Tallinn University of Technology
Department of Cybernetics
Laboratory of Solid Mechanics
Title String collision and sliding against a smooth obstacle in a non-planar vibration setting
Author Dmitri Kartofelev
Tallinn University of Technology, Department of Cybernetics, Tallinn, Estonia
Abstract This paper proposes a model of a plucked ideal string free to vibrate in two mutually perpendicular planes in the presence of a curved obstacle. We consider an obstacle which spans both in the direction of the string at its rest and perpendicular to it. The obstacle, mainly spanning perpendicularly to the string, is essentially a rod with a curved cross-section. The string collisions against the obstacle are modelled using a kinematic model that constrains the string displacement unilaterally. The string sliding against the obstacle surface is modelled using a dry sliding friction model proposed by Jim Woodhouse. The sliding friction model is coupled with the collision model. The virtual reaction force of a collision is used as an input into the dry sliding model. A virtual compression of the obstacle material is used to find the collision force. For simplicity, the material is assumed to be linearly elastic, capable of accepting forces in a single direction only, and lacking internal bulk or shear forces. Numerical simulation results show that the model can generate a rich variety of dynamics all strongly dependent on the selection of initial conditions and system parameters. This nonlinear model is numerically robust due to the used simplifying assumptions. Interactions of a vibrating string with spatially distributed barriers, such as fretboards, play a significant role in the physics of many stringed musical instruments. It is believed that the proposed model can be applied to sound synthesis and physics-based modelling of these musical instruments.
Status The presentation was presented at the 22nd ISNA 2022 conference. The conference was hosted by the St Catherine’s College at Oxford University, Oxford, England, UK on July 4–8, 2022.
Presentation slides
String collision and sliding against a smooth obstacle in a non-planar vibration setting
The slides of the presentation given at the 22nd ISNA 2022 conference:
Slides: Slides.pdf
The original videos embedded into the Slides.pdf file:
Videos: isna22vid1.mp4, isna22vid2.mp4, isna22vid3.mp4, isna22vid4.mp4.
Acknowledgments
This work was supported by the Estonian Ministry of Education and Research (PRG1227), EU through the European Regional Development Fund.    

String collision and sliding against a smooth obstacle in a non-planar vibration setting