Optimal Design, Resonance and Bandgap Formation in Goupillaud-type Elastic Media
April 24, 2012
11:30 a.m.
Dr. George Gazonas
ABSTRACT
This talk will present the derivation of exact analytical optimal design solutions that minimize the amplitude of transient stress waves in Goupillaud-type layered elastic media [1]. The method of characteristics is used to derive a system of layer recursion relations for stress that is solvable for a medium of up to five layers, using symbolic algebra software such as Mathematica. Beyond five layers, the recurrence relations must be simplified by hand calculation using z-transform methods, and written in global matrix form. The z-domain system determinant forms an mth-order palindromic polynomial with real coefficients. The roots of the polynomial that lie on the unit circle and satisfy the optimality conditions, relate to a countable and infinite set of optimal designs. Palindromic polynomials are finding increasing applications in dynamics and pure and applied mathematics, and we illustrate the relation between these optimal designs to problems involving resonance [2], uniaxial impact, and acoustic bandgap formation in Goupillaud-type media. The exact results are supported by numerical experiments and provide a means to verify computationally-based optimal solutions that utilize explicit finite element numerical codes that are linked to optimization algorithms.