboards haptic of design the on Indications ntciesnigi around is sensing tactile in otc nacstetmn cuayo ne oeet npian in movements 2008]. finger Palmer, of and accuracy timing the enhances contact atclrpaoaction. piano particular h ietscsneo atc,wihi tde ee h e i key The here. studied the is on which influence some haptics, leads. have of of may that sense is coating kinestesic action The the The coating. support. plastic heavy a white to screwing a solidly by it action fixing sample t and provided with plate the line stabilized in We technician keyboard. piano piano professional possibilit a regulation by its adjusted to fully mecha respect the with to particularly similar but pianos, purposes demonstration for factory Renner ..Suidaction Studied 1.6. pnlrflxi tleast at is reflex spinal 40 sadacncosrflxi ogrthan longer is reflex conscious a and ms 0.25 o ttcfreand force static a for N 0.5 e.Teato a encare- been has action The ies. o yai oc.The force. dynamic a for N esadrsosre na in observed standards he naalbea finger-key at available on als bevbechange observable mallest o egtdwt key with weighted not s ht e,cvrdby covered key, white a 150 efrac"[Goebl performance" o ..SUIDACTION STUDIED 1.6. aoato ngeneral. in action iano im nuei grand in use in nisms oe sdn this a done is model ms. eln,btnton not but feeling, Tör d uatrdb the by nufactured tt hc metal thick a to it ci,2013]. ocsik, ˝ 15

49 ms

Contents Overview iuain ihmaue ocsaentsfcett valid to r sufﬁcient cal main not are The action. forces piano forces simulated measured 4. of with Chapter comparisons simulations in that presented demonstration model the the of simulation the 2 ..Cnlso ...... Conclusion ...... 2.6...... Discussion ...... 2.5...... Simulations . . . . . 2.4. . . . . models . Simple . . . . 2.3. . . . . Experiments . . 2.2. . . Introduction 2.1. h rsn hpe sa ril ne umsint caAcus Acta to submission under article an is chapter present The oeta hsvrino h ril snttelatest. the not is article the of version this that Note g references. here other are the which references, own its includes article original cin saprn ucs real? success apparent is piano action: the of dynamics the Modelling hpe 2 Chapter h olo hscatri odtriewihiptsol be should input which determine to is chapter this of goal The t yaia oe fthe of model dynamical a ate uae rmposition-driven from culated sl fti hpe is chapter this of esult tee with athered ia The tica. 18 . . . 30 . . 27 . . 25 . . 20 . . 22 . sdfor used 17

48 ms CHAPTER 2. MODELLING THE DYNAMICS OF THE PIANO ACTION: IS APPARENT SUCCESS REAL?

Abstract The kinematics and the dynamics of the piano action mechanism have been much studied in the last 50 years and fairly sophisticated models have been proposed in the last decade. Surprisingly, simple as well as sophisticated models seem to yield very valuable simulations, when compared to measurements. We propose here a too simple model, with only 1-degree of freedom, and compare its outcome with force and motion measurements obtained by playing a real piano mechanism. The model appears either as very good or as very bad, depending on which physical quantities are used as the input and output. We discuss the sensitivity of the simulation results to the initial conditions and to noise and the sensitivity of the experimental/simulation comparisons to the chosen dynamical model. It is shown that only motion-driven simulations should be used for validating a dynamical model of the piano action, contrary to what has been proposed in the literature.

2.1. Introduction The mechanical function of the piano action is to throw the hammer towards the strings. As a human-machine interface, its role is to provide the pianist with a means to perform the following musical task: obtain a given impact velocity of the hammer on the strings at a given instant, with as much as precision as possible. We focus here on the grand piano action but all what is proposed here would apply to the upright piano mechanism. Piano actions are complex systems mostly resulting from engineering during the 18th and 19th centuries, mostly by trial-and-error. The assembly of dozens of pieces is the fruit of a few major inventions (particularly by Cristofori and Érard, for the mechanisms that remained in the 20th century) and many minor refinements. In the resulting sophisticated design, it is not any more obvious to distinguish what are the features due to engineering – economy, ease of manufacturing in given historical conditions, necessity of a silent motion, ease of repair and adjustment, etc.– and those imposed by piano playing requirements: ease and precision of control, compliance with the playing tradition. Actions that are built for digital pianos (sound synthesisers) can be seen as tentatively complying with the latter group of requirements by means of markedly different engineering solutions. Although constantly improving over years, it is interesting to notice that the results are not yet judged as entirely convincing. We focus here on the dynamics of the piano action – the force-motion relationship – as seen from (or felt at) the finger-end of the key. In this paper, the piano action is considered either subject to a given force or to a given motion, which would be imposed by an operator. As of today, the physical quantity controlled by the pianist during the keystroke (or before) in order to perform the musical task has not been identified. In reality, the dynamics of the piano action is coupled to that of the finger/hand/arm/.. . musculoskeletal system which is coupled itself to a neurological system of efferent and afferent nerves. A vast literature is available on various questions pertaining to the pianist control, involving sensory-motor questions as well as the dynamics of the pianist limbs and fingers. This complex question is not analysed here. Since the 60’s, many dynamical or mechanical models of the piano action have been proposed, each of their authors more or less claiming that it emulates successfully the kinematics (usually the angular positions of the key and hammer) or the dynamics of the mechanism as seen at the end of the key. These claims are usually supported by

18 ihafreipt(oeie,teapidfrei osato ide or constant is force applied the (sometimes, input force a with ihu ucs.Rcnl,Bokiau Recently, success. without a xoe yMr oi[97,woapidfre otekywt ca with key the to forces applied who [1997], Mori Mori by exposed was hr w ass(n o h amradoefralteohrparts) other the all for one and hammer the for (one masses two where a e egee l 19]cniee -O oe hr h whi the where model 3-DOF a considered [1995] al. et Berghe den Van nte oe a rpsdb Hayashi by proposed was model Another (key bodies four with model a presented [1992] Cutkosky and Gillespie e lmnaymdl ftepaoato ro oeo t at have parts its of some of or action piano the of models elementary few A ayn ntm)o h e,b esrn hsfre h resulting the force, this measuring by key, the on time) in varying oe sas rpsdb beOo 20] h e n h amraemode are hammer 19 the In and key considered. not The is escapement [2006]. the Oboe t but Oboe in friction, by considered neglecting not proposed are also forces is However, model acceleration. constant driven are a Hayashi’s by forces, all or by to driven Contrary are hammer. which the literature representing the mass free a with model h rtcs,adbtentemaue oc n h predicted the and force measured predi the p can the between they and and motion, measured case, the imposed first between an made the by are driven Comparisons When a well. for key. as motion force the resulting of sep the end on predict the models a measurements at The parameters from apart. Their and taken been mechanism have 2.3). real s the (Section models, on valid simple measurements as very consider considered then be We motion hardly 2.2). the (Section recording it in on and acting levels, mech dynamical key real different a three playing at in consist experiments the paper, this In si observed. the was all kinematics [2007], Lozada and [1999] al. et Hayashi of those va Except of motion the yield simulations Force-driven model. sophisticated aaees tas nldstefis tep fdiigtes the driving a of with attempt model first different the a includes gives also It [2007] Lozada parameters. ignore Lozada pres is Links [2011]. damper [2004]. the Links Hirschkorn i Hirschkorn (5-DOF, lever by model repetition proposed complete The is parameters, A 90s. late [1996]. the Gillespie in in appear models eith complex modelled More discussed. not is therefore input is force a escapement to The response rigid. is assembly n oe scniee o ahsto ieai osrit.Aunid A constraints. kinematic of set each for considered neglect is were model friction and one compliances damping, where hammer) yasrn,rpeetn h nenlflxblt fteaction. the of flexibility internal the representing spring, a by spooe yDjsehi iktrus[95.OeziOezi[19 Oledzki w Oledzki model frictionless [1965]. a Dijksterhuis 1965, Dijksterhuis In by century. proposed 20th is the of half first the oe.Teeprmna eut r eeal bandb imposi by obtained generally are results numerical experimental The and model. measurements experimental between comparisons eosrtn htti snts.T hsed eaayetepred the analyse degree-of-freedom. we one end, only with this mechanism, To m the given so. of a models not validating is for this f irrelevant that given and demonstrating of arbitrary, simu choice less and the or experiments motion-nature, more in a as dynamics or the wh force- controlling known a for not of is motion-profiles more it is Since pianist pieces. the other by of sometimes, and, hammer the of tal. et oiue l 21]hv lopooe rather a proposed also have [2012] al. et Bokiau tal. et aah ta.[99,cnitn na2-DOF a in consisting [1999], al. et Hayashi ihrb osatvelocity, constant a by either mltoswt position, a with imulations ns lotlk pianist, a like almost anism 5 a e Berghe den Van 95, t ueipsdmasses superimposed ith lsd,adteresulting the and alised), dl hsppram at aims paper This odel. ftekyadteforce the and key the of h te iuain in simulations other the d nGlepe[1994], Gillespie In ed. ilsi n Cutkosky and Gillespie gafre(osator (constant force a ng oino h e,that key, the of motion uain eedriven were mulations edrvdfo static from derived re ipeta hycan they that simple o oc nteohrone other the in force r h ieaisin kinematics The er. cin fvr simple very of ictions eut sudb the by issued results ..INTRODUCTION 2.1. iu pieces. rious ae noaccount into taken s 3 tde model a studied 73] nsasmlrmodel similar a ents eitdmtosin motions redicted hpe,jc and jack whippen, , i ae.A2-DOF A paper. his ie oc exerted force given ain a appear may lations ) ihmeasured with d), ltevle fits of values the ll mninlmodel imensional enpooe in proposed been rt icsthat pieces arate te h control the ether irtdweights. librated eeconnected were ppen-lever-jack repolsor orce-profiles ttereacting the ct tal. et lled, 19

47 ms CHAPTER 2. MODELLING THE DYNAMICS OF THE PIANO ACTION: IS APPARENT SUCCESS REAL?

(Section 2.4). The matching between the former appears to be much better than between

the latter, motivating the discussion in Section 2.5. 2.2. Experiments

Figure 2.1 – Experimental set-up. The black-and-white patterns were used for experiments which are not reported here.

The experiments are performed on a single piano key mechanism (Figure 2.1) manufac- tured by the Renner factory for demonstration purposes but similar to the mechanisms in use in grand pianos, particularly with respect to its regulation possibilities. The key action has been carefully adjusted by a professional piano technician in line with the standards observed in a piano keyboard. Compared to normal playing, a few modifications have been introduced. The damper has been removed (which may happen in "normal" playing). It appeared that some experiment-simulation comparisons are sensitive to the precise initial position of the key. Since investigating this question is not important for the object of this paper, the felt supporting the key at rest (left end of the key in Figure 2.1) has been replaced by a rigid support. We consider four phases during a keystroke: the first phase of the motion ends when the hammer escapes, the second phase when it is checked, the third phase lasts until the key is released and the last phase when the key comes back to rest. For a detailed description of the timing of the piano action, see Askenfelt and Jansson [1990]. The position of the key is measured by laser-sensors (Keyence LB12, with LB72 conditioning amplifier) at the end of the key and approximately mid-way between the finger-end

and its rotation centre. Two particular angular positions of the key (θ = θe and θ = θp) and the corresponding times at which they are measured are reported in the ﬁgures of this article by gray dashed-lines and gray continuous lines, respectively. The angular position

θ = θe ≈ 0.035rad has been evaluated in a quasi-static test as the angular position of the key when the jack meets the let-off button. When playing, this position corresponds closely to escapement but not exactly since the felts are compressed, depending on how the key has been played. In fact, escapement occurs slightly after (by a variable margin)

θ(t) reaches θe. For the sake of brevity in formulation, this slight difference is ignored in the rest of the article. The angular position θ = θp ≈ 0.040rad corresponds to the key meeting the front rail punching and has also been evaluated in a quasi-static test.

20 ihBKNxsmaueetapie)gudapoiaeymid approximately glued ampliﬁer) measurement Nexus B&K with ihalgtwih ( light-weight a with h e ceeaini esrdb ih ( light a by measured is acceleration key The h iiu oc ntaigdw-oinadtemxmmfrepre force maximum the and down-motion initiating force minimum The h aaaesmldat sampled are data The force h n fteky.Te r respectively are re They and key). (adding the procedure of end standard the the with estimated been have yabout by iue2.2 Figure rcsos n to and precisions) otnoslnscrepn to correspond lines continuous rcie hoigoeo h te siaino h eoiyhsver has velocity the of results. estimation simulation other the the or one choosing practice, trtos 0 uieain,arglrsto aaee of parameter rest) Char regularisation at regularisation a signal total-variation subiterations, the a 200 of using iterations, value signal, average position integrati the the of estim algorithms: removal be independent (after must two signal velocity by key the numerically viscosity, obtained include models its the that Since so shifted key. (wi also the key is touched the signal not force of has user The end the position. at th rest so the reported shifted to and is factor) key appropriate the the by of multiplied signals motion the follows, what In n h oaincnr fteky h oc xre nteedo h ke the of end the on exerted force The key. the of centre rotation the and F bto)i a in (bottom) F srpeetda ahdln ntebto rm.Tega da gray The frame. bottom the in line dashed a as represented is

2 0.15 − − – Force (N) Acceleration (m/s ) Position (mm) 20 10 10 20 10 esrdposition Measured h ausnral dutdb technicians. by adjusted normally values the N 0 2 4 6 8 0 0 5 ez forte mezzo 0 0 0 0 θ = 1.2 θ 0 p )peolcrcsno Kslr91,wt hreapie 5 ampliﬁer charge with 9211, (Kistler sensor piezoelectric g) . . . m m m y 20 02 20 02 20 02 50 ¨ tekymestefotri ucig respectively. punching) rail front the meets key (the · e e e H ACUB61 yNtoa Instruments). National by USB-6211 (ADC kHz J a a a s s s / esrk.A siaino h nrilpart inertial the of estimation An keystroke. . . . L t . . . y 40 04 40 04 40 04 2 = tp,maue acceleration measured (top), t e ecpmn ftehme,setx o additional for text see hammer, the of (escapement . . . 60 06 60 06 60 06 m F e down T zz i m o . . . ≈ 80 08 80 08 80 08 e 0.4 f ( o 0.70 s )aclrmtr(nec 2250A-10, (Endevco accelerometer g) r ) t e . . . and N 00 10 00 10 00 10 ttezr-auscorrespond zero-values the at F . . . up 5.10 20 12 20 12 20 12 au szr sln sthe as long as zero is value ¨ y ≈ rn 21](ee 30 (here: [2011] trand mdl) esrdforce measured (middle), oigsalmse at masses small moving − no h acceleration the of on 0.38 wybtenteend the between -way n ifrnito of differentiation and 5 td h eoiyis velocity The ated. . . . ml nuneon inﬂuence small y and 40 14 40 14 40 14 ..EXPERIMENTS 2.2. hdlnsadgray and shed-lines etn up-motion venting ,bt exceeding both N, ǫ ( smeasured is y J = . . . hmeasured th / 16 16 16 L 10 2 ) ¨ y − 9 fthe of .In ). 015). 21

46 ms CHAPTER 2. MODELLING THE DYNAMICS OF THE PIANO ACTION: IS APPARENT SUCCESS REAL?

Typical results for mezzo forte playing are displayed in Figure 2.2. The position of the key at the ﬁnger’s location is positive when the key is pushed down. The same convention applies to the force F(t) on the key.

2.3. Simple models

backcheck hammer repet. lever felt material whippen ground jack escapement button key

Figure 2.3 – Grand piano action (without damper).

G3 l32 Hammer θ3 +

G2 l23 W-L-J F (t) θ2 G1 Key l21 θ1 y(t) l 12 L

Figure 2.4 – Simpliﬁed scheme of the grand piano action in three blocks: Key, Whippen- Jack-Lever, Hammer. The damper has been excluded. The position θi of each block is deﬁned as positive when the key is pushed downwards, from 0 at rest. With an counterclockwise convention for angles (and torques), θ1 and θ3 are negative when they leave their rest position. For the sake of clarity, the reaction forces Ri j at the contact points between solids are not represented.

The mechanism (Figure 2.3) consists in several quasi-rigid bodies – key, whippen, jack, lever, hammer (the damper has been excluded) – which are coupled together by felts and pivots. A first simplification consists in considering three blocks in the mechanism: the key, the whippen–lever–jack assembly, the hammer (Figure 2.4). The angular positions of the three blocks shown in Figure 2.4 – {Key}, {Whippen-Jack-Lever}, {Hammer} – are denoted by θ1, θ2, θ3 and their inertia with respect to their rotation axes by J1, J2, J3. The sign convention is counterclockwise for angles and torques. However, for the sake of simplicity in the representation of force and motion, the force F exerted at the end of the key and the motion y of the end of the key are counted positively when the key is pushed down: F = −C/L and y = −Lθ1 = −Lθ. This model does not take into account the compliance between the bodies (key, whippen, etc.) and the contacts with the support (let-off button, drop screw). In line with the spirit of a simple model, it is considered that the variations of the θi are small (see Figure 2.4): the geometrical non-linearities are ignored so that coupling between the parts of the real mechanism does not alter significantly the parameters of the model. Altogether,

22 htfollows. what with ihntefaeo h bv prxmtos h oeto inertia of moment the approximations, above the of frame the Within ihteprmtr ie nTbe2.1. Table in given parameters the with hc iapasa onas soon as disappears which whether hc per when appears which oedatclysmlﬁdmdl(iue25 osssi si a in consists 2.5) (Figure model simpliﬁed drastically more A h cin ftetruseetdo h elmcaimaetrans are mechanism real the on exerted torques the of actions The h oetmo h amri eea ie hto h eto h m the of rest the of that times several is hammer the of momentum The oaigojc ihaglrposition angular with object rotating e seuvln oteoeo h hl ehns if: mechanism whole the of one the to equivalent is key h ml e etbtentemdl ftekyadtepeeo ods forc wood the of as long piece as the least and at F key lasts (in than the This key less of the is more. middle of any neglected the rotation be of between cannot centre felt the we red below Experimentally, small felt the small motion. the frict their of dry prevents compression position, axes rest its whippen’s from the displacements key small very For h uuldpnece ewe the between dependencies mutual the r osdrdaddntdby denoted th and limiting stops considered the by are imposed torques Non-permanent follows. as key ewe h saeetadteceko h amr.Continuati hammer). the 2.4. of Section check in further the discussed mechanis and tha the escapement of from rest the strongly the from between differs dissociated mechanism is hammer whole the when the model of inertia the that ifrn oe utb sd hc spooe further. proposed is which used, be must model different rcig o eotdhr) twsfudta h yaiswsdi was dynamics the that found was it here), reported not tracking, sta fafl.Bsdstevrosfl oeswihhv enp been have which models felt on various validated experimentally the been Besides has that model felt. phenomenological a of that is α 31 = θ L α F 32 a eso oethan more or less was iue2.5 Figure down α 21 yvrosisetoso h oin ftedfeetpee (posi pieces different the of motions the of inspections various By . . θ ( – t          ) ipemdlo h in cin h ipitckey. simplistic the action: piano the of model Simple θ > θ θ 3 2 θ p h opeso law compression The . ( = = C J t s ) = hstru nldsterato fters support rest the of reaction the includes torque This . > ≈ − − J l l l l 1 0.8 θ 0 θ 23 32 21 12 + i n h ecino h rn alpunching rail front the of reaction the and = θ θ r prxmtdb emtia relationships: geometrical by approximated are m eoetefrerahsta hehl,a threshold, that reaches force the Before mm. 2 1 α θ 21 2 1 = = eerdt s"h ipitcky S)in (SK) key" simplistic "the as to referred , J 2 − − + α α 32 21 α L θ θ 31 2 2 1 J = 3 F α ( L 31 θ θ ) 1 + ftefotri punching rail front the of o ntehme’ and hammer’s the in ion ngle-degree-of-freedom loosre htthe that observed also oe ntesimplistic the on posed ftesmlsi key simplistic the of t afwmilliseconds few (a m ..SML MODELS SIMPLE 2.3. htcaso etand felt of class that ple otekey the to applied e caim tfollows It echanism. tntvl different stinctively no h oe is model the of on ooe,w s a use we roposed, oino h key the of motion e J ftesimplistic the of F gr . see 2.1 igure potn it) upporting ( θ t ) y (2.1) (2.2) tion ( LF 23 t )

45 ms CHAPTER 2. MODELLING THE DYNAMICS OF THE PIANO ACTION: IS APPARENT SUCCESS REAL?

geometry Brenon [2002]: F(δ)= kδr + bδ2δ˙ (2.3)

where δ is the compression of the felt. After the hammer check, an additional source of dissipation is located in the back-check: the key (with the whippen block resting on it) becomes also coupled to the support through the hammer and the back-check felt. Hypothetically, the corresponding friction dissipates signiﬁcantly more than the internal compression of the felt of the front rail punching. Therefore, the value of b has been arbitrarily taken 100 times more than that measured by Brenon. The values of k, b and r are given in Table 2.2.

2 i Ji (kg.m ) Cwi (N.m) Cdi (N.m) cvi (N.m.s) Spring Geometry(m) α −3 1 3.36 × 10 −0.0155 0.012 0.022 l12 = 0.129 L = 0.245 −4 −5 2 3.97 × 10 −0.0103 4.93 × 10 κ2 = 0.087 N.m, l21 = 0.060 α21 = 2.15 0 θ2 = 0.42 rad l23 = 0.080 −4 −5 3 1.65 × 10 0.0133 4.93 × 10 l32 = 0.017 α31 = 10.1

Table 2.1 – Parameter values of the grand piano action used in experiments according to Lozada [2007] or measured by us.

Permanent torques independent of the key motion include the action of the pianist C(t)

(kept as is) and the torques Cwi due to the weights of the different parts of the mechanism. Their effects amount to Cw given by Eq. (2.4). Permanent and motion-dependent torques ˙ ˙ due to strains at pivots are modelled by viscous and dry friction: cvi θi and Cdi sign(θi) ˙ ˙ respectively. As before, their effect is written as cvθ and Cd sign(θ). In some mechanisms, including ours, a pre-stressed spring is inserted between the support and the whippen. Its

effect is modelled by a torque κ(θ − θ0). Within the approximation of small angles, the moments of the coupling forces Ri j are given by Ri j li j with the ﬁxed lengths li j represented in Figure 2.4. It comes:

Cw = Cw1 − α21 Cw2 + α31 Cw3  2 2 ⇒ cv = cv1 + α cv2 + α cv3 (2.4)  21 31  2 κ = α21 κ2   At the very beginning of the key motion, when F < Fdown, the force exerted by the key on the whippen is too low to completely overcome the dry friction in the axes of the whippen- lever-jack and of the hammer blocks. In consequence, there is a phase where the key

moves while the hammer does not: LCd1 < F(t) < Fdown. The hammer behaves as if it was ﬁxed to the support and the motion of the key is limited by the compression of coupling felts and springs (in the whippen cushion, etc.). For the sake of simplicity, we model them as one linear equivalent spring. Its stiffness K is estimated as the average ratio between

Fmeas and xmeas for the three different dynamics considered here. This model is denoted below as "the blocked hammer" (BH) model.

24 26 r ie nTbe22 eodetmto of estimation second A 2.2. Table in given are (2.6) smnindi eto .,teieta seto h oe sin is model the of aspect inertial the 2.3, Section in mentioned As hspr rsnssmltoso h oiino h e nresponse in key the of position the of simulations presents part This h yaia qain ftetoscesv oesﬁal read ﬁnally models successive two the of equations dynamical The h aaeeso h oe a eetmtdb en fafwexperi few a of means by estimated be can model the of parameters The hi ausaegvni al ..Teohrvle fteparam the of values other The 2.1. Table in given are values Their saeetadteceko h amr oee,w hs oconti to chose we However, hammer. along. the all of check the and escapement rvnsmltos n ovrey h ecino h e oapr a to key the of simulations). (motion-driven reaction end the conversely, and simulations) driven ..Simulations 2.4. ytesai etdsrbdi eto 2.2: Section in described test static the by siae ymaueet ntesprt lmns sdescri as elements, separate a (2.2) the Eq. on of measurements sides right-hand by the estimated of parameters The surements. al 2.2 Table – aaeeso h KadB oes codn oEs 21,(2 (2.1), Eqs. to according models, BH and SK the of Parameters J 1 θ ¨ J θ + ¨ + c v aaee ueia value Numerical Parameter 1 c v θ ˙ θ ˙ + C C θ θ θ c K κ k J + b r v w p 0 e d C C d d 1 sign sign C C E.(.)yields (2.7) (Eq. w d ( ( − θ θ ˙ ˙ C + ) + ) = = w for for + L L κ K ( 1.6 C θ F F θ 2 F F 0.0273 down down s 0.0221 − ( down down × − θ 0.0397 0.0343 0.1947 0.138 0.039 × C , 0.4 10 θ 60 2 2 θ 10 w ˙ 0 − + L L 1 = ) ) 2.7 9 N.m 10 N.m = ¾ ¶ N.m.s C N.s.m F F kg.m N.m N.m C up up w Iunit SI rad rad rad C C C C w = ( ( ( ( n h siainof estimation the and t t t t ) ) ) ) 0.132 2 -1 -3 ¾ 0 N.m) dE.(.)hv been have (2.4) Eq. nd e nLzd [2007]. Lozada in bed tr fEs 25 and (2.5) Eqs. of eters as: srbdmto tits at motion escribed ogvnfre (force- forces given to ..SIMULATIONS 2.4. u h simulation the nue ai ewe the between valid et n mea- and ments 2 n (2.4). and .2) C d r given are (2.8) (2.5) (2.7) (2.6) 25

44 ms CHAPTER 2. MODELLING THE DYNAMICS OF THE PIANO ACTION: IS APPARENT SUCCESS REAL?

For a given force proﬁle (here: Fmeas(t)), the angular position of the key θsimul(t) (and the displacement ysimul = L θ of the end) has been obtained by solving numerically Eq. (2.5) followed by Eq. (2.6) with C(t) = F(t)meas/L. The link between the two models was done by linear interpolation of the momentum of inertia J˜(θ) and the momentum of

weights C˜w(θ) from their values in the ﬁrst phase to their values in the second phase. The numerical integration has been done by the NDSolve function of Mathematica R , using an Adams method with a maximum step limit of 30000. The results of these force-driven simulations are presented in Figure 2.6 for three different strengths of the keystroke: piano, mezzo forte and forte. We also present the result of the simulation of Eq. (2.6) alone, with initial conditions given ˙ by the observation of θ and θ at t corresponding to F(t)= Fdown. The drift that can be observed in these simulations is discussed in Section 2.5.

piano ) Measurements 10 Simul. (F -driven):

mm BH-SK model ( SK model n o i 5 t i s o P 0 0 0.05 0.10 0.15 0.20 0.25 0.30

mezzo forte ) 10 mm ( n o i 5 t i s o P 0 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

forte ) 10 mm ( n o i 5 t i s o P 0 0 0.02 0.04 0.06 0.08 0.10 Time (s)

Figure 2.6 – Position y of the end of the key during various strokes. Dotted (blue) line: measured position. Dashed (green) line: simulated position according to Eq. (2.5) (BH model). Plain (red) line: simulated position according to Eq. (2.6) (SK model). Dash-dotted (red) line: simulated position according to the SK model starting with initial conditions taken in experimental data (see text). Gray vertical and horizontal lines: see caption of Figure 2.2.

26 seScin22.Terslsaepeetdi iue27frt for 2.7 Figure in presented are results The 2.2). Section (see ovrey oindie iuain yield simulations motion-driven Conversely, 0.8 ie:secpino iue2.2. Figure of caption see lines: iue2.7 Figure eoetesltbtenteds-otdadtepanlns cor lines, plain the t and according dash-dotted position the between simulated split accordin line: the force (red) simulated Before plain line: and (green) Dash-dotted Dashed force. measured n h ceeaino h e aebe esrdadtevelocity the 2.6. and Figure measured been have key the straig are of simulations acceleration such the (2.6), and Eq. and (2.5) Eq. to cording infreeetdb h e o rsrbdmotion prescribed a for key the by exerted force tion mm ti la htteB oe utb sdisedo h Kmode SK the of instead used be must model BH the that clear is it ,

– Force (N) Force (N) Force (N) 10 12 14 16 10 0 2 4 6 8 0 2 4 6 8 0 1 2 3 4 Force 0 0 0 0 0 0 M S F i e m S B ple nteedo h e uigvrossrks otd(blu Dotted strokes. various during key the of end the on applied a K u H s . u l. 20 02 m r m e . ( 50 05 o m m o d . d 20 02 e o e e n t l l i t . o 40 04 s n - d r . i 00 10 v e . 60 06 n . ) 40 04 : T m p i f e m o i F zz a . . r simul 50 15 80 08 e n t o e o ( s f ( o ) t r . = ) 60 06 t . e 00 10 θ . C 00 20 ( ( t t ) ) / (here, towr,oc h position the once htforward, . 20 12 L oE.(.)(Hmodel). (BH (2.5) Eq. to g h poieo h reac- the of opposite the , esm esrksa in as keystrokes same he epnigto responding . 80 08 q 26 S model). (SK (2.6) Eq. o . 50 25 θ . 40 14 ( a enestimated been has t ..SIMULATIONS 2.4. = ) .Ga vertical Gray l. x . . . meas 30 16 10 y / = L )line: e) .Ac- ). θ L 27 ≈

43 ms CHAPTER 2. MODELLING THE DYNAMICS OF THE PIANO ACTION: IS APPARENT SUCCESS REAL?

2.5. Discussion After the escapement, the key hits the front rail punching. This piece does not differ between traditional keyboards and numerical keyboards. Since the latter are not judged as of equivalent quality by pianists, one should infer from this observation as well as from how pianists test and feel a keyboard, that the haptic feedback before escapement is of prime interest for them. Therefore, the discussion is primarily focused on this phase of the motion. A first and basic finding can be deduced from the experimental observations reported in Figure 2.2. As can be seen in the bottom frame of this figure, the dynamics of the mechanism is dominated before escapement by the inertia of its pieces, taken as a whole. The other-than-inertial dynamical effects due the internal degrees of freedom, the various stops that are met or left by the pieces, etc. appear as time variations of the difference F(t) − ÿ(t)J/L2. The corresponding wiggles can easily be distinguished in the bottom frame of Figure 2.2, even though the motion (top frame) is quite smooth. Although not surprising, this elementary observation has important implications with regard to the main point raised in the introduction: in order to validate a dynamical model, should the dynamics be examined as producing a force in response to an imposed displacement or vice-versa? From a purely experimental point of view, it is generally difficult to drive a mechanism with a rapidly changing force or acceleration. In this particular case, it would not be advisable either, since it would generate vibrations in the key that would mask the time-variation of the key displacement that are expected to be characteristic of the dynamics mechanism, considered as an assembly of rigid bodies. Altogether, realistic experimentations would consist in pushing this mechanism with a smooth force-profile, or with a smooth motion- profile similar to the ones reported here. Since the dynamics of this particular mechanism is dominated by inertia until escapement, it follows that the acceleration is generally smooth, possibly displaying some wiggles. When looking at the angular position of the key or displacement of the end of the key (imposed force), these potential wiggles in the acceleration are heavily filtered by the double time-integration: the differences between inertia and the complete dynamics of the system becomes hard, if not impossible, to distinguish. In other words, any model, provided that it is inertia-dominated, is likely to appear as very good when checking its validity by means of comparisons of motion- results obtained in force-driven simulations and tests, before escapement. This lack of sensitivity of the results to the model is represented by a "0" in upper left cell in Table 2.3.

Input F ¨y and y y y¨ Output y F F F

Model 0 + + + Initial cond. + 0 0 + Noise 0 0 + 0

Table 2.3 – Sensitivity of the results of the simulation, driven as indicated in the ﬁrst line, to the elements indicated in the ﬁrst column. "+" means "sensitive". "0" means not or little sensitive.

28 nyamr lbrt oe ol il iuainrslsta would that results simulation yield could model elaborate more a Only h eut eotdhr aebe bandb sn ohpsto and position both using by obtained been have here reported results The 01 locmae h acltdadmaue e’ displace key’s measured and calculated the compares also 2011] aiaetesceso fte"lce e"adte"ipitcky m key" "simplistic the and key" Figure "blocked in the done of be succession can the that validate comparisons experiment/simulation The eysniiet os nteiiildt.Cnesl,using Conversely, data. initial the in noise to sensitive very h rtsto oprsn rsne y[be 06 Fgr 8 is 18) (Figure 2006] [Oboe, by presented comparisons of set first The a enaddbtentebde nodrt euaietebehaviou the regularise in input, to force order given in bodies & the Hirschkorn between added been has rsne nFgr . antb osdrda aiaino the of validation by a look, as considered that be cannot results 2.7 force Figure in yield experim presented The data) counterparts. force experimental their same the from different the of using (instead tests, data experimental motion same the seem with model Simulations "0" any left (upper that obv above remark is described above restrictions this the and Although with circumstances hammer. consistent the is of this check model, the and escapement after left the of cell second the in is "+" dynamics a the by condition that represented initial fact is the the sensitivity to with SK-model succession consistent velocity, the the initial of of issued high-sensitivity those a to denotes compared drift a display curves h eybgnigo h oin h Hmdlsest rdc kinem simulations). predict (motion-driven to dynamics seems as i model well BH interest, as the essential simulations) motion, of the not of Although beginning 2.3. very Table the " in the line by first summarised is the model of the to sensitivity This observations. a aa oiinadvlct o oc lgtyexceeding slightly take force are a that for conditions velocity initial and with position run a th alone are data: to model They tal SK pertains the remark line. on dashed first based given Simulations a Our a conditions. by initial ruled line). 2.6, to are model Figure red models in (plain (SK) diagrams motion key" the certain "simplistic now and analyse (BH) we hammer" mind, in that Bearing hf orsodn oteaglrpsto ftekywe h hamm the when zero key is the of posi hammer position angular relati the angular shifted geometrical the of the to on to motion corresponding relies shift proportional der the becomes it model pieces, and but SK the phase hammer the hammer" of the Since positions of angular motion key. the the in the represent results of not their that did report from We to hammer. chose the authors of many literature, re the these In of synthesis a 2.3. For Table in cells conditions. sponding initial of corresp determination and the drifts generating time-integrations, successive an velocity escapement the before data, time-derivations, successive position by only calculated uses be one If data. experimental e’ n amrsmtosi epnet oc okvr iia om to similar very look force constraints a kinematic different to resp Agai for models response in model. 2-DOF in 3-DOF different motions Using a motions hammer presents hammer’s Berghe and den and Van key key’s model. presents 2-DOF a 19) to (Figure applied set second His al. s -O oe ocmuetehme’ n e’ oiin o a for positions key’s and hammer’s the compute to model 5-DOF a use piano Fgr 2ad1)adin and 14) and 12 (Figure forte Fgr 3ad1) h authors The 15). and 13 (Figure ceeaindt nyrequires only acceleration-data nigt ihsniiiyto high-sensitivity a to onding na/iuaincomparisons ental/simulation oiae yieta This inertia. by dominated nTbe2.3). Table in ms ouni al 2.3. Table in column -most F h euti nw obe to known is result The . et Fgr ) Rubber 9). (Figure ments ilsi Glepee al., et [Gillespie Gillespie , swrhntcn htat that noticing worth is t down oe u rvnb the by driven but model "i h eodcolumn second the in +" em ftekinematics the of terms h ceeainmust acceleration the d fB-Kmdl.This models. BH-SK of estvt fteSK the of sensitivity e rmteexperimen- the from n smlsi e"model. key" "simplistic ,mr rcsl,the precisely, more s, ak,setecorre- the see marks, h corresponding The . hr h "blocked the where osyavr crude very a iously rbgn omove. to begins er orc ne the under correct s ino h e,the key, the of tion ueykinematical. purely tc (force-driven atics . emt fully to seem 2.6 erpeetdby represented re ..DISCUSSION 2.5. oc n il a yield and force nhp between onships etrmththe match better dl,including odels, net force a to onse ftesystem. the of r nte"blocked the in ,teresulting the n, otat very contrast, easurements. vsdirectly ives acceleration 29

42 ms CHAPTER 2. MODELLING THE DYNAMICS OF THE PIANO ACTION: IS APPARENT SUCCESS REAL?

deduce from the similarities with measurements that the model predicts the behaviour of the piano action with reasonable accuracy. Again, the kinematics of the bodies are calculated by Links with a 5-DOF model for a force input (Figure 5.3 to 5.6) and match well the measurements. All these authors (a) compare simulated kinematics in response to a force-profile and (b) find a good agreement with measurements. According to the previous remarks, (b) is quite understandable. It would also be surprising that significantly different models could be valid to the same precision. Our first and main conclusion is that looking at the resulting motion of force-driven simulations cannot discriminate between good and bad models (provided they are inertia-dominated before escapement) and thus is not appropriate as a validation method. Moreover, the sensitivity of the simulated position of the key (or of the hammer) to initial conditions is an other reason for ruling out this choice of simulating the dynamics of a piano action by means of a force input. We recall here that this sensitivity is due to the dominance of inertia in the dynamics, not to the model itself. Some events are very important for the pianist such as the jack/let-off button contact (dashed vertical gray lines in the diagrams) or the escapement of the hammer. The model presented here and other models with 1 or 2 DOF do not take them into account, which probably makes them haptically irrelevant. What matters is that this irrelevance does not appear in the force-driven simulations. On the reverse, the measured force becomes very different from the force calculated by motion-driven simulations with the SK model, starting from the let-off. Almost immediately after escapement, the key meets the front rail punching (plain vertical gray lines in the diagrams), the dynamics of the mechanism is dominated by the corresponding nonlinear and lossy spring (Eq. (2.3)). This can be seen in the right part of the top and bottom frames of Figure 2.2: the position is almost constant and the force on the key has a main constant component with, again, some wiggles. Following the same analysis as above, any model which includes a stop with the appropriate stiffness is likely to appear as very good if one looks at the motion, considered as the output of a dynamical system subject to a smooth force profile. As explained before, the rapidly changing forces represented in the bottom frame of Figure 2.2, or the forces measured in Figure 2.7, are not good candidates for a profile to be used either as a driving experimental force. For the reasons explained previously, they are not good inputs for a dynamical model either. After escapement, the SK model is evidently worse than before escapement: the change in momentum of the hammer is ignored, as is the blocking due to the catch. However, driving the SK model with the measured forces yields an excellent agreement with the measured displacement at the end of the key (Figure 2.6). Conversely, tests where (smooth) position profiles are used as inputs yield simulated forces which differ strongly from the measurements (Figure 2.7). The previous reasoning on the apparent validity of inertia- dominated models can be transposed to stiffness-dominated models, after escapement. The conclusion is the same: matching measured motion and results of force-driven simulations cannot prove that a model is valid.

2.6. Conclusion Observations of the dynamics of the piano action show that it is dominated by inertia before escapement and by stiffness after the key has met the front rail punching. By

30 osbyoeo hs hc aebe ulse,i tmesteforce-c the meets it if published, d been the have render which in may those reflected model of sophisticated is one a possibly dynamics Only internal forces. the measured reaction of the the at complexity looking The and profile motion dynamics. a with mechanism the sufficient Driving be well also escapement. may before model kinematics 1-DOF hammer a t th the models, of of DOF bibliography check) more the or hammer Although 2- after presents chosen. ki (damping arbitrarily the somewhat parameter for be one to account had Only to sufficient along. is all model key w 1-DOF A virtually) or information. ( experimentally kinematic parameter (whether main mechanism the the However, Driving motion. key here. the adjusted was of t model stage seems model first blocked-hammer very the that the is the paper and this instead of used conclusion be minor must Althoug simulations action. position-driven piano noise, the for of to by model given dynamical comp positions a and the validate hammer) cannot that the tions shown or have key the we (of models, positions simple measured too of simulations of means t oc rfieyields profile force a ith ocsb oprd A compared. be forces nmc fteaction, the of ynamics hymyb subject be may they h h ai ige of wiggles rapid the ..CONCLUSION 2.6. aefite years fifteen late e maio test. omparison e1DFmodel 1-DOF he edie simula- ce-driven evldduring valid be o oc eel the reveals force tfns)o this of stiffness) eaiso the of nematics rsnbetween arison o rendering for 31

41 ms

Contents Overview aiaetemdlpeetdi hpe 4. Chapter in presented model the validate e.Teeeprmna eut r oprdt simulation dyn to and compared are bodies results the experimental These all key. of measurements kinematical includes 3 ..Fre...... uncertainties . . . and . . errors . . measurement . . of . . . Summary . . . 3.5. . . . acquisition . and . Trigger . . 3.4. key . the . of Actuation . . 3.3. . Force 3.2. ..Knmtc ...... Kinematics 3.1. ...Aclrmtr...... body . . each . . of . . position . . . the . of . . Tracking . . . . . 3.1.4. velocity . the . of . . Estimation . . . . 3.1.3. . . . Accelerometer . . 3.1.2. . sensors Laser 3.1.1. xeietlset-up Experimental hpe 3 Chapter nti hpe,teeprmna e-po h in action piano the of set-up experimental the chapter, this In eut nCatr6adue to used and 6 Chapter in results mclmaueet fthe of measurements amical 45 ...... sdsrbd It described. is 45 . . . . 36 . . . . 37 . . . . 41 . . 35 . . 36 . 35 . 39 33

40 ms CHAPTER 3. EXPERIMENTAL SET-UP

piano action interface local sensors acquisition

trigger

spot light

video acquisition

high-speed camera

Figure 3.1 – Complete experimental set-up.

piano action

whippen-lever-jack pianist piston key hammer damper

position position force position acceleration Figure 3.2 – Scheme of the measurements.

The motion of the piano action had to be measured in order to be compared with the numerical results of the simulations. We essentially focused on the measurements of the key because that is where the pianist interacts with the action. Complementary measurements on each rigid body were done for a better comparison. Because it was not possible to use directly the displacement of the key as an input for the simulation, a piston was inserted between the ﬁnger and the key. The upper face of the piston had to be measured. More details on the necessity of the piston are given in 5.4.1 page 91. A photograph of the complete set-up is shown in Figure 3.1 and the various measurements are presented in Figure 3.2. The kinematic ones included: • the position of the key (3.1.1); • the position of the upper face of the actuation device (3.1.1);

34 KyneL1 ihL7 mlfiruis eeue,snethey since used, were units) amplifier LB72 with LB12 (Keyence h cuto ftekyi ecie n33 aaaqiiinadsyn and acquisition Data 3.3. in described is key the of actuation The sw r neetdi h yaiso h in cin h reac the action, piano the of dynamics the in interested are we As omauetepsto fteuprfc ftepso n hto the of that and piston the of face upper the of position the measure To h ins’ ne a lomaue (3.2). measured also was finger pianist’s the aiso h in cin hi epnewsstto at set estimated were was and response Their of action. resolution piano the of namics ...Lsrsensors Laser 3.1.1. Kinematics 3.1. uncertaint and errors measurements the All 3.4. in presented nFgr 3.3. Figure in • • • h oiino ahbd sn ihsedcmr n ie trackin video and camera high-speed a using body each of position (3.1.3); the key the of velocity the of estimation an (3.1.2); key the of acceleration the 50 m hi estvte eemaue sn rdae mar graduated a using measured were sensitivities Their µm. 0.398 iue3.3 Figure /mand V/mm ae o h key the for laser – e-pfrataigtekey. the actuating for Set-up 0.449 /m h aeswr oiinda shown as positioned were lasers The V/mm. ae o h silicone the for laser oc sensor force piston 0.15 s hc orsod oa to corresponds which ms, e r umrsdi 3.5. in summarised are ies infreo h e on key the of force tion ontcag h dy- the change not do ..KINEMATICS 3.1. e,lsrsensors laser key, hoiainare chronisation (3.1.4). g iggauge king 35

39 ms CHAPTER 3. EXPERIMENTAL SET-UP

3.1.2. Accelerometer

An Endevco R 2250A piezoelectric sensor was used to measure the acceleration of the key (sensitivity: 0.316 V/(m · s−2)). The accelerometer was placed in the middle of the key, between its the centre of rotation and the piston. This signal was amplified with a Brüel & Kjær Nexus conditioning amplifier and filtered by a band-pass filter (lower and upper cut-off frequencies: 0.1 Hz and 3 kHz, respectively). The mass of this accelerometer is 0.4 g.

3.1.3. Estimation of the velocity

Trials with a Polytec PDV-100 laser vibrometer failed to measure the velocity, as the motion of the key was too slow. Two alternatives were studied. The ﬁrst one uses the Matlab script corresponding to [Chartrand, 2011] and proposed by its author. It consists in minimising the functional 1 F (u)= α |u′| + |Au − y|2 (3.1) [ ] 2 [ ] Z 0,T Z 0,T where y is the position signal, T its temporal length and A is the operator of anti-differentiation

(Au(t)= [0,t] u). The ﬁrst integral controls the regularity of the solution without penalis- ing jumps.R The second integral ensures that u is the differentiation of y. This solution gave acceptable results, see Figure 3.4. The parameters1 were empirically chosen as α = 5 × 10−5, 20 iterations and ǫ = 10−9.

(m) 0.0100 measurements y 0.0075 0.0050 0.0025

Position 0 Time (s) 0 0.2 0.4 0.6 0.8

0.20 )

1 differentiation of position

− 0.15 s integration of acceleration · 0.10

0.05 0 Time (s) 0.2 0.4 0.6 0.8 −0.05 Velocity (m −0.10 ) 2 40 −

s measurements · 20

0 Time (s) 0.2 0.4 0.6 0.8 −20

−40

Acceleration (m −60 Figure 3.4 – Comparison of the integration of the acceleration and the differentiation of the position, piano keystroke.

1ǫ is a numerical parameter to avoid division by zero and α is a weighting factor, see [Chartrand, 2011].

36 lrerslto oaodsaturating). avoid to resolution (large eue iiHC10 ihsedhg-eouincmr,compose camera, high-resolution high-speed HCC-1000 Simi a posit used The We 6). Chapter (see position simulated the to compared was was hc loe ycrnsto ihohrmaueetdvcs(s devices measurement other with synchronisation allowed which l h eua atrsfie ogv cetberslsfor results acceptable give to failed patterns regular the All nipeetto fteKTagrtmwspooe nteMta C Matlab the in proposed was algorithm KLT the of implementation An h eodetmto a oeb nertn fteaclrto a acceleration the of integrating by done was estimation second The h ieaiso l h oispoie nomto ntedynamics, the on information provided bodies the all of kinematics The eoiy eas h cuto eiewsol lgtydissip slightly only was device between actuation results the the because velocity, in appeared differences noticeable no straig Also, worked it because used was differentiation Total-variation omk h rcigese,hg-otatpten eesukt stuck were patterns high-contrast easier, tracking the make To h ieswr rae sn h L rcigalgorithm. tracking KLT the using treated were videos The h aeaaqiiinwstigrdb adaetigrsen trigger hardware a by triggered was acquisition camera The oedtisaegvni Lcse l,18;Tms n Kanade and Tomasi 1981; al., et [Lucas in given are details More tatal orsoddt eowe h e a trs.Integr rest. for at position was the of key derivative the the when with agreement zero to corresponded actually it esr.Tehgetaalbeaqiiinfeunywt re a with frequency acquisition available highest The sensors. t of dynamics the disturb not especial does and which camera, curves high-speed a their using of contac shape the in the measu change a estimate of indicates which purpose to variations, The was signiﬁcant body results. each simulation of of comparisons additional omauetedsac sn the using distance the measure to iiie itnebetween distance a minimises esrks h ceeainwsete auae tolreacce large (too saturated either was acceleration the keystrokes, ...Takn ftepsto fec body each of position the of Tracking 3.1.4. Z rd,dagtors aalllns o rd n irregular and comple grids more dot lines, uncertainties, too parallel measurements or draughtboards, the spotted grids, be reduce to to small order too either In squa were as but such results signs acceptable single gave tested: been have patterns Different choice Pattern oor therefore colour, ta. 2013]. al., et G oorspace colour RGB L tnsfrKnd-ua-oaiwodvlpds-aldKTfe in KLT decomposed is so-called video developed recorded who Kanade-Lucas-Tomasi for stands KLT tracking KLT 2 norcs,tevdo eemd n8lvl fge,btti d this but grey, of levels 8 in made were videos the case, our In h oewihhst etakd h L loih osssi ﬁndi in consists algorithm KLT The tracked. be to has which zone the L algorithm. KLT 923 p hc orsod ooeiaeevery image one to corresponds which fps 2 ∀ e scall us Let . i ∈ [ [ 1, N ]] , F F F n n i ( the : N x L 2 ) [ [ mgs(rms fsize of (frames) images and omwihlast lsia iiiainproblem. minimisation classical a to leads which norm ,1024 1, n t frame. -th F n + ]] 1 ( × x + [ [ piano F ,512 1, h n ) 1.083 sin oaypxlisascae RGB associated its pixel any to assigns , ∀ e o atrfrteudrtnigo the of understanding the for matter not oes esrks e iue34 For 3.4. Figure see keystrokes, x ]] Z ∈ ms. −→ ruhbad,seFgr 3.5. Figure see draughtboards, 1024 h amr t eoiywas velocity its hammer: the aaeadLcsproposed Lucas and Kanade . s h ulttv behaviour qualitative The ts. ative. [ [ e,tinls tr n dots and stars triangles, res, ouinof solution ,256 1, towrl o l keystrokes. all for htforwardly vr ii body. rigid every o h w siain fthe of estimations two the iga + an ding px 1991]. , eain) rinaccurate or lerations), tddt hwdgood showed data ated atrswr tested: were patterns x escin3.4). section ee × trmkn uethat sure making fter ]] epaoaction. piano he oswr measured were ions ag ob accurate. be to large 512 tr rces The trackers. ature 3 RToolbox[Barbacci e sdnt by denote us Let . ..KINEMATICS 3.1. igteposition the ring yteisatof instant the ly 1024 ng ffu CMOS four of d xadcddin coded and px 5 iigedge, rising V h n allowed and ∈ px R × 2 512 which forte 37 px

38 ms CHAPTER 3. EXPERIMENTAL SET-UP

(a) Punctual patterns

(b) Grid patterns (Grid. Draughtboard. Parallel lines. Dotted grid. Irregular draughtboard.)

Figure 3.5 – Tested patterns for KLT tracking. high so that its displacement observed in two consecutive frames could be equal to several periods of the pattern. This made the determination of its correct position impossible: the algorithm could detect minima of the cost function3 in the neighbourhood of the tracked points which were not the right one. This phenomenon could be seen as the tracked points of the hammer shifted from their initial positions and all ended in the same area. On the contrary, this was not observed using irregular draughtboards which probably increase the convexity of the cost function. An overview of the tracking is shown in Figure 3.6 and typical results from the ﬁnal treatment are plotted in Figure 3.7. The comparison of the estimated position of the key using the laser sensor and the high-speed camera shows that the latter gives a relative error of 5 %, which was considered acceptable for the qualitative analysis of Chapter 6.

(a) At rest (b) In motion

Figure 3.6 – Illustration of KLT tracking with the selected patterns.

3The cost function is deﬁned by 2 J (h) = ||Fn(x) − Fn+1(x + h)|| (3.2) x X∈Z

38 qat)gnrtsacag inlpootoa otela acti load the to proportional signal charge a generates (quartz) eas esrdtefreeetdo h e ocpuetedynami the capture to key the on exerted force the measured also We ihalatsursmto,o e eid utbfr h keystroke. the before just periods few a on method, squares least a with ..Force 3.2. 4.403 oc esrwsaKslrMdl91 of 9211 Model Kistler a was sensor force eetdi h oc inl h siltr usnewssubtr was nuisance oscillatory The signal. form force the in detected 3 le ctoffrequency (cut-off filter orce h itrac,seFgr 3.8. Figure see disturbance, the corrected H) rbbydet nufiin nuain eyregular very a insulation, insufficient to due Probably kHz).

CN.Tesga a mlfiduigaKslr51 hreMe Charge 5015 Kistler a using amplified was signal The pC/N). Angle (rad) Angle (rad) Angle (rad) − − − − 0.08 0.06 0.04 0.02 0.01 0.02 0.03 0.04 0.02 0.04 0.06 0.08 0.10 0.1 0.2 0.3 0 0 0 . . . . 0.5 0.4 0.3 0.2 0.1 0 . . . . 0.5 0.4 0.3 0.2 0.1 0 iue3.7 Figure a rcigo h e,wipn ee n akpositions jack and lever whippen, key, the of Tracking (a) θ θ θ θ θ c oprsno h e’ oiinmeasurements position key’s the of Comparison (c) jack lever whippen key hammer ≪ . . . . 0.5 0.4 0.3 0.2 0.1 – 1 nlso h oistakdb L algorithm. KLT by tracked bodies the of Angles b rcigo h amrposition hammer the of Tracking (b) z n eododrlwps le ctoffrequency: (cut-off filter low-pass order second a and Hz) α sin 1.2 ( 100 .We opesd hspeolcrcsensor piezoelectric this compressed, When g. π t + φ ) go h esr(sensitivity: sensor the on ng θ θ 50 key key zsnsia inlwas signal sinusoidal Hz ce yafitn fthe of fitting a by acted rmlaser from tracked so h cin The action. the of cs e ihahigh-pass a with ter ie(s) Time ie(s) Time ie(s) Time ..FORCE 3.2. tperfectly It (3.3) 39

37 ms CHAPTER 3. EXPERIMENTAL SET-UP

0.15 measured force 0.10 corrected force 0.05

Force (N) 0 Time (s) −0.05 0.02 0.04 0.06 0.08 0.10 0.12 Figure 3.8 – Correction of the 50 Hz signal captured in the force measurement.

Typical measurements are shown in Figures 3.9 and 3.10 for piano and forte keystrokes, respectively. The position curves are much smoother than the force curves, as observed in Chapter 2.

0.012 measurements 0.010

0.008

0.006

0.004

Position (meter) 0.002

0 Time (s) 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

measurements 3

2 Force (N) 1

0 Time (s) 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Figure 3.9 – Typical measurements of the force exerted on the key and its position, piano keystroke.

40 ol edie yapsto hsslto a enipeetdi th in implemented been had solution this – fr position the corrector, a PD by a to driven model, the be in key, would the coupling by around worked eeotie hnataigtekyb pligtefredirectly force the applying by key the actuating when obtained were h ancnlso rmCatr2wsta h iuain should simulations the that was 2 Chapter from conclusion main The stepaoato sahgl o-iermcaim tms est be must it mechanism, non-linear highly a is action piano the As oiin h iuainmtosw sdddntofrsc possibil a such offer not did used we methods simulation The position. fwihwscretymdle ihavsolsi a.Mr deta (5.4.1). More 5 law. Chapter viscoelastic s in a of given with piece are modelled a correctly d included was a device which added The of we corrector. experiments, PD the the In to corresponding simulation. the for used program erdcblt ftemto,wihdpnso h klso h playe the of skills the on needed. depends required. which accelerations motion, but the the motor, of and step-by-step Reproducibility a velocities with the key with the deal actuate not to could done were attempts Some 2006]. al., et [Hirschkorn to and o h aeraos eut fti hssaeuulypresent usually are thesis about this respectively of reaching results reasons, a same key the the For of depressing same 1990]. the Jansson, to ha and lead It [Askenfelt not studied. does player being pendulum is a dynamics of proper use the the that us particular, ensure In not does ones. load players’ the of representative are that ..Ataino h key the of Actuation 3.3. keystroke. iue3.10 Figure

forte Position (m) Force (N) 0.002 0.004 0.006 0.008 0.010 0.012 esrks standard A keystrokes. 0 0 2 4 6 8 – .501 .502 .50.30 0.25 0.20 0.15 0.10 0.05 0 .501 .502 .50.30 0.25 0.20 0.15 0.10 0.05 0 yia esrmnso h oc xetdo h e n its and key the on exterted force the of measurements Typical measurements measurements 2 and N forte 10 .Frsmlct,te r eerdt as to referred are they simplicity, For N. orsod oamxmmof maximum a to corresponds n ast pl constant a apply to mass a ing ..ATAINO H KEY THE OF ACTUATION 3.3. dfrdfeetkeystrokes, different for ed l fteP corrector PD the of ils lcn,tebehaviour the ilicone, vc hc ie at aimed which evice de o keystrokes for udied enosre that observed been s ntefresensor. force the on aifcoyresults Satisfactory hto human a of that s ,wside not indeed was r, eedo which of end ee edie ya by driven be 30 t.Ti was This ity. ie(s) Time oiin forte position, ie(s) Time according N computer e h latter the piano 41

36 ms CHAPTER 3. EXPERIMENTAL SET-UP

The viscoelastic material was chosen so that it had a behaviour close to that of a finger, in order to filter the force in a similar way. The Ecoflex R Shore 00-30 hardness A silicone was finally selected. The identification of its viscoelastic properties is set forth in paragraph 3.3.

Addition of a piston between the ﬁnger and the force sensor The silicone was placed inside a piston, the upper position of which was measured by a laser, see Figure 3.11. A photograph of the piston has been shown in Figure 3.3. We denote by ysilicone the upper position of the piston, and by ykey the position of the key.

proﬁle view

metallic plate outer tube inner cylinder silicone force sensor

key

top view

metallic plate

laser beam

Figure 3.11 – Scheme of the piston used to press on the silicone.

Chapter 6 shows that this solution led to good results. Yet, it has some drawbacks: • the force exerted by the pianist had to be symmetrical enough so that the device would not topple over, which in practise also limited the maximum applied force; • a gap between the outer tube and the inner cylinder was required so that they could slide; this gap introduced a bias in the measurement of the upper face’s position of the silicon.

42 ihterslso Srn ta. 98 o h eaiu ftefin the significantl of not is behaviour pianist the the for by felt 1998] is al., what et that considered [Serina we of results the with eemaue.Tevle of values The measured. were where yia euto uha piiaini ie nFgr .2 I 3.12. Figure in given is optimisation an such of result typical A h au ftelna lsi tfns of stiffness elastic linear the of value The h oiino h pe aeo h piston the of face upper the of position The ot nTbe3.1. Table in forth iue31.Teaihei enadsadr eito fthe of deviation standard and mean Identifications arithmetic The forces. 3.13. maximum Figure varying for completed were ments iceace,telna iceatclwi emdaccept about deemed is is law viscoelastic visc linear identified the and discrepancies, measured the between correlation the if Even eana ls spsil otemaueet ftepaoaction piano the of measurements the to possible as close as remain ob iceatc h aaeeso h a eatccoefficien (elastic law the of parameters The silicone viscoelastic. the be of to properties viscoelastic the of Identification etn h nta oiinto position initial the Setting piiainpolmcnitdi finding in consisted problem optimisation sdsrbdi 3.1.3. in described as b 4 eedtrie sn h bv-etoe itno h lme ke clamped the on piston above-mentioned the using determined were ) h ro sdfie ytertoo h nerlo h differe the of integral the of ratio the by defined is error The h nerlo h esrdforces measured the of integral the t

Force (N) i 15 iue3.12 Figure sthe is %. 0 1 2 3 4 5 . . . . . 1.2 1.0 0.8 0.6 0.4 0.2 0 i t esrmn ieand time measurement -th icu coef. viscous lsi coef. elastic es qaefit square least measurements – yia to h iervsolsi eaiu ftepiston. the of behaviour viscoelastic linear the of fit Typical J al 3.1 Table ( a , 0 a b b = ) mada h nta eoiywas velocity initial the as and mm a a ¯ and i X i ¯ b = = = – N 0 = 6.5 envalue mean umr fmeasurements. of Summary b 134 F ( eete dnie sn es qaemethod. square least a using identified then were × t i N 10 ) ( 6.5 t y a − N · 3 silicone , s N h nltm.Tevelocity The time. final the b ( · × y a m ) · 10 m hc iiie h function the minimised which − 1 ( − swl sterato force reaction the as well as 3 t 1 c fteaslt au ftetofre,and forces, two the of value absolute the of nce i N + ) · σ tnaddeviation standard m σ b ( ( b − a ˙ y = ) befrteslcna h error the as silicon the for able 1 = ) ( t si esnbeaccordance reasonable a in is ..ATAINO H KEY THE OF ACTUATION 3.3. i 92 )) 702 eatcfre hw some shows forces oelastic hne yteaddition the by changed y t h iioewsassumed was silicone The N 2 0 ausof values a N · m s n icu coefficient viscous and · · . m oa,tnmeasure- ten total, n · eut r lte in plotted are results m s − e i.Therefore, tip. ger − − 1 1 1 h es square least the , ˙ y a a calculated was ,i re to order in y, oiin(mm) Position and F ftekey the of b r set are (3.4) 43 4

35 ms CHAPTER 3. EXPERIMENTAL SET-UP of the silicon since the ﬁnger and the silicone ﬁlter the force in a similar way, which is necessary for the simulations of the piano action to be relevant. ] 1 ) − 1 270 −

b m ·

m 8000 b 240 s · b b b · average b 210 b b b b (N

6000 b 180 [N

a b b 150 b average b b b 4000 b 120 b b 90 values of a b 2000 60 values of b 30

Elastic coef. 0 0 Viscous coef. 0 2.5 5.0 7.5 10.0 12.5 Force (N) Figure 3.13 – Ten identiﬁcations of the linear viscoelastic parameters of the piston.

Examples of measurements Figure 3.14 shows a comparison between the measured force Fmeas and the estimated normal compression force of the silicone, which should be equal. The discrepancies were mainly due to: • the friction in the piston which invalidates the model;

• the bias of the measurement of ysilicone (see table 3.2); • the differences between the position sensors and the force sensors (and their ampli- ﬁer).

3.5 a ∆x + b ∆x˙ 3.0 2.5 a ∆x 2.0 measured force 1.5 Force (N) 1.0 0.5 0 Time (s) 0 0.1 0.2 0.3 0.4

14 12 a ∆x + b ∆x˙ 10 a ∆x 8 measured force 6

Force (N) 4 2 0 Time (s) 0 0.1 0.2 Figure 3.14 – Comparison of the force measurement and its estimation using the viscoelastic piston. Top: piano keystroke. Bottom: forte keystroke.

44 htgaho h opeestu a enpeetdi iue3.1 Figure in presented been has set-up complete the of photograph A h ycrnsto ewe l h esrmnswsdn sn a using done was measurements the all between synchronisation The l h siae esrmnserr n netite are uncertainties and errors measurements estimated the All atrsadtepso,a ela h oa esr sas give also is sh sensors key local piano the as the well of as photograph piston, A the and 3.15. patterns Figure in described is scheme ti oeotyta hs eaieysalerr npsto ca position in by previously errors as small denoting example, relatively For these force. that noteworthy is It nepee tadtigrdteracquisition. their triggered F d and acquisition it data interpreted. interpreted 6211 natively Instrument National was an the camera by camera the transmitted the both of for acquisition signal the start triggering to sent was edge rising iue3.16 Figure acquisition and Trigger 3.4. ..Smayo esrmn rosand errors measurement of Summary 3.5. piston the and sensor force the lasers, two the accelerometer, acquisition video uncertainties – htgah ftepaoato hwn h rcigpatter tracking the showing action piano the of Photography camera iue3.15 Figure trigger – ceeo h set-up. the of Scheme sensors y key and y silicone ..TIGRADACQUISITION AND TRIGGER 3.4. . ahrdi al 3.2. Table in gathered interface rtelclsnos twas it sensors, local the or h inlo oiino the of position of signal the nFgr 3.16. Figure in n vc oacmue,which computer, a to evice edt ag rosin errors large to lead n h oa esr.The sensors. local the d h corresponding The . wn h tracking the owing rge.A+ A trigger. acquisition sensors s the ns, 5 45 V

34 ms CHAPTER 3. EXPERIMENTAL SET-UP

geometry laser key laser silicone velocity force kinematics (m) (m) (m) (m/s) (N) (rad)

resolution 10−4 5 × 10−5 5 × 10−5 – 5 × 10−4 ¶ 0.02 systematic error 0 2 × 10−4 1 × 10−4 – 0 0 + linearity 0 10−4 10−4 – ¶ 0.1 0 + hysteresis – ¶ 0.1 0

uncertainties 10−3 ¶ 10−4 ¶ 5 × 10−4 – 0.1 ¶ 0.05

Table 3.2 – Summary of all measurement’s errors. key and of the silicone respectively, and k the stiffness of the silicone,

a (1.2 ysilicone − ykey) − a (ysilicone − ykey)= 0.2 a ysilicone (3.5) which is of the order of magnitude of 1 N. Such an amplification of the errors can be seen in Figure 3.14, where the difference between the estimated and the measured forces reaches up to 1 N for the piano keystroke and three times more for the forte keystroke. This is believed to be responsible for a significant part of the discrepancies between the measured and simulated forces presented in Chapter 6. Estimating the errors and uncertainties separately for each body allows to refine them, see Table 3.3. This is because even if the accuracies are about the same for the tracking of each pattern, the corresponding accuracies in angles differ, as the lever arms changes from one body to another.

key whippen jack lever hammer damper (rad) (rad) (rad) (rad) (rad) (rad)

resolution 10−3 5 × 10−3 10−2 2 × 10−2 2 × 10−3 10−2

uncertainty 5 × 10−3 10−2 3 × 10−2 4 × 10−2 10−2 2 × 10−2

Table 3.3 – Details of kinematic errors.

46 Contents hpe 2). Chapter Overview h e n h ne rmapsto-rvnsmlto i a (in simulation position-driven a from ﬁnger the Chapte and in key validated the is model This described. is simulations 4 ..Dnmc:dsrpinadvle ...... values . and . description . Dynamics: . 4.5. . . . . parameters of Classiﬁcation 4.4. ..Pyia lmns...... elements . Physical . 4.3...... Generalities action 4.2. piano the of Characteristics 4.1. ...Ctgre ...... Categories 4.4.1. ...Culn pig ...... 4 ...... springs . . . . . Coupling ...... 4.3.6...... friction ...... Coulomb . . . . . hinges . . in . . . . . 4.3.5. friction ...... dry . . and . . . . Viscous ...... 4.3.4...... forces . reaction . . . . . Felt ...... 4.3.3...... Inertia ...... 4.3.2...... forces . . . External ...... 4.3.1...... description . . Contacts . . . . . 4.2.4. . . . . stiffnesses . . Bodies’ . . . 4.2.3. . . . . Bodies . 4.2.2. . Bidimensionality 4.2.1. ...Gon 66 . . 65 ...... 64 . . . . 6 ...... 62 ...... 61 ...... 60 ...... Ground ...... 4.5.8...... Damper ...... 4.5.7...... Hammer ...... 4.5.6...... Lever . . . . . 4.5.5...... Jack . . . . 4.5.4...... Whippen . . . 4.5.3. . . . . Key . . 4.5.2. . Conventions 4.5.1. Model hpe 4 Chapter nti hpe,tednmclmdlo h in cinused action piano the of model dynamical the chapter, this In si rdcstefrebetween force the predicts it as 6 r crac ihcnlsosof conclusions with ccordance 48 ...... 57 . . . . . 56 . . . . . 53 . . . . . 49 . . . . 52 . . . 49 . . . 52 . . . 51 . . . 52 . . 49 . . 55 . 54 . o the for 56 52 57 3 9 47

33 ms CHAPTER 4. MODEL

4.5.9. Initialpositionsoftheframes ...... 67 4.5.10. Viscoelastic and elastic behaviours ...... 67 4.6. Matrixformulationofthedynamics ...... 68 4.6.1. Equation ...... 68 4.6.2. Detailsoftheterms...... 68 4.6.3. Exampleforthekey ...... 72

A model of the grand piano action is qualiﬁed as complete when each body of the mechanism is modelled, except the ones for which the actions are neglected. Such models stem from a non-phenomenological approach consisting in writing the dynamical equations for every body and coupling them with the reaction forces. Several complete models of the grand piano action have been proposed [Gillespie and Cutkosky, 1992; Hirschkorn et al., 2006; Lozada, 2007]. In [Gillespie and Cutkosky, 1992], no viscous nor dry friction is taken into account. Nevertheless, it is mentioned that friction contributes signiﬁcantly to what is felt by a player. The models proposed in [Hirschkorn et al., 2006] and [Lozada, 2007] are quite similar. They differ mostly by slightly different felt laws, the damper which is considered by Lozada, and by the regularisation of dry friction in [Hirschkorn et al., 2006]. The model described in the present chapter is based on Lozada’s. It is completed by a description of the contact geometries, including that of the backcheck. After introducing the model, all the physical elements involved in the model are presented and detailed for each body. The corresponding values are also given. Most of them were taken from [Lozada, 2007] as described in Chapter 3. All the equations are then written in a synthetic matrix form.

4.1. Characteristics of the piano action The parameters of the piano action model are classified into: • the geometry (positions and lengths); • the inertial properties of the bodies (masses, moments of inertia, centres of inertia); • the material properties of felts and hinges (dry and viscous friction coefficients). All the geometrical quantities (positions and angles) were estimated from two photographs of the action: one of the entire mechanism and the second zoomed on the whippen-jack- lever assembly and the hammer. The photographs were taken at 3 m from the action to reduce the parallax, and the lens distortion was corrected using DxO Optics Pro. Because of the time-worn felts of the used action (see Figure 4.7(b) page 57) and as a double-check, complementary measurements were completed with a dial calliper. The values of the measurements are given in Section 4.5. Some geometrical measurements differed significantly from Lozada’s. In [Lozada, 2007], • the let-off button is 1 mm lower;

• the drop screw was also 2.5 mm lower;

• the whippen spoon was 2 mm closer to the whippen pivot;

48 where osuywehrteflxblt ftesxbde hudb ae into taken be should bodies six the of flexibility the whether study To speiul xlie,tecmlt oe nldstedescr the includes model complete the explained, previously As steae fishnejit r l aall h in action piano the parallel, all are joints hinge its of axes the As h nriladmtra rpriswr ae rm[oaa 20 [Lozada, from taken were properties material and inertial The h emtia netit a siae as estimated was uncertainty geometrical The h ups st siaeteodro antd fteecriteri these of magnitude of order the estimate to is purpose The hs iceace,wihmyse ml oprdt h over the to compared small seem may which discrepancies, These ...Bde’stiffnesses Bodies’ d 4.2.3. the and hammer the lever, the jack, the whippen, the key, the is ...Bodies 4.2.2. bidimensional. therefore is model studied ...Bidimensionality 4.2.1. Generalities 4.2. studied the of because 2007] [Lozada, in measured that than lower h e hc svr ayt iasml.Telte a estima was latter The disassemble. to easy very is which key the fteflssrae,seTbe3.2. Table see surfaces, felts the of iue41rpeet h oiswt h supinta hya they that assumption the with deflection bodies mum the represents 4.1 Figure criterion Static Eu con in material, the isotropic Also, elastic, a linear a sections. bodies as cross the behave to constant supposed assumptions: with strong beams with straight clamped and geometries simplified to rbbymsl u otedfeecsbtentetosuidact studied two measurements. the of between uncertainty the differences to the extent the to and due behaviour mostly dynamical probably the significantly change may action, n w rtrawr considered: were criteria two ing etoswr rectangles. were sections 12 1 h b • • • • 3 h hrceitctm fteitrcinwt h key. c the mode, with normal interaction the the of of period time longest characteristic the the on related criterion dynamic a compa body rigid each of deflection length; the teristic to related criterion static a h asa ce was screw capstan the h hpe uho was cushion whippen the ( E b steYugmdlsadtescn oeto area of moment second the and modulus Young the is and h δ r epcieytewdhadtehih ftergdbd)a ft if as body) rigid the of height the and width the respectively are sgvnby given is 1 mhigher. mm 3 mlower; mm δ = 3 L F ± EI 1 3 m anybcueo h irregularities the of because mainly mm, a ni-ln oin The motion. in-plane an has e-enul kinematics. ler-Bernoulli .Telte eeapplied were latter The a. e as ted pino vr oy that body, every of iption amper. 7,ecp h asof mass the except 07], ecapd h maxi- The clamped. re I saeet hyare They escapement. cin r different. are actions sspoe qa to equal supposed is l ieso fthe of dimension all osadt lesser a to and ions tttv aeilis material stitutive ..GENERALITIES 4.2. con,tefollow- the account, 83 espoe obe to supposed re e oischarac- its to red ,wihis which g, mae to ompared (4.1) 20 49 he %

32 ms CHAPTER 4. MODEL

Figure 4.1 – Approximation of each rigid body as a clamped beam.

Dynamic criterion The modal frequencies are again roughly estimated from a clamped position. The mode of longest period is the ﬁrst mode, of frequency

1.8752 1 EI f1 = (4.2) 2π L2 sρS

where S = b h is the area of the section.

Results Table 4.1 gathers the results of Equations (4.1) and (4.2) for a typical force of 10 N on the key. The characteristic force varies from one body to another because of the different lever arms. τ is the characteristic time during a stroke, estimated at τ = 5 × 10−2 s. The Young modulus of the wood was taken as E = 1010 Pa, its density as ρ = 1000 kg/m3.

δ 1 body L (m) b (m) h (m) F (N) (-) (-) L f1 τ key 0.380 0.015 0.025 10 0.002 0.230 whippen 0.060 0.010 0.008 20 0.005 0.020 jack 0.025 0.005 0.005 30 0.012 0.004 lever 0.030 0.005 0.005 10 0.008 0.007 hammer 0.130 0.005 0.005 30 0.300 0.130 damper 0.070 0.010 0.02 10 0.0002 0.010

Table 4.1 – Order of magnitude of the inﬂuence of ﬂexibility in statics and dynamics.

It appears that most of the bodies can be considered as rigid. The vibration of the key might have an influence when in transitory motion. It is assumed that this influence is sufficiently low in our study. Also, the hammer shank has a low stiffness and its first mode has a long period. This period is even greater when estimated from a punctual mass model, which is likely to model the hammer better, as it consists in reality of an elongated shank and a mass made of felt. This flexibility has been studied in [Askenfelt and Jansson, 1991] and taken into account in [Izadbakhsh et al., 2008] where Izadbakhsh concludes from force-driven simulations that the only observed significant influence of the flexibility of the hammer shank occurs during the contact with the string. As the most interesting phase for haptics ends at the escapement and because the hammer has already escaped from the mechanism when it hits the string, the flexibility of the hammer is not taken into account here.

50 h aaersto ftecnatgoere sdsrbdi 4.5 in described is geometries contact the of parametrisation The h opeemdlhst aeit con h olsosbtendi between collisions the account into take to has model complete The h ocuino hsprgahi htfrteaoeraos l th all reasons, above the for that is paragraph this of conclusion The ifrn emtisfrec otc.Asmlrdsrpino t of description similar 2006]. A comple al., et more [Hirschkorn contact. slightly in is each knuckle, for hammer geometries the t different lifts Also, which matters. jack, t surface the so contact of the friction, of Coulomb direction prec involves tangential (more it the contact because key-hammer differs The contact) hammer contact. ra the large of very of length possibly acteristic circle, one and ke segment was one description using this ideally 5), Chapter (see reasons computational For bodie zone rigid contact between possible distances a estimate to requires This action. ...Cnat description Contacts 4.2.4. bodies. rigid by 1 h otc oei agrta on eas ftemultipl the of because point a than larger is zone contact The contact name KS KS Wkywhippen key KW Ghme ruda string at ground hammer HG Gdme ruda rest at ground damper DG Hkyhammer key KH Dkydamper key KD Hlvrhammer lever LH Glvrgon tstop at ground lever LG JH Gjc ruda button at ground jack lever whippen JG KL Jwipnjack whippen KJ Ljc lever jack JL − + oyA body akhammer jack e ruda e rest key at ground key front key at ground key al 4.2 Table 1 oyB body h emtyo otc ufcsteeoehst edescribed. be to has therefore surfaces contact of geometry The . – ecito ftecnatgeometries. contact the of Description otc geometry contact ere ffreedom. of degrees e fbd A body of egoer ftetppart top the of geometry he ,i h egbuho of neighbourhood the in s, isrltvl otechar- the to relatively dius .1. egoer a done was geometry he ta ipea possible, as simple as pt a h ecito in description the hat oisaemodelled are bodies e .Tbe42lssthe lists 4.2 Table x. fbd B body of geometry contact frn oiso the of bodies fferent sl h backcheck- the isely ..GENERALITIES 4.2. 51

31 ms CHAPTER 4. MODEL

4.3. Physical elements

4.3.1. External forces The external forces provide energy to the piano action. Two types of such forces are considered: • the action of the player on the key, which is the opposite of the reaction force exerted by the key on the player’s ﬁnger; • gravity, which applies to all the bodies.

4.3.2. Inertia From the Latin word iners (literally "without skill"), inertia describes the resistance to a change of the motion. It characterises the dynamics, as opposed to static of quasi-static mechanics where inertia can be neglected. The inertias considered in the further-described model are the rotational inertias of the rigid bodies.

4.3.3. Felt reaction forces The reaction contact forces are smoothed by felts in the thirteen zones of the piano action where contact can occur, as represented in Figure 4.2. These elements are essential for the touch of the piano.

Figure 4.2 – Location of the felts of the grand piano action.

The mechanical behaviour of such felts has been extensively studied, especially that of the hammer which collides with the string. It is highly non-linear and slightly dissipative. The non-linearity is due to the increasing stiffness of felts as their ﬁbres get closer from each other. An elastic model for the hammer was proposed in [Hall and Askenfelt, 1988] as a power law of exponent 2.8 ± 0.6 SI. An additional viscous term was added in [Hunt and Crossley, 1975], such that F(δ)= k δr + b δr δ˙ (4.3) where δ is the compression of the felt and F(δ) the normal component of the corresponding reaction force. A similar modiﬁcation of the power law was proposed for the piano in [Brenon, 2002] in the form F(δ)= k δr + b δ2 δ˙ (4.4) The latter law (4.4) is the one used in the present model and is referred to as the felt law. It is noteworthy that in Equation (4.3), the exponent r is chosen for the dissipative part so that it enables an analytical integration of the equation of motion.

52 seApni ) eoigby Denoting A). Appendix (see hc sstvle in set-valued is which h igso h ifrn in cineeet aefl bush felt have elements action piano different the of hinges The ut iial oteCuobfito ecie ntefollowing the in described friction Coulomb the to similarly Quite ...Vsosaddyfito nhinges in friction dry and Viscous 4.3.4. oelbiatt h amrhnei h amrsosbfr three before stops hammer tech the piano if by used hinge even hammer is the phenomenon to corresp As This lubricant not some does energy. time. equilibrium potential a final long the of the a and of oscillations for damped, free properly highly the hammer, is working dismounted action and any silent on smooth, observed easily be to them enable ehnclsne for sense) mechanical ispto smdle by: modelled is dissipation nhne rsnssnuaiiswe h eoiyconverges velocity the when singularities presents hinges in ipaeet.I swrhntn htti -O rbe required 59). problem (page 1-DOF function this sign that the noting worth is It displacements. ocae ntesg ucindfie by: defined function sign the in concealed n[oaa 07,ti oe aearltv ro ftehme d hammer of the values of error optimal relative the a for gave model this 2007], [Lozada, In • • oainllna icu rcinwihcetsammn ftef the of moment a creates which friction viscous linear rotational a oainldyfito fteform the of friction dry rotational a alddyfito coefficient. friction dry called c v sapstv osattr aldvsosfito coefficient; friction viscous called term constant positive a is iue4.3 Figure 0 θ ˙ hsi ossetwt h nnt fpsil qiirus(in equilibriums possible of infinity the with consistent is This . = c 0 v al . nmrtstetredfeetcsso r friction dry of cases different three the enumerates 4.3 Table . – and sign htgaho h etbsigo h jack. the on bushing felt the of Photograph θ h nl aeb h osdrdrgdbd,ti energy this body, rigid considered the by made angle the ( c θ ˙ d = ) on ymnmsn h ulda itneo the on distance Euclidean the minimising by found              [ c d − − sign ,1 1, 1 1 ] ( θ ˙ ) if if if where , > θ θ < θ ˙ ˙ ˙ = 0 0 0 c d to ns(e iue43 which 4.3) Figure (see ings sapstv osatterm constant positive a is ..PYIA ELEMENTS PHYSICAL 4.3. 0 hs iglrte are singularities These . aarp,dyfriction dry paragraph, pca ramn of treatment special a slcmn of isplacement n oteminimum the to ond oyo h piano the of body iin h add who nicians orm pseudo-periods c v tcnbe can it θ ˙ where , 0.27 (4.5) the 53 %

30 ms CHAPTER 4. MODEL

slip stick slip

external moment M M > cd M ∈ [−cd, cd] M < −cd M M M D D D scheme + + +

moment of dry friction D = −c D = −MD = c D d d

sign of θ˙ θ˙ > 0 θ˙ = 0 θ˙ < 0

Table 4.3 – The three cases of dry friction (M: external moment, D: dry friction). in hinges summed up as:

1 if θ˙ > 0  ˙ ˙ ˙ D(θ)= D(θ) ez = −cd ez  [−1,1] if θ = 0 (4.6)   −1 if θ˙ < 0    The notation = is a misuse (∈ is more appropriate), justiﬁed further.

4.3.5. Coulomb friction

Coulomb friction is a contact model very similar to the above-described dry friction in hinges. In the Coulomb model (1781), the stick-slip threshold of tangential contact force varies linearly with the normal contact force, contrary to dry friction in hinges where the threshold cd is constant. Coulomb friction is considered in the model of the piano key in three zones corresponding to the jack - let-off button contact, the jack - hammer knuckle contact and the hammer - backcheck contact. These three zones are represented in Figure 4.4.

Figure 4.4 – Zones where Coulomb friction is considered.

54 were where hyaemdle slna lsi oainlsrnso oetof moment of springs rotational elastic linear as modelled are They al 4.4 Table h olm rcinlwcnb omltdsmlryt (4.6): to similarly formulated be can law friction Coulomb The al . ie h a o oi ujc oisweight its to subject solid a for law the gives 4.4 Table force). e iue45 h ehns tde eehssc springs. such has here studied mechanism The 4.5. Figure see between: springs rotational have actions piano grand Some springs Coupling 4.3.6. scheme inof sign agnilfito force friction tangential force external • • • θ h ee n h whippen, the and lever the whippen, the and jack the ground, the and whippen the 0 T steeulbimageo h noddsrn and spring unloaded the of angle equilibrium the is and x ˙ – h he ae fCuobfito ( friction Coulomb of cases three The N r h agniladnra otc oc,respectively. force, contact normal and tangential the are iue4.5 Figure T = T T F – m − = x ˙ µ > µ slip oaino h opigsprings. coupling the of Location − > − mg N C mg µ gF mg s ˙ 0              gT mg = F [ κ − ( − ,1 1, θ 1 x 1 − ] θ F 0 ∈ xenlforce, external : ) if if if [ T x x x ˙ ˙ ˙ − m − > < = x stick µ = mg = mg − mg 0 0 0 − mg ˙ 0 . κ T F , F µ sisrttoa stiffness. rotational its is mg ..PYIA ELEMENTS PHYSICAL 4.3. ] x reaction T agnilfriction tangential : F F m < x = C slip − < − mg s µ : T µ mg mg 0 mg (4.8) (4.7) x 55

29 ms CHAPTER 4. MODEL

4.4. Classification of parameters The different physical elements described in Section 4.3, as well as the geometry, introduce numerous parameters. Their values are presented in Section 4.5 using a classification defined here and which will be of further use in the sensitivity analysis (see Chapter 6). The definition of the categories and the classification of the parameters was done a priori.

4.4.1. Categories Four categories are deﬁned: I: adjustable parameters; II: non-adjustable and measurable parameters to be studied; III: non-adjustable and non-measurable parameters to be studied; IV: non-adjustable parameters not be to studied.

Adjustable parameters: category I The adjustable parameters are these which are modiﬁed during a normal regulation of a piano key, see Figure 4.6. A classical regulation procedure is described in Appendix A.

Figure 4.6 – Adjustable geometrical parameters, on a real grand piano action.

Non-adjustable measurable parameters to be studied: category II The non-adjustable measurable parameters to be studied are the parameters for which it is possible to get an estimation by means of measurements and which are considered to potentially influence the simulation results of the model. All the measurable parameters suspected to have a significant influence are gathered in category II. For instance, an error of a tenth of a millimetre in the estimation of the position of the hammer knuckle could significantly change the simulation results, as it is a sensitive part of the action. Therefore the position of the hammer roller is a parameter of category II.

Non-adjustable non-measurable parameters to be studied: category III The category of the non-adjustable and non-measurable parameters to be studied includes the few parameters which are not measurable, but may have a signiﬁcant effect on the simulation results. These are essentially2 the parameters introduced by the description of the contact geometry.

2Some more parameters of category III are introduced in Chapter 5: the masses of the felts which are in motion.

56 egto h e ssalcmae otemmn xre yteplaye the by exerted moment the to compared small is key the of weight h nlso h rm r eoe by denoted are frame the of angles The eeec rm satce oec ii oy sn h follow the using body, rigid each to attached is frame reference A below. presented are paper this in chosen conventions The l h auspeetdhr aebe dnie sdescribed as identified been have here presented values the All h aeoyI nldsalteprmtr hc r assumed are which parameters the all includes IV category The oiswr ae fe hi rtlte.Bi ie eo sa e an as parameters below the given for is conventions 4.5. Table B the in All summarised letter. letter. first appropriate the their by after named were parameters Bodies the of name the for Conventions ovnin o h frames the for Conventions Conventions 4.5.1. ..Dnmc:dsrpinadvalues and description Dynamics: 4.5. suppose is key the becau of mechanism, key gravity the of of behaviour centre overall the the on of influence little position the example, For results. simulation the of influence significant any have ie aaeesntt esuid aeoyIV category studied: be to not parameters Fixed hp fteedo h culjc sntads.Aohreapei i descriptions is 2006]. contact example al., simplified et Another the [Hirschkorn of in disk. positions a irregu me estimated These not not the 4.7. is is in Figure jack see disk actual surfaces, this felts the of of of diameter regularities end the the but of disk, conta a shape in gets by which approximated jack the is of button end the of geometry the instance, For • • h oiotlai sdfie ln andrcino h ii body rigid the of direction main a the along body defined rigid is the horizontal. of axis rotation horizontal of the centre the is frame to; attached each of centre the iue4.7 Figure a akend Jack (a) – xmlso o-esrbeparameters. non-measurable of Examples b reuaiyo et dme e cushion) key (damper felts of Irregularity (b) θ B hr tnsfrtefis aeo h body. the of name first the for stands B where , ..DNMC:DSRPINADVALUES AND DESCRIPTION DYNAMICS: 4.5. aiisipislreinaccuracies large implies larities nCatr3. Chapter in hswsas observed also was This . o oerao,ntto not reason, some for , n conventions: ing ftepaoato are action piano the of twt h escapement the with ct etemmn fthe of moment the se ape elc it replace xample: dcdb h ir- the by nduced ’ finger. r’s srbea the as asurable ohv very have to d rm is frame ls to close 57

28 ms CHAPTER 4. MODEL

generic term meaning

BG centre of gravity of body B

BO centre of rotation of body B

BΠx x-coordinate of point BΠ in the frame of body B

BΠy y-coordinate of point BΠ in the frame of body B

BC centre of the contact zone of body B and body C in the frame of body B

IB,BΠ moment of inertia of the body B around the point BΠ

cdB, cvB dry and viscous coefﬁcient of the hinge joint of body B

mB mass of body B

BCρ, BCλ, BCθ description of the geometry of body B which collides with body C in the frame of body B, see Figure 4.8

νBC coefﬁcient of Coulomb friction between bodies B and C

κBC stiffness of coupling spring between bodies B and C BC θ0 resting angle of coupling spring between bodies B and C

Table 4.5 – Conventions for the notations of the parameters.

For convenience, each contact geometry is deﬁned with respect to the centre of the contact zone P, deﬁned by its coordinates (Px , Py ) in the body B frame. The associated parameters are represented in Figure 4.8.

y y θ P θ P ρ λ

x x (a) circle (b) segment

Figure 4.8 – Parameters for the description of the geometry for the contact zone centred in P.

• If the geometry of the body in the neighbourhood of the contact point P is described by a circle (case 4.8(a)), then the circle is defined by its radius ρ and its centre, which is itself defined by the set of polar coordinates (ρ, θ). • If it is described by a segment (case 4.8(b)), then the segment is defined by its angle θ in the body frame and its length λ.

58 h esrdgoer n h auso h aaeeswihwe which parameters the of values the and geometry measured The ain Catr6 r rsne o ahbd rmFgr .. t 4.5.2 Figure from body each for presented are 6) (Chapter lations ..DNMC:DSRPINADVALUES AND DESCRIPTION DYNAMICS: 4.5. 4.5.8. o eue o h simu- the for used re 59

27 ms CHAPTER 4. MODEL

4.5.2. Key

external forces inertia felt reactions hinge frictions Coulomb friction coupling springs

gravity yes frontrail yes hammer no player whippen back rail hammer damper

Table 4.6 – Elements involved in the key’s dynamics.

−229.5 −220.0 −183.0 −128.7 0.0 241.5 geometry KD 27.6 KH 84.2 KS− 0.0 KW 44.6 KO 0.0 KS+ 0.0

19.0 inertia K m = 0.083 kg I = 3.36 × 10−3 kg.m 2 G 14.0 K K,KO

friction cdK = 0.006 N · m cvK = 0.22 N · m · s

KD KW KG KS+ K KS− O

I KHx , KHy KWx , KWy 4 II cdK cvK 2 III 0

IV KDx , KDy KS-x , KS-y KS+x , KS+y KGx , KGy mK I K,KO 10

Figure 4.9 – Key frame, measurements (lengths in (mm)).

ρ = 2.9 ρ = 100 ρ = 100 circle KH θ = 3.51 KS− θ = 0.0 KS+ θ = 0.0

λ = 20 λ = 80 segment KD θ = 0.0 KW θ = 0.0

KD W

KS+ K KS− O I 0 II 0 III KHρ KWθ 2 IV KDλ, KDθ KHθ KS-λ, KS-θ KWλ KS+λ, KS+θ 8

Figure 4.10 – Contacts description, key (lengths in (mm), angles in (rad)).

60 ...Whippen 4.5.3. III friction inertia geometry I W III circle IV VW IV II II I I rvt e akysn ground no yes jack yes gravity xenlforces external W iue4.12 Figure L L W ρ x , W , L

W ρ θ L W W L c = = θ dW y iue4.11 Figure G O − 100 W

= 1.57 53.0 0.0 0.0 W O W 7.0 L 0.0992 al 4.7 Table O O W K K – ρ θ x inertia , W W W otcsdsrpin hpe lntsin (lengths whippen description, Contacts L K L L W O W N W

L m L θ ρ y W O · W J m = ρ =

16.1 = W , – 1.57 60 2.0 W W 19.8 lever key reactions felt – W W hpe rm,maueet lntsin (lengths measurements frame, Whippen G W W J c K θ vW K K lmnsivle ntewipnsdynamics. whippen’s the in involved Elements x J J , × W W = J J 10 5 1 0 0 L 4.93

O K O ρ θ − y

3 = = 29.8 47.5 kg × 3.44 100 W 10 J − x I , 5 ig frictions hinge W W N , W W · J O K m y

= − · ..DNMC:DSRPINADVALUES AND DESCRIPTION DYNAMICS: 4.5. s 3.97 20.8 59.0 J O W x × , 10 J O W − W y 4 J kg.m olm friction Coulomb

20.8 75.8 m W 2 (mm) I W, J O W W

O 99.1 nlsin angles , 0.0 (mm) c dW opigsprings coupling c ). vW (rad) 10 0 4 0 ). 61

26 ms CHAPTER 4. MODEL

4.5.4. Jack

external forces inertia felt reactions hinge frictions Coulomb friction coupling springs

gravity yes escapementbutton yes escapementbutton lever lever hammerknuckle hammer knuckle whippen

Table 4.8 – Elements involved in the jack’s dynamics.

−13.4 W −8.0 −3.4 0.0 26.8 geometry JW 24 JH 48.8 JL 43.7 JO 0.0 JS 0.0

0.0 inertia J m = 5.0 × 10−3 kg I = 5.0 × 10−6 kg.m2 G 0.0 J J,JO

−4 −5 friction cdJ = 9.9 × 10 N · m cvJ = 4.93 × 10 N · m · s

J H JL

J JO = JG S

I JWx 1 II JHx , JHy JSx , JSy 4 III 0

IV JWy JLx , JLy JGx ,JGy mJ IJ,JO cdJ cvJ 9

Figure 4.13 – Jack frame, measurements (lengths in (mm)).

ρ = 8 ρ = 0.5 ρ = 100 ρ = 2.4 circle JW JH JL JS θ = 1.63replacements θ = 0.92 θ = 3.27 θ = 2.07

JH JH JL JL JS JW

JS JO = JG

I 0 II JHθ 1 III JHρ JSρ, JSθ 3 IV JWρ,JWθ JLρ, JLθ 4

Figure 4.14 – Contacts description, jack (lengths in (mm), angles in (rad)).

62 ...Lever 4.5.5. III friction inertia geometry III segment IV IV II II I I rvt e rudysn jack no yes ground yes gravity forces external L L L W W W iue4.16 Figure λ x y , L L W W L L c L L θ

H H dL G W λ θ x y

= = =

0.0 0.0 iue4.15 Figure − −

L L L L 9.9 8 − 13.3 41.0 H H L L 0.40 θ λ J inertia J x y al 4.9 Table × – 10 W L m L W − otcsdsrpin ee lntsin (lengths lever description, Contacts L L J L λ H 4 S L L = L , x N

O O O W , L λ θ 10 · = J L whippen knuckle hammer jack reactions felt

θ m = = 0.0 0.0 S × L – y L L 8 0.0 H H G 10 – ee rm,maueet lntsin (lengths measurements frame, Lever L L c − L lmnsivle ntelvrsdynamics. lever’s the in involved Elements S S vL L L 3 θ θ J L L J L G J = S kg L S x

H , λ θ 4.93

L 6 1 1 0 = 35.6 = G 2.0 y I 5 1.5 L × , L 10 O m = L − L S 5 ig frictions hinge 5.0 L

N λ θ J

· × = = − I m 47.3 ..DNMC:DSRPINADVALUES AND DESCRIPTION DYNAMICS: 4.5. L 3.0 , 10 8 − L · O s 0.31 − 6 L kg.m c H dL L S 2

54.5 c 0.0 L vL olm friction Coulomb J L (mm) S 9 0 2 1 L W

nlsin angles , − − 13.0 41.0 (mm) opigsprings coupling ). (rad) ). 63

25 ms CHAPTER 4. MODEL

4.5.6. Hammer

external forces inertia felt reactions hinge frictions Coulomb friction coupling springs

gravity yes lever yes key(backcheck) no jack string key (backcheck)

Table 4.10 – Elements involved in the hammer’s dynamics.

−127.9 W −126.4 −16.3 0.0 geometry HK −21.3 HS 54.9 HJ −12.0 HO 0.0

−95.7 inertia H m = 14.24 × 10−3 kg I = 1.65 × 10−4 kg.m2 G 23.7 H H,HO

−4 −5 friction cdH = 9.9 × 10 N · m cvL = 4.93 × 10 N · m · s

HO HG HJ

I 0 H H H H m I c c II Jx , Jy Gx , Gy H H,HO dH vH 8 III 0 IV HKx , HKy HSx , HSy 6

Figure 4.17 – Hammer frame, measurements (lengths in (mm)).

ρ = 6.6 ρ = 18.0 ρ = −3.7 circle HK θ = 0.0 HS θ = −1.57 HJ θ = 1.57

I 0 II HJρ, HJθ 2 III HKρ, HKθ 2 IV HSρ, HSθ 2

Figure 4.18 – Contacts description, hammer (lengths in (mm), angles in (rad)).

64 D O D ...Damper 4.5.7. O III friction inertia geometry III circle IV IV II II I I rvt e e e no yes key yes gravity forces external D D iue4.20 Figure D S S ρ x D G S D , , S S D D D

ρ θ K S S D θ y c D D = = K dD G O iue4.19 Figure 1.57 100

= 50.0 0.0 0.0 D D 0.0 0.0992 K K al 4.11 Table ρ x , , D – D K D inertia K

K otcsdsrpin apr(egh in (lengths damper description, Contacts ρ θ D y θ N m S W = = · D m

1.57 5 − m = 4 0 0 0 22.0 43.0 D – 50.06 etreactions felt ground c aprfae esrmns(egh in (lengths measurements frame, Damper – vD I lmnsivle ntedme’ dynamics. damper’s the in involved Elements D = × , D 4.93 10 O D K − 3

× 78.1 c kg 0.0 dD 10 − 5 ig frictions hinge I N c D vD , · D m O = · s ..DNMC:DSRPINADVALUES AND DESCRIPTION DYNAMICS: 4.5. 0 4 0 4 2.04 × 10 − 4 olm friction Coulomb kg.m 2 (mm) nlsin angles , (mm) opigsprings coupling ). (rad) ). 65

24 ms CHAPTER 4. MODEL

4.5.8. Ground

−308.1 −188.5 96.9 0.0 hinge centres S S S − S OD 39.6 OW 62.06 OH 125.4 OK 0.0

− 265.1 −260.0 −183.3 −93.5 −63.5 242.0 end stops SD 7.2 SS 178.8 SK− 6.16 SL 117.1 SJ 73.52 SK+ −0.38

S OH

SJ S OW S OD S SD OK SK+ SK−

I SLy SJy SK+y 3

II SSy SOW x , SOW y SK−y SOH x , SOH y SJx 7 III 0

IV SOD x , SOD y SSx SK−x SLx SOK x , SOK y SK+x 8 D

Figure 4.21 – Ground frame, measurements (lengths in (mm)).

ρ = 100 circle SL θ = 1.57

λ = 20 λ = 50 λ = 30 λ = 10 λ = 20 segment S S S S S D θ = 0.0 S θ = 0.0 K− θ = 0.0 J θ = −0.21 K+ θ = 0.0

SD SO SK+ SK−

I 0 II 0 III SJθ 1 IV SDλ, SDθ SSλ, SSθ SKλ, SSθ SK−λ, SK−θ SLρ, SLθ SJλ SK+λ, SK+θ 11

Figure 4.22 – Contacts description, ground (lengths in (mm), angles in (rad)).

66 h aaeeso h etlw hc eeue r h nsmeas ones the are used were which laws felt the of parameters The h aaeeso h te et asare: laws felts other the of parameters The h oiin fec rm sdtrie by: determined is frame each of positions The h iceatcpr sepce ob infiatysalrt smaller significantly be to expected is part viscoelastic The h icu atwsntngetda h ai a o ugdsuf judged not was ratio the as neglected not was part viscous The ftehme ncl n ee-rudare: lever-ground and knuckle hammer the of iceatcbhvoro felts of behaviour Viscoelastic behaviours elastic and Viscoelastic 4.5.10. ...Iiilpstoso h frames the of positions Initial 4.5.9. o opeso of compression a for W nta angles initial elastic • • t nl trs,gvni iue4.23. Figure in given rest, at angle its 4.5.8; to 4.5.2 paragraphs in given origin, its h trdeeg n by and energy stored the θ θ D D ( 0 ( 0 ) iue4.23 Figure = ) − 1 0.23 mipsda ipaeetlna ntm in time in linear displacement a at imposed mm θ W ( 0 θ ) W W W – ( 0 viscous elastic nta oiin ftefae (in frames the of positions Initial = ) W                   θ viscous − L k r ( b 0.03 0 αβ αβ αβ k r b = ) H J H J H J θ = H = = Z h isptdenergy, dissipated the Z ( = = 0 = 0 0 2.7 1.6 0.1 5 ) θ 10 3 7 5 L × θ ( − x b × × ..DNMC:DSRPINADVALUES AND DESCRIPTION DYNAMICS: 4.5. 0 J 3 × 10 ( = ) x k 0 10 10 10 ) 2 7 0.40 x ˙ 9 r 7 SI 10 2 d SI SI d x SI t = θ H 22 ( 0 = ) a h lsi at Typically, part. elastic the han 0.40 (rad) cetygetrthan greater ficiently rdb oaa Those Lozada. by ured θ J θ ( ). K 0 ( = ) 0.1 0 ) 0.22 ,dntn by denoting s, θ K ( (4.10) (4.11) 0 (4.9) = ) 67 0.03 1 .

23 ms CHAPTER 4. MODEL

Elastic behaviour of rotational springs The parameters of the linear law of the coupling springs were measured in [Lozada, 2007] and they are given in Table 4.12.

coupling spring κ (N.m) θ0 (rad)

whippen-ground 0.0355 0.419 jack-whippen 0.0299 0.980 lever-whippen 0.1171 0.725

Table 4.12 – Parameters of the linear elastic law of the coupling springs.

4.6. Matrix formulation of the dynamics The last part of this chapter consists in writing the dynamics of the piano action in a single matrix equation. The multiple purposes are: • to give the equations of the dynamics; • to write them in a synthetic form which is easy to analyse and implement; • to introduce the formalism and the notations used in Chapter 5.

4.6.1. Equation The dynamical equations of the rigid bodies are written in a usual form in multibody dynamics:

∂ δ T M(x) x¨ + N(x, x˙) x˙ + c x˙ + c sign(x˙)+ (x) F(δ)+ F⋆(x, t)+ T(δ, x˙)+ κ(x − x )= 0 v d ∂ x 0 (4.12) x is the vector of generalised coordinates,Ç chosenå such that:

T R6×1 x = θK θW θJ θL θH θD ∈ (4.13)

In Equation (4.12), each termï is a generalised force, thatò is here a moment. Each of them is made explicit in 4.6.2. To be compared to the ones in [Lozada, 2007], this equation needs to be completed by the equations of the gaps δ which depends on the contacts’ description (see 4.2.4) and are purely geometrical. The main differences with [Lozada, 2007] are the contacts’ geometry and the writing of Coulomb friction. This equation contains some non-smooth terms, that is functions which are not differen- tiable with respect to velocity: sign and T (related to dry/Coulomb friction). This makes its solving more complicated, as discussed in Chapter 5.

4.6.2. Details of the terms All the physical elements listed in Section 4.3 are included in Equation (4.12). Their mathematical transcription is made explicit in the same order.

68 where where where h eeaie ocso rvt are: gravity of forces generalised The h iertrso inertia of terms linear The h eeaie xenlfre,ie h oeto h externa the of moment the i.e. forces, external generalised The ytepait respectively. pianist, the by are: F Inertia xenlforces External ⋆ ( M x , ( t x ) F F = ) : g gravity ⋆ P steaclrto ftegaiy h eeaie ocsexer forces generalised The gravity. the of acceleration the is ( t                   ) I eoe h oc xre ytepae ntekyat key the on player the by exerted force the denotes K 0 0 0 0 0 0 0 0 0 0 0 0 and , K O F I player ⋆ W F , player ⋆ W F F O gravity ⋆ ⋆ → + ( x key → W M , t O key ( = ) ( ( x W x x ) , , ( M M J x ¨ x t t 2 − ) = ) , L W J W n(.2 r ie ytegnrlsdms matrix mass generalised the by given are (4.12) in I t r h oet ftegaiyado h oc exerted force the of and gravity the of moments the are J 0 0 0 0 0 = ) , J F ( ( O                   x x gravity ⋆ + ) W ) H D                   K L J O O W O K O O O W H K O D J ..MTI OMLTO FTEDYNAMICS THE OF FORMULATION MATRIX 4.6. L O ( G G G K x G G G W ( , ( ( S ( ( ( x x x t L + x x x ) + ) 2 ) ) ) ) ) ( ∧ ∧ x ∧ I ∧ ∧ ∧ 0 0 0 0 0 L ) m , m m F m m m L ∧ O player ⋆ J K L W H D F g g g g g P g M (                   t I → J W ) · J 0                   , e J key O ( z · x ∈ e ( ) z x R ∈ , 6 t M × R ) I L W 1 K L 6 0 0 0 ocsi eoe by denoted is forces l × , P L 1 ( O . x ) e ytepianist the by ted I H 0 0 0 0 , H O I D 0 , D (4.17) (4.14) (4.15) (4.16) O M                   : 69

22 ms CHAPTER 4. MODEL

MWJ and MWL can be calculated from the Lagrangian, similarly to what is demonstrated on the example of the double pendulum detailed in Appendix B.

The other (nonlinear) terms of inertia are N(x, x˙) x˙ where

00 0 000   0 0 NWJ (x, x˙) NWL(x, x˙) 0 0      ( )  0 NJW x, x˙ 0 0 00 N(x, x˙)=   (4.18)   0 N (x, x˙) 0 0 00  LW      00 0 000       00 0 000     The only non-zero components in Equation (4.18) are those concerning bodies which are not in an inertial frame, i.e. the jack and the lever (they are linked to the whippen). NWJ , NWL, NJW and NLW are calculated using the following equation:

1 ∂ Mi j ∂ Mik ∂ Mk j [N(x, x˙) x˙]i = x˙k + − x˙j (4.19) 2 ∂ xk ∂ x j ∂ xi ! Xj,k where Mαβ denotes the component of the α-th line and β-th column of M and x˙k denotes the k-th component of x˙.

Felt reactions

In accordance with the notations of Table 4.2, denoting by δγ the compression length of the felt of the contact γ, the vector δ ∈ R14×1 gathers all the lengths of compression:

δT = δKS+ δKW δKS− δKH δKD δWJ δWL δJG δJL δJH δLG δLH δHG δDG (4.20) where the dependencies with respect to x have been omitted. These dependencies are ï ò purely geometrical.

F(δ) ∈ R14×1 is the vector of the felts reaction forces such that its i-th component is

δ ri 2 ˙ [F( )]i = ki δi + bi δi δi (4.21)

The lever arms of the felt reactions forces are gathered in

∂ δ (x) ∈ R14×6 (4.22) ∂ x and the reaction forces are given byÇ å

∂ δ T (x) F(δ) ∈ R6×1 (4.23) ∂ x

The details of the calculations areÇ given forå the key in paragraph 4.6.3.

70 h arcso icu n r on rcincoefﬁcients friction joint dry and viscous of matrices The olm rcin r nlddi h eeaie forces generalised the in included are frictions Coulomb 5. Chapter icu n r rcini hinges in friction dry and Viscous sign n ini h e-audfnto endby: deﬁned function set-valued the is sign and ignlmtie ie by: given matrices diagonal ti o ucina tasgsmr hnoesaa to scalar one than more assigns it as function a not is It olm friction Coulomb c d stecmoetws infunction: sign component-wise the is =                   c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 dK c dW 0 0 0 0 0 c 0 0 0 0 dJ sign c 0 0 0 dL sign : c dH 0 0 y : T ∈ c ( dD 0 δ R y ,                   7−→ x ˙ ∈ , = ) R ..MTI OMLTO FTEDYNAMICS THE OF FORMULATION MATRIX 4.6. 6              ×                   1 T c T − [ 1 7−→ HK v JG − 1 = T ,1 1, + + 0 0 0 KH                              T T c sign sign sign ] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 vK JH HJ                   if if if . . . ( ( ( c T y y y vW 0 0 0 0 0 6 2 1 ( 0 ) ) ) δ y y y t ramn sdsusdin discussed is treatment Its .            c , < = > v x ˙ c 0 0 0 0 ) vJ ∈ 0 0 0 hc a ewritten be can which R c 6 0 0 0 vL × 6 and c vH 0 0 c c d 0 vD ∈                   R 6 (4.24) (4.25) (4.27) (4.26) × 6 are 71

21 ms CHAPTER 4. MODEL

Each moment can be made explicit, similarly to TKH, by changing the appropriate letters:

δ 0 if ||FKH( )|| = 0 T (x, δ)=  (4.28) KH ∂ δKH  ν (x) F (δ) if ||F (δ)|| > 0  KH ∂ x KH KH   Coupling springs The actions of the two coupling springs are modelled by a generalised force of the form κ (x − x0). This moment can be detailed:

00 0 000 θK 0      WG 0 κWG 0 0 00 θW κWG θ0             0 0 0 00    JW   κJW   θJ   κJW θ0  κ(x − x )=     −   (4.29) 0       0 0 0 κ 0 0  θ   κ θ LW   LW   L   LW 0              00 0 000 θ   0     H                00 0 000 θD   0             

4.6.3. Example for the key In the present paragraph, the dynamics of the key is written explicitly, from Equation (4.12). The ﬁrst line of the matrix equation (4.12) is:

∂ δ T [M(x) x¨] + [N(x, x˙) x˙] + c x˙ + c sign(x˙) + (x) F(δ) 1 1 v 1 d 1 ∂ x " #1 ⋆ δ κ + F (x, t) + [T( , x˙)]1 + (x − x0) = 0 (4.30) 1 Ç å 1 Each term can be calculated as described above: ¨ • [M(x)x¨]1 = JKθK;

• [N(x, x˙) x˙]1 = 0, as the key is directly linked to the ground with a hinge joint; ˙ • cv x˙ 1 = cvK θK ; ˙ • cd sign(x˙) 1 = cdK signθK ; ∂ δ T • (x) F(δ) = ∂ " x #1

∂Ç δKS+ å ∂ δKW ∂ δKS− ∂ δKH (x) F(δKS+ (x))+ (x) F(δKW(x))+ (x) F(δKS− (x))+ (x) F(δKH(x)) ∂θK ∂θK ∂θK ∂θK as explained and detailed in Appendix C ;

⋆ • (F (x, t))1 =(K0KG(x) ∧ mK g) · ez +(KOKS+(x) ∧ F P (t)) · ez δ • (T( , x˙))1 = TKH made explicit in Equation (4.28);

72 • κ ( x − x 0 ) 1 = 0 . ..MTI OMLTO FTEDYNAMICS THE OF FORMULATION MATRIX 4.6. 73

20 ms

Contents Overview action. 5 ..Smltn npatc ...... practice . in . Simulating . 5.6...... simulation . the . for . XDE of . Adjustments . . 5.5...... implementation its . for . model the . of . Adjustments . 5.4. (XDE) Engine Dynamic eXtended 5.3. dynamic non-smooth a as action piano the of Formulation 5.2. ..Lmttoso h euaiigapoc ...... approach regularising the of Limitations 5.1. ...Fl as...... simulation . . . . the . . of . . repeatability . . . . the . . . . Ensuring . . . . . laws . . 5.5.2. felt . . . the . . . of . . Implementation ...... 5.5.1...... 85 ...... laws . . Felt ...... 5.4.2. . key . . the . . of . . Actuation . . . . . 5.4.1...... Results ...... 5.2.5. . . . . scheme . friction Implicit . . dry . with . 5.2.4. pendulum . . a . equat of . non-smooth Example . . the . solve . 5.2.3. to . . used . methods . Numerical . . . . 5.2.2. . . formulation Non-smooth . . 5.2.1...... system . body friction . dry . regularised with . pendulum . a of ODE Example regular a 5.1.2. into (4.12) Equation Changing 5.1.1. iuainmethods Simulation hpe 5 Chapter hscatrpeet h ehd sdt iuaetemodel the simulate to used methods the presents chapter This 76 ...... 94 ...... 91 ...... os....87 . . . . ions 78 . . multi- . al . . . . 93 . . . . . 77 . . . . . 95 . . . . 93 . . . . 88 . . . . ftepiano the of 90 . . . 85 . . 88 . 91 . 92 89 75

19 ms CHAPTER 5. SIMULATION METHODS

The purpose of this chapter is to set forth the methods which were used for the simulation of the model described in Chapter 4. This simulation requires the solving of Equation (4.12) page 68 which is recalled:

∂ δ T M(x) x¨ + N(x, x˙) x˙ + c x˙ + c sign(x˙)+ (x) F(δ)+ F⋆(x, t)+ T(δ, x˙)+ κ(x − x )= 0 v d ∂ x 0

This equation describes the piano’s dynamicsÇ butå does not yield the contact description, included in the function δ. Two terms require a special treatment as they are not functions:

• Coulomb frictions in hinges cd sign(x˙) is a vector of set-valued functions of x˙ (see Eq. (5.1));

• generalised tangential forces associated to Signorini-Coulomb laws T(δ, x˙) is a vector of set-valued functions of x˙ (see (4.7)). It implies that Equation (4.12) is not mathematically correct: its sums (single-valued) functions and set-valued functions. It can be appropriately written by: • redeﬁning the functions as singleton-valued functions instead of scalar-valued functions;

• replacing = 0 with ∋ 0. Nevertheless, the resulting equation is not an Ordinary Differential Equation and cannot be solved using numerical methods for ODEs. The dynamical equation of the piano action can be solved in two ways. It can be approximated as an ODE by regularising it. Such an approach was performed in all the piano action simulations until now. This approach has major numerical drawbacks, see Section 5.1. An illustration of these drawbacks is given on the simple example of a pendulum subject to dry friction. An other way of solving the dynamical equation is to accept the non-smoothness of the equation and use appropriate tools for non-smooth equations, see Section 5.2. This is the solution we chose. An overview of the numerical methods which were used is given. An illustration of such methods for non-smooth dynamical systems in presented for the pendulum with dry friction problem in 5.2.3. The results are compared to that of the regularisation approach. For the non-smooth simulation of the complete model, we used the computer program XDE described in Section 5.3. A few adjustments of the model had to be done (Section 5.4). Also, some additional features had to be implemented in XDE (Section 5.5).

5.1. Limitations of the regularising approach In this section, the solving complications induced by the regularisation of non-smooth terms in the dynamics of the piano action are highlighted. The example of a pendulum with dry friction in the hinge joint is given. The way these complications have been overcome are presented in Section 5.2.

76 ebleei ob ml,a 5mnwr eurdo . H Ccomp PC GHz 2.4 a on required were min 75 as small, be to it believe we Coul of regularisation smooth a is which model, Tucker and Cull a with a ae noconsideration. into taken was ne(.2 a enrglrsd ti osbet ov twt n with it solve to possible is it regularised, been has (4.12) Once a fsligEuto 41)i oapoiaetecmoet of components the approximate to is (4.12) Equation solving of way A h aclto ftehme hn deformation. shank hammer the Cu time of a calculation the the with thesis, treated master does this also 2008] In be al., 2006]. to et [Izadbakhsh, believed in [Izadbakhsh is as it model, used. but steps modelled, time is the friction of [Vyasaraya mention in no adjusted manually is was model The 1999]. Tucker, and frict the 2009], al., et Vyasarayani 2006; al., et [Hirschkorn In instabilities. euaie prxmto of approximation regularised ta. 08 oaa 07 ysryn ta. 09 o h trea the for 2009] al., et Vyasarayani 2007; [Hirschkor Lozada, approaches 2008; such al., al used et now, literature to the Up in actions (ODEs). piano Equations Differential Ordinary continuous on amraclrto n h acltdrato oc ftekyon key the of force of reaction steps time calculated osci the Increasing and w acceleration Simulink. simp hammer in lever also regularised were repetition was contacts the Friction Three easier. nor simulation. the damper simplify the to neither order 2007], [Lozada, In pia published in difficulties observed and friction of Treatment rxmto speetdi iue51b,where pos 5.1(b), One Figure in functions. presented single-valued is with proximation functions, set-valued are euaiigtennsot rcinlaws friction non-smooth the Regularising ...Cagn Equation Changing 5.1.1. sign ( θ (5.1)) Multi-valued (a) ˙ = ) iue5.1 Figure              1 sa elas well as ms [ − − ,1 1, 1 1 sign − – 1 ] 1 ( lt fmulti-valued of Plots sign θ if if if ˙ ) > θ < θ θ ˙ ˙ ˙ 0.1 = ucin(Eq. function sign 0 0 0 (4.12) s hs eaiuswr eivdt edet numerical to due be to believed were behaviours These ms. ntenihorodof neighbourhood the in (5.1) ..LMTTOSO H EUAIIGAPPROACH REGULARISING THE OF LIMITATIONS 5.1. noarglrODE regular a into θ ˙ sign n single-valued and b xml frglrsto of regularisation (5.2)) (Eq. of Example (b) sign fi b stesoeo h piecewise-linear the of slope the is ( θ ˙ = ) − 1 0 / ltoswr bevdi the in observed were llations              b . o aswr approximated were laws ion sign − fi 1 ta. 06 Izadbakhsh 2006; al., et n − 1 b ie omk h solving the make to lified mn ffito,we it when friction, of tment 1 mrclmtosbased methods umerical θ tpi o niae but indicated not is step 1 ˙ ( sign fi oato simulations action no 1 θ iiiyo uha ap- an such of sibility ˙ ie l,20] There 2009]. al., et ni / ladTce friction Tucker and ll ) r mlmne,in implemented, ere if if if b ne iegd for diverged, finger h iuain of simulations the l functions. > θ θ < θ ˙ ˙ ˙ sign m rcin[Cull friction omb o eciehow describe not ∈ [ tr including uter, 1 − − / and 1 1 b / / sign b b T 1 , θ ˙ which , / (5.2) b ] 77

18 ms CHAPTER 5. SIMULATION METHODS

Drawbacks of regularisation

The regularisation of non-smooth laws makes the solving easier, but induces several complications. Firstly, regularising friction does not ensure the convergence to a physical solution: a body subject to regularised friction has only one possible equilibrium position, which is the same as in the frictionless case (see Figure 5.4 for the example of the pendulum). For example, the minimum weight needed to initiate the motion of the key is 70 g, because of gravity and dry friction. Without dry friction or with the regularisation, it would be only of 15 g. In 2000, Stewart afﬁrmed that equations for rigid-body dynamics such as (4.12) (i.e. discontinuous ODEs) needed to be solved using the differential inclusion formalism, if high accuracy was desired [Stewart, 2000]. It is also indicated that the regularisation of non-smooth laws leads to stiff equations. This could explain the computational cost of simulations in [Izadbakhsh, 2006]. Later, more details about regularisation drawbacks are given in [Acary and Brogliato, 2008]. It is reported that regularising singularities in ODEs may impair:

• the efﬁciency, since calculating accurate stick-slip transitions leads to stiff equations which require a very small time step; • the local order of consistency, which is the error in one time step;

• the global order of accuracy (the integer n such that the error varies with the time step to the power n); • stability results.

In 5.1.2, these observations and statements are veriﬁed on an elementary example involving regularised dry friction. This enables the comparison of the regularisation approach with the non-smooth methods presented in Section 5.2, where the same elementary example is studied without regularising the contact law.

5.1.2. Example of a pendulum with regularised dry friction

The free oscillations of a pendulum with regularised dry friction are studied here. The pendulum is studied for parameters’ values of the same order of magnitude as these of the piano action model. To make the discussion easier, the equations are made dimensionless.

The considered pendulum is a compound pendulum of mass m. Its inertia around its centre or rotation O is denoted by I and the distance between its centre of gravity and O is l, see Figure 5.2 where these notations are applied in the case of the hammer. The pendulum is subject to dry friction of coefﬁcient cd and gravity. Its coordinate θ is chosen equal to 0 at the (stable) equilibrium in the frictionless case. The equation of its dynamics is ¨ ˙ I θ + cd sign(θ)+ mgl sin(θ)= 0 (5.3) and the values of the parameters are given in Chapter 4. For example, the case of the m = m I = I l = H H c = c hammer would lead to H, H,H0 , || O G|| and d dH.

78 where with h eiaie nEuto 53 r eiaie ihrespe with derivatives are (5.3) Equation in derivatives The hsscn re ifrnileuto switna arxﬁ matrix a as written is equation differential order second This iesols iei nrdcd endas: deﬁned introduced, is time dimensionless a requivalently or tion: case, frictionless the in pendulum the of period the is ucino h iesols ie and time, dimensionless the of function a hrfr ewritten: be therefore introducing iue5.2 Figure U =    θ θ ˙ ˜ ˜    , –    ceeo h amrsuida reyoclaigpendulum. oscillating freely a as studied hammer the of Scheme θ θ ¨ ˜ ˙ ˜ A    = =       0 0 1 0 0 0 1 0 θ ¨ ˜ Υ +       , · 2    ..LMTTOSO H EUAIIGAPPROACH REGULARISING THE OF LIMITATIONS 5.1. sign F θ θ ˙ ˜ ˜ U = Υ ˙ T (    U = θ = + G ( = ) θ ˙ ˜ τ U A 2 2    + ) π π = − θ    ˙ ˜ s

− Υ + 4 T stedrvtv of derivative the is t l Υ 2 π mgl F mgl c sign ( 2 2 I d U sign sin ) ( ( θ ˙ ˜ θ ( ˜ ) θ U ˜ 0 = ) − 2 steageo h edlmas pendulum the of angle the is ) 0 4 − tt h hscltime physical the to ct 0 π 4 2 π sin s re ifrnilequa- differential order rst 2 sin ( θ θ ˜ ˜ ) qain(.)can (5.3) Equation . (    U 1 )    t Here, . (5.10) (5.8) (5.7) (5.5) (5.4) (5.9) (5.6) 79

17 ms CHAPTER 5. SIMULATION METHODS

Regularisation of dry friction

For the present study, Coulomb friction is approximated by the function sign (see Equa- tion (5.11)) plotted in Figure 5.1(b):

˜ 1 1 if θ˙ > ﬁ  Ψ ˜ 1 1 sign(θ˙)=  Ψ θ˙˜ if θ˙˜ [ , ] (5.11)  ∈  Ψ Ψ  ˜ 1 −1 if θ˙ < −  Ψ   This function is the dimensionlessﬁ approximation of sign corresponding to Equation (5.2), where the parameter Ψ has been introduced:

b Ψ= (5.12) T This parameter somehow measures the degree of the regularisation approximation.

The function sign converges pointwise to a discontinuous (single-valued) function equal to 0 in 0 and |x|/x elsewhere. Numerically, this limit can be identiﬁed to the sign function because θ˙ will never exactly equal 0. This means that the numerical simulation of the regularised modelﬁ converges to that of the original model, when Ψ → ∞. Obviously, increasing Ψ increases the stiffness of the numerical solving as it is going to be proved for the present example.

Linearisation of F at U = T [0 0] for the study of stability and stiffness

To make the study of numerical stability and conditioning easier and analytical, the dynamics of the pendulum is studied for small angles and small velocities. The linearisation of F yields an equation which can be discretised with an implicit scheme, and yet solved without using any root-ﬁnding algorithms such as Newton’s. Indeed, applying an implicit scheme after the linearisation of F is equivalent to applying a linearly-implicit one. It is assumed that the results for the linearised case can be extrapolated to the non-linear case.

0 0 F(U)=   ∼   (5.13) 2 2 2 2 −Υ sign(U2) − 4π sin(U1) −Υ Ψ U2 − 4π U1   0         0     ˜ ˜ because sin(θ˜) ∼ θ˜ and sign(θ˙) ∼ Ψ θ˙. 0 0 Equation (5.9) becomes:

0 1 ˙ U(t)= B · U(t) with B =   (5.14) −4π2 −Υ2 Ψ    

80 51) hyaegtee nTbe51 The 5.1. Table in gathered are They (5.15). he iesols aaeeshv enitoue nEquati in introduced been have parameters dimensionless Three hsparameter This w lmnayoelvltm-iceiain r tde:ex studied: are time-discretisations one-level elementary Two ucinof function a feto rvt.Tevleof value The gravity. of effect umr ftetredmninesparameters dimensionless three the of Summary unit. time physical one simulate to needed are calculations otenmrcloe larger A one. numerical the to Υ ue ehd.Sc iceiain nrdc ueia par numerical a introduce step discretisations Such methods. Euler iese ftetime-discretisation the of step Time nCatr4 h akadtelvrhv infinite have lever the and jack The 4. Chapter in Ψ rotation. of centres their to equal supposed efficiency. prxmto:tehigher the approximation: T stepro ftefitols pendulum: frictionless the of period the is notation a nrdcdfrterglrsto fCuobfito.I gives It friction. Coulomb of regularisation the for introduced was sapyia aaee hc unie o motn rcini friction important how quantifies which parameter physical a is h ee hr iesols aaee sdefined: is parameter dimensionless third a Here, . Θ Υ Ψ Υ expression 2 , al 5.1 Table π Υ body Θ Ψ

T T h stgtyrltdt h opttoa ot si eemnsh determines it as cost, computational the to related tightly is b and mgl c d Θ .11.98 3.51 key . – unie h ubro acltost iuaeoeperiod one simulate to calculations of number the quantiﬁes al 5.2 Table umr ftetredmninesparameters. dimensionless three the of Summary Ψ h etrteapproximation. the better the , Υ whippen ssonfrec oyi al . sn h ausgiven values the using 5.2 Table in body each for shown is Θ motneo rcinwt eadt gravity to regard with friction of importance nrae h cuaybtipistecomputational the impairs but accuracy the increases ..LMTTOSO H EUAIIGAPPROACH REGULARISING THE OF LIMITATIONS 5.1. – T esr fteapoiainof approximation the of measure au of Value = jack = Θ 2 ∞ ∞ π s T h Π Υ mgl lever termsae that states -theorem I o ahbody. each for Υ meaning eas hi ete fgaiyare gravity of centres their because hammer .81.26 1.68 Θ lctadlinearly-implicit and plicit mtrwihi h time the is which ameter eae h hscltime physical the relates damper n 57,(.2 and (5.12) (5.7), ons θ ˜ oprdt the to compared s sign a ewitnas written be can esr fthe of measure a wmany ow (5.15) (5.16) 81

16 ms CHAPTER 5. SIMULATION METHODS

Explicit scheme

In this paragraph, Euler’s explicit scheme is implemented and studied. The vector U˙ is discretised using with the non-dimensional time step 1/Θ:

˙ Un =Θ(Un+1 − Un) (5.17)

This explicit scheme in Equation (5.14) yields:

1 U =(1 + B) · U (5.18) n+1 Θ n

The dynamics of the pendulum can then be solved with some initial conditions U0:

1 n 1 θ0 = n N, = Θ U0   and ∀ ∈ Un  4π2 Υ2 Ψ · U0 (5.19) θ˙ 0 − 1 −     Θ Θ     

The matrix relating Un+1 to Un is denoted by C. The eigenvalues of C are the roots of

1 1 − X 2 2 Θ Υ Ψ 4π det (C − X 1)= det  2 2  =(1 − X ) 1 − − X + 2 (5.20) 4π Υ Ψ Θ ! Θ − 1 − − X  Ü Θ Θ ê   The solutions are:

Υ2 Ψ 1 X = 1 − ± Υ4Ψ2 − 16π2 if ΨΥ2 ¾ 4π  2Θ 2Θ  p  (5.21)   Υ2 Ψ i X = 1 − ± 16 π2 − Υ4Ψ2 if ΨΥ2 < 4π  2Θ 2Θ   p  Equations (5.21) show that A-stability1 requires that ΨΥ2 << Θ. If not, the spectral radius of C would be greater than 1 and the numerical scheme would be unstable, see Figure 5.3 (blue curve). Accurate stick-slip transitions require large Ψ which can lead to very large Θ, especially when dry friction is important (large Υ). The computation can therefore be very costly.

Linearly-implicit scheme

The following implicit discretisation of U˙

˙ Un+1 = Θ(Un+1 − Un) (5.22)

1See [Dahlquist, 1963; Hairer, 2010].

82 gen scluae ihtennsot ueia ehddescr method numerical non-smooth the with calculated is (green) h yaiso h edlmcnte esle ihsm nta con initial some with solved be then can pendulum the of dynamics The h ievle ftematrix the of eigenvalues The lo h odto ubrof number condition the Also, h nl lt r rsne o h xlctadtelinearly-i the and explicit the for presented are plots angle The hsi vnmr ai ftefito ssgicn (large significant is friction the if valid more even is This htteol qiiru oiini h egbuho of neighbourhood the in position equilibrium unconditi only is the scheme that (linearly-)implicit the that illustrates ota o odapoiainof approximation good a for that so n(.4 ed otefloiglnal-mlctscheme linearly-implicit following the to leads (5.14) in for a eesl acltd h aclto yields calculation The calculated. easily be can acltdwtottesalage supin ntefollowin the In assumption. angles small the without calculated h osqec fhvn euaie r friction. dry regularised having of consequence the eut n discussion and Results 2 eoigby Denoting hc stedfiiino h ieryipii ceeadl and scheme linearly-implicit the of definition the is which U n ieryipii cee il h aeresults. same the yield schemes linearly-implicit and icsino hsshm spooe n[eflad 97.H 1987]. [Deuflhard, in proposed is scheme this on discussion ehd iha nta au hsnas chosen value equiv initial is an scheme with linearly-implicit method, The method. c Newton which the equation, algebraic as non-linear a is (5.23) Equation i.e. Θ n h rtse faNwo ehdyields: method Newton a of step first The + 1 i nuhfor enough big from U n f h olwn way: following the h prxmto ftetm-eiaie h mlctEul implicit the time-derivative, the of approximation the U 0 = (    1 θ θ ˙ − 0 0 U    n Θ 1 − B Θ and 1 ) Å D D f ob invertible. be to 1 ( endhr ihteifiiynr n ie by given and norm infinity the with here defined , hc relates which cond U − U n Θ ) 1 n ∀ cond U − U + ..LMTTOSO H EUAIIGAPPROACH REGULARISING THE OF LIMITATIONS 5.1. sign ∂ n n ( n 1 U U + . D f 1 ∈ ( = + ( = ) − U ( N (large Å D n Θ 1 1 , ) 1 ) ã || − f − U Ψ ( D ( U U →∞ n ∼ Θ 1 Θ U || 1 n n = + Ψ + ∂ ∞ n B 1 U 1 + ) ,tetm tphst esal(large small be to has step time the ),     ) − Υ Ψ || f 1 − − ( 4 D Θ nb ovdwt otfidn loih such algorithm root-finding a with solved be an U U to 1 U Θ π 1 − n n · ln oapyn h rtse fteNewton the of step first the applying to alent n 2 ) = ) 2 1 U ã U = || ere, ast 52) numerically-oriented A (5.26). to eads n ( n ∞ 2 0 U Θ 1 1 r titysalrta ,which 1, than smaller strictly are Υ n (5.26): + f nlyAsal.I loimplies also It A-stable. onally + F ). 1 ( − U − a enrglrsds h implicit the so regularised been has U Υ n Θ 1 T U ) Θ 2 rshm osssi estimating in consists scheme er lt,te"xc"solution "exact" the plots, g n [ = Ψ = )     pii ue schemes, Euler mplicit 0 ,0 0, − n bdi eto 5.2. Section in ibed · ] U ditions is 0 [ ,0 0, ] U hc is which , 0 : (5.26) (5.29) (5.27) (5.28) (5.25) (5.23) (5.24) Θ 83 ).

15 ms CHAPTER 5. SIMULATION METHODS

It has been veriﬁed that for small angles3, this solution and the analytical one match perfectly. In Figure 5.3, we can see that:

• the explicit scheme may lead to an instability, even with Θ= 100;

• Θ= 100 leads to an excellent estimation of the solution when Υ= 1.

1.5 solution lin. implicit 1.0 explicit 0.5 t 0 time Angle 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 T −0.5

−1.0

−1.5 ˜ ˙˜ Figure 5.3 – Example of instability of the explicit scheme. (θ0, θ0)=(1,0) and (Θ, Ψ, Υ) = (100,10,1).

Figure 5.4 illustrates the convergence to the only possible equilibrium position for the regularised scheme: θ˜ = 0. Here, the regularisation factor Ψ is low and, in spite of a huge number of time steps (Θ= 10000), the calculated angles using the regularised approach remains unsatisfactory.

1.0 solution lin. implicit explicit

0.5

Angle t 0 time 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 T

−0.5 ˜ ˙˜ Figure 5.4 – Example of non-physical convergence due to regularisation. (θ0, θ0)=(1,0) and (Θ, Ψ, Υ)=(1000,0.1,5).

solution 1.0 lin. implicit explicit

Angle 0.5

t 0 time 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 T ˜ ˙˜ Figure 5.5 – Example of non-convergence due to too small Θ. (θ0, θ0)=(1,0) and (Θ, Ψ, Υ)=(250,10000,5) .

3For small angles, the dynamics of the pendulum can be analytically calculated.

84 ( ( h larger The htti hteigwudntb bevdwt nnlna)impli (non-linear) a with observed be not called would high chattering phenomenon show this well-know that schemes the regularised linearly-implicit observe and can explicit one 5.6, Figure In Figure result. acceptable an get to = small Θ very be must step time iue5.6 Figure eue rmFgr ..T iuaecretytefe oscilla free the of correctly duration simulate To 5.7. Figure from deduced iue5.7 Figure vnuly nodro antd fteaporaetm tpre step time appropriate the of magnitude of order an Eventually, yaishv endsrbd hyhv hnbe lutae nt on th illustrated of been laws then non-smooth have They the described. regularising been of have drawbacks dynamics the 5.1, Section In a as action piano the of Formulation 5.2. ie esfrti problem. this for less times methods, non-smooth with that 0 00,5 10000, 50, · ..FRUAINO H IN CINA O-MOHDYNAMIC NON-SMOOTH A AS ACTION PIANO THE OF FORMULATION 5.2. 4 00,5 10000, , ob oprdt h ausof values the to compared be To

Angle Angle 250 o-mohdnmclmlioysystem multibody dynamical non-smooth h siaino h nl fe v eid shl ftesoluti the of half is periods ﬁve after angle the of estimation the , − − 1.0 0.5 0.5 1.0 0.5 1.0 ) Ψ 0 0 5 . – ) – h etrtefito a sapoiae.Tecmesto is compensation The approximated. is law friction the better the , T ...... 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0 , ovrec fteipii scheme. implicit the of Convergence Θ hteigo euaie schemes. regularised of Chattering a ob hsna eea huad.I eto .,i ilb show be will it 5.2, Section In thousands. several as chosen be to has ...... 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 Υ o ahrgdbd ftepaoato,seTbe5.2. Table see action, piano the of body rigid each for = Θ 20 ssfcet h opttoa oti about is cost computational The sufﬁcient. is ( θ ˜ ( 0 θ ˜ , 0 θ , ˙ ˜ 0 θ ˙ ˜ ( = ) 0 feunyoclain.Note oscillations. -frequency ( = ) in ftepnuu o a for pendulum the of tions explicit implicit lin. solution . hw htee with even that shows 5.5 = Θ = Θ = Θ = Θ solution 1, urdfor quired ,0 1, − i ue scheme. Euler cit esml xml of example simple he 5000 1000 500 100 chattering 10 ) ) and on. and in action’s piano e LMULTIBODY AL 4 time time = Υ (Θ (Θ ohthe Both . , , Ψ 5 SYSTEM Ψ htthe that T T t t , a be can , )= Υ) )= Υ) 100 85 n

14 ms CHAPTER 5. SIMULATION METHODS a pendulum subject to dry friction. The characteristic number of time steps required for a good approximation of the motion of a pendulum, with properties similar to that of the bodies of the piano action, was about a few thousands per period for five pseudo-periods. Here, an overview of the non-smooth methods which were used for the simulation of the piano action is first given. They were implemented in the computer program XDE presented in Section 5.3. Then, such methods are applied to the previous example of the simple pendulum subject to dry friction. The aim is to quantify the efficiency of such methods compared to the regularisation approach for the piano action, without considering collisions.

5.2.1. Non-smooth formulation Equation (4.12)5 is written as a Measure Differential Inclusion (MDI) (see [Acary and Brogliato, 2008]):

M(x) dv = F⋄(x, x˙, t) dt + H(x) di  + +  v =(x˙) (5.30)   (g(x), HT (x).v+, di) ∈ K    The ﬁrst line formulates the dynamics of the piano action. F⋄ gathers all the smooth terms which are not related to contact nor unilateral constraints:

⋄ ⋆ F (x, x˙, t)= −N(x, x˙) x˙ − cv x˙ − F (x, x˙, t) − κ (x − x0) (5.31) where the felt reaction forces have been included in F⋆. dt is the Lebesgue measure, dv and di are vector-valued measures corresponding to the "accelerations" (smooth or not) and the impulses, respectively. Their components are of the form: + − dv = γ dt +(v − v ) dν + dvs (5.32) di = f dt + p dν + dis The measure di includes the reactions associated with non-smooth laws. The prefactors of dt are the smooth acceleration (γ) and the smooth forces (f ) which are not included in F⋄. The measure dν is a countable sum of Dirac deltas, weighted by the value of discontinuity + − in velocity (v − v ) or in force (p). The last parts (dvs and dis) are neglected singular measures, assuming that all the physical quantities involved are regular enough.

The geometric operator HT (x) yields the relative velocities in the contact frame. The third line formulates the non-smooth laws and the unilateral constraints as an inclusion in a set K. As previously, the vector of the compression of the felts is denoted by δ ∈ R14×1. Felts are treated as material compliances so that each felt is modelled as an additional body, which is linked to the body it belongs to with a prismatic joint. Such

5We recall Equation (4.12):

∂ δ T M(x) x¨ + N(x, x˙) x˙ + c x˙ + c sign(x˙) + (x) F(δ) + F⋆(x, t) + T(δ, x˙) + κ(x − x ) = 0 v d ∂ x 0

Å ã

86 where h agnilforces tangential The h r rcini igsi rte sa nlso in inclusion an as written is hinges in friction dry The he ftecnat ewe et n te oisaemdle wit modelle are modelled ones are other the bodies All other 54). and page 4.4 felts Figure (see between law contacts Coulomb the of Three n r uhthat such are and impuls reaction normal The positive. remain gaps the all that so otc,and contact, and nXE qain(.0 ssle sdsrbdi Mrho,2011] [Merlhiot, in described introduced: as are solved tions is (5.30) Equation XDE, In equat non-smooth the solve to used methods Numerical 5.2.2. ota o ahcnatzone: contact each for that so r rte sa nlso otenra oeof cone normal the to inclusion an as written are eal n mlctoso hscoc r ie n5.4.2. in given are choice this of implications and Details e )aewitnfrec otc as contact each for written are 4) ter orencnatzns(e .) h etro h orengp is gaps fourteen the of vector The 5.4). (see zones contact fourteen odto fnnitreerto ewe h oisadteflsis felts the and bodies the between non-interpenetration of condition ramn fteflsad orendgeso reo otesys the to freedom of degrees fourteen adds coordinates felts generalised the of treatment a nadto otefls compressions felts’ the to t addition of In modelling the to inelastic. related are and bodies on the between later described reasons For law. ..FRUAINO H IN CINA O-MOHDYNAMIC NON-SMOOTH A AS ACTION PIANO THE OF FORMULATION 5.2. • • B if if H ( µ g g T r = > ( x N ) 0 0 ) v β , , steds fradius of disk the is 1 r r r h eoiisi h otc frame, contact the in velocities the are N N and ¾ = α 0 0 1 β h omlrato oc spstv fteei contact. is there if positive is force reaction normal the : h omlrato oc szr hnteei ocontact; no is there when zero is force reaction normal the : , 2 r α T ietetneta ocso h same the of forces tangential the give 2 nlddin included , x il h eaietneta eoiiscrepnigt the to corresponding velocities tangential relative the yield steeoecmltdwt h diinlDOFs: additional the with completed therefore is Ä H T x ( x = ) v    ä µ − x α x r d hinges 1 r felts N , i N α g h eei nie eedo h numbering the on depend indices generic The . δ 2 n ie yteCuobfito a seChap- (see law friction Coulomb the by given and N ∈ ( a function gap a , N ∈ ∈ x ) ∈ R ∈ R R 14 B + R 6 14 × ( × µ + 1 ( 1 ( 14 g r    R N ( x ) + ∈ i )) ) 14 (( R µ : c r 20 d steCuobfito coefﬁcient friction Coulomb the is T sign × ) β 1 1 g , β ( 2 sascae oec fthe of each to associated is i x ) t contact. -th ) . es h olwn deﬁni- following The . r rte as written efls h contact the felts, he e.Tevco of vector The tem. N eoe by denoted nlddin included , ihaSignorini a with d ions LMULTIBODY AL Signorini- a h SYSTEM g (5.34) (5.35) (5.36) (5.33) The . p i d -th 87 ν ,

13 ms CHAPTER 5. SIMULATION METHODS

tn+α = tn + α ∆t  ∀α ∈ [0,1],  xn+α(kx ) = xn + α ∆t kx (5.37)   vn+α(kv) = vn + α ∆t kv    and tk+1 r H(x ) di H(x (k )) r (5.38) | n+γ(kx ) ≈ n+γ x Ztk The smooth dynamics and the differential inclusions are time-discretised using a time- stepping scheme as follows:

⋄ M(xn+θ (kx )) kv = F (xn+θ (kx ), vn+θ (kv), tn+θ )+ H(xn+γ(kx )) r   ( )= (5.39)  vn+θ kv kx   T (g(xn+γ(kx ), H (xn+γ(kx )) vn+γ(kv), r) ∈ F    Each iteration is solved in a global Newton loop. First, the contact kinematics and the smooth forces F⋄ are linearised, which leads to an algebraic inclusion with unknowns

(kv, r), known as a One-Step Non-Smooth Problem [Acary and Brogliato, 2008]. This OS- NSP is reformulated using an augmented Lagrangian approach [Studer, 2009] and solved by an iterative projective Gauss-Seidel-like method. More details are given in [Merlhiot et al., 2012].

5.2.3. Example of a pendulum with dry friction

The dynamics of the pendulum described in 5.1.2, where Coulomb friction had been regularised, is now solved with a non-smooth approach. The dynamics equation of the pendulum (5.6) is recalled:

θ¨˜ +Υ2 sign(θ˙˜)+ 4π2 sin(θ˜)= 0

It can be written in a form similar to (5.30):

˜ θ¨ = −4π2 sin(θ˜) − Υ2 λ  (5.40) ˙˜  λ ∈ sign(θ)

 5.2.4. Implicit scheme

Similarly to 5.1.2, the linearly-implicit Euler scheme is applied to θ˙˜:

˙˜ ˜ ˜ θk+1 =Θ(θk+1 − θk) (5.41)

88 where h oprsno h ovrec pesi rsne oeprecis more presented is speeds convergence the of comparison The oprsno h eut acltdwt h euaie and regularised the with calculated results the of comparison A ausof values   writte be  can equations overall the (5.9), Equation to Similarly and tcnb proved be which can It resolution the simplifies g it of but algorithm. effect another, root-finder the to Newton-like because step results one the from on much influence significant no has rgenerically: or othat so oerslsaerpeetdi iue58for 5.8 Figure in represented are results Some Results 5.2.5. oeta h feto rvt a rae xlcty(writte explicitly treated was gravity of effect the that Note lcstepnuu vnfrlow for even pendulum the blocks o-mohshm ilsavr odslto.Terglrsdscheme compu regularised same the The For solution. step. good solution. time chattering very and same a inappropriate the yields with scheme 5.9 non-smooth Figure in given is o ahtm tp tapasthat appears It m the by step. defined time was each solution exact for the and solution calculated each ytennmohapoc,rltvl oterglrsdoe h l the one: regularised the to relatively approach, nonsmooth the by ..FRUAINO H IN CINA O-MOHDYNAMIC NON-SMOOTH A AS ACTION PIANO THE OF FORMULATION 5.2. 6 θ θ ˜ ˙ ˜ aeb ae o example. for case, by Case k k + + proj 1 1    q (Θ = k [ − Υ and 1,1 hc r yia o h in cin h esr ftediffere the of measure The action. piano the for typical are which ] I − stepoetron projector the is W A k 6 λ ) r endby defined are − that: k + 1 1 á ∈ Θ sign    proj      θ θ ˙ ˜ ˜ ( k k θ ˙ ˜ θ λ ¨ ˜    k [ k k − + + + + U 1,1 1 1 1 ) [ k      − = ] ∈ + − Θ ( θ ˙ ˜ 1 ,1 1, α sign θ k 4 hra hswsnvrosre nFgr 5.7. Figure in observed never was this whereas , ˙ ˜ Υ = = ) + k π 1 − ⇐⇒ ] D 2 sago niao fterltv eetoffered benefit relative the of indicator good a is = ( sin endby defined −              θ 4 ˙ ˜ 1 q π k + ( k 1 α − 2 θ 1 Θ ˜ + ) sin 1 k = Υ ) U W − λ ( k k θ ˜ if if if + k Υ λ k + 5 ) > α α < α k 2 F 1 n a e htdyfito actually friction dry that see can One . + − proj ( = ∈ 1 U Υ k [ proj 1 − ) − 2 [ − 1 λ ,1 1, 1,1 k [ 0 − + ] safnto of function a as n 1,1 1 ]

samti equation: matrix a as n ] Υ h o-mohapproach non-smooth the Θ Ç 2 − aiyde o change not does ravity [ l nFgr .0with 5.10 Figure in ely θ ˙ ˜ W q k a ftedifference the of ean k arger k − osntrqiea require not does å ainlcs,the cost, tational 4 Θ π LMULTIBODY AL 2 Υ c between nce sin ed oan to leads h larger the , ( θ ˜ θ ˜ k SYSTEM k .This ). )] (5.45) (5.44) (5.46) (5.43) (5.47) (5.42) !      89 ë

12 ms CHAPTER 5. SIMULATION METHODS

1.0 solution Θ = 5 Θ = 0.5 10 Θ = 20 t 0 time Angle 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 T

−0.5

−1.0

˜ ˙˜ Figure 5.8 – Convergence of the non-smooth scheme. (θ0, θ0)=(1,0) and (Θ, Υ)=(·,5).

1.0 solution nonsmooth regularised 0.5 Angle

t 0 time 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 T ˜ ˙˜ Figure 5.9 – Comparison of the regularised and non-smooth approaches. (θ0, θ0)=(1,0) and (Θ, Ψ, Υ)=(20,10000,5). the relative advantage of the nonsmooth approach is. Also, the nonsmooth approach is systematically more efﬁcient.

50 100 150 200 250 300 350 400 450 500 100 ×××× time step Θ ××××××× ××××××× ××××× × × × × ×××××× × × ut ×××× × × × ×× ×××× × × −1 ×ut × × 10 ××ut × ut × × × ××ut ×× × × ut ×ut ×× ut ut ×× Υ = 1 ut ut ××× ut ut ut ××× ut ××× Υ = 2 ut ut ut ×× ut ut ut ut ×× × −2 ut × 10 ut ut ut ut ut ut × × × ut ut × × Υ = 5 ut ut ut ut × × ut ut × × × ut ut ut ut × × ut ut × Υ = 10 ut ut ut ut ut ut ut ut ut ut ut ut ut ut × regularised ut ut ut Difference to solution ut ut −3 ut ut ut 10 ut ut ut △ non-smooth ut ut ut ut ut ut ut ut ut ut ut ut ut

ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut 10−4

Figure 5.10 – Convergence of the regularised scheme (×) and the non-smooth scheme (△) ˜ ˙˜ for several Υ. (θ0, θ0)=(1,0) and Ψ= 10000.

90 with with D a eeoe tCALS,mil o nutilvrulprot virtual industrial for mainly LIST, CEA at developed was XDE is used we part t The in Engine. Dynamic implemented eXtended were for stands 5.2.2 which in XDE, described methods numerical The h mareto ueia rprisidcdb regularise a by induced properties numerical of impairment The h ocuino hsbifsuyi hti re ocluaethe calculate to order in that is study brief this of conclusion The oeifraini rvddi Mrho,2012]. [Merlhiot, in provided is information More o oois t anfaue hc eeueu o h simulati the for useful were which features are: main Its robotics. for mec interactive intermitte for constraints, kernel kinematic a with systems in multibody consisting rigid kit development C++ a is ..etne yai nie(XDE) Engine Dynamic eXtended 5.3. regular sophisticated more non-sm but has choose friction, approach dry us regularised with simple pendulum make a the to that of enough noteworthy is clear It appeared methods. study, simple this in erdcdfrnmrclraosisedo hsclones. physical of instead reasons numerical for reduced be ieseswihaelreyblwtecaatrsi ieo th of time characteristic the approximately below as largely regularisin are the which mechanism, steps complete time the on as more even matters ehdi infiatymr fcetadrbs hnteregular the than robust pr and the efficient = contacts, more Υ considering significantly without action, is piano method the of bodies the 7 ned h ro ihtennsot ehdgt mle wh smaller gets method non-smooth the with error the Indeed, oscillations. • • • • • • • • 2 = Υ = Θ eeto,wt fcetgoercletmto fcontacts. of estimation for geometrical methods efficient with time-stepping-compatible detection, nonsmooth of implementation the h mlmnaino ieitgainuigtm-tpigmet time-stepping using integration time of implementation the inclusions; differential nons measure with of systems means multibody by for laws, dynamics contact the Sig of formulation laws: nonsmooth the contact nonsmooth with contact laws; frictional Signorini-Coulomb of modelling the h mlmnaino ueia nerto o uhparametri such for integration numerical of implementation the multibody groups; of Lie dynamics the on of formulation the and parametrisation the h oeln n aaersto fmlioysseswt kinema with systems multibody of parametrisation and modelling the fric intermittent with systems methods; multibody numerical rigid of simulation the nrltv ro of error relative an , 15 250 ihtennsot method non-smooth the with ihterglrsdoe For one. regularised the with 1 s(hscrepnsto corresponds (this ms 1 srahdwith reached is % 7 whereas , Θ ≈ = Υ = Θ 1000 ..ETNE YAI NIE(XDE) ENGINE DYNAMIC EXTENDED 5.3. 5 40 h aerltv ro sreached is error relative same the , .Tetm tpwudte aeto have then would step time The ). Υ ihtennsot ehd and method, non-smooth the with ≫ nfito nrae,i hscs ffree of case this in increases, friction en 500 tcnat n r friction. dry and contacts nt srqie tews.This otherwise. required is in cin estimated action, piano e prah sobserved as approach, d sdmdl exist. models ised prahmyrequire may approach g feto r rcinon friction dry of effect tpn n simulation and otyping no h in action piano the of on enue ntestudy the in used been ecmue program computer he aia iuainof simulation hanical sn prah For approach. ising sne non-smooth esented sations; hods; D hsc,which Physics, XDE inuigefficient using tion ohnumerical ooth i constraints; tic ytm based systems oiiand norini collision mooth 91

11 ms CHAPTER 5. SIMULATION METHODS

5.4. Adjustments of the model for its implementation In order to implement the complete model of the piano action described in Chapter 4, some modiﬁcations of the model were required.

5.4.1. Actuation of the key As it was not possible to use the key position as an input of the simulation in XDE, a Kelvin- Voigt viscoelastic model was inserted between the (virtual) key and the (virtual) ﬁnger, as shown in Figure 5.11. It was then possible to control the position of the free end of the Kelvin-Voigt model. imposed position

link to the key

key

Figure 5.11 – Scheme of the Kelvin-Voigt model added for the simulation.

An inﬁnite stiffness in the Kelvin-Voigt model would lead to a position-driven simulation but the numerical stiffness would be too high to be solved properly. In the experiments, a viscoelastic material was added to correspond to a Kelvin-Voigt model. Its choice and the values of its parameters are discussed in 3.3, page 41. The simulated system is composed of the piano action and this PD corrector.

5.4.2. Felt laws There are (at least) two possible ways of implementing the felt laws: 1. as a compliant contact in the normal direction, with a Hertz-like penalisation; 2. as a material compliance, by introducing new bodies corresponding to the felts, new degrees of freedom, and by assigning a Signorini law to each of the contacts. This first option was ruled out for several reasons. The first one is that unilateral constraints are not ensured when a linearly implicit scheme is used together with penalised contacts, because the linearisation can lead to negative reaction forces. Similarly, it is hard to ensure compressive dissipation only. This would be improved using non-linear implicit schemes, but the computational cost is likely to increase significantly. Moreover, for the piano action, several contacts can occur within the relaxation time of a felt (consider for instance, the quick use of the double escapement). The correct estimation of the corresponding dissipation would require to keep the history of the compression of each felt. Such a feature would have to be implemented. We chose to add one body per felt, introducing fourteen new degrees-of-freedom in total. This solution introduces collisions between felts and other bodies. In XDE, collisions are

92 where h oe sdfrtefl ssc htisrato force reaction its that such is felt the for used model The ii oy(iko emn)wsascae oec eti accord in felt each to associated was segment) or (disk body rigid A hsmdli lutae nFgr .2o h xml fteham the of example the on 5.12 Figure in illustrated is model This iue5.12 Figure ...Ipeetto ftefl laws felt the of simulation Implementation the 5.5.1. for XDE of Adjustments 5.5. times. computation f the freedom change of perceptibly degree not does additional total) an in introducing that noteworthy is It contact. hammer-string the ecito seTbe42.Ti diinlrgdbd a given was body rigid additional This 4.2). Table (see description oeldwt nieatccnat ic h aso ahfl i negli felt is each dissipation corresponding of the mass bodies, the rotating Since the of contact. that inelastic an with modelled string. n otecaatrsi aso h etmtra novddu involved material felt the of mass characteristic the to ing otebd tblne owt rsai on.Tefl a a ap was part. law sliding felt its The and joint joint. prismatic prismatic a the with of to part belonged it body the to 8 hsprmtrwsicue nteprmtr aeoyII( III category parameters the in included was parameter This δ stecmrsino h et(e 4.5.10). (see felt the of compression the is – ecito ftecnatwt rsai on n etlw illu law, felt and joint prismatic with contact the of Description δ HS ∀ δ ¾ 0, ∀ δ ˙ a eoecontact Before (a) , b fe contact After (b) m ..AJSMNSO D O H SIMULATION THE FOR XDE OF ADJUSTMENTS 5.5. felt F ( δ , δ ˙ = ) g HS k e 4.4.1). see δ r − b F δ δ ˙ obeys: otc emtyo h hammer the of geometry contact igcollision ring string hammer 2 gible. eylwcmae to compared low very s mass a e oldn ihthe with colliding mer rec et(4DOFs (14 felt each or newt h contact the with ance le ewe h ﬁx the between plied m 8 felt twslinked was It . tae with strated correspond- 93

10 ms CHAPTER 5. SIMULATION METHODS

In XDE, the only implemented law is linear viscoelastic (Kelvin-Voigt model). To implement the felt laws, we change the stiffness klin and the viscosity blin of the linear model at each time step. This also required the addition of a external force flin (joint actuator) at each iteration.

The value of klin, blin and flin were determined from the differentiation of F, which is C 1(R, R) as r > 1. The calculation yields:

∂ F ∂ F ˙ ˙ ˙ ˙ ˙ ˙ F(δn+1, δn+1)= F(δn, δn)+(δn+1 − δn) (δn, δn)+(δn+1 − δn) (δn, δn) ∂ δ ∂ δ˙ r ˙ 2 r−1 ˙ 2 ˙ ˙ =(k δn + b δn δn)+(kr δn + 2b δnδn)(δn+1 − δn)+(b δn)(δn+1 − δn) r r 2 ˙ r−1 2 ˙ = k δn − kr δn − 2b δn δn +(kr δn + 2b δn) δn+1 + (b δn) δn+1 actuator joint effort proportional gain derivative gain (5.48) | {z } | {z } | {z } so that r−1 klin =(kr δn + 2b δn)  2  blin =(b δ ) (5.49)  n  r 2 ˙ flin = k(1 − r) δn − 2b δn δn    An illustration is given in Figure 5.13, where the dissipative part has been omitted.

Felt reaction force F [N]

15 k δr

F(δ ) 10 ≈ n+1 b

F(δ ) 5 n b

δ δ 0 n n+1 Displacement δ [mm] 0.1 0.2 0.3 0.4 0.5

−5 klin

flin −10 Figure 5.13 – Illustration of the implementation of the felt law as a linear model with adjustable parameters (the dissipative part has been omitted).

5.5.2. Ensuring the repeatability of the simulation The simulating program is divided in several elements, which are Orocos Agents (Open RObot COntrol Software, [Orocos, 2013]), see Figure 5.14. Each element is a block with a speciﬁc function: physic engine – Contains the mechanical and the solving parts. clock – Initiates the simulation. It also allows to regulate the speed of the simulation.

94 oesr h eetblt ftesmltos synchronisation a simulations, the of repeatability the ensure To h eut ftesmlto eewitnb yhni te etfil text other in Python by written were simulation the of results The ieanwdt switni hm ntecnrr,gendt sym (factor dots significantly time green simulation contrary, the i down the whether slowed periodically, so On read Doing is data them. the in where written ports i.e. is ports, data whic initial to new r The block th The a the shown. from activates time is which task. signal blocks ports the its i.e. a between ports, finished connexions receive evenemential the has ports 5.14, Figure block input In previous its clock. when the run that only meaning can block each nepee yMATLAB. by interpreted ftx ls e iue51.I oa,tePto cit sdto used scripts lines. Python 2000 than the more total, of made In are action piano 5.15. the Figure of P simulation see with wr communicated files, were It text MATLAB. scripts from of implement Python controlled the additional be After some would framework. simulations 5.5, Python in a described in run ifications was XDE practice, In practice in Simulating 5.6. o-ee mlmnain(ntepyiskernel). physics compensat the completely (in be implementation could low-level time of loss This repeatability. lute law felt export positions joint – input position clock e 5.5.1. see tic esteprmtr ftelna iceatclw ota tm it that so laws viscoelastic linear the of parameters the Sets & iue5.14 Figure mot h oiincmadt eapida h olwn step. following the at applied be to command position the Imports position input xot h eeaie oriae tec iestep. time each at coordinates generalized the Exports y pianist – ycrnzdcnein fOoo agents. Orocos of connexions Synchronized ( t i ) hscengine physic b lin ( t i + 1 k ) lin ( t i + 1 ) δ

..SMLTN NPRACTICE IN SIMULATING 5.6. x ˙ x i + ( ( 1 t t sanwoeo not. or one new a is t i i ) ) ≈ hyaecnetdevery connected are they h 20 δ ˙ i sipeetds that so implemented is + dwt nappropriate an with ed cutreffort actuator ,btesrda abso- an ensured but ), 1 & oto D o the for XDE control on position joint te ota l the all that so itten exportation tr sgvnb a by given is start oie periodical bolises to ftemod- the of ation ddt indicates dots ed to ymeans by ython s hc were which es, rvosone, previous e etlaw felt dl h felts, the odels boolean f lin ( t i + 1 ) boolean 95

9 ms CHAPTER 5. SIMULATION METHODS

➊ ➌

time step, writes reads txt ﬁle sent to XDE input choice

writes reads input data csv ﬁle sent to XDE

writes reads parameters’ values intelligible txt ﬁle sent to XDE

MATLAB scripts text ﬁles Python scripts

➋

calls calls shell scripts

Figure 5.15 – Simulations principle.

96 Contents Overview oprdt h xeietlrslso hpe 3. Chapte Chapter in of explained results as experimental position, the a predicti to to compared in response consist in simulations ﬁnger The the on 5. Chapter in described 6 ..Asniiiyaayi ...... analysis sensitivity A 6.5. ..Psto-rvnsmltos...... simulations . Position-driven action piano 6.2. virtual the of Regulation 6.1. ..Dsuso ...... Discussion . . 6.4. simulations Force-driven 6.3. ...Fredie ...... Force-driven . . 6.4.2. . . Position-driven 6.4.1. ...Dsuso n ocuin...... conclusion . . . . . and . . . . Discussion . . . . . 6.5.6. . . . IV . Category . . . . of . . . Parameters . . . . 6.5.5. . . III . Category . . . of . . Parameters . . . 98 . 6.5.4. . II . Category . . . of . . . Parameters ...... 6.5.3. . . I . category . . of . . Parameters . . . functional . 6.5.2. . cost . the . of . . Choice . . . 6.5.1...... simulations . the . . of . Visualisation . . . 6.2.3. . . . bodies . All . 6.2.2. . . Key 6.2.1. eut n discussions and Results hpe 6 Chapter nti hpe,temdlo hpe ssmltduigthe using simulated is 4 Chapter of model the chapter, this In gterato oc ftekey the of force reaction the ng .Tesmlto eut are results simulation The 2. r 98 . . 98 ...... 104 ...... 113 . . . . . 106 . . . . . 113 . . . . 107 . . . 117 . . . 113 . . . 119 . . . 118 . . . 107 . . 119 . . methods 110 . 102 97

8 ms CHAPTER 6. RESULTS AND DISCUSSIONS

The model of the piano action presented in Chapter 4 is simulated using the methods described in Chapter 5. The ﬁrst results were promising but kinematic observations of the simulations highlighted a few misadjustments of some parameters’ values. Section 6.1 expounds how the virtual piano action has been numerically regulated. Results for piano and forte keystrokes are given in 6.2 and 6.3 for position-driven1 and force-driven simulations, respectively. Testing the effect of signiﬁcant changes led to the same conclusion of Chapter 2. The discussion is focused on the upward phase because the sustain and the release of the key matter less, from the haptical point of view. Finally, a sensitivity analysis of the (simulated) reaction forces, to the parameters of categories I, II and III, is carried out for the piano and forte dynamics.

6.1. Regulation of the virtual piano action First, simulations were run with the values of the parameters given in Chapter 4, for several keystroke dynamics. The comparison between kinematic measurements and the calculated kinematics showed that the virtual escapement occurred a bit too late. As done in real life by piano technicians, this was corrected by regulating the virtual piano action, here by lowering the let-off button. Applying the full procedure (see Appendix A), the regulation of the virtual action yielded the following modiﬁcations:

• the jack regulating screw (jack-whippen contact) was moved horizontally2 by 0.4 mm;

• the backcheck (key-hammer contact) was moved horizontally by −7 × 10−4 m; • the capstan screw (key-whippen contact) was lowered by 0.2 mm; • the let-off button (support-jack contact) was screwed to leave an additional gap of 1.9 mm. Except for the let-off button, these modiﬁcations lie within the margin of uncertainties of the lengths measurements (see Chapter 3). They can be perceived as virtual lutherie: the virtual action is treated as a real action. The results presented in Sections 6.2 and 6.3 were calculated using these adjustments.

6.2. Position-driven simulations The damper is not considered, as it is not a key element in the global dynamics.

6.2.1. Key We remind that, because it was technically not possible to use the key position as an input of the simulation in XDE, a Kevin-Voigt viscoelastic model has been inserted between the (virtual) key and the (virtual) ﬁnger. It was then possible to control the position of the free end of the Kelvin-Voigt model in the simulation (see 3.3 and 5.4.1). A piston has been inserted between the (real) key and the (real) ﬁnger, modelled by the Kelvin-Voigt model. For the key, two different positions were therefore considered in the experiments:

1Contrary to the simulations of Chapter 2, the acceleration measurements are not used in the present chapter. 2As in Chapter 4, the x-axis is positive in the direction of the pianist.

98 a sda niptfrtesmlto.Tesmayo h relati the of summary The simulation. the for input an as used was h oprsn ewe esrdadsmltdqatte are quantities simulated and measured between comparisons The h anrsl stecmaio ewe h iuae reaction simulated the between comparison the is result main The h oprsnbetween comparison The h input The h esrdsignals measured The ae ipaeet ftekyaesmlr e iue62c f 6.2(c) Figure see similar, are key the of displacements lated ins ( pianist respectively. iue .()ad63b,respectively. 6.3(b), and 6.2(b) Figures ifrn hsclqatte sgvni iue6.1. Figure in given is quantities physical different iue6.1 Figure • • h esrdpsto fteuprfc ftepiston the of face upper the of position measured the h esrdpsto ftekey the of position measured the F simu measurements Y – meas .I srpeetdtgte with together represented is It ). ceeo h esrdadsmltdpyia uniis(po quantities physical simulated and measured the of Scheme spotdi iue .()ad63a for 6.3(a) and 6.2(a) Figures in plotted is Y meas y meas and and F y y meas meas meas y simu Y meas y meas eedwsmldat downsampled were isa aigsr htmaue n simu- and measured that sure making at aims ; F meas ..PSTO-RVNSIMULATIONS POSITION-DRIVEN 6.2. o both for simulation Y meas piano or piano piano 2 ; icse nScin6.4. Section in discussed H.Tesignal The kHz. oc ftekyo the on key the of force and nhp ewe the between onships and n .()for 6.3(c) and forte forte sition-driven). esrksin keystrokes F y simu simu dynamics, forte Y meas 99 .

7 ms CHAPTER 6. RESULTS AND DISCUSSIONS

piano 0.012

(m) Y 0.010 meas 0.008 meas Y 0.006 0.004 0.002

Position 0 Time (s) 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35

(a) Input-position of the simulation Ymeas.

5 Fmeas 4 Fmeas

2 Force (N) 1

0 Time (s) 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35

(b) Measured and simulated reaction forces of the key on the pianist’s ﬁnger.

0.012 y 0.010 meas ysimu 0.008 0.006 0.004

Position (m) 0.002 0 Time (s) 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Figure 6.2 – Simulation results for a piano keystroke (input: displacement).

100 Position (m) Force (N) Position (m) iue6.3 Figure 0.002 0.004 0.006 0.008 0.010 0.012 0.002 0.004 0.006 0.008 0.010 0.012 10 12 0 0 2 4 6 8 0 b esrdadsmltdrato oc ftekyo h pi the on key the of force reaction simulated and Measured (b) .501 .502 .50.30 0.25 0.20 0.15 0.10 0.05 0 0.30 0.25 0.20 0.15 0.10 0.05 0 .501 .502 .50.30 0.25 0.20 0.15 0.10 0.05 0 – iuainrslsfra for results Simulation c esrdadsmltdpsto ftekey. the of position simulated and Measured (c) a nu-oiino h simulation the of Input-position (a) forte forte esrk ipt displacement). (input: keystroke ..PSTO-RVNSIMULATIONS POSITION-DRIVEN 6.2. Y meas . ns’ ﬁnger. anist’s F F Y y y meas simu meas simu meas ie(s) Time (s) Time ie(s) Time 101

6 ms CHAPTER 6. RESULTS AND DISCUSSIONS

6.2.2. All bodies Here, we compare the angular positions of each rigid body measured with the video tracking and calculated from the simulation, for the piano blow. The tracking is not very accurate (see Chapter 3) but gives valuable indications. The results are shown in Figure 6.4. For each body, notable events have been marked by a vertical segment: (a): Initial position, at rest. (b): Beginning of the rotation of the jack, which corresponds to the contact between the jack and the let-off button. (c): Contact of the key and the front rail punching. (d): Impact of the hammer and the string. (e): Catch of the hammer by the backcheck. (f): Repositioning of the jack, so that the mechanism is ready to propel the hammer again. These events are used in the discussion 6.4.1. For technical reasons, the impact of the hammer on the string was ﬁrst considered as inelastic. This irrealistic model did not affect much the study of the haptics, because we focused on the phase before the impact of the hammer on string. Nevertheless, at the very end of the thesis, an elastic contact law has been implemented. The angles calculated with ˜ this hammer-string contact model are marqued with a tilde (for instance θK is the angle of the key calculated with this law). The elastic law resulted in a higher velocity of the hammer during its check so that the hammer stopped at a slightly different angle than with the inelastic law. We corrected this by adjusting the backcheck, as done in real life. The kinematic computations, plotted in dashed green in Figure 6.4, show improvements: the hammer trajectory is much closer to the measured one. The implementation of the elastic law resulted in an overestimated computed reaction force of the key on the ﬁnger during the check. This is believed to be due to the position of the backcheck, adjusted to correspond to the measured kinematics, but not to the measured dynamics. Additional investigation would be required to better modelled the catch of the hammer.

102 amrsrn otc law. contact hammer/string piano iue6.4 Figure

Angle (rad) Angle (rad) Angle (rad) Angle (rad) Angle (rad) keystroke. − − − − − 0.10 0.08 0.06 0.04 0.02 0.01 0.02 0.03 0.04 0.05 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.02 0.04 0.06 0.08 0.10 0.12 0.1 0.2 0.3 0.4 0 0 0 0 0 – .501 .502 .503 .504 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 .501 .502 .503 .504 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.45 0 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 .501 .502 .503 .504 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 oprsno h oiino h ois(rcigv simulat vs (tracking bodies the of position the of Comparison Blue .501 .502 .503 .504 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ ˜ ˜ ˜ ˜ ˜ K,simu L,simu J,simu W,simu H,simu H,simu H,meas L,simu L,meas J,meas J,meas W,simu W,meas K,simu K,meas measurements. : (b) (c) (d) (e) b hpe angle. Whippen (b) e amrangle. Hammer (e) d ee Angle. Lever (d) c akangle. Jack (c) a e angle. Key (a) Red piano simulation. : ..PSTO-RVNSIMULATIONS POSITION-DRIVEN 6.2. (f) Green iuainwt elastic with simulation : ie(s) Time ie(s) Time ie(s) Time (s) Time ie(s) Time o)fra for ion) 103

5 ms CHAPTER 6. RESULTS AND DISCUSSIONS

6.2.3. Visualisation of the simulations The visualisation of the simulations offers a good insight on the qualitative behaviour of the mechanism. We compare the real state of the piano action captured by a high-speed ﬁlm and the corresponding simulation, for a piano keystroke. Figure 6.5 presents the states corresponding to events (a), (b), (d), (e) and (f). These events are reported on the position and force results in Figure 6.9.

104 iue6.5 Figure – imadsmlto cenht oprsnfrntbeevents notable for comparison screenshots simulation and Film b einn fescapement of beginning (b) d amrsrn impact hammer-string (d) f akrepositioning jack (f) e hc catch check (e) a trest at (a) ..PSTO-RVNSIMULATIONS POSITION-DRIVEN 6.2. seFgr 6.9). Figure (see 105

4 ms CHAPTER 6. RESULTS AND DISCUSSIONS

6.3. Force-driven simulations For force-driven simulations, the relationship between the different physical quantities is given in Figure 6.6.

ymeas measurements simulation ysimu Fmeas

Figure 6.6 – Scheme of the measured and simulated physical quantities (force-driven).

The measured force Fmeas is used as an input for the simulations which yield a calculated displacement of the key end ysimu. The comparison between ysimu and the measured displacement of the key ymeas are presented for a piano keystroke (Figure 6.7) and for a forte keystroke (Figure 6.8).

piano F 4 meas 3 2

Force (N) 1 0 Time (s) 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35

0.010 ymeas 0.008 ysimu 0.006 0.004 0.002 Position (m) 0 Time (s) 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Figure 6.7 – Simulation results for a piano keystroke (input: force).

forte

10 Fmeas 8 6 4 Force (N) 2 0 Time (s) 0 0.05 0.10 0.15 0.20 0.25 0.30

0.010 ymeas 0.008 ysimu 0.006 0.004 0.002 Position (m) 0 Time (s) 0 0.05 0.10 0.15 0.20 0.25 0.30

Figure 6.8 – Simulation results for a forte keystroke (input: force).

106 hc r h aneeet ftemto ftegadpaoaction: piano grand the of motion the of elements main the are which d:ipc ntestring; the on impact (d): ihsm atrso h oc diagrams. force the of patterns some with h ntnso hs vnsaerpre nFgr . kyposit (key 6.9 Figure in reported are events these of instants The h ntn hntecnatbtentekyadtefotri pun rail front the and key the between contact the when instant The h ntn hntejc et h e-f utn()i olwdb s a by followed is (b) button let-off the meets jack the when instant The b:jc e-f utnpnhn spot contact; (support) punching button let-off - jack (b): eue rmtegoerclmaueet fteky(e Chapte (see key the of measurements geometrical the from deduced mhss hspten hssget that suggests This pattern. this emphasise h esrk yais hsi n ftemjreeet fteregulat the of elements know major and the 2009]) [Porter, of one e.g. (see is systematic commented was This extensively pattern action, This dynamics. force. keystroke the the of decrease sharp rather ocso h e ntepait–bto) hr r oovoscorrelation a obvious (c) no (b), are reverse, There the On bottom). diagrams. – position pianist the in the events on these key the of forces on fve,bte than better view, of point sn h bv-ecie esrmns twspsil ocaptur to possible was it measurements, above-described the Using Position-driven 6.4.1. Discussion 6.4. iuain edrti atr nboth in pattern this render simulations o mltd oprdt h te reuaiis represen A irregularities. other the to compared amplitude low a:satfo h etn position; resting the from start (a): e:ctho h hammer; the of catch (e): c:ky-fotri ucigcontact; punching rail front - key (c): f:rtr ftejc oisiiilposition. initial its to jack the of return (f): Force (N) Position (m) 0.002 0.004 0.006 0.008 0.010 0.012 0 1 2 3 4 5 0 .501 .50.20 0.15 0.10 0.05 0 .501 .50.20 0.15 0.10 0.05 0 (a) (a) F F iue6.9 Figure y y simu meas simu meas F ( t ) – . oal vnsdrn a during events Notable piano piano d d F t and ol ersn,i oewy h haptical the way, some in represent, could forte (b) (b) c (e) (c) (e) (c) (d) (d) n a oeta h upi of is hump the that note can One . piano keystroke. ainof tation os–tp–adreaction and – top – ions d()cnb correlated be can (e) nd lyosre,whatever observed, ally h olwn events, following the e hn c cus was occurs, (c) ching )adcorresponds and 4) r as n ..DISCUSSION 6.4. alhm n a and hump mall d aftertouch d F t ol better would o fthe of ion between s ie(s) Time ie(s) Time The . 107

3 ms CHAPTER 6. RESULTS AND DISCUSSIONS

to a displacement of the key of y = 9.8 mm. It appears on the measured force Fmeas as a local minimum in time: since the hammer has just escaped, the reaction of the key decreases until it meets the front rail. Then, because of the large stiffness of the felt, the reaction increases. The amplitude of this force increase is accurately calculated in piano and forte dynamics. For the piano keystroke, the position of the local minimum is shifted by a about ﬁve milliseconds. After hitting the string, the hammer is caught back by the backcheck. It transfers some of its kinetic energy to the key, pushing the ﬁnger end upwards. Again, the experimental diagram does not exhibit a strong time-variation in F but a strong variation in dF . Since dt the return of the hammer is not simulated realistically (because of the hammer-string impact law, see bottom of Figure 6.4), it is not surprising that this pattern is not visible in the diagram of the simulated reaction forces. As a matter of fact, the simulated force increases during the check, but simulations without the backcheck showed the same increase. It can be deduced from the sensitivity analysis to come (Section 6.5) that the heightening of the force is related to the lever meeting the drop screw. Most of the other irregularities observed in the measurements are also observed in the simulations. In particular, both piano and forte measurements exhibit two peaks and a smaller third one after the contact of the key and front rail punching: from t = 0.14s to t = 0.20s and from t = 0.8s to t = 0.12s, in the piano and forte keystrokes, respectively. This is believed to be due to oscillations of the key, under the reaction of the front rail punching. For the piano keystroke, the frequency of these oscillations is about 70 Hz. The

corresponding force is about Fmeas ≈ 3.5 N. The position of the key is nearly constant during the oscillations. The stiffness of the non-linear felt law of the front rail punching

can therefore be linearised in an apparent linear stiffness klin around the compression δ. From the felt law (4.4) comes:

∂ F k ≈ (δ)= 1.6 × 1010 × 2.7 × δ1.7 (6.1) lin ∂ δ

For F = 3.5 N, the felt law (the dissipative part is neglected) yields δ = 0.26mm. The oscillating mass is composed of two elements: • the key (with the force sensor and the piston); • a part of the musculoskeletal system of the pianist (ﬁnger, hand, arm).

The oscillations’ frequency flin of the linear spring, which approximates the front rail punching non-linear behaviour, is given by:

1 klin flin = (6.2) 2πs meq

The contribution in mass of the pianist is very hard to evaluate. If we ignore it, then meq = 62g and Equation (6.2) gives flin = 121Hz in piano dynamics, which is the same order of magnitude of the frequency of the observed oscillations (70 Hz). Moreover, ignoring

the contribution in mass of the pianist results in an underestimated meq, and therefore in overestimating flin. The similar reasoning can be done for the forte keystroke. The observed frequency is roughly 90Hz, the force oscillates around 9 N which corresponds to a compression of the

108 h ieaishv enpeetdfrthe for presented been have kinematics The h te iceace r eivdt emsl u to: due mostly be to believed are discrepancies other The eysot,a bevdi hpe .Ti ed ovldt h u the validate to tends This 2. Chapter one in measured observed the to as close smooth, very very are key the of positions simulated The ie h prxmtoso hs siain fl’ non-linear (felt’s estimations these of approximations the Given h eaieerru oterlaeo h key the of release the to up error relative The hs iceace eeeautduigtefloigdefini following the using evaluated were discrepancies These etrrglto ftepaoato hc mrvstedsetof descent the improves which action piano the of regulation better h amrwihocr o aewt h nlsi hammer/str inelastic the sim with the late in too captured occurs which accurately hammer are the events the All repeatable. ftemaue oc per ntesmlto eut,frbt dyn both for results, simulation the in appears Al force possible. measured forces the measured of the simul of the irregularities and the kinematics of the decoding of measurements the i both ass validate summary, to In this tend will oscillations), section the following the during in shown action results the Some of model spring-mass ftepaitsmsuokltlsystem. musculoskeletal pianist’s the of u htteerltv roswl erdcdby reduced be will errors relative these that out etof felt η eetees iuae n esrdfre hwsm non-negli some show forces measured and simulated Nevertheless, displacement. a with simulation the drive to 3 h ees ftekyi osdrdt cu at occur to considered is key the of release The = forte • • • • • • • 13 h auso h aaeeso h et n h rcino h hi the of friction they temperature. since the and especially humidity and Lozada, to by felts measured the action the of from parameters differ the of values key; the the of measu ge force length reaction to the the of sensitive in estimation very discrepancies the large is in errors mechanism small the that so geometry: ments, the in upp errors the the of position measured the 3.5); in (see errors piston small of amplification the ag reuaiyocre ntesmlto oc in force simulation the in occurred irregularity large a h mltdsadtetmn ftefrevrain r o alwa not are variations force the of timing the and amplitudes the h ecinfrege akt eofse ntemeasurements. the in faster zero to back goes force reaction the tha smoother is escapement, one; before key, the of descent measured the sured; 0.38 o the for % measure. m h orsodn frequency corresponding The mm. forte n.I eto .,asniiiyaayi scridot n it and out, carried is analysis sensitivity a 6.5, Section In one. η = R | F t meas R = piano | F 0.22 3 meas f − lin sof is 33 o the for s F is | simu esrks h ult fterslsis results the of quality The keystrokes. in % 167 η | = piano z tl goigtecontribution the ignoring still Hz, 21 piano o the for % lto,ecp h ac of catch the except ulation, esr n at and measure ino eaieerror relative of tion n by and piano h infiatvariations significant the l n otc a u tan at but law contact ing lsi eaiu,linear behaviour, elastic t. il discrepancies: gible eo h Dcorrector PD the of se hc a o mea- not was which , h key. the ,adbt uvsare curves both and s, mto splausible. is umption toshv aethe made have ations amics. piano 15 emt esensitive be to seem eet a edto lead can rements saccurate; ys ..DISCUSSION 6.4. in % mtia adjust- ometrical gswihmay which nges h calculated the n t esrk and keystroke rfc fthe of face er = forte 0.13 iha with , o the for s η : turns (6.3) 109

2 ms CHAPTER 6. RESULTS AND DISCUSSIONS

accurate time with the elastic one. Also, the shapes of the position curves are very similar, except for the hammer when the coefﬁcient of restitution of its contact with the string was underestimated. The escapement velocities of the hammer are perfectly estimated in the simulations, see Table 6.1. The escapement velocity for the forte dynamics is usually larger [Askenfelt and Jansson, 1991] because, as mentioned in Chapter 3, forte normally corresponds to higher forces.

measurements simulation

piano 1.01 m · s−1 1.00 m · s−1 forte 1.56 m · s−1 1.56 m · s−1

Table 6.1 – Escapement velocities, measured and simulated.

Time steps and real time applicability The characteristic time of the piano action is about a few milliseconds. All the simulation results presented in the thesis4 have been calculated with a time step of h = 0.5ms (f = 2kHz). The computed forces for different frequencies are shown in Figure 6.10, for a piano keystroke. The results for f = 2kHz and f = 10kHz are very similar. A simulation frequency of f = 2 kHz is therefore assumed to be appropriate. On the reverse, choosing f = 0.4kHz induces signiﬁcant discrepancies.

Simulating 1s of motion of the piano action took about 20s with a laptop computer equipped with an Intel Core i5 processor. This factor 20, between the computation time and the real time, can be improved applying two independent modifications. • Before implementing the felt laws and ensuring the synchronisation (see 5.5.2), which artificially but considerably slowed down the computation, simulations were computed nearly in real time (factor 1.2). It is therefore very likely that the low- level implementation of the felt laws, in XDE’s kernel, will lead to a factor close to 1.2 from real time, reducing the computation time by more than 15 times. • Even if the model of the piano action is plane, XDE works only in 3D. The simulation does not take advantage of this possible simplification. A gain of a factor greater than 3 is expected.

All in all, because the time step of 0.5ms is small enough to simulate realistic forces, and because with a few programming effort, the simulation could run a few times faster than real time, it is very reasonable to consider that the simulations of the grand piano action can be applied to real time tasks.

6.4.2. Force-driven As it can be foreseen according to the conclusions of Chapter 2, the simulated positions are very similar to the measured ones during the attack phase. The return of the key to its resting position is not accurately simulated in piano. This does not seem to be due to a systematic error because the results are far better in forte. The discrepancy could be due to errors in the measurements (see Section 3.5).

4Except those of Figure 6.10, of course.

110 Force (N) Cuobfito sucagd.Sc oie oe sovosynot obviously is model modified a Such unchanged). is friction (Coulomb (in hnte r goe ge uv) uhoclain r o obser not are oscillations such curve), (grey ignored are they When aiaeteasmto f641ta hyoiiae ntefel the in originates they fron that the 6.4.1 of with assumption contact the validate in is key the when forces the of oscillations The h aecmaio a endn o oiindie iuainand simulation position-driven a for done been has comparison same The h eut r oprbet hs n[ishone l,20]wh 2006] al., et [Hirschkorn in those to comparable are results The etoe n641 peri h iuaino iue61 when 6.12 Figure of simulation the in appear 6.4.1, in mentioned reuaiisaentrpoue swl swt h odmodel. good the with as t well of as estimation reproduced bad not a to are leads irregularities model modified the There, 6.12. Figure ii aeil h otcsaeteeoecagdt non-compli to f changed the therefore All are model. contacts the The in made material. is rigid change major similarities a the of simulations, significance driven the further analyse to order presented In mechanism 1-DOF the to afterward comparable better also significantly are and they key, key the the of releasing the before ifrnebtentesmlto fte"od oe (in model "good" the of simulation the between difference aetednmclbhvoro h in cin et r absolute are felts action: a piano for results the The of mechanism. behaviour dynamical the mate 0 1 2 3 4 5 grey iue6.11 Figure Position (m) Force 0.20 (N) 0.15 0.10 0.05 0 ). 0.002 0.004 0.006 0.008 0.010 F F F F 0 0 1 2 3 4 simu simu simu meas .501 .502 0.25 0.20 0.15 0.10 0.05 0 .501 .502 0.25 0.20 0.15 0.10 0.05 0 iue6.10 Figure , , , – f f f = = = iuainrslsfra for results Simulation 0.4 2 10 F y y y kHz meas simul simu meas kHz kHz ihfelts with ihu felts without – piano optdforce Computed piano, esrk r ie nFgr .1 hr svr little very is There 6.11. Figure in given are keystroke F -driven piano F simu esrk ihu et ipt force). (input: felts without keystroke o ifrn iesteps. time different for red ewe esrdadforce- and measured between ftefotrail. front the of t n h oie model modified the and ) .Drn h ecn of descent the During s. h et r considered. are felts the n inrn contacts Signorini ant lsaerpae with replaced are elts efre nparticular, In force. he c r lgtybetter slightly are ich nCatr2. Chapter in e.Ti ed to tends This ved. yesnilt the to essential ly ..DISCUSSION 6.4. alpunching, rail t speetdin presented is ie(s) Time ie(s) Time odt esti- to good 111 ie(s) Time

1 ms CHAPTER 6. RESULTS AND DISCUSSIONS

piano, y-driven 0.012 0.010 Fmeas 0.008 0.006 0.004

Position (m) 0.002 0 Time (s) 0 0.05 0.10 0.15 0.20 0.25

8 7 ymeas 6 5 ysimu with felts 4 y without felts 3 simu

Force (N) 2 1 0 Time (s) 0 0.05 0.10 0.15 0.20 0.25

Figure 6.12 – Simulation results for a piano keystroke without felts (input: displacement).

The conclusion of Chapter 2, according to which the simulation should be driven by the measured position of the key in order to account for the validity of a dynamic model of the piano action, is retrieved in the case of the complete model.

112 se441.Gvntelrenme fprmtr,teranalys their parameters, of number large the Given 4.4.1). (see where h ih aaeeso aeoyIaerpeetdi iue4. Figure in represented are I category of parameters eight The hscs ucinlde o con pcfial o h escapeme the for specifically account not does functional cost This h otfunctional cost The The dnie rmtefrecre n siae as estimated and curves force the from identified ...Prmtr fctgr I category of Parameters 6.5.2. assu later. This discussed pr is good choice well. this a action of gives relevance piano which which the for simulation describes simulation key a a the that of made force is reaction assumption The mer. much. h otfunctional cost the ...Coc ftecs functional cost the of Choice 6.5.1. influential. categ are of they Those how detail. on in depending studied classified, are and I) described (category action a piano are a which of parameters The parameter. each for pre is independently then functional is 4 cost Chapter A in defined parameters parameters. reac the simulated model to sensitivity the the of to sensitivity relatively the study finger, we section, this In analysis sensitivity A 6.5. h simulation: the ftekyi eoe by denoted is key the of t e 5 h term The = ucino h aaees h expression the parameters, the of function • • • • • • 1 0.13 nr a hsnin chosen was -norm S S whippen); the lifts and key the S position); resting the in whippen, the K J K W K J L t W H i y + y x x x stetm fthe of time the is y stevria oiino h e-f button; let-off the of position vertical the is in s stevria oiino h rpscrew; drop the of position vertical the is stehrzna oiino h akrgltn ce wihi i is (which screw regulating jack the of position horizontal the is stehrzna oiino h backcheck. the of position horizontal the is otfunctional cost and stehih ftefotri punching; rail front the of height the is forte K 2 W kHz. y h rqec ftemaueet is measurements the of frequency The . 5 J r h oriae ftetpo h asa ce wihi xdto fixed is (which screw capstan the of top the of coordinates the are J h esrmn inl eedwsmlda h rqec of frequency the at downsampled were signals measurement the , eesto refers sdfie as: defined is J J t b J i ssalcno edt nirlvn amrvlct.The velocity. hammer irrelevant an to lead cannot small is t esrmn.Teisato h einn ftemotion the of beginning the of instant The measurement. -th [ ( F eas eddntwn opnls ag iceace too discrepancies large penalise to want not did we because t J simu b = safnto fasmltdforce simulated a of function a as ] 0 : = ) h ntnso h eesn ftekey the of releasing the of instants The s). otfunction cost t i P = t e t b | F t i meas P = t e t b | ( sused. is F t meas t i e ) − = ( F 0.22 t simu i ) | ( 50 o the for s ..ASNIIIYANALYSIS SENSITIVITY A 6.5. t etd aeoyb category by category sented, i F infreo h e nthe on key the of force tion ) H.Frtecluainof calculation the For kHz. ae5.W eidthat: remind We 56. page 6 simu jse nara regulation real a in djusted | swr nycridout carried only were is When . r IadIIaebriefly are III and II ory endi ...The 6.5.1. in defined tvlct fteham- the of velocity nt piano pinipisthat implies mption J dcino the of ediction scniee sa as considered is esrk,and keystroke, otc with contact n t e (6.4) were 113

0 ms CHAPTER 6. RESULTS AND DISCUSSIONS

The value p of each of these parameters was changed independently, by steps of 0.1 mm 6 around its measured value p0 given in Chapter 4, and up to a range of ±2mm. This corresponds to 82 simulations for each parameter of category I (41 for the piano keystroke and 41 for the forte one). The values of the cost functional were straightforwardly calculated from the simulated force using Equation (6.4). The corresponding plots are represented in Figure 6.13 to 6.20. Cost J (-)

b b 1.0 b piano b b b b b 0.8 b b forte b b b b b b b b b b b b 0.6 b b b b b b b b b bb b b b b b b b b b b b b b b 0.4 b bb b b b b b b b b b bb b b b b b b 0.2b b bb Displacement (mm) −2.0 −1.5 −1.0 −0.5 0 0.5 1.0 1.5 2.0

Figure 6.13 – Sensitivity of J with respect to SK+y .

Cost J (-)

0.3 piano b b b b b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b b b b forte b b b b b b b b b 0.2 b b

b b b b b b b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b b b b b b b

b 0.1 b b b b

b Displacement (mm) −2.0 −1.5 −1.0 −0.5 0 0.5 1.0 1.5 2.0

b Figure 6.14 – Sensitivity of J with respect to KWx . b b b b b b b

b Cost J (-) b b b b

b 1.0 b piano b b 0.8 b forteb 0.6 b

b b 0.4 b b b b b b b b b b b b b b b b b b b b bb b b b b b b b 0.2b b bb b b b b b b b b b b b b b b b b Displacement (mm) −2.0 −1.5 −1.0 −0.5 0 0.5 1.0 1.5 2.0

Figure 6.15 – Sensitivity of J with respect to KWy .

Several shapes of the cost function are observed:

• For SLy and Kwx , the sensitivity of the cost function to a small displacement is negligible. This means that varying these parameters on the piano action does not have a significant effect on the reaction force of the key. It can nevertheless have a significant effect on the functioning of the action, such as an escapement velocity not sufficient for the hammer to reach the string.

6This corresponds to values of p covering exactly

[p0 − 2 mm, p0 − 1.9 mm, ... , p0, ... , p0 + 1.9 mm, p0 + 2 mm]

114 • • − − − − h otfnto s(oal)cne ihrsetto respect with convex (locally) is function cost The httemaue aus(nrd r laso h a ato the of intere part is ﬂat it the on Also, always are action. red) the (in block values measured to the enough that is parameters these of For ipaeeti eylre ti oeotyta a according that noteworthy move is to It key large. the very lea for is which displacement required action force the the of adjustments, settings such to corresponds latter The ities. of 2.0 2.0 2.0 2.0 b b b b b b b b S b b b b b b b b K K b b b b b b b b + W b b b b b b b b y − − − − y b b b b b b b b , orsod otemnmmof minimum the to corresponds 1.5 1.5 1.5 1.5 b b b b b b b b S b b b b b b b b J y b b b b b b b b , b b b b b b b b J − − − − iue6.19 Figure b b b b b b b b b iue6.17 Figure iue6.18 Figure W iue6.16 Figure 1.0 1.0 1.0 1.0 b b b b b b b x b b b b b b b b and b b b b b b b b b b b b b b b b − − − − b b b b b b b b 0.5 0.5 0.5 0.5 K b b b b b b b b H b b b b b b b b Cost Cost Cost Cost x b b b b b b b b 0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1.0 h otfnto sprl a u lohslreirregular- large has also but ﬂat partly is function cost the , – b b b b b b b b – – – b b b b b b b b estvt of Sensitivity estvt of Sensitivity estvt of Sensitivity estvt of Sensitivity . . . 2.0 1.5 1.0 0.5 0 2.0 1.5 1.0 0.5 0 2.0 1.5 1.0 0.5 0 2.0 1.5 1.0 0.5 0 J J J J b b b b b b b b b b b b b b b b b b b b b b b b (-) (-) (-) (-) b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b J b b b b b b b b J J J J h au of value The . b b b b b b b b b b b b b b b b ihrsetto respect with ihrsetto respect with ihrsetto respect with ihrsetto respect with b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b 0.5 S b b b b b b b b K b b b b b b b b b b b b mcag ftevleo one of value the of change mm + b b b b b b b ..ASNIIIYANALYSIS SENSITIVITY A 6.5. y piano forte piano forte piano forte piano forte b b b b b b b and b b b b b b b b L b b b b b b W L J S S y W W L L J W y y is x y ipaeet(mm) Displacement (mm) Displacement ipaeet(mm) Displacement (mm) Displacement . . . oisbokn:for blocking: its to d . y 0.5 h measurement The . otfunction. cost mhge than higher mm oteimposed the to tn onotice to sting 115

-1 ms b

b b

b b b

CHAPTER 6. RESULTS AND DISCUSSIONS

Cost J (-)

0.3 piano

bbbbbbbbbbbbbbbbbb bb b b b b b b b forte 0.2

b bbbbb bbbbbb bbbb b b b b bb b b b b b 0.1

Displacement (mm) −2.0 −1.5 −1.0 −0.5 0 0.5 1.0 1.5 2.0

Figure 6.20 – Sensitivity of J with respect to KHx .

the optimal one. Indeed, the force calculated using this optimal solution yields much better results than with the measured value, see Figure 6.21 for piano and 6.22 for forte. This optimal adjustment, originally found empirically from the sensitivity analysis, is conﬁrmed here. The kinematics corresponding to the optimal setting

of LWy are also given, in Figure 6.23. The analogue kinematic results for the non- optimal value of LWy have been given in Figure 6.4. We recall that the green dashed plots correspond to the computations with an elastic hammer/string contact law. The calculated position of the hammer shows a signiﬁcant improvement and is very similar to the measured one.

5 measurements 4 measured LW y 3 optimised LWy 2 Force (N) 1

0 Time (s) t t 0b 0.05 0.10 0.15 0.20e 0.25 0.30 0.35

time interval [tb, te] for J Figure 6.21 – Comparison of simulation results with measured (red) and optimised (green) values of LWy for a piano keystroke (input: displacement).

12 measurements 10 measured L (N) 8 Wy F 6 optimised LWy 4 Force 2

0 t t Time (s) 0b 0.05 0.10e 0.15 0.20 0.25 0.30

time interval [tb, te] for J Figure 6.22 – Comparison of simulation results with measured (red) and optimised (green) values of LWy for a forte keystroke (input: displacement).

The sensitivity analysis of category I parameters allows to make some distinctions between

the parameters. Some have a signiﬁcant effect on the reaction force of the key (SK+y , LWy ). Some do not (SLy , KWx ) but they may have other functions: SLy is crucial for the double escapement. It does not affect much the sensitivity analysis since during the measurements,

116 ( esrks h aiu au sgvni al . (column: 6.2 Table in given is value maximum The keystrokes. otfnto hp.Frec aaee,terltv differen relative have the to parameter, adjec each likely For qualifying are appropriate shape. and function the measurable cost indicates are and which force) parameters reaction use summar we 6.2 Table parameters, (i.e. flat/irregular). these II or for flat figures (convex, above the introduced all reporting of II Instead Category of Parameters 6.5.3. t of force reaction the significantly t change for not therefore do and but hammer, action, the piano of escapement the for critical are o h esrdvalue measured the for on yipcigtesrn.Ti a owih norcs funct cost our in weight no has This string. the impacting by sound ecinfreo h e pt ie xet u a edt blocking a to lead can but extent, given sign a no to have up parameters key some the of Also, force once. reaction depressed only was key the K K parameter W W y J y W W H H S H H H H L J c J J J m c c J H J aet eajse opeettehme rmbucn n produci and bouncing from hammer the prevent to adjusted be to have , L c c c v H s H y O Ox H v H s S S v s G G O J O , J J J J W S W H H H K H K x ρ y x θ x y x y y y x y x y J O y , J W x [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ , [ [ [ [ [ [ [ [ [ [ − − − − − − − − − − − − − − − K etdrange tested − − − − − − − − − − 1 1 1 H 2 1 1 1 1 2 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10 x mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm .Aan hs aaeesmyhv oeohrfunctions: other some have may parameters these Again, ). % % % % % % % % % % 1 , 1 , 10 , 10 , 10 , 10 , 10 , 1 , 10 , 2 , 1 , 1 , 1 , 1 , 10 , 2 , 1 , 10 , 1 , 1 , 1 , 1 , 10 , 10 , 1 , p mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm 0 % % % % % % % % % % n h iiu au of value minimum the and ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] irregular/linear/irregular irregular/flat irregular/flat flat/irregular flat/irregular irregular/flat irregular/flat irregular/flat irregular/flat irregular irregular ovx- convex ier- linear shape a - flat a - flat a - flat a - flat a - flat a - flat a - flat a - flat a - flat a - flat a - flat a - flat J scluae for calculated is , ..ASNIIIYANALYSIS SENSITIVITY A 6.5. yia lciglength blocking typical ebtentevleof value the between ce ieo h corresponding the of tive ∆ ecnrlofrdb the by offered control he J h characterisations the d ekey. he o.Both ion. 0.4 0.4 0.4 0.2 0.2 0.2 0.4 0.8 0.2 sstoeo Category of those ises otne nnx page next on Continued / niflec nthe on influence an J 4 4 fiatefc nthe on effect ificant ftepaoaction piano the of mm mm mm mm mm mm mm mm mm % % ). piano gasecond a ng S J y and K and W x forte 6.0 3.9 2.7 0.8 0.1 8.9 0.8 4.7 0.6 1.4 0.0 0.0 3.0 6.7 2.5 0.0 0.2 0.0 10 10 18 42 24 26 37 117 and ∆ J W J J J % % % % % % % x % % % % % % % % % % % % % % % % % %

-2 ms CHAPTER 6. RESULTS AND DISCUSSIONS

Table 6.2 – Continued from previous page ∆ J parameter tested range shape typical blocking length J

SK−y [−1 mm,1 mm] convex - 0.0 %

KOx [−1 mm,1 mm] ﬂat/irregular 0.4 mm 2.0 %

KO y [−1 mm,1 mm] convex - 0.0 %

SJx [−1 mm,1 mm] ﬂat - 0.3 %

Table 6.2 – Sensitivity of the cost function with respect to parameters of Category II.

Several results come out of the sensitivity analysis of the cost function J with respect to the parameters of Category II. • Some changes were applied to the values of the model parameters, with large

amplitudes. The cost function was estimated for each set of values. For JOx , JOy , JHx , JHy and LHy , these changes led to signiﬁcantly lower J (between 26 % and 42 % lower, for the worse case between piano and forte). This is due to a mis-setting which has been highlighted previously, in the sensitivity analysis of parameters of Category I. It also shows the interdependencies of the parameters: we saw that J

is a convex function of LWy , now J shows a similar property with the other end of the lever LHy . Also, these parameters are interestingly all related to the escapement, which is known to be very delicate. • For the other parameters, the measured setting is very close to the minimal cost (in the sense of the parameters taken individually) and therefore the optimal setting in terms of reaction force of the key. • Many parameters have no significant influence on the reaction force of the key. This is particularly interesting as it is a step towards a simplification of the model. • Table 6.2 also highlights the high accuracy required for the regulation of the piano action: for many parameters, a variation of 0.2 mm leads to a completely different dynamical behaviour.

6.5.4. Parameters of Category III

∆ J parameter tested range shape typical blocking length J

KHρ [−10%,10 %] ﬂat - 0.5 %

KWθ [0.5 rad,0.5 rad] irregular/convex 0.3 rad 17 %

WKρ [−10%,10 %] ﬂat - 0.8 %

JHλ [−10%,10 %] ﬂat - 0.8 %

JSρ [−10%,10 %] ﬂat - 0.0 %

JSθ [0.5 rad,0.5 rad] ﬂat - 0.0 %

LHθ [0.5 rad,0.5 rad] convex 0.0 mm 18 %

HKρ [−10%,10 %] ﬂat - 0.0 %

HKθ [0.5 rad,0.5 rad] irregular 0.2 rad 0.0 % Continued on next page

118 ntecnrr,tecs ucini a nmn aaees dir parameters’ many in flat is function cost the contrary, the On osdrn h aaee pc and space parameter the Considering h nlsssoe htfrmn emtia aaees chan a parameters, geometrical many for that showed analysis The estvt nlsswscridotfralteprmtr fC of parameters the all for out carried was analysis sensitivity A assumption. the validate to tends This h aaeeso aeoyI r supposed are IV Category of parameters The osvnprmtr rirrl hsnada ifrn sposs as different as and chosen arbitrarily parameters seven to otecretvle fteprmtr.I te od,cmo meth common words, using action other piano In the to applied p parameters. be the c not the at probably this of should directions with optimisation values zero-slope least correct many at has the – function to key cost the the of of measurements of gradient parameters force the and of values position optimal only the find to impossible nearly is scne nafwprmtr’drcin:tehih ftefront the of lever height repetition the the of directions: button parameters’ regulation few a in convex is fec fteeprmtr orsoddt h iiu oti the in cost minimum the to corresponded parameters parameter. these of each of h otc ewe h e n h whippen the and key the between contact the oc ihr hsi nasmto.I a eie tdigthe studying verified was It assumption. an is This either. force ilmtecnla otebokn fteato,tu preventing thus action, the of blocking the properly. to escaping lead can millimetre aiu otwsonly was cost maximum egto h oaino h key the of rotation the of height neednl,uigtecs ucinl(6.4). functional cost the using independently, ...Dsuso n conclusion and Discussion 6.5.6. atiflec nterato oc fteky hswsntkonbfr r before known not n was do This III key. and the II of Categories force of reaction simulations. the parameters on many influence that cant shown been has It IV Category of Parameters 6.5.5. K 7 W n hrfr lotecs ucini locne ntedir the in convex also is function cost the also therefore And et masses felts al 6.3 Table parameter L λ H , y c . ν ν s S ν J KH JH J JS θ and – H S estvt ftecs ucinwt epc oparameters to respect with function cost the of Sensitivity ρ [ [ h otfnto a a nalteeidpnetdrcin a directions independent these all in flat was function cost The . − 0.5 [ [ [ etdrange tested − − − 50 20 20 20 rad % % % % 1000 , 0.5 , al . – 6.3 Table 3 20 , 20 , 20 , ihrta h iiu cost. minimum the than higher % rad % % % % ] ] ] ] ] K O otne rmpeiu page previous from Continued y J n h nl hc hrceie h ieto of direction the characterises which angle the and 7 safnto fteprmtr,tecs function cost the parameters, the of function a as shape lp - slope a < - flat a - flat a - flat a - flat L W y priori a h egto h akri cloth rail back the of height the , K W θ xetfor Except . cino h te n fterptto lever, repetition the of end other the of ection ohv osgicn feto this on effect significant no have to yia lciglength blocking typical ..ASNIIIYANALYSIS SENSITIVITY A 6.5. aito of variation tgre ,I n I taken III and II I, ategories cin.Ti en htit that means This ections. ible: L W alpunching rail h cinmdl from model, action the y h esrdvalue measured the , m itcorresponding oint J h amrfrom hammer the K eo esta a than less of ge fCtgr III. Category of s ucin the function: ost , thv signifi- a have ot . ieto fthe of direction J J K d fconvex of ods , K ihrespect with O nigthe unning , K S S 0.3 4.9 0.9 0.5 K K ∆ G 1.6 + − J x dthe nd J , y y % % % % the , the , % K 119 G y ,

-3 ms CHAPTER 6. RESULTS AND DISCUSSIONS

The choice of the cost functional must be discussed. This one does not reflect whether the escapement actually occurred, nor the escapement velocity, and more generally the kinematics are not directly weighted. Using a cost functional base on the force was intended to reflect the haptical role of the action. It appears that the let-off, known to be essential to pianists’ touch, only appears as a small hump in the force diagram and has therefore a low weight in the cost function. A more appropriate cost functional has yet to be found. This requires a better knowledge of what is felt by the finger.

120 iuainwt lsi amrsrn otc law. contact hammer/string elastic with simulation iue6.23 Figure esrk ihteotmsdvleof value optimised the with keystroke

Angle (rad) Angle (rad) Angle (rad) Angle (rad) Angle (rad) − − − − − 0.10 0.08 0.06 0.04 0.02 0.01 0.02 0.03 0.04 0.05 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.02 0.04 0.06 0.08 0.10 0.12 0.1 0.2 0.3 0.4 0 0 0 0 0 – .501 .502 .503 .504 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 .501 .502 .503 .504 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.45 0 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 .501 .502 .503 .504 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 oprsno h oiino h ois(rcigv simulat vs (tracking bodies the of position the of Comparison .501 .502 .503 .504 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ ˜ ˜ ˜ ˜ ˜ K,simu L,simu J,simu W,simu H,simu H,simu H,meas L,simu L,meas J,meas J,meas W,simu W,meas K,simu K,meas (b) (c) (d) (e) b hpe angle. Whippen (b) e amrangle. Hammer (e) L d ee Angle. Lever (d) c akangle. Jack (c) a e angle. Key (a) W y . piano Blue measurements. : (f) ..ASNIIIYANALYSIS SENSITIVITY A 6.5. Red simulation. : o)fra for ion) ie(s) Time ie(s) Time ie(s) Time (s) Time ie(s) Time Green piano 121 :

-4 ms

Conclusion

The objective of the thesis was to propose a model of the grand piano action, and methods to simulate it, in order to render the subtle dynamics of the piano action from the key’s viewpoint. The first step consisted in an analysis of how this dynamics should be studied. The investigation was based on an elementary single-degree-of-freedom model. The conclusion is that the quality of a model for the action’s dynamics, is given by the quality of the reaction forces of the key that it predicts, for given displacements of the key. On the reverse, the fact that a model predicts accurate displacements of the key, in response to force inputs, does not proof its validity. Such an assertion is a radical change with what has been done for nearly fifty years. Then, a complete model with six degrees of freedom based the one proposed in [Lozada, 2007] has been presented. All the values of its parameters have been measured. The model includes non-smooth friction and contact laws. Instead of regularising them as it has been done up to now, we chose to treat them as such. The reasons of this choice are given, and a quantitative comparison between a regularised approach and a non-smooth approach was illustrated with a pendulum including joint dry friction. The dynamical model of the piano action, as a non-smooth multibody system, was solved with the numerical methods implemented in the computer program XDE. The use of XDE implied a few adjustments. The felt law were manually implemented which slowed down the computations. Also, the position input was imposed in XDE by means of a PD controller placed on the virtual key. In order to make simulations and experiments truly comparable, a piston was inserted between the finger and the real key. The piston had to be adjusted carefully so that its model would correspond to a PD controller. Using key positions as inputs, predicted forces were found to be reasonably similar to the measured ones, in both piano and forte dynamics. Most of the measured irregularities were reflected in the calculations. A sensitivity analysis of the reaction force to the numerous model’s parameters has been proposed. The high-sensitivity of the reaction force to geometrical adjustments, well known by piano technicians, was rendered. The analysis also allowed to detect the misadjustment of one parameter, which is a strong indication of the model’s validity, as well as the way it has been simulated. The results of the sensitivity analysis are largely conditioned by the choice of the cost function. The chosen one, which measured the difference between the simulated force and the measured force, did not render the amplitude of some events which are known to be essential for the touch of pianists, such as the let-off. Better choices of the cost function could lead to more relevant results in haptics. The computations are about twenty times slower than real time, with a standard laptop. The calculation time is expected to be improved by a factor 50 by taking into account the

123 CONCLUSION

speciﬁcities of the model (felt laws, planarity). Integrating them is the last step to make the implementation of the model and its simulation methods usable in haptic devices. The following step is the design of an interface capable of applying a force input faithfully. This is a very reasonable goal, since a lot of fruitful efforts have already been made in that direction. We are conﬁdent that it will open doors to new design methods for grand piano actions, to digital pianos with improved haptics and modulable touch, and to many diverse experiments for the understanding of what, in the piano action dynamics, allows the pianist to control the hammer so accurately.

124 Bibliography

Acary, V. and Brogliato, B. (2008). Numerical methods for nonsmooth dynamical systems: applications in mechanics and electronics, volume 35. Springer. Askenfelt, A. and Jansson, E. V. (1990). From touch to string vibrations. i: Timing in the grand piano action. The Journal of the Acoustical Society of America 88, 52. Askenfelt, A. and Jansson, E. V. (1991). From touch to string vibrations. ii: The motion of the key and hammer. The Journal of the Acoustical Society of America 90, 2383. Barbacci, A., Diener, J. et al. (2013). A robust videogrametric method for the velocimetry of wind-induced motion in trees [submitted]. Agricultural and Forest Meteorology . Blackham, E. D. (1965). The physics of the piano. Scientific American 213, 88–99. Bokiau, B., Poncelet, A., Fisette, P. and Docquier, N. (2012). Multibody model of a grand piano action aimed at understanding and demystifying the escapement principle. The 2nd Joint International Conference on Multibody System Dynamics. Brenon, C. (2002). Mécanique de la touche de piano. Master’s thesis, Université Pierre et Marie Curie, Paris VI, France. Cadoz, C., Lisowski, L. and Florens, J.-L. (1990). A modular feedback keyboard design. Computer music journal 14, 47–51. Chartrand, R. (2011). Numerical differentiation of noisy, nonsmooth data. ISRN Applied Mathematics 2011. Chu, L. (1996). Haptic feedback in computer music performance. In Proceedings of ICMC, volume 96. Clementi, M. (1803). Introduction to the art of playing on the pianoforte. . Closson, E. and Ames, D. (1977). History of the Piano. Scholarly Press. Cull, S. and Tucker, R. (1999). On the modelling of coulomb friction. Journal of Physics A: Mathematical and General 32, 2103. Dahlquist, G. G. (1963). A special stability problem for linear multistep methods. BIT Numerical Mathematics 3, 27–43. Dennerlein, J. T., Mote Jr, C. and Rempel, D. M. (1998). Control strategies for finger movement during touch-typing the role of the extrinsic muscles during a keystroke. Experimental Brain Research 121, 1–6. Deuflhard, P. (1987). Uniqueness theorems for stiff ODE initial value problems. ZIB. Dijksterhuis, P. (1965). De piano. Nederlandse Akoest. Genootschap 7, 50–65. Gillespie, B. (1994). The virtual piano action: Design and implementation. In Proceedings

125 Bibliography

of the international Computer Music Conference. INTERNATIONAL COMPUTER MUSIC ACCOCIATION. Gillespie, B. and Cutkosky, M. (1992). Dynamical modeling of the grand piano action. In Proceedings of the International Computer Music Conference. International Computer Music Association. Gillespie, B. and Cutkosky, M. (1993). Interactive dynamics with haptic display. In 2nd Ann. Symp. on Haptic Interfaces for Virtual Environment and Teleoperator Systems, ASME/WAM, New Orleans, LA, DSC. Gillespie, R. B. (1996). Haptic display of systems with changing kinematic constraints: The virtual piano action. Ph.D. thesis, stanford university. Gillespie, R. B., Yu, B., Grijalva, R. and Awtar, S. (2011). Characterizing the feel of the piano action. Computer Music Journal 35, 43–57. Goebl, W., Bresin, R. and Galembo, A. (2005). Touch and temporal behavior of grand piano actions. The Journal of the Acoustical Society of America 118, 1154. Goebl, W. and Palmer, C. (2008). Tactile feedback and timing accuracy in piano performance. Experimental Brain Research 186, 471–479. Hairer, G. W. E. (2010). Solving ordinary differential equations II. Springer. Hall, D. E. and Askenfelt, A. (1988). Piano string excitation v: Spectra for real hammers and strings. The Journal of the Acoustical Society of America 83, 1627. Hayashi, E., Yamane, M. and Mori, H. (1999). Behavior of piano-action in a grand piano. i. analysis of the motion of the hammer prior to string contact. The Journal of the Acoustical Society of America 105, 3534. Hirschkorn, M., McPhee, J. and Birkett, S. (2006). Dynamic modeling and experimental testing of a piano action mechanism. Journal of computational and nonlinear dynamics 1, 47–55. Hirschkorn, M. C. (2004). Dynamic model of a piano action mechanism. Ph.D. thesis, University of Waterloo. Horváth, P. and Törocsik,˝ D. (2013). Magnetic issues of a haptic keyboard. SzÃl’chenyi IstvÃ ˛an University . Hunt, K. and Crossley, F. (1975). Coefﬁcient of restitution interpreted as damping in vibroimpact. J. Appl. Mech. . Izadbakhsh, A. (2006). Dynamics and control of a piano action mechanism. Master thesis . Izadbakhsh, A., McPhee, J. and Birkett, S. (2008). Dynamic modeling and experimental testing of a piano action mechanism with a ﬂexible hammer shank. Journal of computational and nonlinear dynamics 3. Links, H. (2011). Modeling of a grand piano action mechanism. Master Thesis . Lozada, J. (2007). Modélisation, contrôle haptique et nouvelles réalisations de claviers musicaux. Ph.D. thesis, Ecole Polytechnique.

126 Bibliography

Lucas, B. D., Kanade, T. et al. (1981). An iterative image registration technique with an application to stereo vision. In IJCAI, volume 81. Lutringer-Flecher, M. (2002). Le concept de toucher dans la pratique instrumentale du piano. Master’s thesis, Université Marc Bloch, France. Mamou-Mani, A. and Maniguet, T. (2009). Investigating the history of the piano action using scientiﬁc calculus: The case of an érard grand piano from 1802 . Masoudi, R. (2012). Micromechanics of Fiber Networks Including Nonlinear Hysteresis and its Application to Multibody Dynamic Modeling of Piano Mechanisms. Ph.D. thesis, University of Waterloo. Masoudi, R. and McPhee, J. (2012). A mechanistic model of compression hysteresis in compliant interfaces in multibody dynamic simulation of a piano action mechanism. The 2nd Joint International Conference on Multibody System Dynamics. Merlhiot, X. (2011). On some industrial applications of time-stepping methods for nonsmooth mechanical systems: issues, successes and challenges. In Euromech Colloquium [516]-Nonsmooth contact and impact laws in mechanics. Merlhiot, X. (2012). Contact determination methods for nonsmooth time-stepping schemes. Presented at Summer School on Nonsmooth Mechanics: Modeling and Simulation, Aussois. Merlhiot, X., Le Garrec, J., Saupin, G. and Andriot, C. (2012). The XDE mechanical kernel: Efﬁcient and robust simulation of multibody dynamics with intermittent nonsmooth contacts. The 2nd Joint International Conference on Multibody System Dynamics. Mori, T. (1997). A comparison between upright and grand pianos. volume 83. Journal of the European Acoustics Association. Oboe, R. (2006). A multi-instrument, force-feedback keyboard. Computer Music Journal 30, 38–52. Oboe, R. and Poli, G. D. (2002). Multi-instrument virtual keyboard - the mikey project. Oledzki, A. (1973). Dynamics of piano mechanisms. Mechanism and Machine Theory 7, 373–385. Orocos (2013). The Orocos Project. Pfeiffer, W. (1950). Über Dämpfer, Federn und Spielart, volume 1. Verlag Das Musikinstrument. Pfeiffer, W. (1962). Vom Hammer. Verlag Das Musikinstrument. Pfeiffer, W., Engelhardt, J. and Neupert, H. (1967). The piano key and whippen: An analysis of their relationships in direct blow actions, volume 18. Verlag Das Musikinstrument. Pollens, S. (2002). The gatti-kraus piano action ascribed to bartolomeo cristofori. The Galpin Society Journal 55, 269–278. Porter, L. (2009). The Piano the Pianist. AuthorHouse.

127 Principeaud, N. and Boutillon, X. (2008). Reproducibility of piano playing. The Journal of the Acoustical Society of America 123, 3125–3125. Repp, B. H. (1999). Control of expressive and metronomic timing in pianists. Journal of Motor Behavior 31, 145–164. Rimski-Korsakov, A. (1937). The problem of a piano hammer striking a string (parts 1 and 2). Zhurnal Tehnicheskoi Fiziki 7, 43–74. Rimski-Korsakov, A. and Maveev, P. (1938). Investigation of dynamical properties of the piano key action. SRIMI , 159. Rimski-Korsakov, A. and Samoilenko, K. (1937). Experimental determination of stiffness and friction of hammer piano felt. Zhurnal Tehnicheskoi Fiziki 7, 135–150. Serina, E. R., Mockensturm, E., Mote Jr, C. and Rempel, D. (1998). A structural model of the forced compression of the ﬁngertip pulp. Journal of biomechanics 31, 639–646. Stewart, D. E. (2000). Rigid-body dynamics with friction and impact. SIAM review 42, 3–39. Studer, C. (2009). Numerics of Unilateral Contacts and Friction, volume 47. Springer. Times, N. Y. (1904). Great piano bonﬁre . Tomasi, C. and Kanade, T. (1991). Detection and tracking of point features. School of Computer Science, Carnegie Mellon Univ. Van den Berghe, G., De Moor, B. and Minten, W. (1995). Modeling a grand piano key action. Computer Music Journal 19, 15–22. Vyasarayani, C. P. (2009). Transient Dynamics of Continuous Systems with Impact and Friction, with Applications to Musical Instruments. Ph.D. thesis, University of Waterloo. Vyasarayani, C. P., Birkett, S. and McPhee, J. (2009). Modeling the dynamics of a compliant piano action mechanism impacting an elastic stiff string. The Journal of the Acoustical Society of America 125, 4034. Appendix A Adjustment procedure of the grand A piano action

In this Appendix, the regulation procedure which has been applied to our piano action is presented. It is a transcription of exchanges with two professional technicians. There are several other procedures, either spectiﬁc to given actions or universal. The following one is presented as a typical universal procedure. Main technical terms are given in Figure A.1. The different adjustments are shown in Figure A.2. It is interesting to notice that, according to one of the professional technicians we met, it is not possible to regulate accurately an action without being able to touch it by yourself.

damper head string tuning pin

repetition lever

hammer shank q hammer drop screw jack backcheck let-off button let-off button punching damper lever whippen

spoon capstan screw key

Figure A.1 – Terminology of the grand piano action [after Olek Remesz / Wikimedia Commons].

Preliminary adjustments Key balance rail When lifted, the key must fall down very slowly back to its normal resting position.

I APPENDIX A. ADJUSTMENT PROCEDURE OF THE GRAND PIANO ACTION

Figure A.2 – Location of the regulation settings.

Hammer pivot The friction must be such that when unscrewed and held vertically (downward), the hammer must stop after three pseudo-periods from the maximum angle state1.

General alignment All the bodies must be parallel and aligned.

Adjustments

Jack – Hammer At the resting position, the jack must be aligned with the wooden rectangle of the hammer knuckle as shown in Figure A.3.

Figure A.3 – Appropriate adjustment of the jack regulating screw.

Lever The lever must be in contact with the hammer knuckle. The screw of the lever must be adjusted such that the jack is between 0.1 mm and 0.3 mm away from the knuckle.

Whippen – Key (capstan screw) When the distance between the key and the front rail punching is 10 mm (it has to be between 9.5 mm and 10.5 mm), the distance between the hammer and the string must be of 45 mm to 47 mm. The adjustment is done with the capstan screw.

1More accurate measurements involving a dynamometer may also be used.

II Escapement button The hammer must escape as close as possible to the string, typically of 1.5 mm or less. This is adjusted with the escapement button.

Drop screw When the key is depressed, the drop screw must be adjusted so that the hammer falls outright but not too much.

Backcheck – Hammer The backcheck wire is adjusted so that for forte keystrokes, the hammer is caught a 15 mm of the string. Also, the contact has to occur at the top third of the backcheck.

Damper – Key The spoon can be deformed so that it gets in contact with the key when the hammer is half-way through (23.5 mm).

Mass and inertial of the key The can be changed by adding lead, so that the minimum downward weight of the action without damper is close to 50 g and the maximum upward weight is 25 g. As it is a destructive operation, it is done only if it cannot be avoided. Note that the exact weights depend on the manufacturers and the grand piano model.

III

Appendix B Application of the Lagrangian to a B double pendulum

In this appendix, the Lagrangian is applied to a double pendulum, which is one of the simplest multibody system for which not all the bodies move in a Galilean frame. The goal is to illustrate the efﬁciency of the Lagrangian and the physical meaning of the underlying equations. Furthermore, the dynamics of the double pendulum has similarities with that of the whippen-jack-lever assembly as both the jack and the lever are assembled to the whippen with hinge joints, and the whippen himself is ﬁxed to the ground with a hinge joint, see Figure B.1.

Figure B.1 – Non-inertial frame of the lever and the jack connected to the whippen with hinges joints.

Figure B.2 – Scheme and parametrisation of the double pendulum.

The double pendulum is described in ﬁgure B.2. No external loading are considered, for simplicity. The Lagrangian L is therefore equal to the kinetic energy T .

V APPENDIX B. APPLICATION OF THE LAGRANGIAN TO A DOUBLE PENDULUM

Generalized coordinates The position of the masses in the ground frame is denoted by 4 2 X = [X 1 X 2] ∈ R where X i ∈ R is the position of the mass i in the ground frame. The double pendulum can also be described by the angles θ1 and θ2 ∈ R. These two scalars fully describe its state as the double pendulum is a 2-DOF mechanism. They deﬁne the vector of generalised coordinates x:

θ (t) x(t)= 1 (B.1) θ2(t)   The geometrical link between x and X is given by the following geometric relations:

l1 sin(θ1)

θ1  −l1 cos(θ1)  X(t)= φ(x(t)) with φ : x = ∈ R2 7−→ ∈ R4 (B.2) θ   l sin(θ )+ l sin(θ )  2  1 1 2 2      −l1 cos(θ1) − l2 cos(θ2)     The equations of motion can then be calculated from the Euler-Lagrange equation:

m1 0 0 0 d ∂ L ∂ L 1  0 m 0 0  − = 0 with L = T = XT M X˙ and M = 1 (B.3) dt ∂ x˙ ∂ x 2  0 0 m 0   2     0 0 0 m2     Equation B.3 yields:

(m + m ) l2 θ¨ + m l l cos(θ − θ ) θ¨ − +m l l sin(θ − θ )θ˙2 = 0 1 2 1 1 2 1 2 1 2 2 2 1 2 1 2 2 (B.4)  m l l cos(θ − θ ) θ¨ + −m l l sin(θ − θ )θ˙2 = 0  2 1 2 1 2 1 2 1 2 1 2 1 linear inertia nonlinear inertia  The prefactors of the linear accelerations are gathered in the mass matrix M(x) whereas N(x, x˙) gathers the nonlinear acceleration terms and their prefactors, so that the equation of motion ﬁnally comes as: M(x) x¨ + N(x, x˙) x˙ = 0 (B.5)

Mass matrix M The generalised mass matrix is given by

l2 (m + m ) l l m cos(θ − θ ) M(x)= 1 1 2 1 2 2 1 2 (B.6)  2  l1l2 m2 cos(θ1 − θ2) l2 m2   which is a symmetrical matrix. It is noteworthy that M could also have been calculated from the kinetic energy T :

1 T 1 1 T = X˙ M X˙ = x˙T J (x)T M J (x) x˙ = x˙T M(x) x˙ =⇒ M(x)= J (x)T M J (x) 2 2 φ φ 2 φ φ (B.7)

VI where Jφ is the Jacobian matrix of φ deﬁned in (B.2) equal to

l1 cos(θ1) 0

l1 sin(θ1) 0  Jφ = (B.8) l cos(θ ) l cos(θ )  1 1 2 2    l1 sin(θ1) l2 sin(θ2)    

Nonlinear inertial terms N The calculation of N yields

0 θ˙ N(x, x˙)= m l l sin(θ θ ) 2 (B.9) 2 1 2 1 − 2 ˙  −θ1 0    which actually corresponds to the contribution of the variation of mass in the conservation of the momentum:

d ˙ (M(x) x˙)= M(x) x¨ + M(x) x˙ = M(x) x¨ + N(x, x˙) x˙ (B.10) dt

Also, it can be veriﬁed that: ˘ 1 ∂ M ∂ M ∂ M _ i j ik k j (N(x, x) x˙)i = x˙k + − x˙ j (B.11) 2 ∂ xk ∂ x j ∂ xi ! Xj,k

VII

Appendix C

C Example of the calculation of ∂xδKH

The purpose of the appendix is to make the link between the matricial expression of the δ lever arms ∂x and their geometrical expression. Since

∂ δ T (x) = ∂ x !1 ∂ δKS+ ∂ δKW ∂ δKS− ∂ δKH Ç (xå) (x) (x) (x) 0000000000 (C.1) ∂θK ∂θK ∂θK ∂θK comes ñ ô ∂ δ T (x) F(δ) = ∂ x !1 ∂ δKS+ ∂ δKW ∂ δKS− ∂ δKH Ç (x) Få(δKS+ (x))+ (x) F(δKW(x))+ (x) F(δKS− (x))+ (x) F(δKH(x)) ∂θK ∂θK ∂θK ∂θK (C.2)

The gap functions can easily be evaluated by geometrical calculations. For example,

+ δKS+ (x)= S KS+ · e y (C.3)

key key + S+ + S = KS+ + δ + KS + KS+ felt felt

Figure C.1 – Scheme of the key compressing the front rail punching. Left: Beginning of contact. Right: Stricly positive compression of the felt.

IX APPENDIX C. EXAMPLE OF THE CALCULATION OF ∂XδKH

S+ is the centre point of the top of the front rail punching at rest (see Figure C.1), of ﬁxed coordinates S+ S+ = x (C.4)  + Sy   and KS+ is the point of the key which is the closest to the front rail punching (see Fig- ure C.1). Its coordinates are given by:

KS+ x (0) cos(θK) − sin(θK) KS+ (x)= · (C.5) KS+ y (0) sin(θK) cos(θK)      so that ﬁnally: + δKS+ (x)= KS+ x (0) sin(θK)+ KS+ y (0) cos(θK) − S y (C.6)

Then,

∂ δKS+ (x)= KS+ x (0) cos(θK) − KS+ y (0) sin(θK) (C.7) ∂θK which corresponds to the lever of the reaction force exerted by the front rail punching on the key. ∂ δ T The other components of (x) can be calculated the same way. ∂ x

Ç å

Modéle dynamique non-régulier de la touche de piano à queue

Le mécanisme de la touche de piano à queue sert à propulser le marteau vers les cordes. Ce mé- canisme permet au pianiste de contrôler avec précision la vitesse et l’instant d’impact du marteau sur la corde. Il est raisonnable de penser que c’est le comportement dynamique de la touche qui permet cette contrôlabilité. Avec pour perspective l’amélioration du rendu haptique des claviers numériques, cette thèse propose une méthode de simulation d’un modèle complet du mécanisme. Le son généré par la vibration qui suit l’impact du marteau sur les cordes n’entre pas dans le cadre de l’analyse. Des modèles du mécanisme comportant plusieurs degrés de liberté, des frottements et des contacts intermittents, ont été proposés depuis une quinzaine d’années. Notre approche se distingue de celles suivies jusqu’ici par un changement du point de vue adopté pour valider et pour simuler le modèle. En se fondant sur l’étude approfondie d’un modèle à un degré de liberté, il est en effet montré que la simulation d’un modèle dynamique complet doit se faire à l’aide d’un pilotage en déplacement, tandis que les travaux récents et anciens présentent des simulations pilotées en force. Une analyse des problèmes numériques liés aux discontinuités de vitesses survenant au sein du mécanisme durant l’enfoncement de la touche est présentée. Ils sont résolus par des méthodes de dynamique non-régulière implémentées dans le logiciel XDE. Les résultats sont présentés sous forme de comparaison avec les mesures expérimentales. La plupart des irrégularités des forces mesurées se retrouvent dans les forces simulées, en jeu piano comme en jeu forte. Les simulations rendent également bien compte de la cinématique de chaque élément du mécanisme. Une analyse de sensibilité du comportement dynamique aux paramètres du modèle est enﬁn exposée. Mots-clefs : piano, dynamique multi-corps, dynamique non régulière, modélisation, simulation, haptique.

Non-smooth model of the grand piano action

Cover picture: Capture from the computer program XDE during a virtual keystroke.