Coupling homogeneous chains of damped harmonic oscillators

Abstract

We study vibrational properties of nonhomogeneous materials. We consider idealized materials composed of one dimensional chains of damped harmonic oscillators represented by a sequence of particles, springs and dampers. The system is linear, thus possesses exact explicit solutions, nevertheless the formulas can be very complicated. Homogeneous chains are used as basic building blocks for characterizing global dynamics of the system. In particular, we determine the solutions for a system composed of two distinct homogeneous chains in terms of the original solutions for these two homogeneous chains, and the original parameters, when uncoupled. Two types of coupling are considered, one with a gluing particle, the other with a gluing spring and damper.