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

Abstract

Research highlights: - Nonlinearity characterization of a structure independently from its exciting device. - Experimental modal identification, including damping, in the mid-frequency range. - Boundary conditions, low-and high-frequency regimes clarified in the piano soundboard. - Good match between experimental observations and FEM results at low frequencies. - FEM of a non-regularly ribbed soundboard reveals high-frequency localization in piano.International audienceThe 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 ≈ -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 ≈ 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

Modal analysis of a grand piano soundboard at successive manufacturing stages

“…2. Indeed at high frequency vibration modes are localised between two ribs [18,19], and the mesh size has to be fine enough to represent these mode shapes.…”

“…Two papers ( [19,20]) were published by Ege and Boutillon on the vibroacoustics of piano soundboards in 2013. In [19] the soundboard was excited with a loudspeaker and the non-linear response of the soundboard was estimated to be 30 dB to 50 dB below the linear one.…”

“…In [19] the soundboard was excited with a loudspeaker and the non-linear response of the soundboard was estimated to be 30 dB to 50 dB below the linear one. The measurement was performed for vibration levels of the soundboard corresponding to a musical ff (fortissimo).…”

A Fully Parameterized Finite Element Model of a Grand Piano Soundboard for Sensitivity Analysis of the Dynamic Behavior

“…The forces are transmitted through the bridges (i.e., the bass and treble bridges) (Figure 3.9). To avoid any modeling complexities, most numerical models in the literature (Ege et al, 2012;Mamou-Mani, 2007) The crown is the curved raised area at the center of the board. Its level is typically measured by its radius of curvature which is usually of the order of 15 to 50 m (Conklin Jr, 1996a).…”

“…To model the crowning, the soundboard was assigned the shape of a spherical shell (Ege et al, 2012) as presented in Figure 4.7 2 . The curvature is controlled by a Figure 4.6: Element Type MPC184 (Kohnke, 1999) user-defined radius of curvature.…”