Coupled plate-string vibrations in the presence of a finite bridge: Effect on natural frequencies and harmonicity

Harmonicity of natural frequencies is investigated for a string vibrating against a finite bridge mounted on a flexible plate. This coupled system is akin to the vibration of strings in musical instruments like the sitar, where the top plate is approximated as a clamped circular plate. The bridge is assumed to be parabolic and rigidly attached to the plate at one point. For simplicity, the plate is replaced by linear and torsional springs with frequency dependent dynamic stiffness to account for the continuous nature of the plate. Governing equations of motions are obtained using Hamilton's principle, which are linearized about the static configuration. It is observed that the coupled natural frequencies below the first structural frequencies are inharmonic in a manner opposed to the inharmonicity due to the bending rigidity of the string. A detailed parametric study on the effect of plate thickness, position, mass, and geometry of the bridge on the natural frequencies is performed. It is found that a proper choice of these parameters can significantly increase the harmonic content in the frequencies for real strings. This study guides the proper choice of system parameters to obtain better harmonic relation between the coupled structure-string modes with real strings.