Abstract
A method is presented for reconstructing piano hammer forces through appropriate filtering of the measured string velocity. The filter design is based on the analysis of the pulses generated by the hammer blow and propagating along the string. In the five lowest octaves, the hammer force is reconstructed by considering two waves only: the incoming wave from the hammer and its first reflection at the front end. For the higher notes, four- or eight-wave schemes must be considered. The theory is validated on simulated string velocities by comparing imposed and reconstructed forces. The simulations are based on a nonlinear damped stiff string model previously developed by Chabassier, Chaigne, and Joly [J. Acoust. Soc. Am. 134(1), 648-665 (2013)]. The influence of absorption, dispersion, and amplitude of the string waves on the quality of the reconstruction is discussed. Finally, the method is applied to real piano strings. The measured string velocity is compared to the simulated velocity excited by the reconstructed force, showing a high degree of accuracy. A number of simulations are compared to simulated strings excited by a force derived from measurements of mass and acceleration of the hammer head. One application to an historic piano is also presented.
Highlights
In pianos, precise knowledge of the hammer force in terms of amplitude, duration, and shape is essential since it fully determines the resulting free vibrations of the strings
One commonly used method for deriving the force consisted in multiplying together the measured mass and acceleration of the hammer head.1,2. This method yields an acceptable rough estimate of the hammer pulse, though it suffers from several limitations
II B, an intuitive analysis of the wave propagation on the string is presented, showing that the number of elementary pulses to consider in the measured string velocity to reconstruct the hammer force depend on the ratio between the force width sH and the string period T1
Summary
Precise knowledge of the hammer force in terms of amplitude, duration, and shape is essential since it fully determines the resulting free vibrations of the strings. One commonly used method for deriving the force consisted in multiplying together the measured mass and acceleration of the hammer head.. One commonly used method for deriving the force consisted in multiplying together the measured mass and acceleration of the hammer head.1,2 This method yields an acceptable rough estimate of the hammer pulse, though it suffers from several limitations.. The string is excited by an imposed hammer force pulse This input force has realistic amplitude, time width, and shape for each simulated note. The reconstructed hammer force is compared to the force derived from measurements of hammer head acceleration and mass, and the influence of hammer shank motion is discussed. An example of application of the method to an historic piano with a Viennese action is presented
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