Influence of the edge of the cast iron frame curvature on the spectrum of the piano string vibrations
Dmitri Kartofelev; Anatoli Stulov

Investigation of the boundary condition of vibrating string is a very important problem in musical acoustics. It is well known that the fundamental frequency of piano string is strictly determined by the type of the string termination. The types of the string support in the piano are different for the bass and treble notes. All the far ends of the piano strings are terminated on the bass and treble bridges, which are the rather complicated resonant systems. The nearest ends of the bass and long treble strings begin from the agraffe that can be considered as an absolutely rigid clamp termination. But the most part of the treble strings starts from the edge of the cast iron frame. These strings turn the rigid edge, and its vibration tone depends on the curvature of this termination.
We investigate the vibrations of the ideal flexible string, which one end is rigidly clamped, and another one is terminated on the curved contact surface. The vibrating string touches repeatedly this termination, and this, in turn, causes the modulation of fundamental frequency of the string, and the train of high frequency oscillations is generated. The problem is studied both analytically, and numerically. The effect of the contact nonlinearity and of the shape of the contact surface on of the spectral structure of the string vibration is considered. The influence of the impact amplitude on the vibration spectra of struck string is discussed.