Z-TRANSFORM METHODS FOR THE OPTIMAL DESIGN AND RESONANCE FREQUENCIES OF A FINITE LAYERED ELASTIC STRIP
TITLE:
Z-TRANSFORM METHODS FOR THE OPTIMAL DESIGN AND RESONANCE FREQUENCIES OF A FINITE LAYERED ELASTIC STRIP
DATE:
Friday, March 25th, 2011
TIME:
3:30 PM
LOCATION:
GMCS 214
SPEAKER:
Ani P. Velo, Department of Mathematics and Computer Science at University of San Diego
ABSTRACT:
Despite the widespread use of multilayered structures in various technologies, exact analytical solutions to optimal design problems governed by the wave equation (hyperbolic systems) are rare in the literature. In addition, the exact solutions for the natural modes of vibrations and the resonance response of multilayered elastic media have been primarily limited to the analysis involving only a few layers.
We consider an m-layered Goupillaud-type elastic strip (equal wave travel time for each layer). The strip is subjected to a discrete forcing function at one end and held fixed at the other end. 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 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 and for certain classes of multilayered designs identified by Toeplitz matrices.
For special cases of the discrete forcing function, optimization and resonance results are obtained analytically.
- When the discrete forcing function is the Heaviside loading, optimal layered designs which provide the smallest stress amplitude are identified for up to five layers.
- When the discrete forcing function is sinusoidal and varies harmonically with time, the resonance frequency spectrum is derived The resonance frequency results are then extended to analytically describe the natural frequency spectrum of a free-fixed m-layered Goupillaud-type strip. It appears that the natural frequency spectrum depends only on the layer impedance ratios and is inversely proportional to the equal wave travel time for each layer.
Applications of these results to material design will be also discussed.
This work is jointly made with George A. Gazonas. Army Research Laboratory, MD.
HOST:
Satchi Venkataraman
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