Stability of the Rao-Nakra sandwich beam with a dissipation of fractional derivative type: theoretical and numerical study

TL;DR

This paper investigates the stability of Rao-Nakra sandwich beams with fractional derivative damping, establishing polynomial energy decay and validating it through numerical simulations using an energy-conserving finite difference scheme.

Contribution

It introduces a novel approach by replacing fractional derivatives with diffusion equations, enabling analysis of stability and decay rates in a complex beam model.

Findings

Polynomial decay of energy depending on fractional parameters

Numerical validation confirms decay rates

Finite difference scheme effectively models the system

Abstract

This paper is devoted to the solution and stability of a one-dimensional model depicting Rao--Nakra sandwich beams, incorporating damping terms characterized by fractional derivative types within the domain, specifically a generalized Caputo derivative with exponential weight. To address existence, uniqueness, stability, and numerical results, fractional derivatives are substituted by diffusion equations relative to a new independent variable, $\xi$, resulting in an augmented model with a dissipative semigroup operator. Polynomial decay of energy is achieved, with a decay rate depending on the fractional derivative parameters. Both the polynomial decay and its dependency on the parameters of the generalized Caputo derivative are numerically validated. To this end, an energy-conserving finite difference numerical scheme is employed.