Stabilisation of the generalised Rao-Nakra beam by partial viscous damping

TL;DR

This paper analyzes the stabilization of a complex Rao-Nakra beam model using viscous damping, establishing conditions for strong and polynomial stability, with some cases remaining for future research.

Contribution

It provides necessary and sufficient conditions for strong stability of certain damping configurations in a generalized Rao-Nakra beam model.

Findings

Strong stability conditions derived for specific damping arrangements.

Polynomial stability of certain orders proven.

Some damping configurations remain unstudied for future work.

Abstract

In this paper, we consider the stabilization of the generalized Rao-Nakra beam equation, which consists of four wave equations for the longitudinal displacements and the shear angle of the top and bottom layers and one Euler-Bernoulli beam equation for the transversal displacement. Dissipative mechanism are provided through viscous damping for two displacements. The location of the viscous damping are divided into two groups, characterized by whether both of the top and bottom layers are directly damped or otherwise. Each group consists of three cases. We obtain the necessary and sufficient conditions for the cases in group two to be strongly stable. Furthermore, polynomial stability of certain orders are proved. The cases in group one are left for future study