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a bibo:Document;
dcterms:medium ;
rdau:P60049 ;
rdau:P60050 ;
rdau:P60048 ;
dc:identifier "(DE-101)1329061667";
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dc:identifier "(OCoLC)1437757955";
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rdau:P60049 ;
dc:title "Theory and modeling of bowed strings";
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marcRole:dgg ;
dc:publisher "Universität";
rdau:P60163 "Freiburg";
rdau:P60333 "Freiburg : Universität";
rdau:P60489 "Dissertation, Universität Freiburg, 2024";
dcterms:description "Abstract: In this thesis – titled Theory and Modeling of Bowed Strings – we use a wide spectrum of mathematical methods, most importantly theory of PDEs, modeling of mechanical systems, numerics, acoustomechanics and implementation with C++, together with the theory of Cosserat rods to precisely model and analyze a violin string when bowed by a violin bow over a period of time. In particular we are interested in the initial acceleration of the string resulting from the bow force (where the string is assumed to be at rest at time t = 0) and the torsional effects that appear here. We also analyse which parameters lead to which motion and how to receive a stable Helmholtz motion in a short period of time, that is, after only a few stretch waves. This is essential for good violin sound.
We start with an Introduction in Part I where we briefly lay out the history of models for bowed strings and also the existing discretization models for static and dynamic Cosserat rods where the latter is an active field of research nowadays. The main reason for the choice of the Cosserat rod theory is that we have the possibility to not only control bending and stretching of a string but also torsion which plays a non-neglectable role in bowed strings as we will see in Part II. This part also explains the so called Helmholtz motion (a sort of stick-slip motion) that occurs when bowing a string. The surprising fact is that the extended part of the string (the „Helmholtz corner“) travels on a parabola, but contrary to the bowing direction once the Helmholtz motion is established. This is because the stretch energy in the string leads to a certain whip effect. Real-life experiments for the waveforms done by Bavu et al. in 2005 both for stretch and torsional wave speeds are shown. Furthermore we show some spectrogram plots of real-life experiments done by the author demonstrating natural bowing including bow changes, a roughly bowed string and a flageolet tone. In Part III the Cosserat rod theory is presented – including the important stored energy function and viscous energy function being quadratic in the strains and strain rates, respectively. In Part IV we refer to the work of S. S. Antman and T. I. Seidman done in 2005 and show, by using Grönwall’s lemma, that the total energy over time consisting of the string’s velocity, strains and strain rates – and of applied forces – on the one hand and on the other its analogue consisting of the string’s acceleration and strain accelerations stay bounded. We sketch how this implies existence and uniqueness of solutions in both space and time to the system of equations. The regularity of these solutions is derived. Then, the main focus of this work is set on the discretization and implementation as outlined in Part V. Discrete time-stepping Euler-Lagrange equations combined with the so called Null-Space method applied with a Newton-Raphson solver are used to calculate the Helmholtz motion (in real time). Three different algorithms for the bow-string interaction are presented. This part also includes the so called two-step algorithm that realizes the string’s internal dissipation and pictures the energy evolution over time that behaves in accordance with the theoretical derivation. Part VI shows the results of this work including detailed three- dimensional ParaView snapshots. MATLAB plots show the behaviour of the energy and energy rates over time and finally we present plots of waveform patterns of the stretch waves versus torsional waves measured at different positions of the string and compare them to real-life experiments. Numerical calculations were done both with MATLAB and C++, while MATLAB and ParaView were used for visualization. A parameter study and further application ideas situated in the Appendix show the versatility of our algorithm"@de;
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dcterms:modified "2025-12-26T07:24:01.000"^^xsd:dateTime .
dcterms:issued "2024";
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