# Impact of Damping on Oscillation Patterns on the Plain Piano Soundboard

^{*}

## Abstract

**:**

## 1. Introduction

^{th}century, but were not prolonged as builders had the impression that the sound of the piano was losing brightness or ’bite’ [14,20].

## 2. Methods

#### 2.1. Finite-Difference Model

#### 2.2. Parameter Space

#### 2.3. Driving Mechanisms

#### 2.4. Spatial Analysis

#### 2.5. Damping Estimations

#### 2.5.1. Simulation

#### 2.5.2. Measurements

## 3. Results

#### 3.1. Damping of Piano Soundboards

#### 3.2. Forced Oscillation Patterns vs. Eigenmodes

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Discrete soundboard geometry of a grand piano showing the amount of grid points used in the discrete FDTD model. The points on the geometry indicate the driving points corresponding to the key numbers of the strings acting on the soundboard.

**Figure 2.**Measured T60 decay times for a piano soundboard in two production stages at 14 string/bridge impact points. Circles indicate mean values over all microphone recordings, triangles represent (mean ± standard deviation). (

**Left**) the raw soundboard, (

**right**) the finished soundboard. At the very first stage of production (left) T60 ~ 0.3, while after gluing the soundboard to the piano rim (

**right**) T60 raises up to T60 ~ 0.6 s.

**Figure 3.**T60 of knocking sounds on 14 impact positions at the piano bridge for nine damping strength as used in the simulation, with decreasing strength from left to right. The measured T60 of the raw piano soundboard is around T60 ~ 0.3 s, the T60 for the finished soundboard are around T60 ~ 0.6 s. Therefore, the displayed range is up to the maximum damping of regular piano soundboards. The minimum damping is decreased to below T60 ~ 0.1 s to show the impact of damping on soundboards.

**Figure 4.**Real valued amplitudes of the FDM raw soundboard simulation, driven sinusoidally with 26 Hz at 14 impact points at bridge positions and simulating with nine damping strength values with decreasing damping from right to left. The very left column, therefore, represents the forced oscillation patterns for maximum damping used. The maximum amplitudes of the patterns follow the impact point of driving the soundboard. Leaving the respective impact points leads to a decrease in amplitude of a traveling wave. Each row of the plot shows the change of this pattern when decreasing the damping. With decreased damping, modal patterns appear gradually more and more. Still, even with minimum damping at the very right of the plot, the modal patterns of different impact points are considerably different. Therefore, a certain damping leads to a mixture of the eigenmode with a pattern emphasizing the impact point.

**Figure 5.**Positions of maximum absolute amplitudes on a raw piano soundboard (first three columns) for 14 driving points (colors) for three different frequencies, top row: 26 Hz, middle row: 86 Hz, bottom row: 180 Hz, with respective modal shapes displayed in the very right column. The left three columns show the maximum amplitude positions for three different conditions: sinusoidal driving (left column), knocking with integration of sound from the very start (second left column), and knocking with integration of sound starting 50 ms after knocking onset. In all cases, the simulation case with minimal damping is displayed. With all three frequencies the very left column shows maximum amplitude positions following the impact positions, where the lowest frequency of 26 Hz follows most clearly. When knocking on the top plate with integration of sound from the very start (second left column), a similar behavior can be seen. Still, when knocking and integrating only after 50 ms after knocking (second right column), the maximum points follow eigenmode patterns, again most clear with 26 Hz and less with higher frequencies. Therefore, eigenmode patterns do appear with piano soundboards, still, when driven with forced oscillations, the vibrational patterns also follow the impact point.

Key | 1 | 10 | 15 | 20 | 21 | 23 | 26 | 30 | 34 | 39 | 45 | 53 | 63 | 74 |

x-positon/mm | 395 | 530 | 590 | 645 | 270 | 320 | 400 | 485 | 591 | 680 | 780 | 895 | 1055 | 1220 |

y-position/mm | 1730 | 1570 | 1375 | 1130 | 1560 | 1400 | 1200 | 965 | 735 | 560 | 375 | 225 | 95 | 15 |

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**MDPI and ACS Style**

Bader, R.; Plath, N.
Impact of Damping on Oscillation Patterns on the Plain Piano Soundboard. *Acoustics* **2022**, *4*, 1013-1027.
https://doi.org/10.3390/acoustics4040062

**AMA Style**

Bader R, Plath N.
Impact of Damping on Oscillation Patterns on the Plain Piano Soundboard. *Acoustics*. 2022; 4(4):1013-1027.
https://doi.org/10.3390/acoustics4040062

**Chicago/Turabian Style**

Bader, Rolf, and Niko Plath.
2022. "Impact of Damping on Oscillation Patterns on the Plain Piano Soundboard" *Acoustics* 4, no. 4: 1013-1027.
https://doi.org/10.3390/acoustics4040062