Synthetic description of the piano soundboard mechanical mobility
  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. 
Kerem Ege (LMS), Xavier Boutillon (LMS)
physics.class-ph