Resonance Phenomena in Goupillaud-type Media

Report No. ARL-TR-5384
Authors: George A. Gazonas and Ani P. Velo
Date/Pages: October 2010; 76 pages
Abstract: The discrete resonance frequency spectrum is derived for an m-layered Goupillaud-type elastic medium subjected to a discrete, time-harmonic forcing function at one end, with the other end fixed. Analytical stress solutions are derived from a global system of recursion relationships using z-transform methods, where the determinant of the resulting global system matrix |Am| in the z-space is a palindromic polynomial with real coefficients. The zeros of the palindromic polynomial are distinct and are proven to lie on the unit circle for 1 ≤ m ≤ 5. An important result is the physical interpretation that all positive angles, coterminal with the angles corresponding to the zeros of |Am| on the unit circle, represent the resonance frequency spectrum for the discrete model. The resonance frequency results are then extended to analytically describe the natural frequency spectrum of a free-fixed m-layered Goupillaud-type medium. The predicted natural frequency spectrum is validated by independently solving a simplified version of the frequency equation involving a transformation of the spatial variable. The natural frequency spectrum is shown to depend only on the layer impedance ratios and is inversely proportional to the equal wave travel time for each layer. A sequence of resonance frequencies for the discrete model is found, which is universal for all multilayered designs with an odd number of layers, and independent of any design parameters.
Distribution: Approved for public release
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Last Update / Reviewed: October 1, 2010