Piano string vibration modelling using coupled mobilities and a state- space approach

The two transverse motions of the string in planes parallel and perpendicular to the soundboard are believed to contribute to some of the characteristic effects of piano tones. In this work we aim to model the coupling between a single string and soundboard, considering the two transverse motions. A stiff string model is coupled to the bridge of the soundboard. Modelling approaches are presented in both frequency and time domain, using mobility coupling and state-space equations, respectively. For time domain modelling, the soundboard mobilities are approximated by lumped elements, springs and dampers in this case. In the frequency domain model, the mobilities of the elongated string, detached from the bridge, corresponding to the transverse response to a transverse force, are the same in both directions and the coupling between them is neglected, while the soundboard mobilities are represented by a 22 matrix. The state-space model has the benefit of allowing inclusion of the non-linear hammer excitation. Although the frequency domain model can represent the linear characteristics of the coupling and give an insight into the importance of the cross-terms of the coupled mobility, it is the time domain model that is able to represent typical features of the interaction