Analysis of the vibrational mode spectrum of a linear chain with spatially exponential properties

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 × N Green's function for an arbitrary number N = 2, .., ∞ 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.

Dispersion relations and wave operators in self-similar quasicontinuous linear chains

“…whereT N (h) is the self-similar operator defined in (11). The self-similar analogue of Laplace operator defined by (27) depends on the parameters δ, N, h. We furthermore observe the self-similarity of Laplacian (27) with respect to its dependence on h, namely…”

“…In these papers problems on discrete lattices with fractal features are addressed. Closed form solutions for the dynamic Green's function and the vibrational spectrum of a linear chain with spatially exponential properties are developed in a recent paper [11]. A similar fractal type of linear chain as in the present paper has been considered very recently by Tarasov [7].…”

“…In this limiting case we obtain the oscillator density from [11] (55) where δ is restricted within interval (41). We observe hence that the power 2 δ −1 is restricted within the range 0 < 2 δ − 1 < ∞ for 0 < δ < 2, especially with always vanishing oscillator density ρ(ω → 0) = 0.…”

“…In the sprit of ( 10) and (11) we construct an exactly self-similar function from the second difference according to…”

“…This approximation holds for "small" ǫ ≈ ln N = 0 (0 < ǫ ≪ 1) 6 which corresponds to the limiting case that N s = e v is continuous. In this limiting case we obtain the oscillator density from [11]…”