## Role of modal approach for sound synthesis of nonlinear systems: the case of plates

Ducceschi, Michele and Touzé, Cyril

Proceedings of the Third Vienna Talk on Music Acoustics (2015), p. 133**Abstract:**Time-domain simulation of musical instruments has shown
promising results in recent years. Particularly attractive from a
sound synthesis perspective is the resolution of system displaying
some degree of nonlinearity, because of the richness of the
perceptual information that nonlinearities produce.
In this work, the focus is on one such system, namely a
flat (circular or rectangular) plate which, to a first approximation,
can efficiently simulate the sound of a gong. From
a dynamical perspective, in spite of very different geometries,
plates and gongs behave similarly, meaning that the degree of
nonlinearity is set by how large the amplitude of vibrations of
the flexural waves is. In particular, plates and gongs may attain
linear, weakly nonlinear and strongly nonlinear regimes when
the amplitude of vibrations is, respectively, much smaller, of
the same order of and larger than some defining thickness parameter.
The dynamics of plates is well described by a set of two
coupled Partial Differential Equations (PDEs) known as the von Kármán equations. For rectangular plates, a family of conservative
Finite Difference schemes was developed by Bilbao.
An alternative approach is offered in this work, where the von
Kármán equations are discretised along the modes of the system
in order to reduce the original PDEs to a set of coupled Ordinary
Differential Equations (ODEs). This is approach is referred to
as modal approach, and it used for here in the context of sound
synthesis of nonlinear systems. Salient features of this approach
include
- implementation of complex decay ratios with no extra
effort using modal damping;
- simulation of circular plates without bothering with the
problems related to particular spatial grids (a frustrating
aspect for Finite Difference schemes);
- fast computational times for linear and weakly nonlinear
regimes.
This work intends to show that the modal approach could
be applied to a large class of nonlinear problems, against the
common misconception that modes are only useful in treating
linear problems. Sound examples and videos are shown in order
to complete the presentation.

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**Keywords**

Sound synthesisModal approachNonlinear dynamicsGongsVon Kármán equations**Status**

Invited Paper

not reviewed