Static analysis of violin bow behavior under playing loads.

After discussing the relationship between taper and camber typically used by bowmakers, energy methods are then employed to develop nonlinear boundary value problems describing both the hair tightening problem and static deflection analysis of the violin bow under playing loads, including an analysis of possible hair tension asymmetry by employing a linear springs in series model of the hair. A simple algebraic condition for the distance the frog travels in reaching maximum tension is also presented. Boundary value problems are then applied using tools in scilab, comparing the idealized Tourte taper exhibiting a linear stiffness profile and an alternative design with a significantly different profile, revealing small differences in their behavior under static playing loads and validating previously observed significant increase in hair tension when approaching the tip. While the difference in vertical hair deflection as a function of bow position is extremely small, the Tourte design exhibits somewhat more compliance in the stick when approaching the tip, possibly desirable by players given increasing hair tension. An interesting small loss of hair tension in the lower section of the bow stroke is also discussed, representing additional slight differences in the initial compliance felt by the player when engaging the strings.