Model analysis of a hammer–string interaction

Abstract

A hammer–string interaction for the bass note (A0) of a 6‐ft grand piano is investigated by use of a lumped‐element model. The hammer head is simulated by a mass–spring system with a nonlinear spring characteristic. The string is simulated by many parallel resonant circuits, each of which consists of a mass, a linear spring, and a dashpot. The impulsive force applied by the hammer to the string lasts only approximately 1 ms for a hard‐hit key. The string deformation at the striking point also reaches near its maximum within about this time and retains its amplitude until the wave is reflected back from the closer end at approximately 4 ms. The hammer leaves the string temporarily, since the hammer is slowly moving downward after the first impact. The string reduces its displacement rapidly when the wave is reflected back from the closer end and, consequently, it pushes the hammer farther back. Effect of hammer stiffness, hammer velocity, and inharmonicity on the spectral distribution of partials and the efficiency of energy transmission are also discussed.