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**Computation and Modeling of The Sound Radiation of An Upright Piano Using Modal Formalism and Integral Equations**

Philippe Dérogis, René Caussé
**ICA 95, Trondheim (Norvège), 1995**

## Introduction

### Upright Piano

## Modal Analysis

### Modal Formalism for general viscous damping

## Eigenvalue Problem

where can be writren as :

where are respectively the eigen values
and the eigen vectors of equation(1)

## Modal parameters extraction

The response at the point *r* to a sinusoidal excitation at
the point s can be expressed using the eigen values and the eigen
vectors of equation (1) :

where :

## Computation of the displacement resulting from any excitation

The displacement *x*(*t*) resulting from an excitation can be computed using the formula :

where :

## Mesasurements and Results

**Figure 1:** Modes resulting from the curve-fiting of measurements

## Sound Radiation

### Sound radiation of a baffled plate

The sound pressure *p*(*r*) and the acoustic velocity resulting
from the vibrations of a baffled plate moving with normal acceleration
are given by :

## Sound pressure radiated by the first mode of the soundboard

## Active Intensity

## Normal Active Intensity

The active intensity is the power radiated per unit surface.

It is given by the formula :

It is intersting to calculate the *z* componant of the active intensity
near the soundboard in order to know which regions produce the acoustic
power, in particular :

- If it is positive, the energy travels from the soundboard to the
acoustic field
- If it is negative, the energy travels from the acoustic field to
the sounboard.

Computation of the normal Active Intensity at 5
centimeters of the soundboard

## An example of a loop of active intensity

Active intensity field in a plane orthogonal to the plate
for the mode 1 : 122*Hz*

## Radiation efficiency

### Power radiated

The total power radiated by a source can be computed using :

## Power radiated by the soundboard versus frequency for several driving point

- Computation of the displacement of the soundboard using modal formalism
- Computation of the acoustic field
- Computation of the radiated power

## Results

Radiated power for several points excited by a 1N force

## Radiation efficiency

Radiation efficiency of the soundboard

## Estimated radiation efficiency

Radiation efficiency of a soundboard having eigen values (frequency
and damping) a factor 1.5 higher.

## Conclusion

- The first modes of the soundboard look like the ones of an
isotropic suported plate
- The pressure radiated by the modes depends strongly of the
position of the observer
- The frequencies of the first modes are well below the
coincidence frequency of these modes
- There are loops of active Intesity
- The acoustic power depends on the location of the excitation
point
- The radiation efficiency is about 15-20%

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**Server © IRCAM-CGP, 1996-2008** - file updated on .

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**Serveur © IRCAM-CGP, 1996-2008** - document mis à jour le .