[Verschoben] Jakob Rotter: Theory and Modeling of Bowed Strings including an Algorithm for Viscous Damping using the Cosserat rod theory and the Null-space method

Vortragender/Speaker: Jakob Rotter (Albert-Ludwigs-Universität Freiburg)
Ansprechpartner/Contact: Prof. Dr. Stefan Neukamm
Videostream (BBB): (on request)

Titel/Title:
Theory and Modeling of Bowed Strings including an Algorithm for Viscous Damping using the Cosserat rod theory and the Null-space method

Zusammenfassung/Abstract:
In this thesis we use the theory of Cosserat rods to precisely describe the interaction of a violin bow when bowing a violin string. The main reason for this choice of 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. We explain the so called Helmholtz 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. 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. For the existence of solutions we refer to the work of Antman-Seidman in 2005 and show, by using Grönwall's lemma, that the total energy over time consisting of velocities and strains and strain rates or of accelerations and strain accelerations - and of applied forces - stay bounded and sketch how this implies existence and uniqueness of continuous solutions in both space and time to the system of equations. Then, the main focus of this work is set on the discretization and implementation. 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. Further we present the so-called two-step algorithm that realizes the string's internal dissipation and picture the energy evolution over time that behaves in accordance with the theoretical derivation. In the end we show the results of this work including detailed three-dimensional ParaView snapshots and videos. 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 done by Bavu et al. done in 2005. Numerical calculations were done both with MATLAB and C++, while MATLAB and ParaView were used for visualization. A parameter study and further application ideas show the versatility of our algorithm.