Hyperbolic–Parabolic Coupling and the Occurrence of Resonance in Partially Dissipative Systems - Fluid-Structure Interaction and Biomedical Applications - page 194

Chapter 3
Hyperbolic-Parabolic Coupling
and the Occurrence of Resonance in Partially
Dissipative Systems

Giovanni Paolo Galdi, Mahdi Mohebbi, Rana Zakerzadeh, and Paolo Zunino

Abstract It is well known that elastic solids, when subjected to a time-periodic
load of frequency !, may respond with a drastic increase of the magnitude of
basic kinematic and dynamic quantities, such as displacement, velocity and energy,
whenever ! is near to one of the “proper frequencies” of the solid. This phenomenon
is briefly described as resonance. Objective of our analysis is to investigate whether
the interaction of an elastic solid with a dissipative agent can affect and possibly
prevent the occurrence of resonance. We shall study this problem in a broad class
of dynamical systems that we call partially dissipative , and whose dynamics is
governed by strongly continuous semigroups of contractions. For such systems
we will provide sharp necessary and sufficient conditions for the occurrence of
resonance. Afterward, we shall furnish a number of applications to physically
relevant problems including thermo- and magneto-elasticity, as well as several
liquid-structure interaction models.

Keywords Asymptotic behavior • Elastic solid • Linear magnetoelasticity •
Linear thermoelasticity • Liquid-structure models • Mean ergodic theorem •
Resonance • Stokes equations • Strongly continuous semigroup of contractions

MSC2010: 76D07, 35M99, 35Q61, 74B05, 74F05, 74F10, 74F15

3.1 Introduction

Resonance in elastic solids is a well-known phenomenon occurring in Nature that
can be roughly described as follows: when the frequency of an applied time-periodic
load approaches one (or a multiple of one) of the natural frequencies of oscillations
of the solid, then the basic kinematic and dynamic parameters of the solid, such
as displacement, velocity, and energy become increasingly large, resulting into
damage and even rupture of the structure. In this respect, perhaps the most striking