Analysis of the Vibrational Mode Spectrum of a Linear Chain with Spatially Exponential Properties

TL;DR

This paper derives an explicit frequency-domain Green's function for a linear chain with exponentially varying properties, analyzing its vibrational spectrum and continuum limit, revealing how grading affects frequency band localization.

Contribution

It provides a closed-form Green's function for a discrete exponential chain and links it to the continuum Klein-Gordon model, highlighting effects of spatial grading on vibrational modes.

Findings

Green's function expressed in closed form for any chain size

Frequency band width depends on the grading parameter

Large grading parameter localizes vibrational modes around a single frequency

Abstract

We deduce the dynamic frequency-domain-lattice Green's function of a linear chain with properties (masses and next-neighbor spring constants) of exponential spatial dependence. We analyze the system as discrete chain as well as the continuous limiting case which represents an elastic 1D exponentially graded material. The discrete model yields closed form expressions for the $N \times N$ Green's function for an arbitrary number $N=2,..,\infty$ of particles of the chain. Utilizing this Green's function yields an explicit expression for the vibrational mode density. Despite of its simplicity the model reflects some characteristics of the dynamics of a 1D exponentially graded elastic material. As a special case the well-known expressions for the Green's function and oscillator density of the homogeneous linear chain are contained in the model. The width of the frequency band is determined by the grading parameter which characterizes the exponential spatial dependence of the properties. In the limiting case of large grading parameter, the frequency band is localized around a single finite frequency where the band width tends to zero inversely with the grading parameter. In the continuum limit the discrete Green's function recovers the Green's function of the continuous equation of motion which takes in the time domain the form of a Klein-Gordon equation.