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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 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 ## 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 : 122Hz 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 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