The violin bow: taper, camber and flexibility.

An analytic, small-deflection, simplified model of the modern violin bow is introduced to describe the bending profiles and related strengths of an initially straight, uniform cross-section, stick as a function of bow hair tension. A number of illustrative bending profiles (cambers) of the bow are considered, which demonstrate the strong dependence of the flexibility of the bow on longitudinal forces across the ends of the bent stick. Such forces are shown to be comparable in strength to critical buckling loads causing excessive sideways buckling unless the stick is very straight. Non-linear, large deformation, finite element computations extend the analysis to bow hair tensions comparable with and above the critical buckling strength of the straight stick. The geometric model assumes an expression for the taper of Tourte bows introduced by Vuillaume, which is re-examined and generalized to describe violin, viola and cello bows. A comparison is made with recently published measurements of the taper and bending profiles of a particularly fine bow by Kittel.