Abstract:
In string musical instruments, the sound is radiated by the soundboard, subject to the strings excitation. This vibration of this rather complex structure is described here with models which need only a small number of parameters. Predictions of the models are compared with results of experiments that have been presented in Ege et al. [Vibroacoustics of the piano soundboard: (Non)linearity and modal properties in the low- and mid- frequency ranges, Journal of Sound and Vibration 332 (5) (2013) 1288-1305]. The apparent modal density of the soundboard of an upright piano in playing condition, as seen from various points of the structure, exhibits two well-separated regimes, below and above a frequency flim that is determined by the wood characteristics and by the distance between ribs. Above flim, most modes appear to be localised, presumably due to the irregularity of the spacing and height of the ribs. The low-frequency regime is predicted by a model which consists of coupled sub-structures: the two ribbed areas split by the main bridge and, in most cases, one or two so-called cut-off corners. In order to assess the dynamical properties of each of the subplates (considered here as homogeneous plates), we propose a derivation of the (low-frequency) modal density of an orthotropic homogeneous plate which accounts for the boundary conditions on an arbitrary geometry. Above flim, the soundboard, as seen from a given excitation point, is modelled as a set of three structural wave-guides, namely the three inter-rib spacings surrounding the excitation point. Based on these low- and high-frequency models, computations of the point-mobility and of the apparent modal densities seen at several excitation points match published measurements. The dispersion curve of the wave-guide model displays an acoustical radiation scheme which differs significantly from that of a thin homogeneous plate. It appears that piano dimensioning is such that the subsonic regime of acoustical radiation extends over a much wider frequency range than it would be for a homogeneous plate with the same low-frequency vibration. One problem in piano manufacturing is examined in relationship with the possible radiation schemes induced by the models.

Abstract:
The vibrations of the soundboard of an upright piano in playing condition are investigated. It is first shown that the linear part of the response is at least 50 dB above its nonlinear component at normal levels of vibration. Given this essentially linear response, a modal identification is performed in the mid-frequency domain [300-2500] Hz by means of a novel high resolution modal analysis technique (Ege, Boutillon and David, JSV, 2009). The modal density of the spruce board varies between 0.05 and 0.01 modes/Hz and the mean loss factor is found to be approximately 2%. Below 1.1 kHz, the modal density is very close to that of a homogeneous isotropic plate with clamped boundary conditions. Higher in frequency, the soundboard behaves as a set of waveguides defined by the ribs. A numerical determination of the modal shapes by a finite-element method confirms that the waves are localised between the ribs. The dispersion law in the plate above 1.1 kHz is derived from a simple waveguide model. We present how the acoustical coincidence scheme is modified in comparison with that of thin plates. The consequences in terms of radiation of the soundboard in the treble range of the instrument are also discussed.

Abstract:
An expression of the piano soundboard mechanical mobility (in the direction normal to the soundboard) depending on a small number of parameters and valid up to several kHz is given in this communication. Up to 1.1 kHz, our experimental and numerical investigations confirm previous results showing that the soundboard behaves like a homogeneous plate with isotropic properties and clamped boundary conditions. Therefore, according to the Skudrzyk mean-value theorem (Skudrzyk 1980), only the mass of the structure M, the modal density n(f), and the mean loss factor eta(f), are needed to express the average driving point mobility. Moreover, the expression of the envelope - resonances and antiresonances - of the mobility can be derived, according to (Langley 1994). We measured the modal loss factor and the modal density of the soundboard of an upright piano in playing condition, in an anechoic environment. The measurements could be done up to 2.5 kHz, with a novel high-resolution modal analysis technique (see the ICA companion-paper, Ege and Boutillon (2010)). Above 1.1 kHz, the change in the observed modal density together with numerical simulations confirm Berthaut's finding that the waves in the soundboard are confined between adjacent ribs (Berthaut et al. 2003). Extending the Skudrzyk and Langley approaches, we synthesize the mechanical mobility at the bridge up to 2.5 kHz. The validity of the computation for an extended spectral domain is discussed. It is also shown that the evolution of the modal density with frequency is consistent with the rise of mobility (fall of impedance) in this frequency range and that both are due to the inter-rib effect appearing when the half-wavelength becomes equal to the rib spacing. Results match previous observations by Wogram (1980), Conklin (1996), Giordano (1998), Nakamura (1983) and could be used for numerical simulations for example. This approach avoids the detailed description of the soundboard, based on a very high number of parameters. However, it can be used to predict the changes of the driving point mobility, and possibly of the sound radiation in the treble range, resulting from structural modifications.

Abstract:
Up to around 1.1 kHz, the soundboard of the piano behaves like a homogeneous plate whereas upper in frequency, it can be described as a set of waveguides defined by the ribs. In consequence: a) The acoustical coincidence phenomenon is deeply modified in comparison with that occurring in homogeneous plates since the dispersion curve of a waveguide can present none, one, or two coincidence frequencies. This may result in a nonuniformity of the soundboard radiation in the treble range, corresponding to the so-called killer octave, where a good sustain is difficult to obtain. b) The mobility (mechanical admittance) in the direction normal to the soundboard can be synthesised with only a small number of parameters. It compares well with published measurements (Giordano, JASA, 1998), in particular the step-like falloff of the local impedance due to the localisation of the waves between ribs. c) The synthesised mobility has the same features as those which can be derived independantly, according to Skudrzyk (JASA, 1980) and Langley (JSV, 1994). This approach avoids the detailed description of the soundboard, based on a very large number of parameters. It can be used to predict global changes of the driving point mobility, and possibly of the sound radiation in the treble range, resulting from structural modifications.

Abstract:
Man-made slender structures are known to be sensitive to high levels of vibration, due to their flexibility, which often cause irreversible damage. In nature, trees repeatedly endure large amplitudes of motion, mostly caused by strong climatic events, yet with minor or no damage in most cases. A new damping mechanism inspired by the architecture of trees is here identified and characterized in the simplest tree-like structure, a Y-shape branched structure. Through analytical and numerical analyses of a simple two-degree-of-freedom model, branching is shown to be the key ingredient in this protective mechanism that we call damping-by-branching. It originates in the geometrical nonlinearities so that it is specifically efficient to damp out large amplitudes of motion. A more realistic model, using flexible beam approximation, shows that the mechanism is robust. Finally, two bioinspired architectures are analyzed, showing significant levels of damping achieved via branching with typically 30% of the energy being dissipated in one oscillation. This concept of damping-by-branching is of simple practical use in the design of slender flexible structures.

Abstract:
Usual modal analysis techniques are based on the Fourier transform. Due to the Delta T . Delta f limitation, they perform poorly when the modal overlap mu exceeds 30%. A technique based on a high-resolution analysis algorithm and an order-detection method is presented here, with the aim of filling the gap between the low- and the high-frequency domains (30%

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.

Abstract:
Under the excitation of strings, the wooden structure of string instruments is generally assumed to undergo linear vibrations. As an alternative to the direct measurement of the distortion rate at several vibration levels and frequencies, we characterise weak non-linearities by a signal-model approach based on cascade of Hammerstein models. In this approach, in a chain of two non-linear systems, two measurements are sufficient to estimate the non-linear contribution of the second (sub-)system which cannot be directly linearly driven, as a function of the exciting frequency. The experiment consists in exciting the instrument acoustically. The linear and non-linear contributions to the response of (a) the loudspeaker coupled to the room, (b) the instrument can be separated. Some methodological issues will be discussed. Findings pertaining to several instruments - one piano, two guitars, one violin - will be presented.

Abstract:
On the basis of a recently proposed vibro-acoustical model of the piano soundboard (X. Boutillon and K. Ege, Vibroacoustics of the piano soundboard: reduced models, mobility synthesis, and acoustical radiation regime. \emph{submitted to the Journal of Sound and Vibration}, 2011.), we present several models for the coupling between the bridge and the ribbed plate of the soundboard. The models predict the modal density and the characteristic impedance at the bridge as a function of the frequency. Without parameter adjustment, the sub-structure model turns out to fit the experimental data with an excellent precision. The influence of the elastic parameters of wood is discussed. The model predictions are compared for pianos of different sizes and types.

Abstract:
A method is proposed to identify the mechanical properties of the skin and core materials of honeycomb sandwich. All the elastic coe cients and loss-factors that matter in the dynamics of a panel in the thick-plate approximation are identi ed. To this end, experimental natural modes (i.e. eigenmodes of the damped system) are compared to the numerical modes of a large sandwich panel (lx,y/h 80). The chosen generic model for the visco-elastic behaviour of the materials is E (1 + jη). The numerical modes are computed by means of a Rayleigh-Ritz procedure and their dampings are predicted according to the visco-elastic model. The frequencies and dampings of the natural modes of the panel are estimated experimentally by means of a high-resolution modal analysis technique. An optimisation procedure yields the desired coe cients. A sensitivity analysis assess the reliability of the method.