Stability of Elastic Multi-Link Structures (SpringerBriefs in Mathematics)

✍ Scribed by Kaïs Ammari, Farhat Shel

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 146
• Category: Library

✦ Synopsis

This brief investigates the asymptotic behavior of some PDEs on networks. The structures considered consist of finitely interconnected flexible elements such as strings and beams (or combinations thereof), distributed along a planar network. Such study is motivated by the need for engineers to eliminate vibrations in some dynamical structures consisting of elastic bodies, coupled in the form of chain or graph such as pipelines and bridges.
There are other complicated examples in the automotive industry, aircraft and space vehicles, containing rather than strings and beams, plates and shells. These multi-body structures are often complicated, and the mathematical models describing their evolution are quite complex. For the sake of simplicity, this volume considers only 1-d networks.

✦ Table of Contents

Preface
Contents
1 Preliminaries
1 Introduction
2 Terminology of Networks
3 Spectrum and Resolvents of an Operator
4 Semigroups
5 Hille–Yosida Generation Theorems
5.1 Generation of Semigroups
5.2 Dissipative Operators and Contraction Semigroups
6 Abstract Cauchy Problems
7 Stability
7.1 Strong Stability
7.2 Exponential Stability
7.3 Polynomial Stability
8 Sobolev Spaces in One Dimension
8.1 Definition and First Properties
8.2 Compact Embeddings, H10(Ω) Space
8.3 Some Useful Inequalities
9 Comments
2 Exponential Stability of a Network of Elastic and Thermoelastic Materials
1 Functional Spaces, Existence, and Uniqueness of Solutions
2 Exponential Stability
2.1 First Case
2.2 Second Case
3 Comments
3.1 Comment 1
3.2 Comment 2
3 Exponential Stability of a Network of Beams
1 Functional Spaces, Existence, and Uniqueness of Solutions
2 Exponential Decay
3 Comment
4 Stability of a Tree-Shaped Network of Strings and Beams
1 Abstract Setting
2 Asymptotic Behavior
2.1 Asymptotic Stability
2.2 Exponential Stability
2.3 Polynomial Stability
2.4 Lack of Exponential Stability
3 Comments
3.1 Comment 1
3.2 Comment 2
5 Feedback Stabilization of a Simplified Model of Fluid–Structure Interaction on a Tree
1 Well-Posedness
2 Exponential Stability
3 Two Examples of Non-exponential Stability
3.1 A Circuit (Fig.5.2)
3.2 A Star with Two Fixed Endpoints (Fig.5.3)
4 A Chain with Non-equal Mass Points
6 Stability of a Graph of Strings with Local Kelvin–Voigt Damping
1 Well-Posedness of the System
2 Asymptotic Behavior
Conclusion
References