Vibration of a coupled system is studied where a stiff string (having small flexural rigidity) vibrates against a smooth unilateral parabolic boundary obstacle which is situated on a wooden plate. This is similar to the vibration of string in musical instruments like sitar. The boundary obstacle serves the purpose of a bridge between the string and the wooden top plate of the instrument. The top plate restricts the translational and the rotational motions of the bridge, the phenomenon being modeled by a translational and a torsional spring whose stiffness depends on the frequencies of the string. Flexural rigidity in a string introduces inharmonicity in the natural frequencies. Our study aims to explore the effect of a finite bridge on the inharmonicity of natural frequencies of a stiff string. We have used Hamilton's principle to derive the equations of motion and the boundary conditions which results into a moving boundary problem where we have introduced a dynamic scaling to make it a fixed boundary problem. This in turn introduced nonlinearity into the governing partial differential equations. We have linearized the system of equations and studied the effect of different parameters along with the flexural rigidity of the string on the natural frequencies and harmonicity.

Keywords: Natural frequency, Harmonicity, Moving boundary, Stiff string, Coupled nonlinear system.