Modelling of piano string vibration using coupled mobilities and a state-space approach

In musical acoustics a double decay can occur if there are two decay slopes in the sound envelope with the second part of the time history having a less pronounced slope than the first. In piano strings the double decay can be caused by the differences between the decays of vibration in the two transverse directions of a single string: normal and parallel to the soundboard. Herein we aim to reproduce this phenomenon by coupling a stiff string model and the soundboard, in both frequency and time domain, using mobility coupling and state-space model approaches. The dynamics of a stiff string tensioned between the two end pins but disconnected from the soundboard bridge is first studied. The two transverse directions of vibration are assumed to be uncoupled due to symmetry. On the contrary, the soundboard mobilities at the point of connection with the string can be represented by a multidimensional (2x2 in this work) and fully populated matrix implying coupling between different directions. The string and the soundboard will be connected by means of frequency and time domain models to show the importance of the cross-mobility terms on the vibration response and double decay of the string.