Stability and dispersion analysis of improved time discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string

Abstract

We study the implicit time discretization of piano strings governing equations within the Timoshenko prestressed beam model. Such model features two different waves, namely the flexural and shear waves, that propagate with very different velocities. We present a novel implicit time discretization that reduces the numerical dispersion while allowing the use of a large time step in the numerical computations. After analyzing the continuous system and the two branches of eigenfrequencies associated with the propagating modes, the classical θ-scheme is studied. We present complete new proofs of stability using energy-based approaches that provide uniform results with respect to the featured time step. A dispersion analysis confirms that θ = 1/12 reduces the numerical dispersion, but yields a severely constrained stability condition for the application considered. Therefore we propose a new θ-like scheme, which allows to reduce the numerical dispersion while relaxing this stability condition. Stability proofs are also provided for this new scheme. Theoretical results are illustrated with numerical experiments corresponding to the simulation of a realistic piano string.

Time domain simulation of a piano. Part 1: model description

“…We shall not detail here these analytical computations which are long and tedious but straightforward (see [13]) and shall restrict ourselves to describe the most useful results. As we have a system of two second order equations, it is not surprising that these modes can be split into two families of modes (the following splitting appears naturally in the analytical computations)…”

Vibration of a Timoshenko beam supporting arbitrary large pre-deformation

“…As the present model is valid even for non-asymptotical situation, it looks to be pertinent to establish some intermediate problem. This work may extend other approaches dedicated to the string of musical instruments [23,40].…”

“…In general, three difficulties are present: the nonlinear stress-strain relation (large strain and nonlinear constitutive laws of material), the moving frame induced by large displacement, and the load sensitivity to the transformation (leading to the distinction between dead and followers loads). This explains why, in most of the cases, the methods are dedicated to a particular application that permits simplifications and in some case an analytical formulation [19,23].…”

Mathematical and Numerical Study of Transient Wave Scattering by Obstacles with a New Class of Arlequin Coupling

“…It is then clear that the derivation of the discrete energy relation is independent of the coupling procedure, therefore with some standard arguments (see [8] for instance) we obtain…”