Vibroacoustics of the piano soundboard: (Non)linearity and modal properties in the low and mid-frequency ranges

TL;DR

This study introduces a novel vibro-acoustical method to measure the nonlinearity of piano soundboards, performs high-resolution modal analysis up to 3 kHz, and compares experimental results with finite-element models to understand vibrational behavior.

Contribution

It presents the first quantitative estimate of soundboard nonlinearity, applies a high-resolution modal analysis technique, and provides detailed modal damping and shape data for the piano soundboard.

Findings

Soundboard nonlinearity is approximately -40 dB below linear response.

Modal damping values are consistent with wood properties.

Structural wave behavior varies significantly above and below 1 kHz.

Abstract

The piano soundboard transforms the string vibration into sound and therefore, its vibrations are of primary importance for the sound characteristics of the instrument. An original vibro-acoustical method is presented to isolate the soundboard nonlinearity from that of the exciting device (here: a loudspeaker) and to measure it. The nonlinear part of the soundboard response to an external excitation is quantitatively estimated for the first time, at \approx -40 dB below the linear part at the ff nuance. Given this essentially linear response, a modal identification is performed up to 3 kHz by means of a novel high resolution modal analysis technique (Ege et al., High-resolution modal analysis, JSV, 325(4-5), 2009). Modal dampings (which, so far, were unknown for the piano in this frequency range) are determined in the midfrequency domain where FFT-based methods fail to evaluate them with an acceptable precision. They turn out to be close to those imposed by wood. A finite-element modelling of the soundboard is also presented. The low-order modal shapes and the comparison between the corresponding experimental and numerical modal frequencies suggest that the boundary conditions can be considered as blocked, except at very low frequencies. The frequency-dependency of the modal density and the observation of modal shapes reveal two well-separated regimes. Below \approx 1 kHz, the soundboard vibrates more or less like a homogeneous plate. Above that limit, the structural waves are confined by ribs, as already noticed by several authors, and localised in restricted areas (one or a few inter-rib spaces), presumably due to a slightly irregular spacing of the ribs across the soundboard.