Analytical Solutions for the Resonance Response of Goupillaud-type Elastic Media Using Z-transform Methods

Report No. ARL-RP-353
Authors: G. A. Gazonas; A. P. Velo
Date/Pages: February 2012; 26 pages
Abstract: The resonance frequency spectrum is derived for an m-layered Goupillaud medium subjected to a discrete sinusoidal forcing function that varies harmonically with time at one end with the other end held fixed. Analytical stress solutions are derived from a coupled first-order system of difference equations using z-transform methods. The determinant of the resulting global system matrix in the z-space |Am| 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 d m d 5 and for certain classes of multilayered designs identified by tridiagonal Toeplitz matrices. An important result is the physical interpretation that all the 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. A sequence of resonance frequencies for the discrete model is universal (independent of design parameters) for all multilayered designs with an odd number of layers. The discrete model resonance frequency results are also extended to describe the resonance frequency spectrum for a continuous model. Results show that the natural frequency spectrum depends on the layer impedance ratios and are inversely proportional to the equal wave travel time for each layer.
Distribution: Approved for public release
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Last Update / Reviewed: February 1, 2012