Vibro-impact Dynamics E-Book

Albert C. J. Luo

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 297

Veröffentlichungsjahr: 2013

Beschreibung

Presents a systematic view of vibro-impact dynamics based on the nonlinear dynamics analysis Comprehensive understanding of any vibro-impact system is critically impeded by the lack of analytical tools viable for properly characterizing grazing bifurcation. The authors establish vibro-impact dynamics as a subset of the theory of discontinuous systems, thus enabling all vibro-impact systems to be explored and characterized for applications. Vibro-impact Dynamics presents an original theoretical way of analyzing the behavior of vibro-impact dynamics that can be extended to discontinuous dynamics. All topics are logically integrated to allow for vibro-impact dynamics, the central theme, to be presented. It provides a unified treatment on the topic with a sound theoretical base that is applicable to both continuous and discrete systems Vibro-impact Dynamics: * Presents mapping dynamics to determine bifurcation and chaos in vibro-impact systems * Offers two simple vibro-impact systems with comprehensive physical interpretation of complex motions * Uses the theory for discontinuous dynamical systems on time-varying domains, to investigate the Fermi-oscillator Essential reading for graduate students, university professors, researchers and scientists in mechanical engineering.

Leseprobe

Contents

Cover

Title Page

Copyright

Frontispiece

Preface

Chapter 1: Introduction

1.1 Discrete and Discontinuous Systems

1.2 Fermi Oscillators and Impact Problems

1.3 Book Layout

Chapter 2: Nonlinear Discrete Systems

2.1 Definitions

2.2 Fixed Points and Stability

2.3 Stability Switching Theory

2.4 Bifurcation Theory

Chapter 3: Complete Dynamics and Fractality

3.1 Complete Dynamics of Discrete Systems

3.2 Routes to Chaos

3.3 Complete Dynamics of the Henon Map

3.4 Similarity and Multifractals

3.5 Complete Dynamics of Logistic Map

Chapter 4: Discontinuous Dynamical Systems

4.1 Basic Concepts

4.2 G-Functions

4.3 Passable Flows

4.4 Non-Passable Flows

4.5 Grazing Flows

4.6 Flow Switching Bifurcations

Chapter 5: Nonlinear Dynamics of Bouncing Balls

5.1 Analytic Dynamics of Bouncing Balls

5.2 Period-m Motions

5.3 Complex Dynamics

5.4 Complex Periodic Motions

Chapter 6: Complex Dynamics of Impact Pairs

6.1 Impact Pairs

6.2 Analytical, Simplest Periodic Motions

6.3 Possible Impact Motion Sequences

6.4 Grazing Dynamics and Stick Motions

6.5 Mapping Structures and Periodic Motions

6.6 Stability and Bifurcation

Chapter 7: Nonlinear Dynamics of Fermi Oscillators

7.1 Mapping Dynamics

7.2 A Fermi Oscillator

7.3 Analytical Conditions

7.4 Mapping Structures and Motions

7.5 Predictions and Simulations

Appendix 7.A

References

Index

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ISBN: 9781118359457

Preface

This book is about the dynamics of vibro-impact oscillators. Vibro-impact systems extensively exist in engineering and physics. Such vibro-impact systems possess the continuous characteristics as continuous dynamical systems and discrete characteristics by impact discontinuity. Such properties require an appropriate development of discrete maps for such vibro-impact systems to investigate the corresponding complex motions. The rich dynamical behaviors in vibro-impact systems drew the authors’ attention on nonlinear dynamical systems. In addition, a better understanding of such vibro-impact systems helps one study nonlinear dynamical systems with discontinuity in engineering and physics.

In 1964, Professor Weiwu Deng experimentally studied the lathe vibration reduction through impact dampers, which originated from the flutter reduction of airplane wines in Russia in the 1930s. Professor Deng found the optimal vibration reduction of the lathes is between 0.6 and 0.8 of the impact restitution coefficient with potential maximum energy dissipation. To further understand the dynamical mechanism of such impact dampers and extend applications in engineering, in 1987 Professor Deng invited the first author to work on this problem with him. After literature survey and experimental setup, it was crucial to develop an appropriate mathematical model to describe the impact dampers and to catch all possible complex motions. Since then, the first author has been working on this topic. Herein he would like to share what his group observed during the past 30 years with other scientists and engineers in vibro-impact systems.

This book mainly focussed on analytical prediction and physical mechanisms of complex motions in vibro-impact systems. After literature survey, in the next two chapters, the theory for nonlinear discrete systems is presented from the recent development of the first author primarily, including the Ying-Yang theory of discrete dynamical systems based on the positive and negative maps in discrete dynamical systems. The complete dynamics of nonlinear discrete dynamical systems is discussed and applied to one- and two-dimensional discrete systems, and a geometric method is discussed for the fractality and complexity of chaos in discrete dynamical systems. From the recent development of the first author, in Chapter 4, the theory of discontinuous dynamical systems is presented as a foundation of studying the dynamics of vibro-impact systems. In Chapter 5, bouncing ball dynamics is discussed as one of the simplest problems in vibro-impact systems to show the corresponding physical motions in this simple model. The dynamics for bouncing initiation and impacting chatter vanishing with stick motion is presented for the first time, which is significant in engineering application. After discussing the bouncing ball with the single map, a simple version of an impact damper is presented in Chapter 6 to show how to develop the complex periodic motions analytically. The motion switching from one motion to another is discussed through the gazing phenomena. In Chapter 7, the nonlinear dynamics of the Fermi oscillator is discussed as an application in physics. The methodology presented in this book can be applied to other vibro-impact systems in general, and discontinuous dynamical systems in science and engineering.

Finally, the authors would like to thank their family's support for this work, and this book is also dedicated to Professor Weiwu Deng as a good teacher, colleague and friend. The authors hope the materials presented herein will prove durable in the field of science and engineering.

Albert C. J. Luo Yu Guo Edwardsville, Illinois, USA

2

Nonlinear Discrete Systems

In this chapter, a theory for nonlinear discrete systems will be presented. The local and global theory of stability and bifurcation for nonlinear discrete systems will be discussed. The stability switching and bifurcation on specific eigenvectors of the linearized system at fixed points under a specific period will be presented. The higher order singularity and stability for nonlinear discrete systems on the specific eigenvectors will be presented.