Simulation of three-dimensional motion of a piano string using detailed and simplified computational models

Abstract

It is well known that the decay curve of the piano sound is steep for the first few seconds and followed by slower decay at longer time. The causes of this double decay are also known: the effect of polarization of a string which is considered to be caused by the motion of the bridge which is connected to the string by two bridge pins and interaction among multiple strings for a note. As for the later, it is explained that multiple strings for a note tend to vibrate out of phase each other when they have slightly different frequencies and this suppresses the energy transferred to the soundboard through the bridge. However, to make quantitative evaluations of these causes, it is indispensable to develop a physical model of the piano in which contributions of each factor of these causes can be simulated with a necessary degree of accuracy. In this study, the finite element method and the mode superposition method are employed to simulate the three-dimensional motion of a piano string in which interaction with a bridge on a soundboard is taken into account. The polarization and the longitudinal vibration of a piano string were measured to assess the simulation.