Modeling and simulation of a grand piano

The aim of this thesis is to develop a physical model for vibration phenomena that are at the origin of piano tones. Generation of piano sound is composed of various parts, each one has a submodel in the present work. Chapter 1 deals with partial differential equations modeling waves and vibrations. The equations involved in the model of string vibration (which will be analitically solved in chapter 4) are all hyperbolic pdes, so it is necessary to know the basics ofwave equations and related solution methods. In chapter 2 we present the following physical submodels: interaction between strings and hammers, and the string vibration. It is too difficult to solve analitically the string vibration’s equations, so we analyze a simplified model in chapter 3. Under the hypoteses of "simply supported" boundary conditions (zero displacement and zero torque) we obtain a simplified model, in whichthe piano string is considered as a prestressed Timoshenko beam, that can be solved analitically. We obtain two branches of eigenfrequencies, which correspond to the two propagating waves in the system: flexural and shear. A pratical audio sample showsthe difference between the c2 note (the c note of the second octave of a piano) with or whithout the longitudinal modes. In chapter 4 the matematical model for the soundboard, which is essential to explain the strings-soundboard coupling at the bridge, is presented. Chapter 5 treats structural acoustics and sound propagation, which is the perceptiblepart of the model. The pde system of the full piano model including all the submodels explained in previous chapters is presented. Finally, chapter 6 shows analytically the differences between a real piano soundand a virtually generated one, by using sound spectrum and 3d frequency analysis.