Vibrational and acoustical characteristics of the piano soundboard

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. INTRODUCTION The purpose of this study is to describe the vibration regime of the soundboard of an upright piano in playing condition in a large frequency range [300-2500] Hz with only a few parameters. To this end, we have investigated the modal behaviour of the soundboard by means of a recently published high-resolution modal analysis technique (Ege et al. 2009). Compared to techniques based on the Fourier transform, it avoids the customary frequency-resolution limitation and thus, gives access to a larger frequency-range and to a better precision on damping determinations. In the first section, we study the linearity of the board. Given the essential linear response, we present in the second section the results of two modal identifications of the soundboard from which we derive the modal density and the loss factor up to 2.5–3 kHz. The frequency evolution of the modal density of the piano soundboard reveals two well-separated vibratory regimes of the structure. The low-frequency behaviour (homogeneous isotropic plate) is presented in section 3 and the midand high-frequency behaviour (as exhibited by a set of waveguides) in section 4. LINEARITY Nonlinear phenomena (such as jump phenomenon, hysteresis or internal resonance) appear when the vibration of a bi-dimensional structure reaches amplitudes in the order of magnitude of its thickness (Touzé et al. 2002). In the case of the piano, the soundboard displacement w measured at the bridge remains in a smaller range, even when played ff and in the lower side of the keyboard. Askenfelt and Jansson report maximum values of displacement at the bridge wmax ≈ 6 · 10−6 m in the frequency range [80-300] Hz (Askenfelt and Jansson 1992). This maximum value is less than 10−3 times the board thickness. We can therefore assume that large displacements are far to be reached and the vibrations of the soundboard can be expected as linear to a high level of approximation. The technique In order to quantify experimentally the (non)linearity, we performed measurements on an upright piano soundboard. An exponential sine sweep technique proposed by Farina (Farina 2000), mathematically proved by Rébillat et al. (Rébillat et al. 2010), is used. It gives access both to the linear part of the impulse response of either system and to the nonlinear part of the response, that is the distortion level in the frequency-domain. The technique goes as follows: (a) Let’s consider first a linear system excited by x(t), a sweptsine of duration T with initial and final angular frequencies ω1 and ω2: x(t) = sin [φ(t)] with the instantaneous phase φ(t) = ω1t + ω2−ω1 T t2 2 . The impulse response γimp(t) can be reconstructed by a deconvolution process: the measured signal γmeas(t) (acceleration for example) is convolved with the timereversal of the excitation signal, that is γimp(t) = γmeas(t) ∗ x(−t). (b) For a system with a weakly non-linear behaviour, Farina proposes to use a sine sweep for which the frequency varies exponentially with time – exponential sine sweep – in order to separate the linear and nonlinear parts of the impulse response: x(t) = cos [φ(t)] φ(t) = ω1T ln(ω2/ω1) ( e t T ln(ω2/ω1)−1 ) −π/2 (1) This signal verifies the fundamental property (Rébillat et al. 2010): ∀k ∈ N∗ , cos [kφ(t)] = cos [φ(t +∆ tk)] where ∆ tk = T lnk ln(ω2/ω1) (2) Multiplying the phase of a logarithmic sweep by a factor k shifts it up in time by ∆ tk. Rébillat et al. shown moreover that 1 ha l-0 05 58 18 7, v er si on 1 12 D ec 2 01 2 Author manuscript, published in "20th International Congress on Acoustics, ICA 2010, Sydney : Australia (2010)"