Isospectral Vibrating Systems

Ranjan Ganguli

• Publisher: CRC Press
• Year: 2021
• eBook ISBN: 9781000413496
• Topic: Mathematics
• Subtopic: Civil Engineering
• Index: Mathematics

This book presents an isospectral approach for several important mechanical vibrating systems. Discrete and continuous isospectral systems are discussed using a simple multi-degree of freedom spring-mass system followed by illustration of isospectral beams and their solution through evolutionary computing. Next, it addresses axially loaded Euler-Bernoulli beams and aims to find isospectral counterparts of these systems. The practical application of these isospectral systems for vibration testing and for finding new closed form solutions is discussed. A considerable part of the book is devoted to isospectral rotating beams and their non-rotating analogs including Rayleigh beams. Aimed at researchers and graduate students in mechanical; aerospace; civil; automotive; ocean engineering especially mechanical vibrations, this monograph:

Discusses isospectral vibrating systems to aid vibration testing and computational analysis
Explores isospectral analogs between rotating and non-rotating structures
Provides simpler isospectral beams for vibration testing and for 3D printing
Uses firefly optimization method and electromagnetism inspired optimization method to find isospectral systems
Shows the use of isospectral systems to find new closed form solutions using an indirect approach

1Introduction: Spring-Mass Systems

This chapter introduces the concept of isospectral systems, using the simple and tractable example of a multi-degree of freedom spring-mass system. Isospectral systems have the same frequency spectrum, and so they are dynamically similar in nature. Families of isospectral systems arise as multiple solutions to an inverse problem i.e. the spectral information is insufficient to determine a unique solution. Consider a typical forward problem where model parameters p are used to calculate data d. This modeling process is a forward problem defined by: $d=F(p)$ where F is the forward map. The inverse problem is then given by $p=F^{-1}d$ and is typically not unique, a fact which we will witness in this chapter and throughout this book.

Isospectral systems are often considered to be a type of inverse problem. Isospectral systems play a significant role in mechanics as they allow designers to choose systems with desirable mass and stiffness distributions, which have similar dynamic properties. In this ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Contents
• Preface
• Chapter 1 Introduction: Spring-Mass Systems
• Chapter 2 Discrete Models of Beams
• Chapter 3 Beams with Tip Force - Discrete Models
• Chapter 4 Gravity Loaded Beams - Discrete Models
• Chapter 5 Rotating Beams - Discrete Models
• Chapter 6 Isospectral Beams - Electromagnetism Optimization
• Chapter 7 Isospectral Beams - Closed Form Solutions
• Chapter 8 Isospectral Systems for Testing
• Chapter 9 Beams with Shared Eigenpairs
• Chapter 10 Isospectral Rayleigh Beams
• Bibliography
• Index