Comparison of Finite Element Method and Modal Analysis of Violin Top Plate Ye L

Comparison of Finite Element Method and Modal Analysis of Violin Top Plate Ye Lu Music Technology Area, Department of Music Research Schulich School of Music McGill University Montreal, Canada June 2013 A thesis submitted to McGill University in partial fulfillment of the requirements for the degree of Master of Arts in Music Technology. c ⃝2013 Ye Lu 2013/06/04 i Abstract This thesis presents a study of the vibrational behaviour of the violin top plate and explores the possibility of using composite materials as a substitute for traditional wood in making top plates. Numerical simulations and experimental tests are compared to validate the results. The two most popular methods for numerical and experimental vibrational analysis, the Finite Element Method (FEM) and Experimental Modal Analysis are used, respectively. The vibrational behaviour of a spruce top plate is first studied by the two methods. The results show high coherence. Then the same modeling and testing techniques are applied on two composite plates. Results show that the vibrational behaviour of composite plates differs significantly from traditional wooden plates. Thus, suggestions for further improvement of the composite top plates are given. ii Sommaire Cette th´ ese propose une ´ etude sur le comportement vibratoire de la table d’harmonie du violon et explore la possibilit´ e d’utiliser des mat´ eriaux composites comme un substitut pour le bois traditionnel dans la fabrication des tables d’harmonies. Des simulations num´ eriques ´ et des tests exprimentaux sont compar´ es pour valider les r´ esultats. Les deux m´ ethodes les plus populaires pour l’analyse vibratoire num´ erique et exp´ erimentale, la m´ ethode des ´ el´ ements finis (FEM) et l’analyse modale exprimentale sont utilis´ es, respectivement. Le comportement vibratoire d’une table d’harmonie en ´ epic´ ea est d’abord ´ etudi´ ee par les deux m´ ethodes. Les r´ esultats montrent une coh´ erence forte. Puis la mod´ elisation et mˆ eme des techniques de test sont appliqu´ es sur deux plaques composites. Les r´ esultats montrent que le comportement vibratoire de table d’harmonie composites diff` ere sensiblement de plaques traditionnels en bois. Ainsi, des suggestions d’am´ elioration des tables d’harmonie composites du dessus sont donn´ es. iii Acknowledgments I would like to gratefully thank Prof. Gary Scavone and Prof. Larry Lessard for their supervision, guidance, and expertise during this work. I would like especially, to thank Gary for providing the various equipment to ensure my experimental test worked smoothly and for spending considerable amount of time on all aspects of my thesis. Also I would like to show my great appreciation to Larry for providing me the very helpful violin top plate prototypes and their very detailed manufacturing data. Many thanks to the luthier Peter Purich for offering the spruce plate and the Chladni pattern photos I would like to extend my appreciation to the students, faculty and staffof the Music Technology Area at McGill University for their advice, insight and friendship. In particular, I would like to thank the members of the Computational Acoustic Modeling Laboratory (CAML) for providing me a great environment to pursue my research work. I would also take this opportunity to acknowledge Hossein Mansour at CAML for all the meaningful discussions and for his valued assistance. Many thanks to Darryl Cameron and Harold Kilianski for providing necessary equipment and the technical support during my research work. A special thanks to my girlfriend Jenny Liu for kindly providing the proof reading. Finally, the greatest thanks go to my parents in China, for their financial and moral support in my most difficult times. iv Contents 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Project Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Background 5 2.1 Violin Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 The Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 The Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.3 The Bass Bar and Soundpost . . . . . . . . . . . . . . . . . . . . . 8 2.1.4 The Soundbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Procedures of the Finite Element Method . . . . . . . . . . . . . . 10 2.2.2 Node, Element and Shape Function . . . . . . . . . . . . . . . . . . 11 2.2.3 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Experimental Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.1 Mobility Measurement . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.2 Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.3 Coherence Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.4 Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 Composite Music Instruments . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 FEM and Modal Analysis Set-up 25 3.1 Numerical Model of a Spruce Top Plate . . . . . . . . . . . . . . . . . . . . 25 Contents v 3.1.1 CAD Model of a Top Plate . . . . . . . . . . . . . . . . . . . . . . . 26 3.1.2 Pre-Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1.3 Normal Mode Extraction . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Modal Analysis Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.1 Experiment Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2.2 The Choice of the Hammer . . . . . . . . . . . . . . . . . . . . . . 37 3.2.3 Accelerometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.4 Selection of the Points . . . . . . . . . . . . . . . . . . . . . . . uploads/Societe et culture/ comparison-of-finite-element-method-and-modal-analysis-of-violin-top-plate.pdf

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• Publié le Nov 08, 2021
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