Linear string-soundboard coupling in pianos
Antoine J. Chaigne · 2010
The coupling between strings and soundboard at the bridge is essential in determining the quality of piano tones. The objective of the work is to investigate the coupling properties through time domain modeling of a string triplet coupled to the input admittance. For this purpose, various input admittances were measured on a strung upright piano soundboard and modelled as a parallel set of oscillators. These oscillators are coupled to a linear finite difference modeling of slightly detuned strings. The model accounts for the reaction of the bridge on string waveforms, and thus is different from usual methods were the bridge velocity is obtained through convolution between string forces and admittance impulse response. A comparison between both methods is presented at the end of the paper. Each oscillator of the admittance is defined by three parameters: amplitude, frequency and damping factor from which, equivalently, mass, stiffness and mechanical resistance are derived. Modifying the amplitudes allows studying the effects of variations of the global mobility pattern, without affecting individual frequencies and damping. Simultaneous variations of mass and stiffness can be related to variations of thickness of the soundboard. Variations of selected frequencies simulate selected variations of modes, as in the addition of stiffeners. Finally, variations of frequency-dependent losses in the soundboard material can be simulated through modifications of the set of damping factors. Simulated string and bridge waveforms yield a better understanding on the effects of both string tuning and bridge mobility on the amplitude, duration and