Sound synthesis of a 3D nonlinear string using a covariant Lie group integrator of a geometrically exact beam model

Abstract

We present a three dimensional nonlinear string model based on a geometrically exact beam. The beam model is obtained by applying a variational principle using a covariant Lagrangian formulation; in particular, the equations of motion and the boundary conditions are treated in an unified manner. Following an analogous discrete variational principle, a Lie group variational integrator is given. The energy and momentum conservation properties of the integrator are discussed and illustrated. This geometrically exact beam serves as a basis to formulate a prestressed damped string model with coupled non trivial boundary conditions. Simulation results are discussed and validated against analytical solutions obtained in the context of a small displacement hypothesis.