Effect of Stiffness and Nonuniformity on Vibroscopic Determination of Filament Cross-Sectional Area

Abstract

The presence of stiffness in a stretched string vibrating between fixed supports raises all the natural frequencies over those for a perfectly flexible string. The presence of nonuniformity in cross-sectional area along the string changes some or all of the frequencies from those for a perfectly uniform string, the amount and direction of change depending on the form of the nonuniformity. These changes in frequency may introduce error into the calculation of the average cross-sectional area of a fiber by means of the vibroscopic technique, wherein the mechanical resonance of a stretched fiber is observed under oscillating mechanical or electrostatic forces. In this paper the natural frequencies are calculated with both the stiffness and the nonuniformity taken into account to first-order terms. The relation between the average cross-sectional area and the natural frequencies is analyzed. Consideration is given to the manner in which the changes in frequency and shifts of the nodes, due to nonuniformity, determine the detailed form of the area variation. The characterization of nonuniformities in terms of certain arbitrary parameters is shown to lead to the possibility of calculating the average area by measuring the natural frequencies in only the first and second modes and of calculating the maximum or minimum area by measuring also the nodal shift in the second mode.