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When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given e.g. AUTHOR (year of submission) "Full thesis title", University of Southampton, name of the University School or Department, PhD Thesis, pagination http://eprints.soton.ac.uk University of Southampton Faculty of Engineering, Science and Mathematics Institute of Sound and Vibration Research (ISVR) A Modal Method for the Simulation of Nonlinear Dynamical Systems with Application to Bowed Musical Instruments by Octávio José Patrício Fernandes Inácio Thesis for the degree of PhD in Sound & Vibration July 2008 UNIVERSITY OF SOUTHAMPTON ABSTRACT Bowed instruments are among the most exciting sound sources in the musical world, mostly because of the expressivity they allow to a musician or the variety of sounds they can generate. From the physical point of view, the complex nature of the nonlinear sound generating mechanism – the friction between two surfaces – is no less stimulating. In this thesis, a physical modelling computational method based on a modal approach is developed to perform simulations of nonlinear dynamical systems with particular application to friction-excited musical instruments. This computational method is applied here to three types of systems: bowed strings as the violin or cello, bowed bars, such as the vibraphone or marimba, and bowed shells as the Tibetan bowl or the glass harmonica. The successful implementation of the method in these instruments is shown by comparison with measured results and with other simulation methods. This approach is extended from systems with simple modal basis to more complex structures consisting of different sub-structures, which can also be described by their own modal set. The extensive nonlinear numerical simulations described in this thesis, enabled some important contributions concerning the dynamics of these instruments: for the bowed string an effective simulation of a realistic wolf-note on a cello was obtained, using complex identified body modal data, showing the beating dependence of the wolf- note with bowing velocity and applied bow force, with good qualitative agreement with experimental results; for bowed bars the simulated vibratory regimes emerging from different playing conditions is mapped; for bowed Tibetan bowls, the essential introduction of orthogonal mode pairs of the same family with radial and tangential components characteristic of axi-symmetrical structures is performed, enabling an important clarification on the beating phenomena arising from the rotating behaviour of oscillating modes. Furthermore, a linearized approach to the nonlinear problem is implemented and the results compared with the nonlinear numerical simulations. Animations and sounds have been produced which enable a good interpretation of the results obtained and understanding of the physical phenomena occurring in these system. FACULTY OF ENGINEERING, SCIENCE & MATHEMATICS INSTITUTE OF SOUND AND VIBRATION RESEARCH Doctor of Philosophy A Modal Method for the Simulation of Nonlinear Dynamical Systems with Application to Bowed Musical Instruments by Octávio José Patrício Fernandes Inácio CONTENTS LIST OF FIGURES ......................................................................................................iv LIST OF TABLES.......................................................................................................xii LIST OF ACCOMPANYING MATERIAL.............................................................. xiii DECLARATION OF AUTHORSHIP........................................................................xiv ACKNOWLEDGEMENTS.........................................................................................xv DEFINITIONS AND ABBREVIATIONS USED .....................................................xvi PRESENTATION..........................................................................................................1 1. INTRODUCTION .................................................................................................4 1.1. Friction-excited vibrations and sound ..........................................................4 1.2. Simulation Methods and Dry Friction Models..............................................6 1.2.1 The Stick-Slip Phenomenon .................................................................10 1.2.2 Friction vibrations in Industrial components........................................11 1.2.3 Friction vibrations in Musical Instruments...........................................11 1.3. Research on Bowed Musical Instruments ...................................................15 1.3.1 Bowed bars...........................................................................................16 1.3.2 Bowed shells.........................................................................................16 1.3.3 Bowed strings.......................................................................................17 1.4. Aims and contributions................................................................................19 1.4.1 Contributions on the modelling techniques..........................................19 1.4.2 Contributions on the results..................................................................20 2. GENERIC SIMULATION METHODOLOGY ..................................................23 2.1. Introduction.................................................................................................23 2.2. Nonlinear computations in the time domain using a modal basis ..............23 2.2.1 Strings...................................................................................................25 2.2.2 Bars with constant cross section...........................................................25 2.2.3 Bars with variable cross section ...........................................................26 2.2.4 Axi-symmetric structures (bodies of revolution)..................................26 2.3. Friction model.............................................................................................28 2.3.1 Classical Coulomb model.....................................................................28 2.3.2 Spring-dashpot true adherence model ..................................................30 2.3.3 Pseudo-adherance with a regularized near-zero velocity model ..........31 2.3.4 Other friction models............................................................................33 2.4. Numerical aspects .......................................................................................34 2.4.1 Integration algorithm............................................................................34 2.4.2 Comparison with a classical approach..................................................35 2.5. Advantages and disadvantages of the method.............................................45 3. BOWED STRINGS.............................................................................................49 3.1. Introduction.................................................................................................49 3.1.1 Bow Width............................................................................................49 3.1.2 String/Body Coupling...........................................................................52 3.2. Computational Model..................................................................................54 3.2.1 Formulation of the string dynamics......................................................54 3.2.2 Friction Model......................................................................................58 3.2.3 Formulation of the Body Dynamics .....................................................62 3.2.3.1 Incremental Convolution Formulation..............................................62 3.2.3.2 Modal Formulation ...........................................................................63 3.2.3.3 Discussion of the Body Dynamics Formulation Methods .................64 3.2.4 Formulation of the String/Body Coupling............................................64 i 3.3. Numerical Results .......................................................................................65 3.3.1 Influence of the string inharmonicity ...................................................65 3.3.2 Influence of the string torsion...............................................................66 3.3.3 Influence of the bow width...................................................................69 3.3.3.1 Influence of the number of bow pseudo-hairs ...................................69 3.3.3.2 Influence of the width of the bow ......................................................70 3.3.3.3 Input parameter dependence.............................................................77 3.3.3.4 Motion Regimes.................................................................................79 3.3.3.5 Flattening effect.................................................................................82