Abstract
In this paper we analyse longitudinal wave propagation in exponentially tapered rods from both a theoretical and an experimental perspective. The tapering introduces significant changes to the behaviour of the rod. The longitudinal wave does not propagate from zero frequency, its cut-off frequency depending on the coefficient in the exponent. The analytical description of this phenomenon is well established, however little experimental work has been published to date. After a brief review of the classical solution of the exponential rod equation, we derive a methodology allowing the wavenumbers to be estimated from a set of equally spaced dynamic responses. Our approach is verified numerically against a finite element simulation and validated experimentally, both showing very good agreement. To further explain the results and provide an outlook for future work, we present a finite element model of the tapered rod embedded in an infinite solid medium. We conclude with a discussion on the effects of the surrounding medium on the behaviour of the structure and resulting characteristic features of the wavenumber.
Highlights
After a brief review of the classical solution of the exponential rod equation, we derive a methodology allowing the wavenumbers to be estimated from a set of spaced dynamic responses
To further explain the results and provide an outlook for future work, we present a finite element model of the tapered rod embedded in an infinite solid medium
Waves in tapered rods have been widely studied throughout the years and many classical solutions are available in the literature with the acoustic horn as the foremost example [1]
Summary
Waves in tapered rods have been widely studied throughout the years and many classical solutions are available in the literature with the acoustic horn as the foremost example [1]. Stevenson [2] used the WKB approximation to formulate solutions for horns of an arbitrary shape enabling the identification of transmission and attenuation zones. Nayfeh and Telionis [3] employed the multiple scales method to study wave propagation in ducts with a varying cross-section. Longitudinal wave propagation and vibration were described using Bessel functions [4, 5]. Langley [6] performed an extensive investigation into wave propagation in non-homogeneous waveguides (including both longitudnal and flexural motion) with the aid of perturbation methods
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