An approach for wave velocity measurement in solid cylindrical rods subjected to elastic impact

Abstract

Certain cross-sectional resonances of a long, solid, cylindrical rod, excited by transverse, elastic impact loading, may be measured by an experimental technique. The values of these resonance frequencies can be predicted knowing the material characteristics of the rod, but it is of greater interest to inversely solve for the material characteristics of the tested material from the experimentally obtained frequency values. In the case of portland cement concrete testing specifically, the bulk shear wave velocity of the material is important to know but difficult to measure. In this paper, the governing resonance equation will be manipulated and inverted, ultimately resulting in an expression of bulk shear wave velocity in terms of the nth ordered resonance frequency, Poisson's ratio, and cross-sectional solid rod radius. The operation is not tractable when performed symbolically, however, because of the presence of Bessel functions; therefore, this novel inversion will be achieved through the approximation of Bessel functions within the resonance equation with 2nd order Taylor series, resulting in a quadratic equation in normalized resonance frequency Ω. The roots of the quadratic equation may then be solved explicitly, resulting in two symbolic expressions for Ω, one of which is selected as the appropriate approximation. Manipulation of the selected root expression results in the desired symbolic expression for bulk shear wave velocity. With numerical examples from the literature, it is demonstrated that use of the series provides good approximation of the roots of the original resonance equation across a significant span of coefficient values and allows for sufficient inverse calculation of bulk shear wave velocity based on experimental results. The symbolic form of the inverted expression for bulk shear wave velocity is given in the Appendix.