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

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.