Simscape Model - calculate inharmonicity in piano string.

SImscape models the flexural vibration of a piano string including the wire bending stiffness. Modal frequencies are a good match to theory.

The vibrating “string” of a piano is a thin steel wire that is tensioned to tune it to the intended playing frequency. Sufficient tension in the wire gives it the properties approximately of a flexible string, with its multiple resonance frequencies approximately at harmonic intervals. However the wire bending stiffness modifies the resonance frequencies by a small amount. It is well known that the additional stiffness modifies the resonance frequencies such that where and is the fundamental frequency of a similar ideal flexible string. A Simscape model of the vibrating string demonstrates this behavior, and its solver has sufficient resolution to match this theoretical prediction.