Analytical investigation of the flattening effect: The reactive power balance rule
Xavier Boutillon · The Journal of the Acoustical Society of America · 1991
McIntyre et al. [J. Acoust. Soc. Am. 74, 1325–1345 (1983)], through comprehensive time-domain analyses of the bowed string mechanisms and various numerical simulations, have first pointed out that the hysteretical character of the sticking–slipping process necessarily induces, under certain end conditions, a flattening effect (the note plays flat as the bow pressure exceeds a given level). Taking advantage of a frequency-domain analysis, it is demonstrated that the area of the hysteresis cycle has an analytical relation to various physical parameters of the bowed string, leading to the concept of reactive power. The flattening effect appears as a consequence of this relation. Another possible consequence is the observed dependence of the spectrum and intensity of the string vibration on the bow pressing force. Based on realistic assumptions, an algorithm is proposed to connect the flattening effect or the intensity variations to the physical characteristics of the string. Finally, the variations of the flattening effect with playing parameters such as bow position or bow velocity are evaluated.