Linearised models are often invoked as a starting point to study complex dynamical systems. Besides their attractive mathematical simplicity, they have a central role for determining the stability properties of static or dynamical states, and can often shed light on the influence of the control parameters on the system dynamical behaviour. While the bowed string dynamics has been thoroughly studied from a number of points of view, mainly by time-domain computer simulations, this paper proposes to explore its dynamical behaviour adopting a linear framework, linearising the friction force near an equilibrium state in steady sliding conditions, and using a modal representation of the string dynamics. Starting from the simplest idealisation of the friction force given by Coulomb's law with a velocity-dependent friction coefficient, the linearised modal equations of the bowed string are presented, and the dynamical changes of the system as a function of the bowing parameters are studied using linear stability analysis. From the computed complex eigenvalues and eigenvectors, several plots of the evolution of the modal frequencies, damping values, and modeshapes with the bowing parameters are produced, as well as stability charts for each system mode. By systematically exploring the influence of the parameters, this approach appears as a preliminary numerical characterisation of the bifurcations of the bowed string dynamics, with the advantage of being very simple compared to sophisticated numerical approaches which demand the regularisation of the nonlinear interaction force. To fix the idea about the potential of the proposed approach, the classic one-degree-of-freedom friction-excited oscillator is first considered, and then the case of the bowed string. Even if the actual stick-slip behaviour is rather far from the linear description adopted here, the results show that essential musical features of bowed string vibrations can be interpreted from this simple approach, at least qualitatively. Notably, the technique provides an instructive and original picture of bowed motions, in terms of groups of well-defined unstable modes, which is physically intuitive to discuss tonal changes observed in real bowed string.
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March 2017
March 23 2017
Linear modal stability analysis of bowed-strings
V. Debut;
V. Debut
a)
1Instituto de Etnomusicologia, Música e Dança, Faculdade de Ciências Sociais e Humanas,
Universidade Nova de Lisboa
, 1069-061 Lisboa, Portugal
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J. Antunes;
J. Antunes
b)
2Centro de Ciências e Tecnologias Nucleares, Instituto Superior Técnico,
Universidade de Lisboa
, Estrada Nacional 10, Km 139.7, 2695-066 Bobadela LRS, Portugal
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O. Inácio
O. Inácio
3
Instituto Politécnico do Porto
, ESMAE, Musical Acoustics Laboratory, Porto, Portugal
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a)
Also at: Centro de Ciências e Tecnologias Nucleares, Instituto Superior Técnico, Universidade de Lisboa, Estrada Nacional 10, Km 139.7, 2695-066 Bobadela LRS, Portugal. Electronic mail: [email protected]
b)
Also at: Instituto de Etnomusicologia, Música e Dança, Faculdade de Ciências Sociais e Humanas, Universidade Nova de Lisboa, 1069-061 Lisboa, Portugal.
J. Acoust. Soc. Am. 141, 2107–2120 (2017)
Article history
Received:
October 14 2016
Accepted:
January 22 2017
Citation
V. Debut, J. Antunes, O. Inácio; Linear modal stability analysis of bowed-strings. J. Acoust. Soc. Am. 1 March 2017; 141 (3): 2107–2120. https://doi.org/10.1121/1.4976092
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