Lectures on Nonlinear Dynamics (Understanding Complex Systems)

✍ Scribed by José Roberto Castilho Piqueira (editor), Carlos Eduardo Nigro Mazzilli (editor), Celso Pupo Pesce (editor), Guilherme Rosa Franzini (editor)

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 352
• Category: Library

✦ Synopsis

This book presents a compilation of lectures delivered at the São Paulo School of Advanced Sciences on Nonlinear Dynamics, categorized into four groups: parametric resonance, nonlinear modal analysis and model reduction, synchronization, and strongly nonlinear dynamics. Interwoven seamlessly, these groups cover a wide range of topics, from fundamental concepts to practical applications, catering to both introductory and advanced readers. The first group, consisting of chapters 1 and 2, serves as an introduction to the theory of parametric resonance and the dynamics of parametrically excited slender structures. Chapters 3, 4, and 5 form the second group, offering insights into normal forms, nonlinear normal modes, and nonlinear system identification. Chapters 6 and 7 delve into asynchronous modes of structural vibration and master-slave topologies for time signal distribution within synchronous systems, respectively, representing the third group. Finally, the last four chapters tackle the fourth group, exploring nonlinear dynamics of variable mass oscillators, advanced analytical methods for strong nonlinear vibration problems, chaos theory, and dynamic integrity from the perspectives of safety and design. This book harmoniously combines theoretical depth and practical relevance to provide a comprehensive understanding of nonlinear dynamics.

✦ Table of Contents

Foreword
Preface
Acknowledgements
Contents
Contributors
1 Brief Introduction to the Theory of Parametric Resonance
1.1 Pendulum with a Moving Support
1.2 Poincaré Map
1.3 Linearized System and the Floquet Matrix
1.4 Linear Discrete-Time Dynamical System
1.5 Multiple Eigenvalues and Jordan Chains
1.6 Stability Theory
1.7 The Meissner Equation
1.7.1 Floquet Matrix
1.7.2 Stability Conditions
1.7.3 Stability Diagram
1.8 Mathieu Equation
1.9 Physical Interpretations and Applications
1.9.1 Stabilization of Inverted Pendulum
1.9.2 Ion Traps
References
2 An Introduction to Parametrically Excited Systems and Their Importance on the Dynamics of Slender Structures
2.1 Introduction
2.2 Floquet Theory and Application to the Mathieu's Equation
2.2.1 An Overview on the Floquet Theory
2.2.2 Application to the Mathieu's Equation
2.3 Asymptotic Analyses Using MMS
2.3.1 Transition Curves of the Undamped Mathieu's Equation
2.3.2 Post-critical Amplitude of a Class of Nonlinear Mathieu's Equation
2.4 Use of HBM
2.5 Bifurcation of Periodic Orbits
2.6 Examples of Application on the Dynamics of Slender Structures
2.6.1 A Brief Overview of the Literature
2.6.2 Reduced-Order Modeling and Application of the Presented Techniques
References
3 Normal Forms
3.1 Introduction
3.1.1 Normal Forms in the Case of a Diagonal Jacobian bold upper JJ
3.1.2 Jordan Blocks
3.1.3 Matrix Normal Forms
3.2 Normal Forms for Hamiltonian Systems
3.2.1 Preliminaries
3.2.2 Perturbation Approach
3.2.3 Lie Transform
3.3 Nonlinear Normal Modes
3.3.1 NNMs as Invariant 2-Dimensional Manifolds ch3ShawPierre1994
3.3.2 Nonlinear Normal Modes for Conservative Systems
3.3.3 Analysis of NNMs Using Group Theory (ch3Golubitsky:Stewart:SchaefferspsII)
3.3.4 Discrete Symmetries and Circular Systems
3.3.5 Hopf Bifurcations for Circular Systems
3.4 Conclusions
References
4 Nonlinear Normal Modes and Reduced Order Models
4.1 Introduction
4.2 Brief Literature Review
4.3 Definition of NNMs
4.4 Use of Poincaré Sections to Identify NNMs
4.5 Asymptotic Method
4.6 Galerkin Based Procedure
4.7 Multi-mode—Asymptotic Approach
4.8 Computation of NNMs Using Numerical Continuation Techniques
4.9 Illustrative Examples
References
5 An Introduction to Nonlinear System Identification
5.1 Introduction
5.2 Testing and Data Collection
5.2.1 Testing
5.2.2 Choosing the Sampling Period
5.3 Choice of Model Class
5.4 Structure Selection
5.4.1 The ERR, SRR and SSMR Criteria
5.4.2 Other Criteria
5.5 Parameter Estimation
5.5.1 Underlying Issues
5.5.2 Classical Estimators
5.5.3 The Danger of Overparametrization
5.6 Model Validation
5.6.1 Residual Tests
5.6.2 Dynamical Invariants
5.6.3 Synchronization
References
6 Asynchronous Modes of Vibration
6.1 Introduction
6.2 Linear Asynchronous Modes About the Undeformed Equilibrium Configuration
6.2.1 2DOF Ziegler's Column Under Sub-critical Follower Force
6.2.2 3DOF Ziegler's Column Under Sub-critical Follower Force
6.2.3 One-Storey Portal Frame
6.2.4 Three-Storey Shear Building
6.2.5 3DOF Pre-tensioned Heavy Chain
6.2.6 NDOF Pre-tensioned Heavy Chain
6.2.7 Simply-Supported Beam with a Cantilever Extension
6.3 Linear Asynchronous Modes About the Deformed Equilibrium Configuration
6.3.1 2DOF Ziegler's Column Under Super-Critical Follower Force
6.4 Nearly-Asynchronous Non-linear Normal Modes About the Undeformed Equilibrium Configuration
6.4.1 3DOF Pre-tensioned Heavy Chain
6.4.2 Pre-tensioned Beam on a Winkler Foundation
6.4.3 Beam on a Winkler Foundation with Unilateral Contact
6.5 Exactly-Asynchronous Non-linear Normal Modes About the Undeformed Equilibrium Configuration
6.5.1 2DOF Pre-tensioned Heavy Chain
6.6 Concluding Remarks
6.6.1 3DOF Pre-tensioned Heavy Chain Coupled to a Piezoelectric Element and Subject to Parametric Instability
References
7 Comparing Master-Slave Topologies for Time Signal Distribution
7.1 Introduction
7.2 Single Node Simulations
7.2.1 Adjusting the Node
7.2.2 Phase Step
7.2.3 Phase Ramp
7.3 One-way Master-Slave Network
7.3.1 Adjusting the Network
7.3.2 Phase Step Perturbation
7.3.3 Phase Ramp Perturbations
7.4 Two-way Master-Slave Network
7.4.1 Adjusting the Network
7.4.2 Phase Step Perturbation
7.4.3 Phase Ramp Perturbations
7.5 Conclusions
References
8 Nonlinear Dynamics of Variable Mass Oscillators
8.1 Introduction
8.2 The Analytical Dynamics of Variable Mass Systems
8.2.1 Readdressing the Lagrange's Equation for Variable Mass Systems
8.2.2 The Generalized Hamilton's Principle and the Extended Lagrange's Equation for a Non-material Volume
8.3 Statistical Methods Applied to Variable Mass Oscillators Under Random Excitations
8.3.1 The Method of Statistical Linearization (SL)
8.3.2 SL Applied to the Water Column Dynamics Forced by Random Free Surface Waves
8.3.3 Higher Order Procedures: The Statistical Quadratization
8.3.4 Simulations
8.4 Conclusion
References
9 Generalized Krylov-Bogoliubov Method for Solving Strong Nonlinear Vibration
9.1 Introduction
9.2 Mathematical Procedure
9.3 Modified Krylov-Bogoliubov Method and the Truly Nonlinear Oscillator
9.3.1 Exact Solution of the Generating Equation
9.3.2 Period of Vibration
9.3.3 Approximate Solution of the Perturbed Equation
9.3.4 Van der Pol Oscillator
9.4 Oscillator with Time Variable Parameter
9.4.1 Special Case
9.4.2 Oscillator with Slow Variable Mass
9.5 Exact Steady States of Periodically Forced Oscillator
9.5.1 Exact Nonlinear Fundamental Resonance
9.5.2 Special Case
9.5.3 Approximate Solving Procedure for the Perturbed Oscillator
9.6 Chaos in Truly Nonlinear Oscillator
9.6.1 Example: Oscillator with Quadratic Nonlinearity
9.7 Concluding Remarks
References
10 Chaos Theory
10.1 Introduction
10.2 Dynamical Systems: Background
10.2.1 Stability
10.3 Chaos
10.3.1 Chaotic Attractors
10.4 Route to Chaos
10.5 Lyapunov Exponents
References
11 Dynamical Integrity and Its Background
11.1 Introduction
11.2 Dynamical Systems
11.2.1 A Continuous Time Dynamical System
11.2.2 A Discrete Time Dynamical System
11.2.3 Poincaré Section and Poincaré Map
11.2.4 Different Kinds of Motion
11.3 Stability
11.3.1 Stability of Equilibrium Points
11.3.2 Stability of Fixed Points
11.3.3 Stability of Generic Solutions
11.3.4 Local Analysis Around an Equilibrium Point
11.3.5 Local Analysis Around a Fixed Point
11.4 Dynamical Integrity
11.4.1 Safe Basins
11.4.2 Integrity Measures
11.4.3 Integrity Profiles
11.4.4 On the Regularity of Integrity Profiles
11.5 Conclusions and Further Developments
References