Taper Design and Wood Selection for Reverse Engineering of Bows for String Instruments

Abstract

After discussing the difficult conflict between stiffness and density in producing professional quality bows, we derive the new taper required to produce the same local stiffness as the target bow from a blank of possibly different modulus of elasticity. Furthermore, we prove that by selecting wood with the same value of the parameter  as the target bow, both the mechanical bending and mass properties between the head and forward frog position of the target can be reproduced exactly. This parameter replaces the speed of sound measurement as the general basis for wood selection as a measure of wood quality. In outlining a procedure for reverse engineering, modeling of the target bow, geometry and methods for estimating the modulus and density of bow or blank are presented together with a derivation of the associated camber and its relationship to maximum hair tension and proximity to full camber. The Tourte taper is discussed with particular emphasis on its close approximation to linear stiffness as a function of position along the stick. Data on modulus, density and damping ratio from a sample of 50 violin bow blanks is also presented.