Piano hammers and their force compression characteristics: Does a power law make sense?

Abstract

We have studied the force characteristics of a collection of piano hammers, through studies of the acceleration of the hammer head, the force due to the compression of the hammer felt, Fh, and the associated bending of the hammer shank which occurs when a hammer strikes a rigid object. By integration of the acceleration one can estimate the compression of the felt, and thus obtain a force-compression Fh(z) relationship; our results for this function are compared with previous experiments and theoretical models. Close examination of our findings suggests that bending of the hammer shank, and also the time required for the force to be transmitted from the outer edge of the hammer to its core, play significant roles in the hammer dynamics. The data are used to estimate the quantitative impact of these effects on the derived Fh(z) relation. The implications for physical modeling of a piano tone are briefly discussed.

Physical Modeling of the Piano

“…With that in mind, we now consider the quality of the tones calculated with the current version of the model. In our initial modeling, we employed power law hammers described by (1) with parameters based on type I hammer experiments by our group [31]. The results were disappointing-it is hard to accurately describe the tones in words, but they sounded distinctly plucked and somewhat metallic.…”

“…Figure 1 shows the force-compression characteristic for a particular hammer (a Steinway hammer from the note middle C) measured in two different ways. In the type I measurement, the hammer struck a stationary force sensor and the resulting force and felt compression were measured as described in [31]. We see that for a particular value of the felt compression, y f , the force is larger during the compression phase of the hammerstring collision than during decompression.…”

“…This force determines the initial excitation and is thus a crucial factor in the composition of the resulting tone. Considerable effort has been devoted to understanding the hammer-string force [1,2,3,4,5,6,7,27,28,29,30,31,32,33]. Hammer felt is a very complicated material [34], and there is no "first principles" expression for the hammerstring force relation F h (y f ).…”

Experimental investigation of the piano hammer-string interaction

“…Algebraic derivation of indirect quantities, such as F h from a h , is immune from distortion, so obtaining compression from measured displacement is recommended. Uncertainty about integration constants is also problematic, and methods to estimate them, such as assuming the simultaneity of maximum force and compression, 3 introduce bias and accumulated error.…”

“…Contact identification may be given little consideration, 2,26 or defined by inspection, for example by (1) examining the force signal, 3,15 (2) using an arbitrary force threshold value above noise level, 13,14 or (3) determining when agraffe pulses have dissipated in the string velocity signal to define end of contact. 19 Due to the very shallow initial rise of contact force, identifying a contact start moment, or compression zero, is ambiguous and imprecise by inspection of force or acceleration signals.…”