Chapter36 Vibrations of Discretely Layered Structures Using a Continuous Variation Model Arnaldo J. Mazzei and Richard A. Scott Abstract Recently, there has been a large body of work directed towards the use of non-homogeneous materials in controlling waves and vibrations in elastic media. Two broad categories have been studied, namely, media with continuous variation of properties and those with discrete layers (cells). Structures with both a finite and infinite number of cells (periodic layout) have been examined. For the former, direct numerical simulation or transfer matrix methods have been used. The current work focuses on one-dimensional cases, in particular a two-layer cell. Transfer matrix methods require writing solutions for each layer of the basic cell and then matching them across the interface, a process that can be quite lengthy. Here an alternate strategy is explored in which the discrete cell properties are modeled by continuously varying functions (here logistic functions), which has the advantage of working with a single differential equation. Natural frequencies have been obtained using a forced motion method and are in excellent agreement with those found using a transfer matrix approach. Mode shapes for the continuous variation model have been obtained using a finite difference scheme and compare well with those obtained via the transfer matrix approach. Keywords Waves in non-homogeneous media • Layered structures vibrations Nomenclature Bi Constants of integration ci Wave speed, ci DpE= i cr Numerical parameter, cr Dc1/c2 E Young’s modulus (Ei, Young’s modulus for i-th material) f1, f2 Non-dimensional material functions H(x) Logistic function Heaviside(x) Step function L Length of rod, LDL1 CL2 mr Numerical parameter, mr D 1/ 2 Si Shape function (i-th) t Time u Longitudinal displacement of the rod w Non-dimensional longitudinal displacement of the rod x Longitudinal coordinate xd Non-dimensional longitudinal coordinate A.J. Mazzei ( ) Department of Mechanical Engineering, C. S. Mott Engineering and Science Center, Kettering University, 1700 University Avenue, Flint, MI 48504, USA e-mail: amazzei@kettering.edu R.A. Scott Department of Mechanical Engineering, G044 W. E. Lay Automotive Laboratory, University of Michigan, 1231 Beal Avenue, Ann Arbor, MI 48109, USA R. Allemang (ed.), Topics in Modal Analysis II, Volume 8: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-04774-4__36, © The Society for Experimental Mechanics, Inc. 2014 385
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