Modelling of Sympathetic String Vibrations

String instruments are usually composed of several strings connected to a vibrating body allowing efficient sound radiation. For some special string tunings, sympathetic vibrations can occur: if one string is excited, some others are also excited via the body. In order to investigate this phenomenon, an analytical model of a simplified generic string instrument has been developed. The body of the instrument is represented by a beam clamped at both ends, to which several strings are attached. The state vector formalism and the transfer matrix method are used to describe the propagation of bending and extensional waves in each sub-structure (strings and beam). Coupling conditions between sub-structures take into account the angle formed by the beam and the strings. This leads to a linear system from which the normal modes of the assembly are computed. Numerical computations are carried out in the case of the beam-2 strings assembly and mode shapes are classified thanks to a criterion, the kinetic energy ratio (KER). Four kinds of modes may be identified: beam modes, string modes, beam-string modes and string-string modes. The latter are responsible for sympathetic response. An experimental modal analysis carried out on a clamped beam equiped with two strings confirms our theoretical results.