Dependence of the Lowest Thickness Longitudinal Mode Resonant Frequencies of Long Rectangular Bars on the Width and Thickness Dimensions

Abstract

A first-order theory of the dependence of the lowest thickness longitudinal frequency constant (ft) of long bars on width to thickness ratio (w/t) is proposed. The essential idea of this theory is that the frequency constant (ft) of long bars is a function of the width for w/t less than 1 and that (w/t) (ft) is a function of the thickness for w/t greater than 1. The dependence of the frequency thickness product on the dimensions in each case is the same as the velocity of the lowest longitudinal elastic mode of the infinite plate on the plate thickness. Experimental resonant frequency measurements on long bars of three ceramic transducer materials, namely, barium titanate, lead zirconate titanate, and sodium potassium niobate, are compared with those calculated from the proposed theory for values of w/t from 0.2 to 6.0. The fit between experiment and theory is optimized by a suitable choice of isotropic elastic constants for the resonator. These “best fit” values of the elastic constants are then compared with the values found by the usual methods. In this way, it is shown that the proposed theory is consistent with the dependence of the lowest resonant frequency of the bar on the dimensions and the elastic constants. The application of the theory to a method of determining both isotropic elastic constants using the thickness longitudinal mode resonant frequencies of long bars with finite widths and thicknesses will be described.