Piano string excitation V: Spectra for real hammers and strings

Abstract

New measurements of piano hammer properties are reported, as well as hammer accelerations during impact and resulting string motion on a grand piano. Comparison with extant theory on string–hammer interaction requires discussion of limitations due to: (1) nonlinear mode coupling for finite amplitude, (2) string stiffness and the resulting dispersion, (3) soundboard admittance, (4) finite hammer width, and (5) nonlinearity in the hammer. Items (1) and (4) are shown to be relatively unimportant for measurements made promptly after striking, while (3) is significant mainly for the uppermost treble octave. Item (2) becomes important above roughly 1 kHz for notes at the bass end, 5 kHz in midrange, and 10 kHz at the treble end. Item (5) is the most important limitation, and leads us to consider either using an ‘‘effective compliance’’ to obtain an approximate spectrum or performing numerical integrations to handle the nonlinearity. With these limitations in mind, an examination is made of how well the theory can explain the measured string vibration spectra. The agreement is moderately good in the bass and midrange while, at the treble end, the measured spectra fall off significantly faster than the predictions. Further progress apparently requires methods capable of fully integrating the nonlinear hammer compliance into the theory.